KUT, May 2015
On the Quasi-probability associated with the Weak Value
Izumi Tsutsui
Institute of Particle and Nuclear Studies
High Energy Accelerator Research Organization (KEK)
Plan of Talk
1. Weak trajectory and quasiprobability
2. Physical value in HVT and quasiprobability
3. Postselected measurement and quasiprobability
Review of time symmetric
formalism
2.1 Time symm
lained by the interaction which
makesTime
correlation
2.1
symmetric formalism
projective
is explained
bsystem
calledmeasurement
ancilla. The interaction
called von by the interaction which makes corre
The
oneobservable.
of the time symmetric
of quantum
mechanics was brough
A is aand
measured
nbetween
as!
Hint = Ap
thewhere
system
the
subsystem
calledformalisms
ancilla.
The interaction
calle
byare
Y. eigenstates
Aharonov [8].
First, this time symmetric formalism is evoked though th
c
|a
⟩
with
{|a
⟩}
which
for
as
1.
Weak
trajectory
and
quasiprobability
i
i
i
i
=
Ap
where
A
is
a
measured
obser
Newmann
interactionidea
is written
as
H
int
that what is predicted
by
quantum
mechanics
is
not
results
of
measurement
!
epare the initial
ancilla
state
as
|q
=
0⟩
localized
The
one time
of the
time symm
!
but
only
probability
distributions
which
are
expressed
symmetrically
(con
c
|a
⟩
with
{|a
⟩}
which
are
eigenstat
The
initial
state
|ψ⟩
is
written
as
i
i
i
i
by the
interaction.
em becomes i ci |ai ⟩|q = ai ⟩stant
by
Y.
Aharonov
[8].
First,
t
under
changing symmetrically both
initial
and final states).
Let
|ψ⟩ an
2.1
Time
symmetric
formalism
}
of
A.
If
we
prepare
the
initial
ancilla
state
as
|q
=
0⟩
loc
eigenvalues
{a
i
know is macroscopic
state.
When
we
know
which
|φ⟩
be
initial
and
final
state
in
the
system.
The
probability
amplitude
of
weak value as a new ‘observable’ of
mechanics
!quantum
idea that
what is predicted th
b
from the initial
state to the final
state⟩|q
under
some
interactions
is writte
pse
of thestate
eigenstates.
c
|a
=
a
⟩
by
the
intera
at toq the
= one
0, the
ofprocess
all system
becomes
i
i
i
i
w of time symmetric
lism
but
only probability
The one of the time symmetric
formalisms
of quantumdistribu
mechani
me
symmetric
For example,
we
con- is[8].
The
ancillaformalism.
state which
webycan
know
macroscopic
state.
we know
Y. Aharonov
First,
time
symmetric
formalism
is evok
ti )|ψ⟩
(2.1
⟨φ|Uthis
(tf −
stant
underWhen
changing
symm
as isthe
idea that
what
predicted
by quantum
mechanics is not results of
ment
of the
observable
A which
have
{|wi ⟩}
ancilla
state
is, the
state
collapse
to
one
of
the
eigenstates.
postselection
and
final
of theand
system.
Generall
where ti and tf represent the time of initial
|φ⟩
be
initial
final
state
but
only
probability
distributions
which
are
expressed
time
symm
integers
mplicity, the eigenvalues of we
{|winterpret
i ⟩} are!
this process as the unitaryprocess
evolutionfrom
of thethe
initial
state and
th
initial
state
stant
under
changing
symmetrically
both initial
and final
states
We
now
go
back
to
the
time
symmetric
formalism.
For
example,
we
c
|w
⟩
the
initial
state
and
the
final
state
be
projection
measurement
to
the
final
state.
However
this
process
is
also
inte
i
i
i
he time symmetric formalisms |φ⟩
of quantum
mechanics
was
brought
be
initial
and measurement
final state into
the
system.
The
probability
am
as,
−
t
)|φ⟩.
These
tw
preted
as
the
projection
the
state
U
(t
easurements.
The
initial
state
prepared
at
t
=
−T
i
f
sider
the
projective
measurement
of
the
observable
A
which
have
{|w
ov [8]. First, this time symmetric
formalism
isinitial
evoked
though
thequantum
process
from
the
state
to the
final
state under
some
interact
interpretations
imply
that
the
probability
of
the
process
is
unchange
ted at t = +T . The interaction Hamiltonian Hint
is predicted by(Fig
quantum
mechanics
is notinitial
results
of
measurements
as,
⟩} are
in
eigenstates
2.1).
For
simplicity,
eigenvalues
of {|widirectly
under
exchanging
andthe
final
state. Quantum mechanics
predict
!
interaction.
timeasurement
)|ψ⟩tf represent
⟨φ|U
(tft−
the
probability
of quantum
processes, notwhere
the
values
of
results.th
W
and
ability
distributions
which
are
expressed
time
symmetrically
(coni
i = ·!
· · − 2, −1, 0, 1, 2,have
· · ·no. reason
Let the
initial
state
and
the
final
state
be
i
to
determine
how
measurements
affect
the
system.
The
proba
preselection
hanging symmetrically
both initial
final
states). Let
|ψ⟩
and
we
interpret
this
as
tf represent
the time
of initial and
finalprocess
of the syste
whereand
ti and
(2.8)
Hint = jδ(t)qA
c′j |wj ⟩ by projective
measurements.
The
initial
state
prepared
at
t
and
bility of quantum processes has time symmetric characteristics. If so, it mus
we
interpret
this process
as the
evolution of the initial
and final state in the system.
The
probability
amplitude
ofunitary
the
projection
to
be reachable
to
construct a time
symmetrical
formalismmeasurement
of quantum mechanic
and
thestate
finaltostate
is post-selected
atinteractions
t = +Tto. The
interaction
Hamiltonia
projection
measurement
the
final
state. However
this proces
he
initial
the
final
state
under
some
is
written
as
thestate
projection
me
they reached to a timepreted
symmetric
formulation
of quantum
cilla state. Since the observableRecently,
Apreted
has discrete
−
t
as
the
projection
measurement
to
the
U
(t
i
f )|φ
is
given
as
the
von
Newmann
interaction.
useful
for
a
deeper
understanding
of
quantum
phenomena
mechanics,“Two-state
of quantum
mechanics” [9].imply
They formalize
interpretations
th
cal variable p takes discrete localized
states which formalism
interpretations
imply that the probability
of the quantumthat
proces
thetphysical
states by
−
)|ψ⟩
(2.1)
⟨φ|U
(t
f
i
the interaction.applications
The state |mfor
that
it’s
under
exchanging
initial
and
final
state.
Quantum mechanics
di
i ⟩ means
under
exchanging
initial and
precision measurement
†
the
probability
of
quantum
processes,
not
the
measureme
)|ψ⟩⟨φ|U
(t −
tf )values
(2.2
℘(t)
ˆ
:
=
U
(t
−
t
⟩.
rf simplicity,
The
ancilla
states
prepared
in
|m
i
=
δ(t)qA
H
0
represent the time of initial and final ofint
the system. Generally,
the probability
of of
quantum
p
have of
nothe
reason
to determine how measurements affect the system
⟩.
There
is
no
Hamiltonian
all
te
as
|m
1
his process as the unitarywhere
evolution
of the
initial
and
|ψ⟩ is initial
state,
|φ⟩ is state
final state
of the
the
quantum
This stat
have
no
reasonprocess.
to determine
bility
of
quantum
processes
has
time
symmetric
characteristics.
n between ancilla
and
system
at the
measuring
asurement
to
the
final
state.
However
process
isa time
also
interbility
of quantum
processes
be reachable
to
construct
symmetrical
formalism
of
quant
where q is the position of
ancillathis
state.
Since
the
observable
A
has
di
as,
me symmetric formalism
weak value
projective measurement is explained by the interaction which makes correlation
between the system and the subsystem called ancilla. The interaction called von
Newmann interaction is written as!
Hint = Ap where A is a measured observable.
The initial state |ψ⟩ is written as i ci |ai ⟩ with {|ai ⟩} which are eigenstates for
eigenvalues {ai } of A. If we prepare the initial
! ancilla state as |q = 0⟩ localized
at q = 0, the state of all system becomes i ci |ai ⟩|q = ai ⟩ by the interaction.
The ancilla state
we can know
macroscopic
state. When we know which
Y. Aharonov,
P. G.which
Bergmann,
J. L. isLebowitz
(1964)
ancilla state is, the state collapse to the one of the eigenstates.
time-symmetric formulation of quantum mechanics
We now go back to the time symmetric formalism. For example, we consider the projective measurement of the observable A which have {|wi ⟩} as
eigenstates (Fig 2.1). For simplicity, the eigenvalues of {|wi ⟩} are!integers
i = ·!
· · − 2, −1, 0, 1, 2, · · · . Let the initial state and the final state be i ci |wi ⟩
and j c′j |wj ⟩ by projective measurements. The initial state prepared at t = −T
and the final state is post-selected at t = +T . The interaction Hamiltonian Hint
is given as the von Newmann interaction.
... the result of the measurement at t has
implications not only for what happens after
t but also for what happened in the past ...
Hint = δ(t)qA
(2.8)
where qwith
is the all
position
ancilla state. Since
observable
A has discrete
consistent
the ofpredictions
madetheby
the standard
description
integral eigenvalues, the canonical variable p takes discrete localized states which
are described as {|mi ⟩} after the interaction. The state |mi ⟩ means that it’s
of QM
momentum p is equal to i. For simplicity, The ancilla states prepared in |m0 ⟩.
no Hamiltonian
of the allbefore
We select
the final
state as
|m1 ⟩. There is that
shed new
lights
on ancilla
quantum
phenomena
were missed
system without the interaction between ancilla and system at the measuring
Let thevalue,
measurement
performed
at ttunneling
= 0. First, we
see the evolution
(such point.
as weak
statebereduction,
etc.)
!
of the retarded state of all system i ci |wi ⟩|m0 ⟩. When the retarded state arrives
at the measurement
point, the ancilla
state makes a correlation with the system
may suggest
generalizations
of QM
state.
"!
c |w ⟩|m ⟩ (−T < t < 0)
(2.9)
|ψ(t)⟩ = !i i i 0
c
|w
⟩|m
⟩
(0
<
t
<
+T
)
i
i i i
w
w
2 := ⟨φ|UA (s)|ψ⟩
−igA
K(s)
2.1
Time
sym
(g)|ψ⟩|
U
(g)
=
e
P
(g)
=
|⟨φ|U
A
between the system and
the subsystem A
called
The
2 ancilla.
2
2 interaction called von
2
p P (g) = |⟨φ|(1 − igA − (g /2)A + · · ·)|ψ⟩|
2 A is a measured
2
=|⟨φ|ψ⟩|
Ap
where
observable.
2
Newmann interaction
as!
Hint
2 2(13)
−isA
,(s)
H = PPis
Pwritten
(0)
h
|pU
(T )|
(g)
/2)A
+
·
·
·)|ψ⟩|
|⟨φ|(1
−
igA
−
(g
f
ii
(s)|ψ⟩|
U
(s)
=
e
P
(s)
=
|K
=
|⟨φ|U
A
A
!
"
The
of the timefor
sy
=:=
p
=
⟨φ|U
(s)|ψ⟩
2 with 22{|a ⟩} 2which
3 one
w
A
c
|a
⟩
are
eigenstates
The initial state |ψ⟩2m
isPK(s)
written
as
=
1
+
2gImA
+
g
|
−
Re(A
)
)
|A
+
O(g
i
i
i
w K(s)
w⟨φ|UA (s)|ψ⟩ w
:=picture’
(0)offer
|⟨φ|ψ⟩|
K(s)
:= an
⟨φ|U
h} f:=
|U (T[8].
)| Firs
A
(1)
WV
asi(s)|ψ⟩
relative
correction
towhere
transition
amplitude
Can
the weak
‘intuitive
?
ii
defined
by thevalue
initial
state
|ψ⟩
of
the
meter,
{Q,
P
by
Y.
Aharonov
!
"
2−isAstate
2 2 as |q =20⟩23 localized
of A.
IfPwe
prepare
the
initial
ancilla
2
2
ere m eigenvalues
is the mass{a
ofi }the
particle,
and
p
is
a
momen2
(s)
/2)A
+
·that
·|K(s)|
·)|ψ⟩|
|⟨φ|(1
−
isA
−
(s
2
−isA
2 what
2=
−isA+ g
2 ⟨φ|U
(s)|ψ⟩|
U
(s)w
=|have
e(s)
P (s)
=
P (s)
= (s)
|⟨φ|U
!
K(s)
:=
(s)|ψ⟩
1
+
2gImA
−
Re(A
)
|A
+
O(g )is predict
A
A
A
w
w
idea
(s)|ψ⟩|
U
(s)
=
e
P
(s)
=
|K(s)|
PK(s)
(s)
|⟨φ|U
QP= +
P
QA=is
theA⟨φ|U
anticommutator.
We
also
⟨φ|A|ψ⟩
(s)|ψ⟩|
U
=
e
P
=
|K(s)|
P
(s)
|⟨φ|U
A
:=
(s)|ψ⟩
A
=
A
2 =
−igA interaction.
c
|a
⟩|q
a
⟩
by
the
at q =x 0,
the stateThe
of all
system
becomes
Aw= =
m alongside
direction.
unitary
evolution
of
this
2
i
i
i
(g)|ψ⟩|
U
(g)
=
e
P
(g)
|⟨φ|U
i
2
2
2
2
−isA
2
A
A
P
(0)
|⟨φ|ψ⟩|
but
only
probability
dist
P (s)2
/2)A
+ · · ·)|ψ⟩|
|⟨φ|(1
− isA
−⟨φ|ψ⟩
weak value under
dynamical
evolution
U
(s)
=
e
P
(s)
=
|K(s)|
=
|⟨φ|U
2 P (s)
2(sA (s)|ψ⟩|
A
h
|pU
(T
)|
P (s) which
/2)A
+macroscopic
· · ·)|ψ⟩|
|⟨φ|(1we
− isA
−
(s
= is
!
f
±
!
"
±
The
ancilla
state
can
know
state.
When
we
know
which
2
2
2
2
ticle
is
described
by
2
−isA
2
2
2
2
=
P
(0)
|⟨φ|ψ⟩|
⟨φ|A|ψ⟩
2isA
2
P/2)A
(g)2 |⟨φ|(1
/2)A
+
·w·"·)|ψ⟩|
−
igA
− (g
(s)
+
·
·
·)|ψ⟩|
|⟨φ|(1
isA
! e(s
2pstant
2=under22 changing sy
(s)
=−
P
(s)
=
|K(s)|
) = |⟨φ|UAP(s)|ψ⟩|
Ad∆
!
P (0) U−
|⟨φ|ψ⟩|
P
(s)
/2)A
+
· · 3·)|ψ⟩| )w + O(s
|⟨φ|(1
−
−
(s
P
=
1
+
2sImA
+
s
|
−
Re(A
|A
=
=
A
2
2
2
w
w
w
!
(T )|
=
=
ancilla state is, the state collapse
one
of
the
eigenstates.
2Re(A
=to
1t+2the
2sImA
+
s·(x)
|ww
−
)|φ⟩
)h f |U
|A
O(sinitial
!(0)
"ImA
±i
w2
w
w2 +
=
2Var
(ψ)
,
P
|⟨φ|ψ⟩|
(t)
=
̸
x
(t)
t
=
0
=
T
+
V
x
be
and
final
s
2
3
P
cl
⟨φ|ψ⟩
2
P (0) )w + O(s
|⟨φ|ψ⟩|
!
= 1 + 2sImA
+
s
|
−
Re(A
)
|A
P (0)
|⟨φ|ψ⟩|
w
w
!
"
iHt 2g=0
" 3
2 dg
2
2
22 !
2 2 ) from
2 O(g
3
=
1
+
2gImA
+
g
|
−
Re(A
)O(s
|A
+
process
initial
s
"
P (s)
/2)A
+2 ·!· ·)|ψ⟩|
|⟨φ|(1
−
− (s to
=† (14)
1formalism.
+ s For
− Re(A
)w the
)
|Aw | example,
+we
w+ 2sImA
w
We U
now
go
back
the .time
symmetric
conww
(t)isA
= exp
2
3
U|(t)|ψ⟩
U (T −2t)|φ⟩
t+
AO(s
We
considered
time
evolu
w (t)
=
=
1
+
2sImA
+
s
−
Re(A
)
)
|A
}
†
w
w
w
as,
†
2
observable
A
which
have
{|w
Uof
(Tk ⟩⟨φ|A|ψ⟩
−x|ψ⟩
t)|φ⟩
t
A
(t)
Umeasurement
(t)|ψ⟩
tR
(t)U
P (0) sider the projective
|⟨φ|ψ⟩|
Rt)|φ⟩
I Aw
CI+
|φ
i ⟩} as
w
(t)
=
̸
x
(t)
t U=(T
0−
tC(t)|ψ⟩
=the
T
VU (x)
w
cl
†
at tt =
0 and a po
(t)|ψ⟩
Ustate
(T − t)|φ⟩
Aw (t)
where
!
"
=
A
w
integers
eigenstates
(Fig+2.1).
For
simplicity,
eigenvalues
of= 1/2
{|w
2
2
2 |φthe
3= ⟨φ|ψ⟩
2 |φ
i ⟩} are!
2
⟩
|ψ⟩
|φ
⟩
⟩
⟩
α
0,
1
α
|φ
1
1
2
3
weakly
measure
a
mome
|U
(T
)|
i|
=
e transition
probability
is
|h
=
1 + 2sImA
s
|
−
Re(A
)
)
|A
+
O(s
≪
|φ⟩ Ψ(Q)
1 (Ex
Φ(Q) Q
i −
w
w :=f ExPg|A
w |w
2 (ψ)
Var
(ψ)
(ψ))
P
P
†Let the initial state and the final
tf represen
where
ti and
ci |w
·!
·if· we
− 2,take
−1,U0,
1,
2,
·
·
·
.
state
be
ipreselection:
2 m xfix=
i⟩
(t)|ψ⟩
U
(T
−
t)|φ⟩
t
A
(t)
i
|
≪
|φ⟩
Q
g|A
1
Ψ(Q)
Φ(Q)
between
(1)
and
(20)
are
(10)
and
(11)
as
preand
postw
:
H
⊗
H
→
C
I
⟨A⟩
:
H
→
R(A)
⊂
R
I
A
w
w| ≪ 1 |φ⟩ Ψ(Q) Φ(Q) Q
g|A
} T
′
w
we
interpret
thist =
proces
R
C
R
I
C
I
|φ
⟩
|ψ⟩
(t)
=
̸
x
(t)
t
=
0
t
=
T
+
V
(x)
x
k
c
|w
⟩
by
projective
measurements.
The
initial
state
prepared
at
−T
and
w
cl
j
j
j
#g|A
t
)|
i
=
|
i.
Because
th
cted state, respectively.
The
interference
pattern
does
|
≪
|φ⟩
Q
1
Ψ(Q)
Φ(Q)
i
⟨φkX|A|ψ⟩
(Φ) − ExX (Ψ) i s|A|
∆w
X =2 Ex
†
projection
measurement
= ⟨ψ|A|ψ⟩
|⟨φw
kt|ψ⟩|
postselection:
and
the
final state
is post-selected
at
=
+T
.
The
interaction
Hamiltonian
H
U
(t)|ψ⟩
U (T
− t)|φ⟩
t
A
(t)
int
⟩
|ψ⟩
|φ
⟩
|φ
⟩
|φ
⟩
α
=
0,
1
α
=
1/2
|φ
⟨φ
|ψ⟩
k
1
1
2
3
preted
the projection
∆X = ExX (Φ) k− Ex⟨φ|A|ψ⟩
ExX (Ψ)ass|A|
∆X = ExX (Φ) −
X (Ψ) s|A|
is given as the von NewmannAinteraction.
=
R C R
I
CI2
|φk ⟩ |ψ⟩
|φ⟩: w
Q
g|Aw | ≪ 1 ⟨A⟩
Ψ(Q)
Φ(Q)
imply th
(Ψ)
⟨Ψ|X|Ψ⟩/∥Ψ∥
X
=interpretations
Q,H
P →C
ExXR(A)
⟨φ:=
:
H
⊗
I
H→
⊂R
I
A
1 |A|ψ⟩
⟨φ|ψ⟩
w
|b⟩
⟩
|ψ⟩
|φ
⟩
|φ
= 0, 21 exchanging
α = 1/2
|φ
1
1
2 ⟩ |φ3 ⟩ α under
time-dependent (dynamical) weak
value
initial
⟨φ=
1 |ψ⟩
Ex
(Φ)
−
Ex
(Ψ)
s|A|
∆
2
(Ψ)
:=
⟨Ψ|X|Ψ⟩/∥Ψ∥
X
=
Q,
P
Ex
X
X
X
#
X
X ⟨φ
= kQ,
P
ExX (Ψ) := ⟨Ψ|X|Ψ⟩/∥Ψ∥
|A|ψ⟩
=
δ(t)qA
(2.8)
H
#
int
2
the
probability
of
quantu
Q
g|Aw | ≪ 1 |φ⟩ Ψ(Q) Φ(Q)
#
⟨A⟩
→2 |A|ψ⟩
R(A) ⊂ R
I = ⟨ψ|A|ψ⟩
Aw : H ⊗ H → CI
d∆
|⟨φ
Q: #H⟨φ
k |ψ⟩|
= ⟨φ
ReA|ψ⟩
w + C(Ψ) · ImAw ,
#
k
#
dg
have no reason to determ
#
g=0⟨φ2 |ψ⟩
k
#
⟨φ
|A|ψ⟩
#
d∆2Q # k
(Φ)
−
Ex
s|A|
∆
# ofX
X =
X (Ψ)
=
⟨ψ|A|ψ⟩
|⟨φ
=
ReA
+ C(Ψ)
·A
ImA
d∆
⟨φ|U
(T
−
t)AU
(t)|ψ⟩
(TQEx
−
t)
·
A
·
U
(t)|ψ⟩
k |ψ⟩|
where q is the ⟨φ|U
position
ancilla
state.
Since
the
observable
has
discrete
w
w,
bility
of
quantum
proce
#
#
dg
⟨φ
|ψ⟩
=
ReA
+
C(Ψ)
·
ImA
,
|A|ψ⟩
⟨φ
k
w
w
n
g=0
22
=
Aw (t) =
#
⟨φ
k 1 |A|ψ⟩
dg
(Ψ)
:=
⟨Ψ|X|Ψ⟩/∥Ψ∥
X states
=toQ,construc
Pwhich
Ex
=
g
[ReA
+
C(Ψ)
·
ImA
]
+
O(g
)
∆
Q
w
w
g=0
be
reachable
X
integral eigenvalues, the canonical
variable p takes discrete
localized
|b⟩
⟨φ|U
(T
−
t)
·
U
(t)|ψ⟩
⟨φ|U
(T
)|ψ⟩
⟨φn |ψ⟩
∆X = ExX (Φ) − ExX (Ψ) s|A|
⟨φ1 |A|ψ⟩
⟨φ
1 |ψ⟩
2
2
=
2gVar
(ψ)
· gImA
++
O(g
)|m
∆
[ReA
C(Ψ)
· Recently,
ImA
]
+
O(g
)that reac
∆
⟩
means
it’s
are described as {|mi ⟩} after the interaction.
|b⟩
P
P Q =The
wstate
w
w
they
i
(g)|ψ⟩
K(g) = ⟨φ|UA⟨φ
2 1 |ψ⟩
O(g )
∆Q = g [ReAw + C(Ψ) · ImA
2 w] +
2
⟨φ
momentumEx
pX
is(Ψ)
equal
i. For simplicity,
The
in |m0 ⟩.
fo
:=to⟨Ψ|X|Ψ⟩/∥Ψ∥
P Pstates
=Q,
2gVar
(ψ)mechanics,“Two-state
· ImAprepared
∆=
# 2X|A|ψ⟩
Pancilla
w + O(g )
2σx ⟨φ2 |A|ψ⟩
#
+
σ
z
d∆
=
2gVar
(ψ)
·
ImA
+
O(g
)2 |ψ⟩
∆
the
physical states
byall
P
P
w
Q
⟨φ
⟩.
There
is
no
Hamiltonian
of
the
We select the final ancillaAstate
as
|m
(Ψ)
− 2Ex
· Ex
C(Ψ)
:=
Ex
Q (Ψ)
P (Ψ),
1
{Q,P
}
π
#
√
=
σ
:=
⟨φ2 |ψ⟩
= ReA
2
w + C(Ψ) · ImAw ,
ppens to be the time developed pre-selected state. As such, for the unitary time
like the expectation value, the dynamical weak value satisfies
ment U (t) = exp(−iHt/!) governed by the Hamiltonian H, the weak value Aw (t)
e equation,
d
i ⟨ψ|U (T − t) [A, H] U (t)|φ⟩
Aw (t) = −
dt
!
⟨ψ|U (T )|φ⟩
i
= − [A, H]w (t),
(3)
!
analogues to one stipulated by the Ehrenfest theorem for the expectation value,
hat Aw (t) is complex in general.
Ehrenfest theorem for the weak value
t follows we consider the system of a particle of mass m under the non-relativistic
p2
nian H = 2m + V (x). If, in particular, the particle is free V (x) = 0, by putting p
the observable A in (3) we find that the momentum weak value pw and the position
ue xw obey
d
pw (t) = 0,
(4)
dt
1
d
xw (t) = pw (t).
(5)
dt
m
ollows from (4) and (5) that xw (t) is a linear (complex) function of t. This implies
t) is a real function during the entire interval t ∈ [0, T ] if and only if both the initial
†
assume
that
the
of
the
system
can
⟨φ|U (tHamiltonian
⟨φ|U
(tiobject
) U †|x
(tfi)|ψ⟩
f ) U (ti )|ψ⟩
i
+
|
x
i
i
The
interference
is caused by the phase
ch represents
a
point
on
the
screen,
⟨φ|A|ψ⟩
p
=and
| have
.
(11) K
TESTthe
SPACE
ii A
disturb
object
system
and
will
a
physical
sigbetoseparated
in
the
x
and
y
direction
is
chosen
not
=
tween
K
′
′
† ′
†
w
+ (↵) and K (↵). In the case
2
2
U
(t,
t
)
=
U
(t
−
t
)
=
U
(t)U
(t
)
A(t)
:=
U
(t)
A
U
(t)
⟨φ|ψ⟩
arPnificance.
(ψ) := Ex|P Einstein
(ψ)
−|x(Ex
P (ψ))
et
al.
said
about
an
element
ofthe
physi
=
i,
(12)
f
f
ple,
post-selected
state is eigenstat
to vanish the Utransition
amplitude
between
two
points,
†
U
(t, tf )|φ⟩
⟨φ|U
(t,-‘weak
tway
(t, ti )|ψ⟩
†
trajectory
in
weak
value
f)
trajectory’
ical
reality
in
[11]:
If,
without
in
any
disturbing
a
tion
|x
i.
Because
u
(↵) is the transl
f
ppost-selected
We
put
a
point
eigenstate
at
t
=
T
as
a
and
then
a
state
in
the
y
direction
does
not
contribute
†
coincident
with
the(i.e.,
the with
post-selected
state
| (t
fi
1 U †certainly
† probabil†
⟨φ|U
)
A(t)
(t
)|ψ⟩
)
A(t)
U
(t
)|φ⟩
⟨ψ|U
(t
system, we⟨φ|A|ψ⟩
can
predict
with
f
i
i
f
u
(↵)|x
i
=
|x
↵i,
u
√ which
fon thef screen, p (↵) moves the
(|1⟩
+
|2⟩represents
+ |3⟩)
|ψ⟩
=the
p its definition.
state
a
point
+(1−α)
=
α
toity
the
weak
value
in
x
direction
due
to
ment
atλ(A)
the
point
P
.
(Tthe
− t)AU
(t)|ψ⟩
⟨φ|U
(T − t) · Aquantity,
·U
(t)|ψ⟩
3 † (t )|ψ⟩
†⟨φ|U
Aw =to unity)
equal
the
value
of
a
physical
then
⟨φ|U
(t
)
U
⟨φ|U
(t
)
U
(t
)|ψ⟩
x
↵
alongside
screen.
The weak va
f
i
i
f
f
=
(t)
=
A
⟨φ|ψ⟩
w
free
particle
lWe
put will
the Hamiltonian
ofdependence
this
system⟨φ|Uon
+ put
ignore
any
the
yU (t)|ψ⟩
axis.pwWe
a
1
(T
−
t)
·
⟨φ|U
(T )|ψ⟩
tum
,
p
and
p
are
there exists an element
of
physical
reality
corresponding
w
w
|φ⟩ = √ (|1⟩ + |2⟩ − |3⟩)
i=
|xf i,
(12)
| fas
3
′
′
†
′
†
superposition
of
two
position
eigenstates
a
pre-selected
2
to this physical
We
may (t
treat
a weak
U (t, t ) =
U (t − t ) =
U (t)U
) A(t)
:= Uvalue
(t) Aas
U (t)
pquantity.
, Pi =reality
Hof
= physical
xf + ixi ta
|i⟩⟨i| i =
1, 2, 3 (13)
h f |pU (T )| i i
state,
an
element
because,
in
weak
measure+
σ
σ
(T
−
t)AU
(t)|ψ⟩
T − t) · A · U (t)|ψ⟩2m ⟨φ|U
†
z
x state
pw)|==f|xi coincident
= m the the
with
where
the
post-selected
π
U
(t,
t
)|ψ⟩
U
(t,
t
)|φ⟩
⟨φ|U
(t,
t
√
A
=
σ
|ψ⟩
:=
⟩
|φ⟩
=
|x
⟩
=
i
f
f
By
using
new
i
f
h
|U
(T
)|
i
T
ment,
without
disturbing
a
system,
we
can
obtain
the
f
i
4
Uis(Tthe
− t)
·
U
(t)|ψ⟩
⟨φ|U
(T
)|ψ⟩
2point P .
measurement
at
the
mass of the particle,
and
p
is
a
momen(xf − xi )t |xi i1 + |
x
i
tion
amplitud
weak value[12].
i
=
x
x
(t)
=
(t)
x
(t)
=
x
(t)
w
cl p
w
cl|3⟩)
gside x direction.
The
unitary
evolution
of|2⟩
this
√
i
=
|
.
(11)
(|1⟩
+
+
|ψ⟩
=
We
will
put
the
Hamiltonian
of
this
system
T
i
Let
us
consider
the
weak
value
of
the
position
x
in
√ w h f |pU (T )| ± i
xf ⌥ x i
1+1
3 2
±
s described
σz + σx by
=m
.
=
pprepare
π)
√
(σ
=
+
=
2
[−1,
+1]
w
w
√ of the
|ψ⟩
=
|x
= terms
⟩
|φ⟩
=
|x
⟩
element
of
physical
reality.
we
will
i
f
4
TEST SPACE
K
TEST SPACE
h
|U
(T
)|
i
T
2
−igA σ
f
±

2
1
2
g  σ1 ΨMD (0, 0) e
ΨMD (0,t0)
p
=
0
t
=
T
t
=
0
t
=
T
the
preand
post-selected
state
as
(10)
and
(11),
respeciHt
|φ⟩ = √ at
(|1⟩t +=|2⟩T− as
|3⟩)
We put
a
point
eigenstate
a
post-selected
,
(13)
H
=
iφ (14)
U (t) = exp
.
|σconsidered
(θ, φ) = cos3θ |0⟩ + e sin θ |1⟩ ⟨+x
ΨMD = ΨMD
x |+z⟩
We
time evolution
and
prepare
2m
tively.
}
√
The
interferen
=
+1
)
=
(σ
state 1which
represents
a
point
on
the
+1
x w screen,
state|+z⟩
at
t|ψ⟩
= 0 and a post-selected state
⟨+x
)w = √ = + A2w := [−1,
+1]
A =P
Bi +=C|i⟩⟨i|
Aw =R
B
+
Cw
C
R
I
C
I
|φ
⟩
w
i
=
1,
2,
3
k
Cm R
Iist)xU
CI (t)|
|φk ⟩i of
|ψ⟩ the particle, and p is
(↵)
tween
K
2mass
+
his R
the
a
momenweakly
measure
a
momentum
p
at
t =
|U
(T
)|
i|
=
nsition 2probability where
|h (T
f |U
i
trajectory
of
particle
in
f
i
Aiw=
=1B
Cw
xwA(t)
⟩|x
|φ1 ⟩ a|φ
= 1/2
|φ
w := = A = B|C
w |ψ⟩
i ∈
2 ⟩R |φ3 ⟩ α = 0, 1 α(12)
f xi
i,
f
f
⟩
|ψ⟩
|φ
⟩
|φ
⟩
|φ
⟩
α
=
0,
1
α
=
1/2
|φ
between
(1)
and
(20)
areple,
A = the
p,of| this
i=|
if
we
take
(10)
and
(11)
as
preand
post1 tum h
1 f |U (T
2 )| 3x
alongside
direction.
The
unitary
evolution
⟨+x |σx |+z⟩
fpost
i
terms
of
the
weak
value
i
T
| 3+
x⟩
σz ⊂σRIxt )|(σAzwi):=
+1
= +1
+1
=
(σx )w
wH=
⊗ i.
H→
CI(σx )w the
: )H
R(A)
=
(P2 )|w+=z⟩
1 ⟨A⟩
(P
= does
−1
(P1 )=
w
w →
|
Because
imaginary
part
tate,
respectively.
The
interference
pattern
xfH
xii→ C
i
tion
|x
i.
B
m
⟨+x |+z⟩⟨A⟩particle
:
H
⊗
I
:
H
→
R(A)
⊂
R
I
A
f
is
described
by
w
x
(t
T
)
tan
xf t= a σstate
i
!
} ⟨φTk |A|ψ⟩with the the
coincident
where the post-selected
|
+
a
σ
+
O(a)
fai
†
3σ
3 |ψ⟩|2
= !(x
+1fi 1− xi2⟨φ
(23)
)t2k |A|ψ⟩
=
⟨ψ|A|ψ⟩
|⟨φ
coincides
with
classical
trajectory
k
u
(↵)|x
i
=

f
2
p
⟨φ
|ψ⟩
=
x
(t)
=
x
+
(t)
x
(t)
=
x
(t)
x
T
T
k
w at
i
w
cl
|⟨φk |ψ⟩|P .
† the
measurement
point
k cl = ⟨ψ|A|ψ⟩
−igA⊗P
iHt
⟨O(a)⟩
=
Ψ
O(a)Ψ
=
2g
(a
ImA
−
a
ReA
)
+
a
T
⟨φk |ψ⟩
up3 to quadratic
potentialsxfby
↵ alongs
w
2
w
x⟩ σz σx (σz )wMD= +1k MD
(σx )w = 1+1
⟨φ|e
|ψ⟩
=
exp aEhrenfest
.theorem
(14)
⟨φ
Weprepared
will putathe
Hamiltonian
We
pre-selected
state of
atUthis
t(t)
=1 |A|ψ⟩
0system
and
post+
|b⟩
}
tum
p
,
p
⟨φ1 |A|ψ⟩
⟨φ1 |ψ⟩
w w an
selected state at t = T and weakly measure
x
at
time
t.
|b⟩
2
⟨φ
|ψ⟩
−igA
σ
2
w 1⟨φ2 |A|ψ⟩
g  σ(1)
ΨMD(23)
(0,10)
eA = x,
⟨φ|e−igA⊗P
|ψ⟩
p
Relations
between
are
UΨ(t
T )|0)f iis= |h |U (T )| i|2
1 and
MD (0,
=
The Htransition
probability
i
⟨φ2 |ψ⟩
,
(13)f
= ⟨φ2 |A|ψ⟩
h
2
w
1
assume
that
the
Hamiltonian
of
the
object
system
can
|x
i
+
|
x
i
i
i
The
interference
is caused by the phase
ch represents a point on the screen,
⟨φ|A|ψ⟩
p
=and
| have
.
(11) K
ii A
disturb thein
object
and
will
a
physical
sigbetoseparated
the system
x and
y
direction
is
chosen
not
=
w tween K
+ (↵) and K (↵). In the case
2
2
⟨φ|ψ⟩
arPnificance.
(ψ) := Ex|P Einstein
(ψ)
−|x(Ex
P (ψ))
et
al.
said
about
an
element
physi
=
i,
(12)
f
f
ple,
post-selected
state is eigenstat
to vanish the transition amplitude betweenofthe
two
points,
†
trajectory
in
weak
value
-‘weak
trajectory’
ical
reality
in
[11]:
If,
without
in
any
way
disturbing
a
tion
|x
i.
Because
u
(↵) is the transl
f
ppost-selected
We
put
a
point
eigenstate
at
t
=
T
as
a
and
then
a
state
in
the
y
direction
does
not
contribute
with the(i.e.,
the with† probabilpost-selected
state
| f i coincident
†
system, we⟨φ|A|ψ⟩
can predict
with
certainly
u
(↵)|x
i
=
|x
↵i,
u
fon thef screen, p (↵) moves the
p its definition.
state
which
represents
a
point
toity
the
weak
value
in
the
x
direction
due
to
ment
at A
the
point
P
.
⟨φ|U (Tthe
− t)AU
(t)|ψ⟩
⟨φ|U
(T − t) · Aquantity,
·U
(t)|ψ⟩
w =to unity) the value of
equal
a
physical
then
x
↵
alongside
screen.
The weak va
f
=
(t)
=
A
⟨φ|ψ⟩
w
lWe
put will
the
Hamiltonian
ofdependence
this system⟨φ|Uon
non-free
particle
ignore
any
the
yU (t)|ψ⟩
axis.pwWe
put
a
CE
(T −
t) ·corresponding
⟨φ|U
(T )|ψ⟩
tum
, p+
and
p
are
there exists an element of physical reality
w
w
i=
|xf i,
(12)
| fas
superposition
of
two
position
eigenstates
a
pre-selected
2
to this physicalpquantity. We may treat a weak value as
telement
= 0 Htof
=physical
T , + V (x)
(13)
=
xf + ixi ta
h f |pU (T )| i i
state,
an
reality
because,
in
weak
measure+
σ
σ
(T
−
t)AU
(t)|ψ⟩
T − t) · A · U (t)|ψ⟩2m ⟨φ|U
z
x state
pw|==f|xi coincident
= m the the
with
where
the
post-selected
π
TEST SPACE
√
A
=
σ
|ψ⟩
:=
⟩
|φ⟩
=
|x
⟩
=
By
using
new
i
f
h
|U
(T
)|
i
T
ment,
without
disturbing
a
system,
we
can
obtain
the
f
i
4
Uis(Tthe
− t)
·
U
(t)|ψ⟩
⟨φ|U
(T
)|ψ⟩
2point P .
measurement
at
the
mass of the particle,
and
p
is
a
momeni V+(x)| xwx(t)i i̸= xcl (t)
tion amplitud
weak value[12].t = 0 t = T|xi+
gside x direction. The
unitary
evolution
of this
p
i
=
|
. position√x ofin this
(11)system
We
will
put
the
Hamiltonian
i
|ψ⟩ value of the
R Let
C usR
I consider
CI
|φ
the
k ⟩ weak
xf ⌥ x i
1+1
± w h f |pU (T )| ± i
2
s described
σz + σx by
=m
.
=
pprepare
π)
√
(σ
=
+
=
2
[−1,
+1]
w
w
√ of the
|ψ⟩
=
|x
= terms
⟩
|φ⟩
=
|x
⟩
element
of
physical
reality.
we
will
i
f
TEST SPACE
K
TEST SPACE4
h f |U (T )| ± i
T
2
2
ψ⟩ |φ21 ⟩ |φ2 ⟩ |φ3 ⟩ R
α =C0, 1RI t =
αCI0= 1/2
⟩ T|ψ⟩
|φt(10)
t = 0 (11),
t =p
T
k=
the
preand
post-selected
state
as
and
respeciHt
We put
a
point
eigenstate
at
t
=
T
as
a
post-selected
,
(13)
H
=
U (t) = exp
.
(14) ⟨+x
|σconsidered
x |+z⟩
We
time evolution
and
prepare
2m
tively.
}
⟩
|ψ⟩
|φ
⟩
|φ
⟩
|φ
⟩
α
=
0,
1
α
=
1/2
|φ
:
H
⊗
H
→
C
I
:state
H
→1which
R(A)
⊂
R
I
A
1
1
2
3
√
w
The
interferen
=
+1
)
=
(σ
represents
a
point
on
the
+1
x w screen,
state|+z⟩
at
t|ψ⟩
= 0 and a post-selected state
⟨+x
)w = √ = + 2
[−1, +1]
R
C
R
I
C
I
|φ
⟩
k
R|U
C
IRIist)xU
CI Aw(t)|
|φ⊗
|ψ⟩CIthe particle, and p is
:H
→
⟨A⟩ : where
H
→
R(A)
⊂R
!
(↵)
tween
K
k ⟩H
2mass
2probability
+
h
(T
i
m
the
of
a
momen⟨φ
|A|ψ⟩
weakly
measure
a
momentum
p
at
t =
|U
(T
)|
i|
=
nsition
is
|h
f
i
k
trajectory
of
particle
in
2
f
i
=
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
xw (t) |φ=⟩ !
= 1/2
|φ1 ⟩|x|ψ⟩i, |φ1 ⟩ |φ2 ⟩ |φ3 ⟩ α = 0, 1 α(12)
k
f xi
i
=
|
f
f
⟨φ
|A|ψ⟩
|ψ⟩
|φ
⟩
|φ
⟩
|φ
⟩
α
=
0,
1
α
=
1/2
between
(1)
and
(20)
areple,
A = the
p,of| this
i=|
if
we
take
(10)
and
(11)
as
preand
postk
1 tum h
1 f2|U (T
2 )| 3x
alongside
direction.
The
unitary
evolution
⟨+x |σ⟨φ
fpost
i
k |ψ⟩
x |+z⟩
terms
of
the
weak
value
i
T k
=
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
k | + z⟩
|+
σz ⊂σRIxt )|(σAzwi):=
+1
= +1
=
+1
)
=
(σ
wH=
⊗ i.
H→
CI(σx )w the
: Hx⟩
→does
R(A)
x
w
⟨φk⟨A⟩
|ψ⟩
|
Because
imaginary
part
tate, respectively.
The
interference
pattern
xfH
xii→ C
i
tion
|x
i.
B
k: H → R(A)
m
⟨+x |+z⟩⟨A⟩particle
:
H
⊗
I
⊂
R
I
A
f
is
described
by
w
x
(t
T
)
tan
t state
i
!
}2 ⟨φTk |A|ψ⟩with the the
i
coincident
where the
|
⟨φ1post-selected
|A|ψ⟩ = xf!
f
†
+
i
(23)
⟨φ
|A|ψ⟩
=
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
1
k
|b⟩
u
(↵)|x
i
=
⟨φ
|A|ψ⟩

k |b⟩ T
f
2
p
⟨φ
|ψ⟩
T
k
⟨ψ|A|ψ⟩
|⟨φ⟨φ
|ψ⟩|
A. Tanaka, PLA (2002)
⟨φ1 |ψ⟩semiclassical
kstudy
of
trajectories
measurement
at
the
point
P
.
kthe=
−igA⊗P
iHt
1 |ψ⟩
⟨φk |ψ⟩
x⟩ σz σx (σz )w = +1k (σx )w = +1
⟨φ|e
|ψ⟩
x
↵
alongs
f
U
(t)
=
exp
.
(14)
A.
Matzkin,
PRL
(2012)
⟨φ
1 |A|ψ⟩
Weprepared
will ⟨φ
put
the
Hamiltonian
of
this
system
We
a
pre-selected
state
at
t
=
0
and
a
post⟨φ
|A|ψ⟩
+
2
|b⟩
}
2 |A|ψ⟩
tum
p
,
p
⟨φ⟨φ1 |A|ψ⟩
⟨φ1 |ψ⟩
w w an
|ψ⟩
selected state
at
t
=
T
and
weakly
measure
x
at
time
t.
2
|b⟩
2
⟨φ
|ψ⟩
2
⟨φ
|ψ⟩
2
1
⟨φ(t
2 |A|ψ⟩
⟨φ|e−igA⊗P
|ψ⟩ (1) and (23)
p
Relations
between
are
A
=
x,
U
T )| f iis= |h |U (T )| i|2
⟨φn |A|ψ⟩
=
The
transition
probability
f
i
⟨φ
|ψ⟩
,
(13)
H
=
2
⟨φ
|A|ψ⟩
h
⟨φ |A|ψ⟩
2
2
creen. for
Orange
and
bluethe
lines
areare finite
to the
weaktwo
value
in theslits
x direction
due
(massive)
particle
passing
through
narrow
S± and later
ability
the case
where
slits
in size and
the particle
lanes.
by a Gaussian
distribution
around
S± . out
will ignore
any
on analy
the y
to form
pattern
(see FIG.1).
make
both
real, the
index (5)
turns
to bean interference We
i To
= |x
i, our
| dependence
ak trajectory and which path information
f
f
of two position
eigenstates
our state at t = superposition
0 by the superposition
of two
localized
" m xchoose
#
x
state,
where
the post-selected state | f i coincident
xi tan ! fT i
onearbitrary
particle
system,
the most
tangible
of physical
quantity
is. (2015)
1will
to disturb
the source
object
system
and
have
a physical
sigfor
, the
.
(11)
=
Idouble
= Imxpfw
slitmlefthand
experiment
measurement
at
the
point
P
T.
Mori
and
I.T.,
Quant.
Stud.
xperiment
√ (|x
=about
+ | − x|xiof
⟩)i .+
T nificance. Einstein et al. |ψ⟩
i ⟩element
said
an
| thisxisyste
i
i physbly the trajectory of the particle, so let us examine
howput
the
weak
value
2the
We will
Hamiltonian
of
p a
| ii =
.
ical
reality
in
[11]:
If,
without
in
any
way
disturbing
the
is by
just
imaginary
partisImdone
pConsider
2
our2index
argument
thethe
double
slit experiment.
adiverges
ete position
xI varies
with
time.
This
by
setting
formally
the
prew , which
2
2
Our
post-selection
at time
tcertainly
= T is then
madeprobabilby the positio
(0)|
|K+ (0)|
system,
we
can
predict
with
(i.e.,
with
p
+
eed
passing
through
two
narrow
slits
S
and
later
hits
the
screen
rence
becomes
completely
K(0)=→U0(t)|φ⟩
and vanishes
±
pwby the
state
retardeddestructive
state
|φ(t)⟩
andput
the
post-selected
, at
Hquantity,
=
2
2
x
=
x
,
We
point
eigenstate
t = T a
ity
equal
to
unity)
the
value
of
a
physical
f
(0)|
|K(0)|
2m then
erence constructive.
pattern (see FIG.1). To make our analysis simpler, we
mally
by
the
advanced
state
|ψ(t)⟩
=
U
(t
−
T
)|ψ⟩
for
0
≤
t
≤
T
.
The
resultant
state
which
represents
a point on the sc
x
x
there
exists
an
element
of
physical
reality
corresponding
i
f
m
t
=
0
by
the
superposition
of
two
localized
states,
xat
tan
i
|φ⟩treat
= |xafof⟩,weak
} T
where m
the
mass
the particle,
to this physical quantity.
Weismay
value as and p
value,
.
(22)
T
1
imeasure= |x
| fThe
tumreality
alongside
x direction.
unitary
evo
f i,
an element of physical
because,
in weak
|ψ⟩ = √ (|xi ⟩ + | − xi ⟩) . corresponding
(6)
to the
point where
the particle
is spotted.
tory and which
path
information
particle
is
described
by
2
⟨φ|U
(T
−
t)xU
(t)|ψ⟩
⟨ψ(t)|x|φ(t)⟩
ment, without disturbing a system, we can obtain the
the variation
of
interference
Ignoring the dynamics
in the
the post-selected
y-direction(12)
which
plays
no e
i
coinci
where
state
|
xw (t) :=
=
f

weak
value[12].
he
imaginary
the
weakby the
n at
time t =part
T is of
then
made
position
eigenstate
at measurement
⟨ψ(t)|φ(t)⟩
⟨φ|U
(T )|ψ⟩
the
following
discussion,
we just
take
account
of the
at
the
point
PiHt
. dynamics
cle
system,
the
most
tangible
source
of
physical
quantity
is
Let
us
consider
the
weak
value
of
the
position
x
w in.2
U (t) = exp
interference fringes correspond in the x-direction described
We
will
put
thewe
Hamiltonian
this s
by
the
Hamiltonian,
H}= pof /2m,
TEST
SPACE
ajectory
of
the
particle,
so
let
us
examine
how
the
weak
value
terms
of
the
element
of
physical
reality.
will
prepare
TEST
SPACE
f the weak
value pw .
general
complex-valued,
but itmass
canofbe
readily
seen
that,
if both |φ⟩along
and x. The time ev
the
particle
and
p
the
momentum
t==T0 (7)
t the
= T state
tpre= 0 and tformally
|φ⟩
= |x
the
post-selected
as (10) and (11), respecf ⟩,. The
x
varies
with
time.
This
is
done
by
setting
pre2
part
of
the
weak
value
p
w xw (t) becomes real andThe
p
re +position eigenstates,
agrees
with
the
classical
|U (T
transition Uprobability
is |h fand
given
by
the
unitary
transformation
(t)
=
exp
(−iHt/!),
tively.
,
H
=
ythe
retarded
state
|φ(t)⟩
= U (t)|φ⟩ and the post-selected
2 m xf xi
and
pwhere
, which
expresses
pthe
w
w
point
the
particle
is
spotted.
cos
ifreads
we take (10) and
2m(11) as pr
tory.
process
the
transition
probability
} T
en
two eigenstates
of=the
point,
anced
state
|ψ(t)⟩
U (t
−which
T )|ψ⟩
for no
0RI ≤
tCI≤
The
resultant
Ressential
C TR
I.|φ
C
Iselected
|φ
|ψ⟩
k ⟩ state,
dynamics
in
the
y-direction
plays
role
in
weak
trajectory
respectively.
The interference
h
|U
(T
t)xU
(t)|
R
C
⟩
|ψ⟩
f
i iwe find
k
"
!
ow,
if
we
instead
have
the
superposition
state
for
|φ⟩
given
in
(6),
where
m
is
the
mass
of
the
particle,
an
x
(t)
=
x
x
m
m
of
the
classical
motion.
The
w
f
i
2
2
cussion, we just take account of the |φ
dynamics
of
free
motion
|φ3 ⟩ h (T
αf=
0,(T
1 )|α==
|U
1 ⟩ |ψ⟩ |φ1 ⟩ |φ2 ⟩ |⟨ψ|U
)|φ⟩|
cos
.
i i 1/2
⟩
|ψ⟩
|φ
⟩
|φ
⟩
|φ
⟩
α
=
0,
1
α
=
1/2
|φ
tum
alongside
x
direction.
The
unitary
2
1
1
2
3
"
#
ue
p
is
equal
to
the
average
w
π!T m xf xi ! T
n described by the Hamiltonian,
H = p /2m, where mmisxthe
f xi
+
HTdescribed
→
CI
⟨A⟩
→
R(A) ⊂ R
I xf t particle
Awx: iH
x−iR(A)
(t
−: H
T
)I tan
pwx
+p
t :,x.H
(tCI⊗ is
) tan
by
⟨φ|U
(T
t)xU
(t)|ψ⟩
+ and p ⟨ψ(t)|x|φ(t)⟩
fw
!
T
} T
the
momentum
along
The
time
evolution
is
then
and
p
,
Re
[p
]
=
pcle
:
H
⊗
H
→
⟨A⟩
→
⊂
R
A
w
w
+ .i (12)
(23)and is
(13) fringes
xw (t) == 2 + i This picture
of =
interference
yields undiminished
ww(t) := w
!
⟨φ
|A|ψ⟩

kT
T
ary
transformation
= exp
and for
the
2present
⟨ψ(t)|φ(t)⟩
⟨φ|U (T )|ψ⟩
T of(−iHt/!),
T
is equivalent
to theU (t)
average
!
=
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
k
|A|ψ⟩
simplistic,
perhaps
one can render it more realisticiHt
by cons
2 ⟨φkand
⟨φ
|ψ⟩
k
ition
probability
reads
= ⟨ψ|A|ψ⟩
|⟨φ
assical path.
U (t)
= exp
.
k |ψ⟩|
k
We
prepared
a
pre-selected
state
at
t
=
0
and
a
post⟨φ
|ψ⟩
sian
distribution
for
the|φ⟩
initial
state
(see FIG.1).
However,
k
} it w
real
imaginary
hen notice that
real
part
x
(t)
corresponds
to
the
average
of
the
mplex-valued,
butthe
it
can
be
readily
seen
that,
if
both
and
"
!
k Re
w
m
⟨φsufficient
selected
state
at
t = T and
weakly measure
x at time
t.the wea
i
1 |A|ψ⟩
2
2 m xf x
our
picture
is
to
capture
the
key
feature
of
|b⟩
|⟨ψ|U
(T
)|φ⟩|
cos
.
(8)
=
eigenstates,
xw (t)
becomes
andthe
with
the
classical
lassical
trajectories
coming
from
two
slits
S(1)
(see(23)
FIG.
2).
Although
± and
⟨φagrees
⟨φ1 |ψ⟩
1 |A|ψ⟩
Relations
between
are
A
=
x, probability
U (t T )| f i =
π!T
!real
Twave-particle
The
transition
is |h f
duality.
|b⟩
LUE
OF THEwith
POSITION
xf xin
i
s
consistent
the
real
part
of
the
momentum
Re
p
(9),
being
2 m
⟨φ
|ψ⟩
i
=
|
i,
respectively.
The
real
part
of and
the
|
i,
and
U
(t)|
w
1
i
⟨φ
|A|ψ⟩
cos
if
we
take
(10)
a
2
nterference yields undiminished fringes and
is
admittedly
too
To put the present case in
the
general
context,
we (11)
introd
} T
the imaginary part
w may be regarded act an indicator of interference effect,
T choose our state at t = 0 by the
of two#localize
bly the
trajectory
of the
particle,
let us examine m
how!thesuperposition
weak "
value
shown
to be
valid in a more
general
context so
[11].
m 2xf xi
2
|⟨xis
|Udone
(T
)|φ⟩|
=diverges
3 1+ 2 cos
the
is just
the
imaginary
part
Im
pwfor
, which
ourindex
observation
on the
weak
trajectories
the setting
simple
twoformally
cases
fmade
etofposition
xI varies
with
time.
This
by
pre6π|ψ⟩
!T = √ (|xthe
!− xiT⟩) .
⟩
+
|
i
rence
becomes
completely
destructive
K(0)=→
and vanishes
be
examined
by the
cases
where more
general|φ(t)⟩
selections
are
considered.
We
now
ed
state
by
retarded
state
U0(t)|φ⟩
and
the
post-selected
2 "
#&
$
%
2
xfTx.i The resultant
m xi
triple
slitadvanced
case, beforestate
going |ψ(t)⟩
to the multiple
slit
later.
mally
by
theconstructive.
= U (tTransition
− Tcase
)|ψ⟩
for 0 ≤ m
t≤
triple slit experiment
+ 4 cos
. the positi
Our post-selection
at time t = T cos
is then made by
! T
! 2T
value,
x = xf ,
le Slit Experiment
Note
that
thediscuss
transition
probability
oscillates
as slits
a function of xf on the scre
ptory
toward
generalization,
we
the
triple
slit
experiment
where
the
and
which
path
information
⟨φ|U (T − t)xU (t)|ψ⟩
⟨ψ(t)|x|φ(t)⟩
|φ⟩ = |xf ⟩,
xeach
=case
:= Thisdouble
previous
slitby
it does
not necessarily
even at the mo
equally fromthe
other.
is realized
choosing
the pre-selected
state as vanish(12)
w (t)
⟨ψ(t)|φ(t)⟩
⟨φ|U (T x)|ψ⟩
f +
1 tangible
cle system,
the most
source
of physical
quantity
is
corresponding
to=the
point where
the particle is spotted.
interference
points
(see
Fig.4).
(13)
|ψ⟩ = √ (|xi ⟩ + |0⟩ + | − xi ⟩)
Im x how
Ignoring
in the
y-direction
which
plays no
ajectory
of
the
particle,
so
let
us
examine
thedynamics
weak
value
3
(t)the
can
be
obtained
an analogous
manner
as
Now, the weakbut
trajectory
xwreadily
general complex-valued,
it can be
seen
that,
if in
both
|φ⟩ and
the
following
discussion,
we
just take account of the dynamic
x
varies
with
time.
This
is
done
by
setting
formally
the
prethe
post-selected
state
|x
⟩
as
before.
Assuming
again
the
free
Hamiltonian,
f
Re is
x
re position
xw (t)
becomes
real and agrees with the classical
sliteigenstates,
case, and the
result
the x-direction
described by the Hamiltonian, H = p2 /2m
y the retarded
(t)|φ⟩
and the post-selected
ansition
probability,state |φ(t)⟩ = Uin
ctory.
! mass "of the particle
# ⟨xand
p(Tthe
momentum
along x. The time
|U
−
t)
x
U
(t)|φ⟩
anced
state
|ψ(t)⟩
=
U
(t
−
T
)|ψ⟩
for
0
≤
t
≤
T
.
The
resultant
f
TEST
SPACE
TEST
SPACE
2x
x
m
m
i
f
2
4 |⟨x
The
weak
value
xw (t)superposition
as
of state
the =
post-selection
xf forin
a (6),
fixed we
t forfind
(t)
weak
trajectory
ow,Fig.
if we
instead
have
for
|φ⟩
given
= the
3 +agiven
2function
cos x
w
f |U (T )|φ⟩|
by
the
unitary
tT=transformation
0 ⟨x t |U
= T )|φ⟩U (t) = exp (−iHt/!), an
0different
t = behaviors
!t =
T
ffrom(T
0 ≤ t < T . Both Re xw and 6π
Im!xTw show distinctively
the double slit
"
#& transition
" m xfprobability
#probability
process
the
reads
"
#
$m
%
2
x
i
case, yielding finite peaks at
thexflocations
where
the
transition
becomes
minimal.
x
m
x
i
i ) tan
− T(t)|ψ⟩
t !
−i (t
t)xU
⟨ψ(t)|x|φ(t)⟩
f t (T x
"
+ 4 cos x⟨φ|U
cos
.
(14)
! T t
x
x
m
m
g(x|φ
= Cx|φfRIk ⟩ +
⟩ f )|ψ⟩ 1 − (13)
.CI(T
xw (t) == ! T+ i R !C2T RI R
f i
(12)
2 ,
i ,k2x
w (t) :=
C
I
|ψ⟩
|⟨ψ|U
)|φ⟩|
cos
.
=
T
T
⟨ψ(t)|φ(t)⟩
⟨φ|U
(T
)|ψ⟩
T
T
a complexascoefficient
function
given
where g probability
= g(xi , xf ) isoscillates
! T
transition
a function
on the
|φby
|φ2 ⟩ but
|φ3 ⟩ unlike
α = 0, 1π!T
α = 1/2
|φof
1 ⟩ xf|ψ⟩
1 ⟩screen,
|φ1 ⟩ |ψ⟩ |φ!1 ⟩ x x|φ"2 ⟩ # |φx3 ⟩$ α = 0, 1 α = 1/2
7the
m
double
slit case
it
does
at
most
sin
sin
2x
hen
notice
that
the
realnecessarily
Re
xseen
(t)Teven
corresponds
to
mplex-valued,
but
it not
can
bepart
readily
that,
both
and
This
ofm!if→the
interference
yields
i vanish
w!picture
2T
: Hundiminished
⊗ H →of
CI the fringes and i
⟨A⟩
:H
R(A)
⊂|φ⟩
R
I destructive
Awaverage
$
$⊗
Re g =
, H→C
I
⟨A⟩#:mH2x→
R(A) ⊂! mR
I x x " Aw# m: H
x
x
oints
(see
Fig.4).
simplistic,
and
perhaps
one
can
render
itstraight
more realistic
eigenstates,
xw (t) becomes
and
agrees
with
the
classical
line
by con
lassical
trajectories
coming
from
the
two
slits
S
(see
FIG.
2).still
Although
!
3 + 2real
cos
+
4
cos
cos
± !2 ⟨φ
T
2Tk |A|ψ⟩
!
! T
=state
⟨ψ|A|ψ⟩
|⟨φ
k |ψ⟩|
(t)
can
be
obtained
an
analogous
as
in
the
double
ak
trajectory
x
|A|ψ⟩
#⟨φmanner
% in!
$&for
w
sian
distribution
the
initial
(see FIG.1).
However,
it
k
!
!
"
"
2
|ψ⟩
s consistent with the real
part
of
momentum
p
in
(9),
being
as
function
of
time
x the
x Re
x ⟨φ
x
m x|⟨φ
m
m
k
w
=
⟨ψ|A|ψ⟩
|ψ⟩|
2xi cos ! T k + cos ! k2T
sin ! T
the
result
is
⟨φ
|ψ⟩
the
key feature of the we
#our
$ it
Im=
g =
− particular),
, capture
nstead
superposition
state
forpicture
|φ⟩
given
in (6),# mwexto$find
!ismksufficient
k
veragehave
(Rethe
xw (0)
0 in
cannot
be
regarded
x x " ⟨φreasonably
m 2x x
cos1 |A|ψ⟩
3 + 2 cos
+ 4 cos
Probability
w
w
f
i
f
f
f
2
i
i
f
i
i
2
i
i
2
i
f
f
i
i
2
i
T
T
!duality.
! 2T
⟨xf |U (T − t) x"wave-particle
U! (t)|φ⟩
|b⟩
#
⟨φ
|A|ψ⟩
xw (t) =
⟨φ1 |ψ⟩
m xf xi 1
(16) context, we intro
xi (t −⟨x
Tf)|U
tan
|b⟩
xf t
(T )|φ⟩
To
put
the
present
case in the general
! T
⟨φ
1 |ψ⟩
+i
.
(13) of post-selected states V †
xw (t) =
"
#
⟨φ2 |A|ψ⟩
U
(T
)|φ⟩
and
consider
the
family
for the real T
and imaginary
t parts,Trespectively. t
p (α)|x
⟨φ2 |ψ⟩ (15)
= xf + g(xi , xf ) 1 −
,⟨φ2 |A|ψ⟩
the imaginary part
actnow
an indicator
of our
interference
effect,
We
illustrate
argument
by the double slit experime
w may be regarded
T
!thetwo
"
#
bly the
trajectory
of the
particle,
so
let
us examine
how
weak
value
shown
to be
valid in a more
general
context
[11].
(massive)
particle
passing through
narrow
slits
S
±i and later
2x
x
m
m
f
2
|⟨x
|U
(T
)|φ⟩|
=diverges
3 +FIG.1).
2 cos
t
the
index
I
is
just
the
imaginary
part
Im
p
,
which
our observation
on the
weak
trajectories
made
for by
the setting
simple
twoformally
cases
fan
toThis
form is
interference
pattern
(see
To make our analy
e ofposition
x varies
with
time.
done
the prew
6π !T
! T
rence
becomes
completely
destructive
K(0)
and
be
examined
by the
cases
where more
general
selections
considered.
now
choose
our
state
at
t =vanishes
0and
by We
the
superposition
of
two localized
ed
state
by
retarded
state
|φ(t)⟩
=→are
U0(t)|φ⟩
the
post-selected
"
#&
$m x x %
2
m
x
triple
slit
case,
before
going
to
the
multiple
slit
case
later.
mally
constructive.
i
by the advanced state |ψ(t)⟩ = U (tTransition
− T )|ψ⟩ for 0 ≤ t ≤fT .i The resultant
triple slit experiment
value,
le
Slit Experiment
Probability
+ 4 cos |ψ⟩ = √1 cos
.
(|x
⟩
+
|
−
x
⟩)
.
! T 2 i ! 2T i
Note
that
thediscuss
transition
probability
oscillates
as
a is
function
of xby
the
scre
f on
ptory
toward
generalization,
we
the
triple
slit
experiment
where
the
slits
and
which
path
information
Our
post-selection
at
time
t
=
T
then
made
the
positio
⟨φ|U (T − t)xU (t)|ψ⟩
⟨ψ(t)|x|φ(t)⟩
xeach
=
:= Thisdouble
previous
it does
not necessarily
even at the mo
equally fromthe
other.
is realized
choosing
the pre-selected
state as vanish(12)
w (t)
xslit
=by
xfcase
,
⟨ψ(t)|φ(t)⟩
⟨φ|U (T x)|ψ⟩
f +
1 tangible
cle system,
the most
source
of physical= quantity is
interference
points
(see
Fig.4).
(13)
|ψ⟩ = √ (|xi ⟩ + |0⟩ + | − xi ⟩)
|φ⟩ = |xf ⟩,
Im
x
w
ajectory
of
the
particle,
so
let
us
examine
how
the
weak
value
3
(t) can seen
be obtained
an analogous
Now, the weakbut
trajectory
xwreadily
general complex-valued,
it can be
that, if in
both
|φ⟩ and manner as
xthevaries
witheigenstates,
time.
done
by
setting
formally
the
prepost-selected
state This
|xf ⟩ asis
before.
Assuming
again
the
free
Hamiltonian,
corresponding
to
the
point
where
is spotted.
Re
x
re
position
x
(t)
becomes
real
and
agrees
withthe
theparticle
classical
slit case, and thew result isw
y the retarded
and the
thedynamics
post-selected
ansition
probability,state |φ(t)⟩ = U (t)|φ⟩
Ignoring
in the y-direction which plays no e
ctory.
!
"
# ⟨x |U (T
−
t)
x
U
(t)|φ⟩
anced
state
|ψ(t)⟩
=
U
(t
−
T
)|ψ⟩
for
0
≤
t
≤
T
.
The
resultant
f
TEST
SPACE
TEST
SPACE
2x
x
m
m
the
following
discussion,
we
just
take
account
of the dynamics
i
f
2
4 |⟨x
The
weak
value
xw (t)superposition
as
a 2function
of state
the =
post-selection
xf forin
a (6),
fixed we
t forfind
(t)
x
weak
trajectory
ow,Fig.
if we
instead
have
the
for
|φ⟩
given
|U
(T
)|φ⟩|
=
3
+
cos
w
f
tT= 0 ⟨x tby
=T
0different
t described
= behaviors
!t = T
|U
(T
in distinctively
the x-direction
the
Hamiltonian,
H = p2 /2m,
ffrom
0 ≤ t < T . Both Re xw and 6π
Im!xTw show
the)|φ⟩
double slit
"
#& " x x #
" along
# x. The time ev
$m
%
2
i p the becomes
fprobability
m
case, yielding finite peaks at
thexflocations
where
the
transition
minimal.
mass
of
the
particle
and
momentum
x
m
x
i
i ) tan
− T(t)|ψ⟩
t
t
−i (t
t)xU
⟨ψ(t)|x|φ(t)⟩
f t (T x
+ 4 cos x⟨φ|U
cos
.
(14)
!
T
unitary
transformation
= exp
and
g(x|φ
= Cx|φ
1−
,
R
I⟩ +
⟩ f )U
|ψ⟩(t)
.CI|ψ⟩
(13)(−iHt/!),
xw (t) == ! T+ igiven
(12)
!by
2Tthe
i ,kx
fR
w (t) :=
R
C
R
I
C
I
k
T
T
⟨ψ(t)|φ(t)⟩
⟨φ|U
(T )|ψ⟩
T
T
process
the
transition
probability
reads
,
x
)
is
a
complex
coefficient
function
given
by
where
g
=
g(x
i f
transition probability
oscillates as a function|φof1 ⟩ xf|ψ⟩
on the
|φ ⟩screen,
|φ2 ⟩ but
|φ ⟩ unlike
α = 0, 1 α = 1/2
0, 1 3 α = 1/2
|φ1 ⟩ |ψ⟩ |φ!1 ⟩ x x|φ"2 ⟩ # |φx32⟩1$ α =
!m x x "
i
7
m f even
m the
i
double
slit case
it
does
at
most
destructive
m
sin
sin
2xi vanish
hen
notice
that
the
realnecessarily
Re
xseen
(t)
corresponds
to
the
average
of
the
mplex-valued,
but
it not
can
bepart
readily
that,
if
both
|φ⟩
and
f i
w! T⟨A⟩ : H !→2TR(A) ⊂ R
I
A2w : H ⊗ H → CI2
(T$ ,)|φ⟩| =
cos
.
$
Re g =
I
⟨A⟩#:mH2x→
R(A) ⊂! mR
I xf xi " |⟨ψ|U
Aw# m: H
x2i ⊗ H → C
oints
(see
Fig.4).
f xi
eigenstates,
xw (t) becomes
and
agrees
the
still
straight
! line
T
lassical
trajectories
coming
from
slits
S
(see
FIG. 2).π!T
Although
!
3 + 2real
cos
4 cos with
cos± !classical
Tthe+ two
T
2Tk |A|ψ⟩
!
!
2 ⟨φ
= ⟨ψ|A|ψ⟩
|⟨φk |ψ⟩|
(t)
can
be
obtained
an
analogous
as
in
the
double
ak
trajectory
x
|A|ψ⟩
#⟨φmanner
% in!
w
k
2 $&
!
!
"
"
2
s consistent with the real
part
of
momentum
pw in
beingof
as (9),
function
timeand is
xi the
xiRe
xf⟨φ
fpicture
m x|⟨φ
m xinterference
k |ψ⟩
This
yields
undiminished
fringes
⟨ψ|A|ψ⟩
k |ψ⟩|
+ cos of
2xi cos
sin =m
k2Ti
T
T
!
!
!
the result
is
⟨φ
k |ψ⟩
# k for |φ⟩
$ it
Im=
g =
− particular),
,
nstead
superposition
state
given
in "(6),#one
wex2i $find
!
veragehave
(Rethe
xw (0)
0 in
cannot
be it
regarded
simplistic,
and
perhaps
can
render
more realistic by consi
m 2xf xi
m xf xi ⟨φreasonably
m
|A|ψ⟩
1
cos
3
+
2
cos
+
4
cos
T
! T
! 2T
⟨xf |U (T − t) x" U! (t)|φ⟩
|b⟩
#
sian
distribution
for
the
initial
state
(see FIG.1). However, it w
⟨φ
|A|ψ⟩
xw (t) =
⟨φ1 |ψ⟩
m xf xi 1
(16)
xi (t −⟨x
Tf)|U
tan
|b⟩
xf t
(T )|φ⟩
!
T
our"picture
capture the key feature of the wea
⟨φ
|ψ⟩sufficient⟨φto
+i
. 1is
(13)
xw (t) =
#
2 |A|ψ⟩
for the real T
and imaginary
t
t parts,Trespectively.
wave-particle
⟨φ2 |ψ⟩ (15)
= xf + g(xi , xf ) 1 −
,⟨φ2duality.
|A|ψ⟩
†
†
A(t)
U in
(tithe
)|ψ⟩
)
A(t)
U
(t
)|φ⟩
⟨ψ|U
(t
(tby
ch ⟨φ|U
we
shall
following
sections.
f ) discuss
i
f
erence
becomes
completely
destructive
K(0)
→
0
and
vanishes
⟩
|ψ⟩
|φ
⟩
|φ
⟩
|φ
⟩
α
=
0,
1
α post-selected
= 1/2
|φ
ed
state
the
retarded
state
|φ(t)⟩
=
U
(t)|φ⟩
and
the
1 +(1−α)1
2
3
=α
|φ2resultant
⟩ |φ3 ⟩
|φ 0⟩ †≤
mally
constructive.
1t )|ψ⟩
(tf ) U †state
(ti )|ψ⟩
(t|ψ⟩
i ) 1U
f
by the⟨φ|U
advanced
|ψ(t)⟩ = U (t − T⟨φ|U
)|ψ⟩(tfor
≤ |φ
T 1. ⟩The
:H⊗
Hi ⟩→+CI| − xi ⟩) .
⟨A⟩ : H → R(A) ⊂ R
I |ψ⟩ =
Aw √
(|x
evalue,
Slit Experiment and the Reality ⟨A⟩
Condition
: H → R(A)2⊂ R
I
Aw : H ⊗ H → CI
!
multiple
slit
or
general
case
′
† TEST
′
† 2 ⟨φk |A|ψ⟩
SPACE
= Ugeneral
(t−t ) =
U (t)U
(t )there
A(t)
:=
(t) A
Ux(t)
|φ
|ψ⟩
=x⟨ψ|A|ψ⟩
|⟨φN
|ψ⟩|
k.⟩|A|ψ⟩
kU
!
the
case
where
are
slits
at
=
,
x
,
.
.
,
xmade
we
1
2
N . Here
⟨φ
tory and which Our
path
information⟨φ|U
⟨φ−k |ψ⟩
k
2is then
post-selection
at
time
t
=
T
by
the
position
(T
t)xU
(t)|ψ⟩
⟨ψ(t)|x|φ(t)⟩
k
=
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
k
t = 0= t =†T + V (x) xw (t) ̸= xcl (t)
x
(12)
(t)
:=
preselection
w
nding pre-selected
state,
⟨φ
|ψ⟩
postselection
k
U⟨ψ(t)|φ(t)⟩
(t,
t
)|φ⟩
⟨φ|U
(t,
t
)
U (t, ti )|ψ⟩x =
xf , f
f k(T )|ψ⟩
⟨φ|U
⟨φ
|A|ψ⟩
1
cle system, the mostN tangible source of physical quantity
is
|b⟩
!
⟨φ
⟨φ1 |A|ψ⟩
1
1 |ψ⟩
ajectory
of
the
particle,
so
let
us
examine
how
the
weak
value
R
C
R
I
C
I
|φ
⟩
|ψ⟩
k
general
but
it
beCcan
that,
both
|φ⟩ =if|x
f ⟩, |φ⟩ and
√ cweak
ted
statecomplex-valued,
as |x
trajectory
thenseen
be written
as
⟩,
c+
,readily
(17)
|φ⟩
(|1⟩
|2⟩can
|3⟩)
|ψ⟩
=The
f ⟩.=
n |x
n+
n ∈
⟨φ1 |ψ⟩
x varies with time. This
by setting
the pre|A|ψ⟩
|φ1 ⟩ |φ2⟨φ
⟩formally
|φ1 ⟩ |ψ⟩
2 |φ
3 is done
3 ⟩ α = 0, 1 α = 1/2
re position eigenstates,
n=1 xw (t) becomes real and agrees with the classical
y the retarded state
|φ(t)⟩ = ⟨A⟩
U (t)|φ⟩
andpoint
the
post-selected
⟨φ
|ψ⟩
corresponding
to
the
where
the
2
→2 |A|ψ⟩
CIparticle is spotted.
:
H
→
R(A)
⊂
R
I
Aw : H ⊗ H
⟨φ
ctory.
⟨x
|U
(T
−
t)
x
U
(t)|φ⟩
1 (t −f T )|ψ⟩
8
vanced statex|ψ(t)⟩
=
U
for
0|3⟩)
≤ t ≤⟨φT |A|ψ⟩
. The resultant
!
(t)
=
√
|φ⟩
=
(|1⟩
+
|2⟩
−
w
Ignoring
the
dynamics
in
the
y-direction
which
plays
no
es
⟨φ2 |ψ⟩
⟨φ
k
2state
ow, if we
instead
have
the
superposition
for
|φ⟩
given
in
(6),
we
find
n |A|ψ⟩
=
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
weak trajectory
⟨xf |U (T )|φ⟩
k
3
⟨φk |ψ⟩
⟨φ
|ψ⟩ just
k
the following discussion,
take
account of the dynamics
nwe
"
#
|A|ψ⟩
⟨φ
n
N
x
x
f i
m!
N
2
!
x
(t
−
T
)
tan
⟨φ
|A|ψ⟩
t
x
⟨φ|U
(t)|ψ⟩
Pi⟨ψ(t)|x|φ(t)⟩
= |i⟩⟨i| ini the
= 1,
2, 3(Tn−i t)xU
Pi K(α)
=
|c1 i=⟩⟨c
|
fx-direction
x
→x
described
by
the
Hamiltonian,
H
=
p
/2m, w
i
n
f
!
T
⟨φ|u
(α)|ψ⟩
|b⟩
⟨φ
|ψ⟩
A
n
+
i
.
(13)
x
(t)
=
=
(12)
(t)
:=
xw⟨φ(t)
ω n xw
(t)
w =
w
ω
x
(t),
(18)
n
1 |ψ⟩=
w
⟨ψ(t)|φ(t)⟩
⟨φ|U
(T )|ψ⟩ and
T the
T p the momentum along x. The time evo
mass of
particle
Kk (α) = ⟨φ|u
n=1K(α)
= ⟨φ|uA (α)|ψ⟩
n=1 ⟨b|Pi |a⟩
A (α)|χ
k ⟩⟨χk |ψ⟩
⟨φ2 |A|ψ⟩
(P
)w by
=part
given
theRe
unitary
transformation
U (t)
= expof(−iHt/!),
and f
ireal
⟨φ2if
|ψ⟩both |φ⟩
hen
notice
that
the
x
(t)
corresponds
to
the
average
the
mplex-valued, but it can be readily
seen
that,
and
w
⟨b|a⟩ |χ1 ⟩ |χ2 ⟩ K|χ
p ⟨φ|uA (α)|χk ⟩⟨χk |ψ⟩
3⟩ =
(α)
k
the
transition
reads
roduced
the xweight
factor,
⟨φprobability
eigenstates,
(t)process
becomes
real
andthe
agrees
with
classical
n |A|ψ⟩the
−igA
‘weak
quasiprobability’
with
lassical
trajectories
coming
from
two
slits
S
(see
FIG.
w σ12). Although
w
±
ΨMD
(0,⟨φ|U
0) (T e)
ΨMD (0,
0)
g  σ1 ⟨φ(T
)|
⟨φn=|ψ⟩
"
!
⟩ in
|χ3(9),
⟩ p being
|χ1 ⟩ Re|χp2m
s consistent with the real part of the momentum
w
x
x
m
f xi
2
2
K(α)
=
⟨φ|u
(α)|ψ⟩
iφ
A|ψ⟩
=
a|ψ⟩
A|φ⟩
=
a|φ⟩
A
c
⟨x
|U
(T
)|x
⟩
|⟨ψ|U
(T
)|φ⟩|
cos
|ψ⟩
⟨φ(T
)|En |ψ⟩ .
⟨φ(T
)|x
nstead
superposition
state
for
|φ⟩
given
in
(6),
we=
find
n
n
f particular),
n ⟩⟨x+
nbe
=
Ψ
(θ,
φ)
=
cos
θ
|0⟩
e
sin
θ
|1⟩
Ψ
veragehave
(Rethe
xωw (0)
=
0
in
it
cannot
reasonably
regarded
MD
MD
= !(19)T
⟨φ(T )| .= ⟨φ|U (T )
π!T
n → ⟨xn |ψ⟩
n = c"
(α)
=
⟨φ|u
(α)|χ
⟩⟨χ
|ψ⟩
K
k
A
k
k ⟨φ(T )|ψ⟩
⟨φ(T )|ψ⟩
"(T
c
⟨x
|U
)|x
n
xf xni ⟩#
f
m
n
⟨φ |x ⟩⟨x |ψ⟩
⟨A⟩ := ⟨ψ|A|ψ⟩
n n
n
xi (t −
T
)
tan
xf t This
"
#
:=
A
=
B
+
C
A
=
B
+
C
A
=
B
A
!
T
⟩
|χ
⟩
|χ
⟩
p
|χ
c
→
⟨x
|ψ⟩
U
(T
)
→
1
1
2
3
picture
of
interference
undiminished
is a
w
w
w
w
w fringes
w Cand
w
n
n yields
+i
.
(13)
xw (t) =
!
1
∂
1
⟨φ|ψ⟩
2
2
T
T
n
lim
|K(α)|
. ⟩,
I
:=
−
|K
(α)|
⟨φ(T
)|
=
⟨φ|U
(T
)
k
simplistic,
and
perhaps
one
can
render
it
more
realistic
by#consid
weak trajectory
with
the
pre-selected
state
|x
i.e.,
on xw (t) is just the
2
n
"
α→0
2
|K(α)|
∂α
Aw := A = B C A
=
B
C
a
∈
R
w
w
w
i
x
k
!
|ψ⟩
⟨φ(T
)|E2n |ψ⟩
⟨φ|x
n ⟩⟨x
n∂
1 state
1⟨φ|U
that the real part sian
Re xwcdistribution
(t)
corresponds
to
the
average
of
the
for
the
initial
(see
FIG.1).
However,
2 it wi
=
→
⟨x
|ψ⟩
⟨φ(T
)|
=
(T
)
n
n
|K(α)|
I := lim
−
|Kk (α)| .
⟨φ|ψ⟩
⟨φ(T
)|ψ⟩
2
α→0
jectories coming
the
S±
(see ⟩2FIG.
Although
|K(α)|
⟨xfx|U
− slits
t)isx U
(t)|x
(P
)capture
= feature
−1
(P
our
sufficient
to22).
the3 )#w
key
of the weak
−
)t(Ttwo
"
n (xffrom
1 )wn=
w = 1 ∂α(P
ipicture
k
| ↑⟩1
| ↓⟩2
what we have learned so far:
|R⟩1
|L⟩2
| ↓⟩2
|R⟩1
|L⟩2
1
√
(| ↑⟩1 + i| ↓⟩1
|R⟩
=
1
1
|R⟩1 = √ (| ↑⟩1 + i| ↓⟩12)
2
1
|L⟩2 = √ (| ↑⟩2 − i| ↓⟩2
1
|L⟩2 = √ (| ↑⟩2 − i| ↓⟩22)
2
it is given by the average over respective ‘classical’ weak
trajectories weighted with ‘weak quasiprobability'
xf
xf
= Re[xw (T )]
imaginary part describes the degree of interference
(more next)
Rex̃ (t)
= Re[xw (T )]
Rex̃±
w (t)
Rex̃+
w (t)
Rex̃−
w (t)
Re x̃±
w (t)
further example: Lloyd’s mirror
Re x̃+
w (t)
Re x̃−
w (t)
Rex̃±
w
Rex̃+
w
x
Rex̃−
w
Re xw
lized around (xi , y
S+
0
y
=
xf +
Im xw
Im xw (0)
±
w
Rex̃+
w (t)
Rex̃−
w (t)
Re x̃±
w (t)
Re x̃+
w (t)
Re x̃−
w (t)
Rex̃±
w
Rex̃+
w
Rex̃−
w
Im [xw ]
(13.34)
Re [xw ]w
(13.35)
Im xtw
(13.36)
Re x
Im xTw (0)
Im [xw ]
(13.37)
0
Re [xw ]
(13.38)
Im [xw ]
S+
Re [xw ]
S−
t
j(x; t) (13.34)
1
⟨x|p
=
Re
ẋj(x;
= v(x;
t)
≡
(13.39)
t
t)
1 P (x;
⟨x|p|ψ(t)⟩
t)
m = 1⟨x|
Im [x
= Re
R
ẋ w
=] v(x; t) ≡
(13.35)
m ! ⟨x|ψ(t)⟩
m
(13.40)
TP (x; t)
Re [xw ]
(13.36)
! j(x; t) = m Im[⟨ψ(t)|x⟩∇⟨x|ψ(
j(x; t) = 0 Im[⟨ψ(t)|x⟩∇⟨x|ψ(t)⟩]2
m P (x; t) = |⟨x|ψ(t)⟩| (13.37)
t
P (x; t) = |⟨x|ψ(t)⟩|2
T
(13.38)
: Lloyd’s mirror experiment. The particle is localized around S+ at time t = 0
T
d at xf at time t = T . In classical optics, the light can take the path toward the
0
the wall (mirror) and straight forwardly toward the
screen. This particle cause
0 for a number of different
Figure 4.8: The weak value for position xw (t) in the complex plane
e on the screen, and the orange filled curve represents
the interference
fringes
(the to the transition probability |⟨xf |U (T )|xi⟩|2.
post-selections
with the density
proportional
real and imaginary
are described in orange and green lines and projected on the
probability) especially when one choose the position The
eigenstate
as theparts
pre-selected
bottom and the left-back planes, respectively.
Y. eigenstates
Aharonov [8].
ichbyare
forFirst, this time symmetric formalism is evoked though the
idea that what is predicted by quantum mechanics is not results of measurements
tebut
as |q
= 0⟩ localized
The one of the time symmetric formalisms of quantum
only probability distributions which are expressed time symmetrically (conai ⟩stant
by the
interaction.
byinitial
Y.formalism
Aharonov
[8]. Let
First,
this time symmetric formalis
under
changing symmetrically both
and final states).
|ψ⟩ and
2.1
Time
symmetric
When
knowand
which
|φ⟩ bewe
initial
final state in the system.
of the
ideaThe
thatprobability
what is amplitude
predicted
by quantum mechanics is not
weak
value
a final
correction
to⟨φ|U
transition
probability
process from
the initial
state as
to the
state under
some
interactions
is written
tates.
K(s)
:=
(s)|ψ⟩
A
but
only probability
distributions
which are expressed t
formalisms
of quantum
mechanics was brought
as, The one of the time symmetric
For example,
we con- [8]. First,
2under changing
by Y. Aharonov
time
formalism
is evoked
though
the 2 initial and fin
−
ti )|ψ⟩symmetric
(2.1)P (s)
⟨φ|Uthis
(tA
stant
symmetrically
both
f (s)|ψ⟩|
UA (s)
= e−isA
= |K(s)|
P (s) = |⟨φ|U
as is predicted by quantum mechanics
idea that
what
is not resultstransition
of measurements
which
have
{|w ⟩}
ormalism
amplitude
postselection
and
final
of the
system.
Generally,
where ti and tf i represent the time of initial
|φ⟩
be
initial
and
final
state
in
the
system.
The
proba
2
2
2
but
probability
distributions
which
expressed
time
symmetrically
(conP (s)
/2)A
+ · ·and
·)|ψ⟩|
|⟨φ|(1
− isAare
− the
(s
areonly
of we
{|winterpret
i ⟩}
this process as
the =
unitary
evolution
of
initial
state
the
!integers
process
from
theand
initial
state toLet
the|ψ⟩final
state under som
K(s)
:=
⟨φ|U
(s)|ψ⟩
stant
under
changing
symmetrically
both
initial
final
states).
and
2
A
c
|w
⟩
nal
state
be
projection
measurement
to
the
final
state.
However
this
process
is
also
interP (0) was brought |⟨φ|ψ⟩|
K(s) := ⟨φ|UA (s)|ψ⟩
i i i mechanics
alisms |φ⟩
of quantum
initial
and measurement
final state into
the
system.
amplitude
of the
! probability
"
as,
preted
as be
the
projection
the
state U (t2The
te
prepared
at
t
=
−T
i − tf )|φ⟩.
2 These two
22
3
2
−isA
ymmetric
formalism
is
evoked
though
the
=
1
+
2sImA
+
s
|
−
Re(A
)
)
|A
+
O(s
2
−isA
(s)|ψ⟩|
U
(s)
=
e
P
(s)
=
|K(s)|
(s)
=
|⟨φ|U
wstate under
wsome
w
process
the
initial
state
final
interactions
is
written
A from
A to the
interpretations
imply
that
the
probability
of
the
quantum
process
is
unchanged
(s)|ψ⟩|
U
(s)
=
e
P (s
P
(s)
=
|⟨φ|U
−
t
⟨φ|U
(t
tion Hamiltonian Hint
A
A
f
i )|ψ⟩
m mechanics
is notinitial
results
measurements
transition probability
under as,
exchanging
andof
final
Quantum
mechanics
directly predicts
:= ⟨φ|U
2state.K(s)
2
2A (s)|ψ⟩
(s)
+
·ft· −
·)|ψ⟩|
|⟨φ|(1
− isA
− (s /2)A
2of
2
2 of
t
)|ψ⟩
(2.1)
⟨φ|U
(t
the
probability
of quantum
processes,
not
the
values
of
results.
We
and
t
represent
the
time
initial
and
final
where
imeasurement
chP
are
expressed
time
symmetrically
(conP
(s)
/2)A
+
·
·
·)|ψ⟩|
|⟨φ|(1
−
isA
−
(s
i
f
g|A2w | ≪ 1 |φ⟩ Ψ(Q)
Φ(Q) Q
=
−isA
2
2
=
have
no
reason
to
determine
how
measurements
affect
the
system.
The
proba(s)|ψ⟩|
U
(s)
=
e
P
(s)
=
|K(s)|
P
(s)
=
|⟨φ|U
P
(0)
|⟨φ|ψ⟩|
A
A
preselection
both initial
final
states). Let
|ψ⟩
and
2
we
interpret
this
as
the
unitary
evolution of t
tf represent
the time
of initial
and
finalprocess
of the system.
Generally,
whereand
ti and
P
(0)
|⟨φ|ψ⟩|
(2.8)
bility of quantum processes has time
If so, it must
! symmetric
" A (s)|ψ⟩
K(s)characteristics.
:= ⟨φ|U
2 as the
2unitary
2 2 of the 3
! the
"
we
interpret
this process
initial
statefinal
and
ystem.
The
probability
amplitude
of
the
2evolution
2 the
projection
measurement
to
state.
However
th
2
2
2
be reachable
to
construct
a
time
symmetrical
formalism
of
quantum
mechanics.
=
1+
2sImA
+
s
|
−
Re(A
)
)
|A
+
O(s
P
(s)
/2)A
·
·
·)|ψ⟩|
isA
−
(s
w |⟨φ|(1 − w
w=+
1+
2sImA
s inter| −
Re(A
)w
|Aetwal.,
projection
measurement
to to
the
final
state.
However
this
process
w )is+also
2
−isA
2
=
al
state
under
some
interactions
is
written
=
Ex
(Φ)
−
Ex
(Ψ)
s|A|
∆
Dressel
RMP
(2014)
relative
change
due
the
unitary
action
(for
small
Q
asQUformulation
the
projection
measurement
to the state U (
to a (s)|ψ⟩|
timepreted
symmetric
=Qeof quantum
P (s) = |K(s)|
Pthey
(s)reached
= |⟨φ|U
rvableRecently,
A has discrete
2(s)
A state U (ti − tf )|φ⟩. These two
preted as P
the
projectionAmeasurement
to the
(0)
|⟨φ|ψ⟩|
mechanics,“Two-state
of quantum
mechanics”
[9].imply
They formalized
!
" is2probability
interpretations
that
the
of the quantu
localized
states which formalism
interpretations
imply
that
the
probability
of
the
quantum
process
unchanged
2
2
2
3
2
2
g|A
|
≪
1
|φ⟩
Ψ(Q)
Φ(Q)
Q
thetphysical
states
by (s)= 1 +
wP
2sImA
+ sexchanging
| −mechanics
Re(A
)w
)
|Aw
+|φ⟩
O(s
/2)A
+w· |· ·)|ψ⟩|
|⟨φ|(1
−w(2.1)
isA
−Quantum
(s
−
)|ψ⟩
g|A
≪
1
Ψ(Q)
Φ(Q) mec
Q
f|m
i
⟩
means
that
it’s
under
exchanging
initial
and
final
state.
directly
predicts
i
under
initial
and
final
state.
Quantum
2
=
ExQnot
†(Ψ) := ⟨Ψ|Q|Ψ⟩/∥Ψ∥
2 of measurement
the
probability
of
quantum
processes,
the
results. We
)|ψ⟩⟨φ|U
(t −
tf )values
℘(t)
ˆ
:
=
U
(t
−
t
⟩.
ates
prepared
in
|m
i
P
(0)
|⟨φ|ψ⟩|
0
initial and final of the system. Generally,
the probability
of
quantum(2.2)
processes,
not the values of m
!
"
have of
nothe
reason
to determine how measurements
affect the system. The
probaHamiltonian
all
2
2
2
3
arywhere
evolution
of
the
initial
state
and
the
g|A
|
≪
1
|φ⟩
Ψ(Q)
Φ(Q)
Q
|ψ⟩ is initial state, |φ⟩
is1final
state
of# the
quantum
process.
This
state how
have
no
reason
to
determine
measurements
affect
w
=
+
2sImA
+
s
|
−
Re(A
)
)
|A
+
O(s
bility
of
quantum
processes
has
time
symmetric
characteristics.
If
so,
it
must
=
Ex
(Φ)
−
Ex
(Ψ)
∆
w
w
w
Q
Q
Q
tem at the
measuring
Extime
− ExQ (Ψ) charac
∆
# of quantum processes
Q =has
Q (Φ) symmetric
d∆
state.
However
this
process
is
also
interQ
bility
formalism
of quantum
#
= ReA
+ C(Ψ)
· ImA , mechanics.
, we seebe
thereachable
evolutionto construct a time symmetrical
w
w
7
#two
−
t
)|φ⟩.
These
t to the state
U
(t
dg
i
f
to construct
a time
symmetrical formalism
g=0
Recently,
time symmetric
formulation
of quantum
e retarded state
arrives they reached tobea reachable
g|A
|≪
1 Q|φ⟩
Ψ(Q)
Φ(Q)
Q
ility of mechanics,“Two-state
the quantum processformalism
is unchanged
wQ
=
Ex
(Φ)
−
Ex
(Ψ)
∆
of
quantum
mechanics”
[9].
They
formalized
2
Q
ation with the system
with
a
constant,
2
(Ψ)
:=
⟨Ψ|Q|Ψ⟩/∥Ψ∥
Ex
Recently,
they
reached
to
time symmetric
fo
Q
ExQ (Ψ) :=a ⟨Ψ|Q|Ψ⟩/∥Ψ∥
te. Quantum
mechanics
the physical
states
bydirectly predicts
mechanics,“Two-state
formalism of quantum mechanic
not the values of measurement
results.
We
†
C(Ψ) := Ex
(Ψ) − 2Ex (Ψ) · Ex (Ψ),
#
}
(t − tfQ)
℘(t)
ˆ : = U (t −{Q,P
ti )|ψ⟩⟨φ|U
P
(2.2)
I :=
⟨ψ|A|ψ⟩
|⟨φk |ψ⟩|⟨φk |ψ⟩
R
I
CI
|φk ⟩ |ψ⟩
⟨φ
|ψ⟩= C
kR
⟨φ
|ψ⟩
k
⟨φ
|A|ψ⟩
k
k
⟨φ
|A|ψ⟩
1
2
the measurement is negligible.
Therefore,
the parameter
k
limit
of⟩ ↵ ! 0 .
|b⟩ |φ ⟩the
|φ
⟩
|ψ⟩
|φ
⟩
|φ
1
1
2
3
⟨φ
|ψ⟩|φ1discuss
1will
⟩
|ψ⟩
|φ
⟩
|φ
⟩
|φ
"
↵ is also infinitesimal.
From
here,
because
we
⟨φ
|A|ψ⟩
1
2
3 ⟩⟩
⟨φ1⟨φ
|A|ψ⟩
⟨φ⟨φ|E|ψ⟩
1
⟨ψ|E|φ⟩
2 |ψ⟩
|φ
⟩
|ψ⟩
|φ
⟩
|φ
⟩
|φ
|A|ψ⟩
"
#
1
1
2
3
n
1
|b⟩
|b⟩
X
|b⟩
+⟨φ(1
−
α)
µ(E)the
= contribution
tr(W E) = α
|K
⟨φ→
⟨φ
|ψ⟩
about
from
interference
e↵ect,
we
interk (0
2 |A|ψ⟩
|ψ⟩
k
⟨A⟩
:
R(A)
⊂
R
I
A
:
H
⊗
H
→
C
I
1H
!
1
w
⟨φ
|ψ⟩
1
I
A2w⟨φ: H
⊗ H → CI ⟨A⟩
1 ⟨A⟩ : H →
1 R(A)
Im A
AAwww: :HH⊗⊗HH
A→
→
R(A)
⊂⊂R
IR
ICI
⟨φ|ψ⟩
⟨ψ|φ⟩
2⟨A⟩: H
w→C
† ∂ ⊂ R
:
H
→
R(A)
I
|A|ψ⟩
⟨φ
|ψ⟩
n
2
|K(0
as the variation
of
post-selected
state
lim pret u2A (↵)| i|K(s)|
.
−
|Kthe
⟨φ
|A|ψ⟩
k (s)|
⟨φ2⟨φ
|A|ψ⟩
!
2
k=1
2 |A|ψ⟩
!
k |A|ψ⟩
!| i and∂s
2 ⟨φ⟨φ
|A|ψ⟩
2 s→0 |K(s)|
!
⟨φ
|ψ⟩
⟨φ
|A|ψ⟩
observe
the
changing
of
the
transition
probabilkk
⟨φ
|A|ψ⟩
n
2 2 Jordan,
==⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
n
⟨φ
|A|ψ⟩
k
k
⟨φ
|ψ⟩
2
Dressel,
PRA
(2012)
⟨φ
|ψ⟩
2
2⟨φ2 |ψ⟩ |⟨φk |ψ⟩|
⟨ψ|A|ψ⟩
k
=
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
=
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
⟨φ
|ψ⟩
k
k
k
⟨φ
|ψ⟩
⟨φ
|ψ⟩
ity.
n
⟨x|A|ψ⟩
k
k
⟨φ
|ψ⟩
transition
amplitude⟨φ
through
intermediate
states
k
k
k |ψ⟩
T.
Mori,
I.
T.,
PTEP
(2015)
|A|ψ⟩
⟨φ
k
|A|ψ⟩
⟨φ
(g)|ψ⟩
K(g)
=
⟨φ|U
n
A(λ)
=
k To treat quantum
n
|A|ψ⟩
⟨φ
A we decompose
n = ⟨φ|U
interference,
the
tranK(g)
(g)|ψ⟩
A
The variation of cross
⟨x|ψ⟩ ⟨φn⟨φ|ψ⟩
⟨φ⟨φ
⟨φ
|ψ⟩
1 |A|ψ⟩
n
|A|ψ⟩
|ψ⟩
n
sition amplitude K(↵) by using an orthonormal set of ⟨φ1 1 |A|ψ⟩
imaginary part relates to interference
Kk (s) = ⟨φ
⟨φ|U
1 |A|ψ⟩
A (s)|χ|b⟩
k ⟩⟨χk |ψ⟩
between the imaginar
⟨φ⟨φ
|ψ⟩
1⟨φ
"A#
# |K (0)|2
P k |Kk
(g)|ψ⟩
K(g)
=
⟨φ|U
states
1 |ψ⟩
(g)|ψ⟩
K(g)
=
⟨φ|U
" {| i}.
|ψ⟩
A
(g)|ψ⟩
K(g)
=
⟨φ|U
"
1
A
k
k
⟨ψ|A|φ⟩
⟨φ|A|ψ⟩
!
1! 1 ∂
A
Aw |K(0)|2 .
Aw |K
1 |ψ⟩
+
(1
−
α)
a1i⟨φ
µ(E
)
=
tr(W
A)
=
α
λ(A) =
X
I
:=
lim
P
(s)
−
P
(s)
1
∂
k
|A|ψ⟩
⟨φ⟨φ
i
2⟨φ
2 2 s→0 P (s) ∂s 2
|A|ψ⟩
2
|A|ψ⟩
2
1
2
3
⟨φ|ψ⟩
⟨ψ|φ⟩
lim
|K(s)|
.
I :=
−
|K
(s)|
tribution
from the eac
I
=
|
ih
|
(4)
intermediate
states
k
"
k
"
## #
k
k
i 2 s→0
#
⟨φ2|K(s)|
|A|ψ⟩ 2 ∂s 1 1 1 1 1 1∂" ∂ ∂"
⟨φ⟨φ
|ψ⟩
2⟨φ
!
!!
22|ψ⟩
2 |ψ⟩
from
!
2
k
2
2 2 the variation of to
k
2
P
(0)
k
lim
|K(s)|
I
:=
−
|K
(s)|
k
lim
|K(s)|
I :=
−
|K
(s)|
|K(s)| −A |K
I := lims→0
k . . .
k k (s)|
2 ∂s
−
=
Im
A
2
2
w
⟨φ2 |ψ⟩
s→0
w
2
|K(s)|
2 2 s→0
|K(s)|
∂s ∂s
|K(s)|
because we can expres
|A|ψ⟩
P (0)k k⟨φk⟨φ
|A|ψ⟩
n⟨φ
|A|ψ⟩
nn
total probability
|χ ⟩
|χ ⟩
|χ ⟩
p
⟨φ(T )| = ⟨φ|U (T )
k
The absolute square of the transition amplitude
K(↵) is
value
in terms of the l
⟨φ
|ψ⟩
⟨φ
|ψ⟩
n
⟨φ
|ψ⟩
n
x
|A|ψ⟩
⟨φ
n
K(s)
⟨φ|UAtransition
(s)|ψ⟩ |K(0
†
† :=
!
! U ⟨φ(T
⟩⟨x
|ψ⟩
|ψ⟩
⟨φ(T
)|E
)|x
2
ability
n
n
(t
)|ψ⟩
)
A(t)
U
(t
)|φ⟩
⟨ψ|U
(t
⟨φ|U (tf n) A(t)
n
i
i
f
∗ ! !!!
X
!!!
=
xn |ψ⟩ λ(A)
(T
)
P⟨φ(T
(s)==)|
P⟨φ|U
K
(s)K
(s)
P
(s)
P
(s)
P
(s)
+(1−α)
α = |K(↵)|
⟨φ+
|ψ⟩
(α)|ψ⟩
K(α)
=
⟨φ|u
∗
∗ ∗∗(s)
(α)|ψ⟩
K(α)
=P
k (s)
k
1
2
3
2n P (s)
(α)|ψ⟩
K(α)
=2⟨φ|u
⟨φ|u
A
j
A
21†
−isA
ence
pattern
A
P
(s)
=
P
(s)+
(s)K
P
(s)
P
(s)
PN will make
P
(s)
=
P
(s)
+
K
(s)K
(s)
P
(s)
P
(s)
P
=
P
(s)
+
K
(s)K
(s)
P
(s)
P
(s)
P
(s)
P
(s)
=
P
(s)
+
K
(s)K
(s)
P
(s)
(s)
P
k
k
1
2
3 (s)
†
k
2
3 (s)
j
=
h
|u
(↵)|
ih
|
i
k
k
1
2
3
k
k
1
3P(s)
j
j
j
A
k
k
(s)|ψ⟩|
U
(s)
=
e
P(
P
(s)
=
|⟨φ|U
⟨φ(T
)|ψ⟩
⟨φ(T
)|ψ⟩
⟨φ|U
(t
)
U
(t
)|ψ⟩
⟨φ|U
(t
)
U
(t
)|ψ⟩
A
A
f j̸=k i
i
f
k
k
j̸
=
k
j̸
value
enormous.
It sh
k k k
j̸=kj̸=k
k
K
(α)
=
⟨φ|u
(α)|χ
⟩⟨χ
|ψ⟩
K
(α)
=
⟨φ|u
(α)|χ
⟩⟨χ
|ψ⟩
K
(α)
=
⟨φ|u
(α)|χ
⟩⟨χ
|ψ⟩
K(g)X
= ⟨φ|UA (g)|ψ⟩
k kk
A AA 2 k kk 2k kk
2
‘off-diagonal’2 P (s)
e↵ect
expressed
in term
/2)A
+
·
·
·)|ψ⟩|
|⟨φ|(1
−
isA
−
(s
⟨φ|A|χ
k ⟩ k ih2 k | i|
=
|h |uA (↵)|
=
k
|χ⟩1k1⟩|χ
⟩ |χ
|χ
|χ
|χ
|χ
|χ
values.
|χ
⟩2⟩|χ|χ
2
K
(s)
=
⟨φ|U
|ψ⟩
|χ
⟩
⟩
⟩
2
1
2
333⟩⟩⟩ p |χN ⟩
‘intensity’ of A
interference
k= ⟨φ|U
A (s)|χ
k1⟩⟨χ
2
3
=
P
(0)
|⟨φ|ψ⟩|
(s)
=
⟨φ|U
(s)|χ
⟩⟨χ
|ψ⟩
K
(s)
(s)|χ
⟩⟨χ
|ψ⟩
K
(s)
=
⟨φ|U
(s)|χ
⟩⟨χ
|ψ⟩
K
w k
k k k
A A Ak k kk k k
"
#
(s)
=
⟨φ|U
(s)|χ
⟩⟨χ
|ψ⟩
K
! = ⟨φ|U
"
⟨φ|χ
⟩
A
k
k
k
X k
⟨φ(T
)|
(T
)
2
2
2
†1
|Aw | − Re(A )w
⟩|χ1 ⟩2=
|χ
⟩(↵)|
⟩pi.
⟩3|χ
pw + s
1
⟩ih1|χ
⟩
⟩32sImA
2|χ
3|χ
2+
3 ⟩p (5)
+1 ∂
h |uA (↵)| k!
ih k | ih |χ
| 1j|χ
|u
j A|χ
x
P (s)
−|χ2 ⟩ P|χ
⟩⟨xnn|ψ
|ψ
⟨φ(T
)|x)|E
⟨φ(T
⟨φ(T⟨φ(T
)|x)|n ⟩⟨x
k (s)
n
n |ψ⟩
|χ
⟩
⟩
p
III.
THE
DOU
1
3
→
⟨x
|ψ⟩
=
⟨φ|U
(T
)
c
$ 2 s→0 P (s)
j6=ckn
∂s→ ⟨xn |ψ⟩
n=
)| ⟨φ|U
=
)(T ) † †
=⟨ψ|U⟨φ(T
⟨φ(T )|⟨φ(T
=⟨φ(T
⟨φ|U
(T⟨φ|U
)|
=)n(T
⟨φ|U
)|⟨φ(T
)(T
)|ψ⟩
k
†
)
A(t)
U
(t
)|ψ⟩
)A(t)
A(t)
(t)ii)A(t)
⟨φ|U
(t
)
U
(t
)|ψ⟩
UU
⟨ψ|U
(t
⟨φ|U
(t
f A(t)
i )|ψ⟩
⟨φ(T
⟨φ(T
)|ψ⟩
f
i
)
A(t)
U
(t
)|ψ⟩
U
⟨ψ|U
(t
⟨φ|U
(t
f
i
i
#
"
2
TEST
SPACE
+(1−α)
λ(A)
=αα
g|A
|)|x
≪
1)|x
|φ⟩
Ψ(Q)
Φ(Q)
Q
+(1−α)
λ(A)
=
x x
w
x )|E
+(1−α)
λ(A)
=
α
†)|x
†
⟩⟨x
|ψ⟩
|ψ⟩
⟨φ(T
)|E
⟨φ(T
A(λ)ρ(λ)
ρ(λ)
=
|ψ(x)|
λ
x
⟨A⟩dBB = dλ
⟨φ(T
)|
=
⟨φ|U
(T
)
Before
proceeding
further,
we
will
introduce
new
states
†
⟩⟨x
|ψ⟩
|ψ⟩
⟨φ(T
⟨φ(T
n
n
⟩⟨x
|ψ⟩
|ψ⟩
⟨φ(T
)|E
⟨φ(T
⟨φ|U
(t
)
U
(t
)|ψ⟩
⟨φ|U
(t
)
U
n
n
n
⟩
⟨φ|A|χ
n i )|ψ⟩
n
n(tii) U†† ((
! Pkc(0)
†
f
i
n
⟨φ|U
(t
)
U
(t
⟨φ|U
k
k
f
c
=
→
⟨x
|ψ⟩
⟨φ(T
)|
=
⟨φ|U
(T
)
⟨φ|U (ti ) as
U (t
= section,
→|ψ⟩
⟨x
⟨φ(T
)| ⟨φ|U
= (T
⟨φ|U
In this
a
nc⟨x
n n |ψ⟩⟨φ(T )|
= ⟨φ|U
)(tf(T) )Uk ⟩(ti )|ψ⟩ Aw==
nkn
n →
⟨φ|A|χ
⟨φ(T
)|ψ⟩
⟨φ(T
)|ψ⟩
i
and
the
transition
amplitudes
K
(↵)
for
simplicity.
|
⟨φ(T
)|ψ⟩
⟨φ(T
)|ψ⟩
k
⟨φ(T )|ψ⟩
⟨φ(T )|ψ⟩
Aw
= Im k Aw −
k =
x ⟨φ|χka⟩ weak value
A
consider
R
C
R
I
C
I
|φ
⟩
|ψ⟩
w
P (0)
|ψ⟩
|ψ⟩:= U†† (t) A U (t)
⟨φ(T
⟨φ(T
k† † ′ )|E
′ )|xn ⟩⟨x
′ ⟩
′
n
n
′
′
⟨φ|χ
U
(t,
t
)
=
U
(t−t
)
=
U
(t)U
(t
)
A(t)
k
k
′ ) A(t) := U
† (t) A U (t)
=
U3(t−t
)=
U
(t)U
cn → ⟨xn |ψ⟩
=
⟨φ(T )|| =
⟨φ|U (T|)(t,
!† (t(t
Young’s
double
slit
3′ ) 3=
t|′ )tk=
U (t−t
U
(t)U
)
A(t)
:=
U
(t) A U
(t)exp
iU·(t,
i)⟨φ(T
(6)
k i := h k U
=
Ex
(Φ)
−
Ex
(Ψ)
s|A|
∆
)|ψ⟩
⟨φ(T
)|ψ⟩ ρ(λ)
Q
Q
Q
2
3
†= |ψ(x)|
⟨A⟩
=
dλ
A(λ)ρ(λ)
λ
dBB
bated.
We
will
show
th
!
†
U
(t,
t
)|ψ⟩
U
(t,
t
)|φ⟩
⟨φ|U
(t,
t
)
|φ
⟩
|ψ⟩
|φ
⟩
|φ
⟩
|φ
⟩
i
f
f
†
1
1ti )|ψ⟩ U
2U
U (t,
⟨φ|U(t,
(t,t tf) )
Kk (↵) := h |uA (↵)|
(7)
f3)|φ⟩ ⟨φ|U
k i U (t,
ti )|ψ⟩
(t,(t,
tft)|φ⟩
f
2
valueλforxa momentum
⟨A⟩dBB =
dλ
A(λ)ρ(λ)
ρ(λ)
=
|ψ(x)|
!
!
3
⟨A⟩QM = ⟨ψ|A|ψ⟩
I :=
lim
1 11|A|ψ⟩
|φ
⟩ |φ
P2(s)
∂s
⟨φ
1 ⟩ |ψ⟩ 2|φs→0
1 ⟩ |φ
3⟩
|b⟩
|b⟩
|b⟩
k 2
⟨ψ|φ⟩
⟨φ
|A|ψ⟩
⟨φ
|ψ⟩
A⟩
:
H
→
R(A)
⊂
R
I
A
:
H
⊗
H
→
C
I
⟨φ⟨φ
2
⟨A⟩
:
H
→
R(A)
⊂
R
I
A
:
H
⊗
H
→
C
I
1
1 |ψ⟩
w
|ψ⟩
w
!w +AAC(Ψ)
=R(A)
g [ReA
·H
ImA
] +#O(g )
∆Q→R(A)
1⟨A⟩ : H
⟨A⟩
IR
AAww: :HH⊗⊗H
ICI
w
⊂⊂R
IR
:
H
⊗
→
C
I
w
⟨A⟩: H
:⟨φH2→
→R(A)
R(A)⊂⊂R
I
H→
→C
⟨A⟩ : H→
I
:
H
⊗
H
→
C
I
|ψ⟩
[ReA
∆Q =wg"
⟨φ2 |ψ⟩
w + C(Ψ) · Im
⟨φ
|A|ψ⟩
P
(0)
φ2⟨φ
|A|ψ⟩
!
⟨φ
|A|ψ⟩
2
⟨φ
|A|ψ⟩
!
k
2 22|A|ψ⟩
⟨φ
k 2
2 |A|ψ⟩
⟨φ
|A|ψ⟩
!
k |A|ψ⟩
!
k
2 ⟨φ⟨φ
2
|A|ψ⟩
!
⟨φ
|A|ψ⟩
A
−
=
Im
A
⟨φ
|A|ψ⟩
!
k k |A|ψ⟩
k k |A|ψ⟩
2 2 ⟨φ
⟨φk∆
|A|ψ⟩=2 22gVar
w
==⟨ψ|A|ψ⟩
=(ψ)
⟨ψ|A|ψ⟩
|⟨φ
w
n|ψ⟩|
⟨φ
k |ψ⟩|
⟨φ2n|⟨φ
|ψ⟩
·
ImA
+
O(g
)
⟨φ2⟨φ
|ψ⟩
⟨φ
|ψ⟩
⟨φ
|ψ⟩
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
P
P
w
2
2
⟨φ
|ψ⟩
=
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
k
2 |ψ⟩
2
k
P
(0)
=
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
⟨φ
|ψ⟩
=
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
⟨φ⟨φ
k |ψ⟩
k
k |ψ⟩
⟨φ
n |ψ⟩ k
⟨φn |ψ⟩
|ψ⟩
∆P =k 2gVarP (ψ) · ImAw
kk |ψ⟩
k ⟨φ
kk
k
⟨φ
⟨φ
|ψ⟩
k
k
|A|ψ⟩
⟨φ
k
|A|ψ⟩
φn⟨φ
|A|ψ⟩k unitary family of postselections
n
⟨φ⟨φ
n|A|ψ⟩
n |A|ψ⟩
n |A|ψ⟩
n
K(g)
= ⟨φ|UA (g)|ψ⟩
K(g) = ⟨φ|UA (g)|ψ⟩
⟨φ
|A|ψ⟩
⟨φ⟨φ
⟨φ
|ψ⟩
φn⟨φ
|ψ⟩
1
⟨φ
|ψ⟩
n
⟨φ
|ψ⟩
1 |A|ψ⟩
⟨φ
|A|ψ⟩
|ψ⟩
⟨φ
n
n n |ψ⟩
n
⟨φ1 1 |A|ψ⟩
⟨φ1 |A|ψ⟩
1 |A|ψ⟩
isp
isp
†
⟨φ⟨φ
|ψ⟩
⟨φ⟨φ
|ψ⟩
1⟨φ
1⟨φ
"A (g)|ψ⟩⟨ψ|A|φ⟩
#
ψ⟩
|ψ⟩
|ψ⟩
=A⟨φ|U
"
#
|φ(s)⟩
=
e
|φ⟩
=
e
|xf ⟩ = |xf − s
1
(g)|ψ⟩
⟨φ|U
|ψ⟩
(s)|φ⟩
U
(g)|ψ⟩
⟨φ|U
1
(g)|ψ⟩
K(g)
=
⟨φ|U
|ψ⟩
(g)|ψ⟩
)(g)
= ⟨φ|U
(g)|ψ⟩
K(g)
=
⟨φ|U
1
A
AA
A
1
A
!
!
1+ (1
∂ − α)
1
1 ∂
†
m⟩
P (s) −⟨φ⟨φ
P
(s)
k
I
:=
lim
P
(s)
−
P
(s)
|A|ψ⟩
⟨ψ|φ⟩
UA (s)|φ⟩
k
|A|ψ⟩
⟨φ⟨φ
2⟨φ
2⟨φ
0 P (s) ∂s
2 |A|ψ⟩
|A|ψ⟩
2 |A|ψ⟩
|A|ψ⟩
s→0 P (s) ∂s
2
2
2
k
" ""
## #
"
#
""
k#
#
#
⟨φ
|ψ⟩
#
"
⟨φ
|ψ⟩
!
2
!
⟨φ
|ψ⟩
! ⟨φ
2⟨φ
!
!
222 |ψ⟩
22|ψ⟩
1 1 ∂∂
11
∂ ∂ ! 2Pk2(0)2 ! ⟨φ
222|ψ⟩
2 2
!
2
2
2
P
(0)
|K(s)|
. . . I I:=:= lim
− −A−k|K|K
lim
|K(s)|
− k |K
|K
(s)| . ..
(s)|
k (s)|
|K(s)|
−−
(s)|
|K(s)|
|K
k (s)|
k(s)|
k|K
kk
2 |K(s)|
− |K(s)|
A
2
2
2
w
A
−
=
Im
A
w
)|∂s
∂s
w
s→0
s→0
w|A|ψ⟩
2 2 |K(s)|
|K(s)| ∂s∂s
|A|ψ⟩
P (0)k k⟨φk⟨φ
|A|ψ⟩
n⟨φ
Pk k(0)
|A|ψ⟩
"
"
|A|ψ⟩
n⟨φ
k ⟨φ⟨φ
nn
|A|ψ⟩
nn
k
k
∗
P
(s)+
K
(s)K
(s) P1 (s) P2 (s
P
(s)
=
⟨φ
|ψ⟩
†
|ψ⟩
|ψ⟩
n⟨φ
k ⟨φ⟨φ
k
|ψ⟩
|ψ⟩
nn
n⟨φ
|ψ⟩
j
n
n
U ⟨φ
(t
)|φ⟩
⟨ψ|U (ti ) A(t)
K(s)
:=
⟨φ|U
(s)|ψ⟩
f
A
!
k
j̸=k
!
!
!
!
!
!
†
∗
∗
(α)|ψ⟩
K(α)
=
⟨φ|u
∗
∗
(α)|ψ⟩
K(α)
=
⟨φ|u
∗
∗
(α)|ψ⟩
K(α)
=
⟨φ|u
∗
(α)|ψ⟩
K(α)
=
⟨φ|u
∗
(α)|ψ⟩
K(α)
=
⟨φ|u
(α)|ψ⟩
K(α)
=
⟨φ|u
A
−isA
2PP
A
⟨φ|U
)P1U
(t2 P(s)
)|ψ⟩
A
A
(s)
P(s)
(s)
PN P
A
(s)K
(s)
P121(s)
(s)
P2U
(s)
P
s)K
(s)
P
P
P
(s)
K|⟨φ|U
(s)K
(s)
(s)
P
(s)
i (s)
f2P
(s)
+
K
(s)K
(s)
P
(s)
P
kj(s)K
3 (s)
(s)
=
P
(s)+
K
(s)K
(s)
P
(s)
P
(s)
(s)
=
P
(s)K
(s)
P
(s)
P
(s)
P
(s) P3 (s) PN
2A
3 (s)
P
=
P
(s)
+
K
(s)K
(s)
(s)
P
(s)
P
(s)
jj(t
1
3(s)
kK
3PP
k
1
2
3
k
k
1
2
k
k
1
2
3(s)
k
k
1
2
3
j
j
j
j
(s)|ψ⟩|
=
e
P
(s)
=
|K(s)|
j
A
A
− α)
|φ1 ⟩|b⟩
1 |A|ψ⟩ |b⟩
|b⟩
⟨φ2 |A|ψ⟩
⟨φ
|ψ⟩
1
φψ⟩
|ψ⟩
1
k
|ψ⟩
|φ1 ⟩
|φ2 ⟩
|φ3 ⟩
equivalent picture
kk
(α)|χ
⟩⟨χ
|ψ⟩
K
=⟨φ|u
⟨φ|u
(α)|χ
⟩⟨χ
KK
==⟨φ|u
⟩⟨χ
|ψ⟩
k (α)
AA
k
k
k (α)
k
k
k (α)
A (α)|χ
k
k
2
2 |ψ⟩
k
=k
j̸=j̸k
=
j̸=k
(α)|χ
⟩⟨χ
|ψ⟩
K(α)
(α)
=⟨φ|u
⟨φ|u(α)|χ
(α)|χ⟩⟨χ
⟩⟨χ|ψ⟩
|ψ⟩
KK(α)
==⟨φ|u
k kk
A AA
k kk k kk
|⟨φ|(1 − isA − (s /2)A + · · ·)|ψ⟩|2
=
=
|χ⟩1k1⟩|χ
⟩ |χ
|χ
|χ
|χ
|χ
2
|χ⟩1k1⟩⟩⟨χ
⟩K
|χ
|χ
p |χ
⟩
|χ
|χ
⟩|χ|χ
|χ
s)
=
⟨φ|U
|ψ⟩
|χ
|χ
⟩2(s)
|χ
⟩|ψ⟩
⟩2 2⟩2⟩|χ|χ
⟩333⟩⟩⟩ p |χN ⟩ KK(s)
k
AN(s)|χ
k ⟩⟨χk |ψ⟩
1
|χ
⟩
|χ
⟩
⟩
A (s)|χ
k1⟩⟨χ
Kk (s)
= ⟨φ|U
(s)|χ
|ψ⟩
1
22
333⟩⟩⟩⟨φ|U
2|χ
3|χ
1
2
3
A
k
|⟨φ|ψ⟩|
=A
⟨φ|U
⟩⟨χ
|ψ⟩
⟨φ|U
(s)
=
⟨φ|U
(s)|χ
⟩⟨χ
|ψ⟩
|U
(s)|χ
⟩⟨χ
|ψ⟩
⟨φ|U
(s)|χ
⟩⟨χ
|ψ⟩
A (s)|χ
k ⟩⟨χ
k
Ak(s)|χ
k
k
=
⟨φ|U
(s)|χ
⟩⟨χ
|ψ⟩
k
A
k
k
k
k
AA
kk kk
!
"
⟨φ(T
2 )| = ⟨φ|U
2 (T )
2
3
⟨φ(T )| = ⟨φ|U (T )
=
1
+
2sImA
+
s
|
−
Re(A
)
)
|A
+
O(s
3
|χ
⟩
|χ
⟩
p
⟩
|χ
⟩
|χ
⟩
p
|χ
⟩
|χ
⟩
|χ
⟩
p
w
w
w
3
χ1 2 ⟩ 2 |χ2 3 ⟩ 3 p3
|χ1 ⟩1 |χ|χ
|χ ⟩ |χ ⟩ |χ ⟩ p |χ ⟩
2 ⟩22 ⟩ |χ|χ
3 ⟩33 ⟩ p p
1
2
3
N
⟨φ(T )|Enx |ψ⟩
⟨φ(T )|xn ⟩⟨xn |ψ⟩
⟨φ(T )|E
⟨φ(T )|xn ⟩⟨xn |ψ⟩
=
→
⟨x
|ψ⟩
⟨φ(T
)|
=
⟨φ|U
(T
)
cT
=
→
⟨x
|ψ⟩
⟨φ(T
)|
=
⟨φ|U
(T
)
c
n
n
n==
n(T
)| ⟨φ|U
=
)(T ) † †
⟨φ(T
⟨φ|U
(T
))
φ(T
)| ⟨φ|U
= (T
⟨φ|U
=
)(T
=⟨φ|U
⟨φ|U
(T
⟨φ(T
)A(t)
†† (t)|)|
† † (t )|ψ⟩
†† (t )|φ⟩
⟨φ(T
)|ψ⟩
⟨φ(T
)|ψ⟩
⟨φ(T
)|ψ⟩
⟨φ(T )|
†
†
)
A(t)
U
(t
)|ψ⟩
)
A(t)
U
)|φ⟩
⟨ψ|U
(t
⟨φ|U
(t
†
†
)
U
)
A(t)
U
⟨ψ|U
(t
⟨φ|U
(t
)
A(t)
U
(t
)|ψ⟩
)
A(t)
U
(t
)|φ⟩
⟨ψ|U
(t
⟨φ|U
(t
f
i
i
f
)
A(t)
U
(t
)|ψ⟩
)
A(t)
U
(t
)|φ⟩
⟨ψ|U
(t
⟨φ|U
(t
f
i
i
f
f
i
i
f
)
A(t)
U
(t
)|ψ⟩
)
A(t)
U
(t
)|φ⟩
⟨ψ|U
(t
|U
(t
f
i
i
f
)
A(t)
U
(t
)|ψ⟩
)
A(t)
U
(t
)|φ⟩
⟨ψ|U
(t
⟨φ|U
(t
f
i
i
f
i
i⟨φ|U (T ) f
⟨φ(T
)|xxx=
+(1−α)
α
TEST
SPACE
g|A
|)|x
≪
1)|x
|φ⟩
Ψ(Q)
Φ(Q)
Q
+(1−α)
λ(A)
=αα f
+(1−α)
x x λ(A)
+(1−α)
=
w
x )|E
+(1−α)
†
†
+(1−α)
λ(A)
=
α
⟩⟨x
|ψ⟩
|ψ⟩
⟨φ(T
)|E
⟨φ(T
)|x
†
⟩⟨x
|ψ⟩
|ψ⟩
⟨φ(T
)|E
⟨φ(T
)|x
†
†
⟩⟨x
|ψ⟩
|ψ⟩
⟨φ(T
⟨φ(T
n
n
⟩⟨x
|ψ⟩
|ψ⟩
⟨φ(T
)|E
⟨φ(T
†)|x
††(t
n
⟩⟨x
|ψ⟩ (t
⟨φ(T
)|E
⟨φ(T
⟨φ|U
(t
U(t
⟨φ|U
(t)ii)U
)UU
(t
)|ψ⟩
n n(t
⟩⟨x
|ψ⟩
|ψ⟩
⟨φ(T
)|E
⟨φ(T
)|x
n
⟨φ|A|χ
n(t
nn k n
⟨φ|U
(t
)
U
)|ψ⟩
⟨φ|U
(t
)
U
(tf)|ψ⟩
)|ψ⟩
† (t
† (t
n)|ψ⟩
n |ψ⟩
⟩
⟨φ|A|χ
f)
i )|ψ⟩
f=
n(t
nn
n⟩⟨φ|U
⟨φ|U
(t
)
U
)|ψ⟩
(t
)|ψ⟩
k⟨φ|U
†
†
n
f
i
i
f)|ψ⟩
n
⟨φ|U
(t
)
U
(t
⟨φ|U
)
U
k
f
i
f
k
f
i
i
=
⟨φ|U
(T
)
⟨φ|U
(t
)
U
)|ψ⟩
(t
)|ψ⟩
c
=
→
⟨x
|ψ⟩
⟨φ(T
)|
⟨φ|U
(T
)
=
=
⟨φ|U
(T
)
⟨φ|U
(t
)
U
(t
)|ψ⟩
⟨φ|U
(t
)
U
(t
=
|U (T ) f
n
n
i
i
f
c
=
→
⟨x
|ψ⟩
⟨φ(T
)|
=
⟨φ|U
(T
)
c
=
→
⟨x
|ψ⟩
⟨φ(T
)|
=
⟨φ|U
(T
)
f
i
i
f
=
A
A = ⟨φ(T )|ψ⟩
n n ⟨φ(T
nn
w
⟨φ(T
)|ψ⟩
⟨φ(T
)|ψ⟩
⟨φ(T
)|ψ⟩
⟨φ(T
⟨φ(T
)|ψ⟩
⟨φ(T
⟨φ(T
)|ψ⟩ w
⟨φ(T
⟨φ(T
)|ψ⟩
⟨φ(T
⟨φ|χk)|ψ⟩
⟩ )|ψ⟩)|ψ⟩
⟨φ|χ
⟩)|ψ⟩
k)|ψ⟩
⟩⟨x
⟨φ(T
)|x
R
I
CI′
|φk†⟩† ′ ′ |ψ⟩
n
R
C
R
I
C
I
|φ
⟩
|ψ⟩
†
k
′
′
†
′
†
′
†
′ t )c
′ ) |ψ⟩
=UU
(t−t
(t)U
A(t):=
:=U†U (t)
(t)AAUU
(t)
|φ
⟩(t−t
|ψ⟩
U
=
(t−t
=UU(t)U
(t)U
(t′ ) ) A(t)
A(t)
:=
(t)
A)UU(t)
(t)
|φkk⟩⟩ |ψ⟩
|ψ⟩
→
⟨x
)|
=
⟨φ|U
(T
′ ) )==UU
† (t
′ (t) ) A(t)
k⟩U
′(t,
′|ψ⟩
† † (t
′ ⟨φ(T
†U† (t)
(t−t
(t)U
(t)
|φ
n
n
U
(t,
t
)
=
U
)
=
:=
U
A
|φ
!
k
!
3
3
(t−t
)
=
U
(t)U
(t
)
A(t)
:=
U
(t)
A
U
(t)
|φ
⟩
|ψ⟩
(t, t ) = Uk(t−t
)
=
U
(t)U
(t
)
A(t)
:=
U
(t)
A
U
(t)
|φ
⟩
|ψ⟩
3∆Q = ExQ (Φ) − ExQ (Ψ) Us|A|
3
3
3
k
⟨φ(T
)|ψ
2
2
†
⟨A⟩
=
dλ
A(λ)ρ(λ)
ρ(λ)
=
|ψ(x)|
λ
x
†
⟨A⟩
=
dλ
A(λ)ρ(λ)
ρ(λ)
=
|ψ(x)|
λ
x
ψ⟩ U|φ(t,dBB
⟩ti )|ψ⟩
|φ ⟩U (t,
|φtf ⟩)|φ⟩ ⟨φ|U† (t, tf ) |φ ⟩ |ψ⟩ U|φ(t,dBB
⟩t )|ψ⟩
|φ ⟩U (t,
|φt ⟩)|φ⟩ ⟨φ|U† (t, t )
1ti )|ψ⟩ U2U
U (t,
⟨φ|U† (t,
(t,t tf) )
f3)|φ⟩ ⟨φ|U
U (t,
ti )|ψ⟩
(t,(t,
tft)|φ⟩
f
A) ⊂ R
I
⟨ψ|A|ψ⟩
⟨A⟩⊗
QMH=→
H
CI
1
i
f)|φ⟩ ⟨φ|U
1ti )|ψ⟩
U (t,
(t,t tf)f)
f3
U (t,
ti )|ψ⟩
U2U
(t,(t,
tft)|φ⟩
⟨φ|U † (t,
f
Aw :
⟨A⟩ :2H → R(A) ⊂ R
I
Aw :
ExQ (Ψ) := ⟨Ψ|Q|Ψ⟩/∥Ψ∥
"!
⟨φ |A|ψ⟩
⟨φ |A|ψ⟩
⟨φ|A|χk ⟩
=
⟨ψ|A|ψ⟩
⟨A⟩
k
QM
H ⊗ HA
→w CI=
⟨φ|χk ⟩
"
1 ∂
1
lim!
P (s) − # Pk (s)
I := (α)|ψ⟩
K(α)
=
⟨φ|u
A 2 s→0 P (s) ∂s
!
⟨φ
|A|ψ⟩
k 1
k
"
2
∂
1
|⟨φkI|ψ⟩|
#
!= ⟨ψ|A|ψ⟩
lim
P
(s)
−
:=
P
(s)
k
"
⟨φ
|ψ⟩
K
(α)
=
⟨φ|u
(α)|χ
⟩⟨χ
|ψ⟩
k
ex.) double slit
experiment
Pk (0) k
A P (s) k∂s k
2 s→0
kk
k
−
=
Im
A
w
#P (0) Aw
!
k
"
|χ
⟩
|χ
⟩
|χ
⟩
p
P
(0)
generator
of translation
1
2
3
k
⟨φ
|A|ψ⟩
k
1
nd the Wave-Particle Duality
5
Aw
= Im Aw −
(0)
⟨φ1 |ψ⟩ k Pisp
isp
|φ(s)⟩
=
e
|φ⟩
=
e
|xfthe
⟩ = |xf − s⟩
ur choice of selections, on the screen at t = T the weak value of
⟨φ
2(p|A|ψ⟩
pw and those for the partial processes
± )w are given
isp
isp by
"
|φ(s)⟩ = e |φ⟩ = e |xf ⟩ = |xf −#
s⟩
! m xf x!i "!
⟨φ2 |ψ⟩
1
∂
1
#
2
2
x
+
i
x
tan
⟨ψ|
p
|φ(T
)⟩
i
f
I
:=
lim
.
|K
(α)|
"
k
"
1 !unitary
∂ T− family
1" |K(α)|
pw = 2 α→0 |K(α)|
= m 2 ∂α
,
(10)
of postselections
∗ Pk (s)
⟨ψ|φ(T
)⟩
T
lim
P
(s)
−
I
:=
T SPACE
TEST SPACE
|A|ψ⟩
P
(s)+
K
(s)K
P (s) = 2⟨φ
kk
n kP (s) ∂s
j (s) P1 (s) P2 (s) P3
s→0
=T0)⟩ t = Tx ∓ x
t⟨ψ|
= 0p |φ tt=
k
(T
i
±
f k"
j̸
=
k
"
#
!
⟨φn. |ψ⟩" ∗
=m
(11)
(p± )w =
Kk (s)KPj k(s)
P
(s)
P
(s)
P
(s)
P
P (s)
=)⟩ Pk (s)+
⟨ψ|φ
T
± (T
1
2
3
N
(0)
k
⟩
⟨φ|A|χ
k
kk ⟩Im|ψ⟩
Aw
−
=
A
|φ
w
R C R
I R CIC RI|φkkC
⟩I |ψ⟩
j̸
=
k
A
=
P
(0)
(α)|ψ⟩
K(α)
=
⟨φ|u
w
‘intensity’
of
interference
A
are both real, the index (5) turns out to be
Transition
Probability
⟨φ|χ K⟩ (s) = ⟨φ|U (s)|χ ⟩⟨χ |ψ⟩
k
A
k
k
"I
Hm
⊗H
⟨A⟩ : H → R(A) ⊂ R
I
Aw : !
x C
x →
K
: (α)
Hx⊗
H=
→ ⟨φ|u
CI! TA (α)|χk ⟩⟨χk |ψ⟩
⟨A⟩ : H → R(A) ⊂ R
I
A
i tan
wk
!
|φ1 ⟩
|ψ⟩
!
|φ1 ⟩ |ψ⟩ |φ1 ⟩
|φ1 ⟩
|φ2 ⟩
|φ3 ⟩
|φ2 ⟩
|φ3 ⟩
α = 0, 1
α = 0, 1 α = 1/2
f
m
I = Im pw =
k |A|ψ⟩
2 ⟨φ
|⟨φk |ψ⟩|
|⟨φ |ψ⟩|
k
2 ⟨φk |A|ψ⟩
k
⟨φk |ψ⟩
α = 1/2kk
i
∝|χIm
x(s)|χ
. ⟨φ|U
(12)
w|χ2 ⟩k ⟩⟨χ
⟩
|χ
⟩
p
|χ
⟩
K
(s)
=
|ψ⟩
1
3
N
k
A
k
= ⟨ψ|A|ψ⟩
T
k |ψ⟩
=⟨φ⟨ψ|A|ψ⟩
|χ1 ⟩
|χ2 ⟩
|χ3 ⟩
Re xw
xf +
=
Im xw
which
that the index I is just⟨φthe
imaginary
part
p⟩w , |χ
⟨φ(T
(T )
|χ2theorem
p)| =|χ⟨φ|U
|χEhrenfest
1 ⟩ Im
3 ⟩ diverges
N ⟩when
1 |A|ψ⟩
|b⟩
Fig.
weak value x (t) as a2function of the post-selection x for a fixe
2 3 The
isp
isp
⟨φ
|A|ψ⟩
⟨φ
|ψ⟩
1
ence becomes completely
|K(T
0)|
1 destructive
|φ(s)⟩ =
e ;|φ⟩
==thick
e |⟨ψ|U
|xrepresents
|xxf while
−→
s⟩
the0,
thin line represents Im x . The
T . The
line
Re
f ⟩(T=)|φ⟩|
|b⟩
⟨φ1 |ψ⟩
diverges
at the
transition
probability,
|ψ⟩
)|x
s when it is maximally
constructive.
⟨φ(TIm)|x =
⟨φ|U
(Tlocations
) where the⟨φ(T
n ⟩⟨xnindicated
⟨φ2 |A|ψ⟩
k
w
f
w
⟨φ2 |A|ψ⟩
cn → ⟨xn |ψ⟩
⟨φ |ψ⟩
2
w
w
filled line,)|
vanishes.
⟨φ(T
= ⟨φ|U (T )
⟨φ(T )|ψ⟩
3
1
|φ⟩ = √ (|1⟩ + |2⟩ − |3⟩)
3
= |i⟩⟨i|
2. Physical value in HVT and quasiprobability
Pi = |ci ⟩⟨ci |
i = 1, 2, 3
⟨b|P
TEST
i |a⟩vsSPACE
expectation
value
weak
value
(P ) =
i w
R
⟨b|a⟩
expectation value
⟨A⟩ := ⟨ψ|A|ψ⟩
ACE
R
weak value
C
C
R
I
CI
⟨A⟩ : H → R(A) ⊂ R
I ‘one-state’
Aw : value
H⊗H→
within the (real) range of spectrum
R
I
CI
‘two-state’ value
A⟩ : H → R(A) ⊂ R
I
Aw : H ⊗ H → †CI
:=
⟨ψ|U (ti )
⟨φ|U (tf ) A(t) U (ti )|ψ⟩
(xf −Axwi )t
+(1−α)
=λ(A)
xcl (t)= α xwentire
xi +
(t) = x
range
complex
numbers
cl (t) of
T
⟨φ|U (tf ) U † (ti )|ψ⟩
⟨φ|U (
… the result
of
measurement
of
a
spin
component
of
′
′
†
′
†
†
†
U
(t,
U
(t
−
t
)
=
U
(t)U
(t
)
A(t)
:=
U
)
A(t)
U
(t
)|ψ⟩
)
A(t)
U
(t
)|φ⟩
⟨ψ|U
(t
φ|U
(t
−igA
σ1 t ) =
w
f
i
i
f
spin
1/2
particle
can
turn
out
to
be
100
…
ΨMD (0, 0)
σ1 ΨMD (0, 0) e
+(1−α)
† (t )|ψ⟩
† (t )|ψ⟩
⟨φ|U|ψ⟩
(tf ) U|ψ⟩⟨ψ|
⟨φ|U
(t
)
U
i
f L. Vaidman (1988) †
Y. Aharonov,i D. Z. Albert,
Γk
Γs
′
̸=
U (t, ti )|ψ⟩
†
′
†
U (t, tf )|φ⟩
⟨φ|U (t, tf )
TEST SPACE
property of weak value
⟨b|Pi |a⟩
(Pi )w =
⟨b|a⟩ ⟨b|Pi |a⟩
(Pi )w =
Aw :=
⟨A⟩ := ⟨ψ|A|ψ⟩
(analogous to expectation value)
⟨b|a⟩
⟨A⟩ := ⟨ψ|A|ψ⟩
|a⟩
⟨b|P
⟨b|P
i(x
if|a⟩
−
x
)t
i
1) agrees with
)wi )=
=
eigenvalue
if the preselected
ithe
w
(xf=−xxcli )t
x(P
(t)
x=i ⟨b|a⟩
+⟨b|a⟩
(t)(or postselected)
xw (t) = xcl (t)
w(P
= xcl (t)
xw (t) = xTi +
xw (t) =
SPACE
state is TEST
an eigenstate
|ψ⟩ T |ψ⟩⟨ψ|
Γk A Γ
=
̸
TEST
SPACE
TEST
SPACE
s
A
:=
=
a
A
=a
A|ψ⟩
a|ψ⟩ orA|φ⟩
A|φ⟩
a|φ⟩
A|ψ⟩
==
a|ψ⟩
=w=a|φ⟩
w
−igAw σ1−igA
(0,
g  σ1 Ψg Â
wσ
1
MD
ΨMDe(0, 0) e ΨMD
Ψ0)
σ1(0, 0)
MD (0, 0)
2)
O
O
O
3
k
s
⟨A⟩
:=
⟨ψ|A|ψ⟩
⟨A⟩ := ⟨ψ|A|ψ⟩
iφ
iφ
=
Ψ
(θ,
φ)
=
cos
θ
|0⟩
+
e
sin
θ
Ψ
fulfills sumArule
:=
A
=
B
+
C
A
=
B
+
C
A
:=
A
=
B
+
C
A
=
B
+
C
MD
MD
=
θw|0⟩ + ew |1⟩
sin θ |1
w
w ΨMD = ΨMDw(θ, φ) w
w cos w
O = A(O1 ⊗ O2 ⊗ O3 ⊗ · · · ⊗ Os )A
C
A
=
B
+
C
AA
:= AA==B
B+
+
C
=
B
+
C
w w:=
w
w
w
w
ww Aw =
V1 V2 V3
but does not fulfill
|ψ⟩
|ψ⟩⟨ψ|
A
:=
A
=
a
A
=
a
A
:=
A
=
a
A′w)w
=Baw Cw ai ∈ Rai ∈
A
:=
A
=
B
C
A
=
−
x
)t
(x
A
:=
A
=
B
C
A
=
B
C
w
w
w
w
−
x
)t
(x
f
i
w
w
w
w
f
i
(Γ,
V
rule = =
+
x(t)
=
x
(t)
w (t)
cl (t)
w (t)
cl
xclx(t)
==
xi x+iproduct
xΓ
=
x
(t)
xwx(t)
w
cl
Γ
=
̸
!
k
s
T
T
(P−1
3 )w = −1
(P12))ww==(P12 )w(P=3 )1w =
(P1 )w = (P
C(a, b) = O3dλOρ(λ)
A(a,
λ) B(b, λ)
O
k
s
C(Ψ) := Ex{Q,P } (Ψ) − 2ExQ (Ψ) · ExP (Ψ),
TESTC(Ψ)
SPACE
Exinitial
− |ψ⟩
2Exof
ExC(Ψ)
Q (Ψ)
P (Ψ),
{Q,P } (Ψ)
defined by:=
the
state
the· meter,
where
:=− 2ExQ (Ψ) · ExP (Ψ),
:= Ex{Q,
(Ψ)
{Q,PP}}
R meter,
C also
R
Ihave{Q,CIP } :=
QPby+the
P Qinitial
is thestate
anticommutator.
We
defined
|ψ⟩ of thedefined
where
by
the the
initial
state |ψ⟩ of the
meter, where {Q,
link
between
the
weak
value
and
expectation
value
2
!
QP + P Q is the anticommutator.
WeQP
also
have
+
P
Q is−
the (Ex
anticommutator.
We also have
!
2
d∆
(ψ)
:=
Ex
(ψ)
(ψ))
Var⟨A⟩
P
P : !H →
P⊗ H → C
P
! R(A)
:
H
I
⊂
R
I
A
=
2Var
(ψ)
·
ImA
,
w
P
w
!
where
!
d∆P !! dg g=0
d∆P !!
=
2Var
(ψ)
·
ImA
,
P
w
!
dg !
g=0
⟨φ |A|ψ⟩ dg !
|⟨φk |ψ⟩|
2
k
2
= 2VarP (ψ) · ImAw ,
g=0
= ⟨ψ|A|ψ⟩
(Ex
(ψ))
VarP (ψ) := ExP 2 (ψ) −⟨φ
P
|ψ⟩
where
k
where
k
2
2
VarP (ψ) := ExP 2 (ψ) − (ExP (ψ))
VarP (ψ) := ExP 2 (ψ) − (ExP (ψ))
TEST SPACE
w
⟨φ|A|ψ⟩
average
of weak Avalues
w = over postselections
TEST
SPACE
R C⟨φ|ψ⟩
R
I
CI
|φk ⟩ |ψ⟩
⟨φ|A|ψ⟩
⟨φ|A|ψ⟩
Aw =
"
†
†
A
=
w
⟨φ|ψ⟩
C1U
R
I 2(t
CIi )|ψ⟩
|φk ⟩ |ψ⟩
⟨φ|U (tTEST
)
A(t)
U
(t
)|φ
⟨ψ|U
(t
⟨φ|ψ⟩
f )RA(t)
i
f
SPACE
|φ2 ⟩ |φ3 ⟩
Aw|ψ⟩ |φ1 ⟩+(1−α)
Aw|φ1 ⟩"
λ(A) = α
"
†
† (t )|ψ⟩
1
2
⟨φ|U (tA⟨A⟩
)
U
(t
)|ψ⟩
⟨φ|U
(t
)
U
1
2
|φ
|ψ⟩
⟩⊂
|φ
⟩ : H ⊗1H →2CI
f
i|φ1C
i
f
1:⟩A
2 ⟩ |φ
3w
H
→
R(A)
R
I
A
w
w
⟨A⟩
A
A
R
R
I
C
I
|φ
⟩
|ψ⟩
|φk ⟩ |ψ⟩
kw
w
w
w
! ⊂R
→ CI
⟨A⟩ : H → R(A)
I 2 ⟨φA
w :H⊗H
average
k |A|ψ⟩
= t)AU
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
k|φ
|φ
|φ2 ⟩ (t)|ψ⟩
|φ3 ⟩
⟩ |φ2 ⟩ |φ3 ⟩ ′
1 ⟩ |ψ⟩
1 ⟩−
′ ·U
†
′
†
⟨φ|U
(T
⟨φ|U (T !
− t) · A
(t)|ψ⟩
⟨φ
|ψ⟩
k
⟨φ
|A|ψ⟩
U (t,
t )== U (t −|⟨φt )k|ψ⟩|
=2 Uk(t)U
(t
) A(t) := U (t) A U (t)
Aw (t)
=
=
⟨ψ|A|ψ⟩
k· U (t)|ψ⟩
⟨φ|U
⟨φ|U (T
)|ψ⟩
⟨φ|U
(T
−
t)AU
(t)|ψ⟩
⟨φ|U
(T C
· A−
· Ut)(t)|ψ⟩
:· H
⊗UH
→ CI ⟨φ|U (T − t)AU (t)
⟨A⟩
: H ⟨φ
→
R(A)
⊂
R
I
A
H→
I− t) (T
Aw : H ⊗
w
|ψ⟩
⟨φ|U
(T
−
t)
A
·
(t)|ψ⟩
k
Aw (t) =average of the
= A (t)
k weak values
|A|ψ⟩
1=
assigned
to all possible
processes
=
⟨φ|U (T − t) · U (t)|ψ⟩ ! w ⟨φ
⟨φ|U
(T
)|ψ⟩
†
⟨φ⟨φ|U
(T⟨φ|U
− t) · U(t,
(t)|ψ⟩
⟨φ|U (T )|ψ⟩
A|ψ⟩
k |A|ψ⟩
U
(t,
t
)|ψ⟩
U
(t,
t
)|φ⟩
t
)
2|ψ⟩
i
f
f
⟨φ
1
⟨φ1|⟨φ
|A|ψ⟩
= ⟨ψ|A|ψ⟩
= ⟨ψ|A|ψ⟩
gives the expectation
value
k |ψ⟩|
+
σ
σ
z
x
⟨φk =
|ψ⟩|xf ⟩
k |ψ⟩
|ψ⟩
=
|x
⟩
|φ⟩
A = σ π4 := √
i
k⟨φ1 |ψ⟩
⟨φ2 |A|ψ⟩
2
σ z + σx
1
σ z + σx
π :=
√
|ψ⟩
=
|x
=
|x
⟩
A
=
σ
i ⟩ |φ⟩
f
⟨φ
|A|ψ⟩
⟨φ
|ψ⟩
A|ψ⟩
π
4
:= +√|3⟩) |ψ⟩ = |xi ⟩ |φ⟩ = |xf ⟩
A 2=+
σ14 |2⟩
√2 |A|ψ⟩
(|1⟩
2 |ψ⟩ = ⟨φ
2
√
⟨φ |ψ⟩
1+1
⟨φ|A|ψ⟩
A =
⟨φ|ψ⟩
"
A
A
⟨φ|U (T − t) · A · U (t)|ψ⟩
=
⟨φ|U (T − t)AU (t)|
⟨x|A|ψ⟩
dλ ρ(x) Re
= ⟨ψ|A|ψ⟩
⟨x|ψ⟩
TEST SPACE
“interpretation” as physical value in HVT (ontological model)
⟨φ|A|χk ⟩
= ⟨φ|A|χk ⟩
⟨φ|χk ⟩
w =
⟨A⟩QMTEST
= ⟨ψ|A|ψ⟩
SPACE
k
w
k
⟨φ|χk ⟩
de Broglie-Bohm theory
Akw =
2
ρ(λ) ≥ 0
ρ(λ) =
!
dλ ρ(λ) = 1
"
A!
µ(E)
A ρ(λ)
= ≥ 0 aρ(λ)
⟨φ|A|χ
i Ei= dλ ρ
k ⟩ = tr(W E) W
i
σ z = ? σx
⟨φ|χ|ψ⟩
k ⟩ = |+z⟩|−z⟩+|−z⟩|+z⟩ v(A) = λ(A)
2
hidden
variable
λ)ρ(λ)
ρ(λ)
|ψ(x)|
λ)ρ(λ) ρ(λ)
==
|ψ(x)|
λ xλ
x
#
$
λ = (λ1 , λ2 , λ3 , . . .)
A = A(λ)
"
"
⟩
⟨φ|A|χ
k
!
|ψ⟩ = |+z⟩|−z⟩+|−z⟩|+z⟩
v(A) = λ
k
A
=
µ
Ewi 2 = ⟨φ|χkµ(E
i ) {Ei }
⟩
⟨A⟩dBB =k dλ⟨φ|A|χ
A(λ)ρ(λ)
ρ(λ) = |ψ(x)| λ x λ = (λ1 , λ2 , λ3 , . . .) A
k⟩
i
i
Aw =
QM
C=
! A + B v(C) = v(A) + v(B) v(A) = ±1, v(B) = ±
⟨φ|χk ⟩
QM
2
⟩
⟨φ|A|χ
k=
=dBB
⟨ψ|A|ψ⟩
k⟨A⟩QM
⟨A⟩
dλ
ρ(λ)A(λ)
ρ(λ)
=
|ψ(x)|
A = σx B = σy A + B = σx + σy λ x
A
=
w
!
C√=⟨φ|A|ψ⟩
A+B
v(C) = v(A) + v(B)
⟨φ|χk ⟩
A =
v(A + B) ̸= v(A) + v(B)
2 v(A + B)w= ± 2 ⟨φ|ψ⟩
⟨A⟩dBB = dλ A(λ)ρ(λ) ρ(λ) = |ψ(x)| ⟨A⟩
λ QM
x = ⟨ψ|A|ψ⟩
A = σ x B = σy A + B
√
!
⟨x|A|ψ⟩
v(A
+ ′B)
2′ , b)| v(A
+
|C(a
, b′ )=+±C(a
≤ 2 + B)
|C(a, b) − C(a, b′ )|⟨x|A|ψ⟩
A(λ) =
2
"
! x
⟨x|ψ⟩
=
⟨ψ|A|ψ⟩
⟨A⟩dBB = ⟨A⟩dλ
A(λ)ρ(λ)
ρ(λ)
=
|ψ(x)|
A(λ) =λRe
QM
A
a
E
µ(E)
=
tr(W
E)
W
A
=
C(a, b) = dλ⟨x|ψ⟩
ρ(λ) A(a, λ) B(b,
i ′ λ)
†
†
i
)|
+ |C(a′ , b′ )
|C(a,
b)
−
C(a,
b
†
†
i
i
f
i
)
A(t)
U
(t
)|φ⟩
⟨ψ|U (t
⟨φ|U (tf ) A(t) U (ti )|ψ⟩
i
f
!
A(a,
λ)
=
±1
B(b,
λ)
=
±1
+(1−α)
λ(A) = α
$
#
† ⟨ψ|A|ψ⟩
⟨A⟩
=
†i
†
QM
†
† (t )|ψ⟩
C(a, b) = dλ ρ(λ)
A(a,
i
f
⟨φ|U
(t
)
U
(t
)|ψ⟩
⟨φ|U
(t
)
U
"
"
2
f
ii ⟩
f
⟨x|A|ψ⟩
i
ii
fA |ϕi ⟩ = ai |ϕ⟨φ|E|ψ⟩
Prob(A = ai ) = |⟨ϕi⟨ψ|E|φ⟩
|ψ⟩|
A(λ) =
=
µ
Ei= α
µ(Ei +
) A(a,
{E
i }=α)
(1
−
µ(E)
=
tr(W
E)
′
λ)
±1
B(b, λ
†
⟨x|ψ⟩
C(a,
b)
−
C(a,
b
)
⟨φ|ψ⟩
⟨ψ|φ⟩
i
i
i
f i!
probability distribution
==⟨ψ|A|ψ⟩
⟨ψ|A|ψ⟩
expectation value
⟨x|A|ψ⟩
we require
λ) = if⟨x|A|ψ⟩
⟨x|ψ⟩
λ) =
⟨x|ψ⟩
(t )|ψ⟩
⟨ψ|U (t ) A(t) U (t )|φ⟩
+(1−α)
‘local
expectation
value’
)|ψ⟩
⟨φ|U
(t
)
U
(t
)|ψ⟩
(t )|ψ⟩
⟨ψ|U (t ) A(t) U (t )|φ⟩
)|ψ⟩
+(1−α)
⟨φ|U (t ) U (t )|ψ⟩
⟨φ|A|χk ⟩
k
Aw =
I
⟨A⟩ : H → R(A) ⊂ R
I
Awk:⟩kH
⟨φ|χ
⟩ ⊗H→C
⟨φ|A|χ
k
Aw =
!
⟨ψ|A|φ⟩
⟨φ|A|ψ⟩
⟨φ|χk ⟩
!
A
⟨φ
|A|ψ⟩
−
α)
ai µ(Ei ) =recall
tr(W A) = α
k
! + (1
2
|⟨φ
|ψ⟩|
⟨φ|ψ⟩k
⟨ψ|φ⟩= ⟨ψ|A|ψ⟩ 2
⟨φk |ψ⟩ ρ(λ) = |ψ(x)|
=
dλ
ρ(λ)A(λ)
⟨A⟩dBB
!
k
⟨A⟩dBB = dλ ρ(λ)A(λ) ρ(λ) = |ψ(x)|2
†
φ|U (tf ) A(t) U † (ti )|ψ⟩
)
A(t)
U
(tf )|φ⟩
⟨ψ|U
(t
i
!
+(1−α)
†
⟨φ|U (tf ) U (ti )|ψ⟩
⟨φ|U
(tf )|ψ⟩
2 †⟨x|A|ψ⟩
† (ti ) U
λ
x
λ
x
†
=
⟨ψ|A|ψ⟩
dx
|ψ(x)|
⟨φ|U (tf ) A(t)
U
(t
)|ψ⟩
)
A(t)
U
(tf )|φ⟩
⟨ψ|U
(t
!
i
i
⟨x|ψ⟩
λ(A) = α
+(1−α)
⟨x|A|ψ⟩
2
†
† (t )|ψ⟩
|φk ⟩ →
|x⟩ (tf ) Udx(t|ψ(x)|
⟨φ|U
)|ψ⟩
⟨φ|U
(t
)
U
=
⟨ψ|A|ψ⟩
i
i
f
!
⟨x|ψ⟩
†
†
⟨x|A|ψ⟩
φ|U (tf ) A(t) U (ti )|ψ⟩
)
A(t)
U
(tf )|φ⟩ = ⟨ψ|A|ψ⟩
⟨ψ|U
(t
i
dλ
ρ(x)
Re
+(1−α)
!′
′
′
†
†
††(t
⟨x|ψ⟩
⟨φ|U (tf )UU(t,
(tit)|ψ⟩
⟨φ|U
(t
)
U
)|ψ⟩
)=
U
(t
−
t
)
=
U
(t)U
(t
)
A(t)
:=
U
(t) A U (t)
i
f
⟨x|A|ψ⟩
or
dλ
ρ(x) Re
= ⟨ψ|A|ψ⟩
′
† ′
†
⟨x|ψ⟩
U (t−t ) = U (t)U (t )U (t,
A(t)
:=
U
(t)
A
U
(t)
|ψ⟩† (t, t )
ti )|ψ⟩ U (t, tf )|φ⟩|φk ⟩⟨φ|U
f
= ⟨ψ|A|ψ⟩
U (t, ti )|ψ⟩ U (t, tf )|φ⟩ ⟨φ|U † (t, t⟨A⟩
f ) QM
1
|ψ⟩ = √⟨A⟩(|1⟩
+=|2⟩
+ |3⟩)
1
⟨ψ|A|ψ⟩
QM
|ψ⟩ = √ (|1⟩ + |2⟩ + |3⟩)
3
‘local
expectation "
value’ in dBB theory
3
A
µ(E)
=
tr(W
E)
W
A
=
a
E
i i
1
1
|φ⟩ = √ (|1⟩ + |2⟩ |φ⟩
− |3⟩)
= √ (|1⟩ + |2⟩ − |3⟩) "
i
A
3
µ(E) = tr(W
E)
W
A
=
a
E
3
i i
$
#
i
Pi = |i⟩⟨i| i = 1, 2, 3
Pi = "
|ci ⟩⟨ci |
"
Pi = |i⟩⟨i|
i =E1,
2,=3
Pi i=
|c{E
i ⟩⟨c
i|
µ
µ(E
)
}
$
#
i
i
⟨b|Pi |a⟩
"
"
(Pi )w =
i
⟨b|Pi |a⟩
⟨b|a⟩
1 j=1 ontological
modelmodel
と複
1 ontological
!
"
ntological
model
と複素確率
M
A
ogical
model
と複素確率
1 ontological
model
% model
1 ontological
と複素確率
bj ,と複素確率
Ei ψ
2
状態 ψ において物理量
が値 ai
状態 ψ において物理量
A が値 aAである確
in general
|⟨bj , ψ⟩| .
⟨b
,
ψ⟩
j
において物理量 A が値 ai である確率 pj=1
(A = ai |ψ ) は, Born の公式より
=
i
て物理量
A
が値ψaiにおいて物理量
である確率
p (A a=
aである確率
)は
, である確率
Bornp
の公式より
iin|ψmeasuring
probability
of obtaining
observable
は状態
, Bornψの公式より
において物理量
A が値
(A
= api |ψ
状態
A
が値
a
(A)=はa,i Bo
|ψ
)
i
i
p (A
2
! B の値が
" b である確率と解釈できる
p
(A
=
a
|
のとき, |⟨bj , φ⟩| は状態 φ
において物理量
;
A
i
j
p (A = ai |ψ ) =
! ψ, EAi ψ" .
# A
$N (1)
!
"
!
"
p
(A
=
a
|ψ
)
=
ψ,
E
ψ
.
(1)
ここで
,
E
=
|a
⟩
⟨a
|
は
物理
i
i
i
#
$
i
i
i=1A
N A
# A
$N
#
$
(1)
M
A
2
B
p
(A
=
a
|ψ
)
=
ψ,
E
ψ
.
p
(A
=
a
|ψ
)
=
ψ,
E
ψ
.
i
i
ここで
,
E
=
|a
⟩
⟨a
|
は
物理量演算子
Ei = |ai$
⟩ ⟨ai | i=1 は 物理量演算子
A
のスペクトル測度
.
ここで
,
E
=
|b
⟩
⟨b
|
i
i i=1
p (B = bj |ψ ) = |⟨bj ,iψ⟩|
. i
j
j j=1
j iB のスペクトル測
を 物理量演算子
#
$
N
M
B
#
$
M A のスペクトル測度. ここで, E = |bj ⟩ ⟨bj |
=
|ai ⟩ ⟨aiB
|#i=1
は
B 物理量演算子
#
$
j ;
量演算子
のスペクトル測度とすると
, 物理量演算子
式
(1)
は次のように変形できる
$
j=
N
を
B
のスペクトル測度とす
度.!ここでif," E
=
|b
⟩
⟨b
|
for
some
observable
N
j
j
A
j
A
j=1
ここで
,
E
=
|a
⟩
⟨a
|
は
物理量演算子
A
のスペクトル
A
A
ここで
,
E
=
|a
⟩
⟨a
|
は
物理量演算子
A
のスペクトル測度
.
こ
i
i
i
i
i
のスペクトル測度とすると
,
式
(1)
は次のように変形できる
;
た子
, bBj , E
φ
/
⟨b
,
φ⟩
は
pre-selection
φ
および
post-selection
b
における
E
の弱値
i
i=1
j
j
i=1
p (A
i
i = ai |ψ
!
"
ように変形できる
;
A
p
(A
=
a
|ψ
)
=
ψ,
E
(2)は次
ある
このような射影演算子の弱値を次のように書くとする
; , 式 (1) は次のように
を 物理量演算子
B のスペクトル測度とすると
, 式 (1)
i ψ
を. 物理量演算子
Bi のスペクトル測度とすると
! M A "
ψ,%
Ei! ψ
p (A = ai |ψ ) =
p (A = ai |ψ ) =
(2)
!A " A "
B
(2)
(3)
ψ, Ej Eiφ,ψEi ψ !
=
!
"
"
A
Mai |ψ, φ ) :=
A
.
(4)
c (Ap=
%
p
(A
=
a
|ψ
)
=
ψ,
E
ψ
"
!
j=1
i
(A = ai B|ψ )A ⟨φ,=ψ⟩
ψ, Ei ψ
i
=
=
(3)
ψ,! Ej E
"ψ
Ai
M
%
(3)
M
b
,
E
ψ
M
j
2
i
%
j=1
"
!
%
"
!
|⟨b
,
ψ⟩|
.
=
j
B A
の表記を採用すると, 式 (2) は
B
A
⟨b
,
ψ⟩
!
"
ψ,
E
=
j
ψ, E E ψ j Ei ψ
=
Mj=1
A
j
i
% bj , Ei ψ
2
j=12
=
|⟨bj , ψ⟩|
= %
M
j=1.
2
このとき, |⟨bj , φ⟩| !は状態 φ におい
⟨b
,
ψ⟩
ψ⟩|
j
"
き
, |⟨b.j , φ⟩|2 は状態
φ
において物理量
B
の値が
b
である確率と解釈できる
;
!
"
=
b
|ψ
)
(5)
p (A = ai |ψ ) =j=1 c (A = ai |ψ,j bj ) p (B
M
j
A
M
%
A bj , E ψ
%
b
,
E
ψ
i
j=1
j
2|⟨b
i
=
|⟨b
,
ψ⟩|
.
j
2 =
p
(
2
2
⟨b
,
ψ⟩
j
p
(B
=
b
|ψ
)
=
|⟨b
,
ψ⟩|
.
j
j
⟨b
,
ψ⟩
, φ⟩| は状態 φ において物理量 B
の値が b, j|⟨b
である確率と解釈できる
;
j
このとき
は状態
φ
において物理
j=1
j , φ⟩|
j=1
書くことができる.
ある確率と解釈できる
;
⟨bj , ψ⟩W. Spekkens は量子力学の隠れた変
で, 2010 年に Nicholas Harriganj=1
と Robert
らは ontological model と呼ぶ) を分類するという仕事をした [1]. それは以下の
2
,である
φ⟩| compare
は状態
φ において物理量
B の値が bj である確率と解釈できる;
;
with
年に Nicholas Harrigan
と Robert
states {λ, 2010
} の混合であった考える
. すなわち
,量 W
preparation) ψ が実は onticところで
k k
2
(
彼らは
ontological
model と呼ぶ) を分類すると
数理論
ontological
model
by
Harrigan
&
Spekkens
(2010)
p (B = bj |ψ ) = |⟨bj , ψ⟩| .
ψ で出力 A = ai を得る確率が
;
ようなものである
%
"
A
p (Apre-selection
= ai |ψ ) =state
p (preparation)
(A = apost-selection
|λ
)
p
(λ
|ψ
)
(6)
φ / ⟨bj , φ⟩ は
φ および
b
における
E
の弱値
i
k
ψk が実は
‘onticstates
states’
j ontic
k }k の混
i {λ
k
ような射影演算子の弱値を次のように書くとする
子論を状態 ψ で出力 A =; ai を得る確率が
!
"
な ontic states {λk }k がすべての物理量に対して存在するという理論であると考
%
A
φ, Ei ψ
= aquasiprobability'
|ψ ) =. この
p (Aontic
= (4)
ai |λk
i という
る. このように量子論を考えるモデルを
model
.p (A
c (A = ai |ψ, φ ) := ontological
‘weak
⟨φ, ψ⟩
k
隠れた変数に対応する. この文脈において, 確率 p (λk |ψ ) (状態 ψ が実は ontic
あった確率
)は
state と呼ばれる. この量子論の ontological model を
用すると
,式
(2)epistemic
は
となるような
ontic states
{λk }k がすべての物理量に対し
appears naturally
in the ontological
interpretation
of QM,
tate を用いて分類したのが上記の仕事である. (そのうえで EPR-Bell の議論を
えるのである
. このように量子論を考えるモデルを
ontol
modulo the state
dependence
(and complex-valuedness)
M
%
いる（あまりちゃんと読んでいないけれど
...）)
が隠れた変数に対応する
λk =
c (A
ai |ψ, bj ) p (B = bj |ψ ) . この文脈において
(5) , 確
p (A = ai |ψ ) = state
j=1 λk であった確率) は epistemic state と呼ばれる. こ
state
epistemic1 state を用いて分類したのが上記の仕事である
できる.
したりしている（あまりちゃんと読んでいないけれど...）
? しかし, これは c (A = ai |ψ, φ ) に明
は弱測定を用いて実際に測定できる量である
• c (A = ai |ψ, φ )かもしれない
|ψ, φ ) は弱測定を用いて実際に測定できる量である
.
• ac i(A
c (A =
|ψ,=φa)i の性質
?きてはじめて達成することかもしれない,
M. Ozawa, AIP Conf. Proc. (2011)
properties
) の性質? 1. ai !→ c (A = ai |ψ,
) は複素測度
=φSteinberg,
a)i |ψ,
φ )Phys.
の性質
は弱測定を用いて実際に測定できる
• cφ(A
=c a(A
A|ψ⟩
=Rev.
a|ψ⟩
i |ψ,
A.
A?(1995) A|φ⟩ = a|φ
!
A = ai |ψ, φ )c2.
は複素測度
1. ai !→ c (A = a⟨A⟩
φ )⟨ψ|A|ψ⟩
は複素測度
(A
=)φaquasiprobability
φ?) =?1, 規格化
c
(A==
の性質
i |ψ,:=
(A
|ψ,
φ
の性質
i) |ψ,
i |ψ,
iacia
complex
weak
!
A
"
#
ψ,E
ψ
⟩
⟨
a
→
!
c
(A
=
a
|ψ,
φ
)
は複素測度
= ai |ψ, φ ) =3.1,1.c1.
規格化
A
(Acψ=
a=i |ψ,
φi)φは複素測度
(A =
a=
φ)=
= a1,i |ψ
規格化
iai=!→
i) =
i |ψ,
i c?
= ψ,
E
ψ
p
(A
).
(A
aic|ψ,
(A
ai⟨ψ,ψ⟩
|ψ,
)2.の性質
i
A
!"
A !
"
#
"
#
ψ,E
ψ⟩
ψ
⟨
⟩
⟨ψ,E
2.
c
(A
=
a
|ψ,
φ
)
=
1,
規格化
A
i
A
i a1.
i 2.
probability
B
A
cψ,
(A
=
|ψ,
φ
)
=
1,
規格化
i =ito
a
→
!
c
(A
=
a
|ψ,
φ
)
は複素測度
i
=
ψ,
E
3.
c
(A
=
a
|ψ,
ψ
)
=
i
i
=
E
ψ
=
p
(A
=
a
|ψ
).
|ψ, ψ ) = reduction
i
|ψ,
b
)
p
(B
=
b
|ψ
)
=
ψ,
E
E
ψ
=:
q
(A
=
a
,
B
=
4.
c
(A
a
(x
i
i
i
i
i
j f i− xi )t ⟨ψ,ψ⟩
⟨ψ,ψ⟩
i i
=
x
x
(t)
=
x
+
(t)
x
A
w
i
cl
w
"
#
!
ψ,E
ψ
⟩
⟨
"
A
"a |ψ, ψB)2.=Afunction);
#c ⟨(A
"
#
ability
(Kirkwood
A= 1, 規格化
T
iψ,E ψ ⟩
=
a
|ψ,
φ
)
=
ψ,
E
ψ
=
p
=
a
|ψ
).
3.
c
(A
=
A bi (A
i
i
i ic (A =i ai |ψ,
)
p
(B
=
b
|ψ
)
=
ψ,
E
4.
i
⟨ψ,ψ⟩
i
|ψ, bi ) p (B = b3.
= =ψ,aiE|ψ,
E
ψ
=:
q
(A
=
a
,
B
=
b
|ψ
)
,
joint
quasiprob
=
ψ,
E
ψ
=
p
(A
=
a
|ψ
).
ψ
)
=
i |ψc)(A
i
j
i
j
i
⟨ψ,ψ⟩
i
" ⟨ψ,E
# " function);
A
#
ψ
ability
(Kirkwood
⟩
B
A
A
"
#
Kirkwood function);
q
(A
=
a
,
B
=
b
|ψ
)
i
i
j
|ψ,
b
)
p
(B
=
b
|ψ
)
=
ψ,
E
E
ψ
=:
q
(A
=
a
,
B
=
b
4.
c
(A
=
a
=
ψ,
E
ψ
=
p
(A
=
3.
c
(A
=
a
|ψ,
ψ
)
=
i
i
i
i
ja
j
i
i
B
A
relation
to
joint
quasiprobability
(Kirkwood
function)
⟨ψ,ψ⟩
i
.
|ψ,
b
)
=
c
(A
=
a
q (Aw σ=1 ai ,
4. c (A = ai |ψ, bi ) p (B = bi |ψ
i ) =i ψ, Ej Ei ψ =:
−igA
(B
ability (Kirkwood function);g  σ1 ΨMD (0,p0)
" =e bBi |ψA) # ΨMD
ability (Kirkwood
function);
aaii,|ψ,
B b=
|ψ =
) bi |ψ ) = ψ,cE(A
pj(B
ψi |ψ,
=: bqi (
4.q c(A
(A==
i )b
j E
i a
=
. (θ, φ) = cos θ |0⟩ + eiφ
c (A = ai |ψ, bi ) =
Ψ
=
Ψ
MD
c (A = ai |ψ, φ ) はability
conditional
p (B(Kirkwood
= bMD
|ψ ) function);
q (A = ai , BA=と
bjB
|ψが可換
)
i quasiprobability[2].
.b |ψ
c (A = ai |ψ, bi ) =
q
(A
=
a
,
B
=
i )
j
p
(B
=
b
|ψ
i
,
B
=
b
|ψ
)
なので
,
c
(A
=
a
|ψ,
b
)
は通常の条件付き
p
(A
=
a
j
ii ) C
=) iはAconditional
(Ac=
Avalue
:=
Aa=
+
+quasip
Cwai ,
weak value asi expectation
i |ψ,
(A
aBi b|ψ,
φ
w c
w = Bw
q
(A
=
p
(B
=
b
|ψ
)
|ψ, φ ) は conditional quasiprobability[2]. A と B が可換のとき
,
q
(A
a
,
B
=
i
i
|ψ,
b
)
=
c
(A
=
a
i
i
!c (A = a |ψ, φ ) は conditional
wpquasiprobability[2].
,
B
=
b
|ψ
)
なので
,
(A
=
(A
=
a
pc (B
=
i
j
A
と
B
が可換の
i
⟨A⟩
value
Awφψ
:=, expectation
A = B C →Aweak
, B = bj |ψ )5.なので
(A==aai i|φ,
|ψ,ψb)i )=は通常の条件付き確率になる
w = .B
w Cw
i a,i cc(A
c (A
は) conditional
Aw と B が
= bφj)|ψ
なので,!
c (A quasiprobability[2].
= ai |ψ, bi ) は通常の条件付き確
p (A
==
ai ,aB
i |ψ,
c (A = ai5.
|ψ, φ )i(P
は
quasiprobability[2].
A
ai1cconditional
(A
=
a
ψ
)
=
⟨A⟩
,
expe
i |φ,
w
φψ
)
=
(P
)
=
1
(P
)
=
w = a |ψ,
2 wb ) は通常の条件
3 w
law
of
, B = b j→
|ψ weak
) なので
,
c
(A
(A,total
=
aiprobability
A = ai |φ, ψ ) 6.
= ⟨A⟩
expectation
value
!pφψ
i
i
w
⟨A⟩dBB = dλ A(λ)ρ(λ)
λ x
⟩ ⟩ = |ψ(x)|
⟨φ|A|χ
⟨φ|χ
kρ(λ)
k
k
+ iWI ) is a arbitrary operator.
Aw =
ther conditions to the evaluation⟨φ|χ
function
on projection operk⟩
!
We
now
get
another
expression
for
linearity
conditions.
We
settle
proper
ar in the Gleason’s theorem.
⟨A⟩QM = ⟨ψ|A|ψ⟩
2
!
=
dλ
A(λ)ρ(λ)
ρ(λ)
=
|ψ(x)|
λ xvalue assignment
tion on ⟨A⟩
the
two-state
value.
First,
we
define
physical
3.dBBPostselected measurement and
quasiprobability
2
⟨A⟩
=
dλ
A(λ)ρ(λ)
ρ(λ)
=
|ψ(x)|
λ space
x sat- H.
Gleason’s
The dimension
the Hilbert
space
H
ion µ on theorem).
thedBB
projection
operators of
P(H)in
a Hilbert
the
function
λ : P(H) → C, according
"
⟩
⟨φ|A|χ
⟩
⟨φ|A|χ
3. If the function µ : P(H)
→
R
allows
following
conditions
k
k
k
⟨A⟩QM A
=k⟨ψ|A|ψ⟩
A
complex
probability
measure
(extending
Gleason)
as
same
form
of (3.1).
A
=
=
a
E
µ(E)
=
tr(W
E)
W
A
=
w
w
⟨A⟩
=
⟨ψ|A|ψ⟩
i i⟩
nition 3.2 (The two-stateQM
value on projection
operators).
⟨φ|χk ⟩ ⟨φ|χ
kThere are another conditions to th
i
0
≤
µ(P
)
≤
1
(3.42)
T. Morita
and
I.T. (2012)
unction µ : P(H) → C satisfies following condition.
ators
which
appear
in the Gleason’s th
"
#
! $ !
"
A
Ai E
"
"
a
µ(E)
=
tr(W
E)
W
A
=
theorem). T
i
ai Ei Theorem
µ(E) = tr(W E) W A =
23.3 (Gleason’s
2
⟨A⟩
=
dλ
A(λ)ρ(λ)
ρ(λ)
=
|ψ(x)|
λ
x
=
dλ
A(λ)ρ(λ)
ρ(λ)
=
|ψ(x)|
λ
x
⟨A⟩
dBB
=
µ
E
µ(E
)
{E
}
dBB
isfies
dim(H)
≥
3.
If
the
function
µ
i
i
i
)
=
1
(3.43)
µ(
i
artial linearity
・condition
i
i
# i $$
#
0≤µ
#
"#⟨A⟩
" ! ⟨A⟩
"
"
=
⟨ψ|A|ψ⟩
=
⟨ψ|A|ψ⟩
"
"
QM
QM
"
"
µ
EEi i == µ(E
µ
µ(E
i ) i ){E{E
i } i } mutually orthogonal
Pi orthogonal
µ (Pi ) (3.44)
(3.50)
=
Pi =
µ (Pi ) {P
} :µmutually
ii i
i ⟨φ|A|ψ⟩
µ(
i
=
A
w i
i
i
⟨φ|ψ⟩ i
"
"
!
#
A
"
"
ntation
of this
function
µ isorthogonal
written
as
⟨φ|A|ψ⟩
a
E
µ(E)
=
tr(W
E)
W
A
=
trace-class
a
P
µ(P
)
=
tr(W
P
)
W
A
=
where {P
}
are
mutually
projection
operators.
i
⟨φ|A|ψ⟩
i Pii =
iµ
i
AA
µ (Pi )
w ==
w
⟨φ|ψ⟩
i
i
i
i
⟨φ|ψ⟩
⟨ψ|E|φ⟩
⟨φ|E|ψ⟩
µ(P ) = tr(ρP )
(3.45)
+ (1 − α)then, the representation of this functio
µ(E) =condition
tr(W E)#= α $
me
symmetric
e representation of W is " ⟨φ|ψ⟩ "
⟨ψ|φ⟩
⟨ψ|E|φ⟩
⟨φ|E|ψ⟩
initial
and
final=operator.
state
isµ|ψ⟩,
states
not
orthogonal,
i.e.,|ψ⟩
⊥
̸
tive
self-adjoint
⟨φ|A|ψ⟩
µ(P )
preselection
+ (1
−µ(E
α) are
µ(E)
tr(W E)
= α |φ⟩.
=
Ei these
)
{E
}
⟨ψ|E|φ⟩
⟨φ|E|ψ⟩
i
i
A
=
⊥ = tr(W
⊥
|ψ⟩⟨φ|
|φ⟩⟨ψ|
⟨φ|ψ⟩
⟨ψ|φ⟩
w
+
(1
−
α)
µ(E)
E)
=
α
⟩
⊥
|ψ⟩,
∀|φ
⟩
⊥
|φ⟩.then,
|φ⟩. set
∀|ψ
postselection
i
W =α
+ i(1 ⟨φ|ψ⟩
− ⟨x|A|ψ⟩
α)arbitrariness
, ⟨φ|ψ⟩
α⟨ψ|φ⟩
∈ ρCis a positive self-adjoint
(3.20)
where
opera
µ : P(H) → R is expected
to
be
a
provability
measure
on
the
⟨φ|ψ⟩A(λ) =
⟨ψ|φ⟩
⟨x|A|ψ⟩
he condition (3.42) is positivity
of
the
function.
Positivity
keeps
The
function µ : P(H)(3.51)
→ R is exp
⟨x|ψ⟩
)
=
µ(|ψ⟩⟨ψ|)
=
1
µ(P
ψ
A(λ) =
⟨φ|A|ψ⟩
dBB
TEST SPACE
⟨φ|A|ψ⟩
⟨φ|A|ψ⟩
⟨x|A|ψ⟩ A =
w
A⟨x|A|ψ⟩
A(λ) =
w =
⟨x|ψ⟩⟨φ|E|ψ⟩ ⟨φ|ψ⟩ ⟨A⟩ ⟨ψ|E|φ⟩
⟨φ|ψ⟩
QM = ⟨ψ|A
⟨A⟩
QM = ⟨ψ|A|ψ⟩ µ(E) = tr(W E) = α
A(λ)
+
(1
−
α)
R
C
R
I
C
I
"
⟨φ|ψ⟩
⟨x|ψ⟩
⟨ψ|A|φ⟩⟨ψ|φ⟩
⟨φ|A|ψ⟩
physical observable
⟨x|A|ψ⟩
+ (1
− α)
λ(A) =
ai µ(Pi ) = tr(W A) = α
A(λ)
=
⟨φ|ψ⟩
⟨ψ|φ⟩
⟨ψ|E|φ⟩
⟨φ|E|ψ⟩
i
⟨x|ψ⟩
"+ (1 − α)
µ(E) = tr(W E) = α ⟨φ|A|ψ⟩
⟨ψ|A|φ⟩
|φ=
|ψ⟩E) |φW
|
1 ⟩tr(W
1⟩ A
⟨x|A|ψ⟩
A
µ(E)
⟨ψ|φ⟩
µ(P=
= tr(W
ai+
Ei(1"− α)
(E)
E) A)W= αA⟨φ|ψ⟩
=
i ) tr(W
⟨ψ|A
⟨φ|A|ψ⟩
A(λ)
=
+ (1 − α)
= tr(W⟨x|ψ⟩
A) = α
⟨φ|ψ⟩λ(A) = ai µ(Pi )⟨ψ|φ⟩
⟨φ|ψ⟩
⟨ψ
†
⟨A⟩⟨ψ|U
:H
R(A)
⊂
R
I
A(t)
U
(t
)|φ⟩
(ti )→
⟨φ|Ui(tf ) A(t) U †i(ti )|ψ⟩
f
$
#
+(1−α)
λ(A)
=
α
$
#
† (t )|ψ⟩
⟨x|A|ψ⟩
"(tf ) U † (ti )|ψ⟩
"⟨ψ|A
"
⟨φ|U
⟨φ|U
(t
)
U
⟨φ|A|ψ⟩
i
f
A
!
A(λ)
= =
"
"
+
(1
−
α)
a
µ(E
)
=
tr(W
A)
=
α
λ(A)
⟨φ
|A|ψ
i
=
µ
E
µ(E
k
i
i
2
⟨x|ψ⟩
⟨φ|ψ⟩
⟨ψ
µ
Ei =
µ(Ei ) {E
|⟨φ
|ψ⟩|
†
†
i}
k
)
A(t)
U
(t
)|ψ⟩
)
A(t)
U
⟨ψ|U
(t
⟨φ|U
(t
i
f
i
i
i
i |ψ⟩
†
†
⟨φ
+(1−α)
λ(A)
=
α
k †
) A(t)
U (ti )|ψ⟩
)
A(t)
U
(t
)|φ⟩
⟨ψ|U
(t
(tf"
†
i
i
i
f
k
(ti )|ψ⟩
⟨φ|A|ψ⟩
† ⟨φ|U (tf ) U ⟨ψ|A|φ⟩
† ⟨φ|U (ti ) U (tf
A
+(1−α)
(ti )|ψ⟩
⟨ψ|U (ti ) A(t) U (tf )|φ⟩
⟨φ|U
(tf ) A(t) U +
(1
−
α)
a
µ(E
)
=
tr(W
A)
=
α
=
†
†
i
+(1−α)
|U (tf ) U (tii )|ψ⟩λ(A) = α ⟨φ|U (t
⟨φ|U
(t
)
U
(t
)|ψ⟩
i
f
†
† (t )|ψ⟩
⟨φ|A|ψ
⟨φ|ψ⟩
⟨ψ|φ⟩
)
U
(t
)|ψ⟩
⟨φ|U
(t
)
U
f
i
i
f
⟨φ
|A|ψ⟩
1
i
†
†
=
A
w
⟨φ|A|ψ⟩
U
(
⟨ψ|U (ti ) A(t)
⟨φ|U (tf ) A(t) U (ti )|ψ⟩
⟨φ|ψ⟩
Aw =
λ(A)
=α ′
† +(1−α)
†
⟨φ
|ψ⟩
′
† ′(t
†)|ψ⟩
1
†
†
)
A(t)
U
(t
)
A(t)
U
⟨ψ|U
(t
⟨φ|U
f
i
i
U⟨φ|ψ⟩
(t, t ) = U (t−tλ(A)
) ⟨φ|U
=U
(t)U
(t
)
A(t)
:=
U
(t)
A
U
(t)
|φ
⟩
|ψ⟩
(tf ) U (ti )|ψ⟩
⟨φ|Uk(ti ) U (tf )
+(1−α)
=α
† (t )|ψ⟩
† (t
⟨φ|U
(t
)
U
⟨φ|U
(t
)
U
f
i
i
f
Note: most of
the
properties
of
the
weak
value
(and
the
quasi
†
†
|A|ψ⟩
⟨φ
⟨φ|E|ψ⟩
2
)
A(t)
U
(t
)|ψ⟩
)
A(t)
U
(t
)|φ⟩
⟨ψ|U
(t
⟨φ|U
(t
2
†
†
f (t )|ψ⟩ i
i
f
2
)
A(t)
U
)
A(t)
U
(t
)|φ⟩
⟨ψ|U
(t
(t
µ(E)
=
tr(W
E)
=
α
probability)
mentioned
earlier
hold
even
with
the
parameter
!
f
i
i
f
+(1−α)
=α
′
′
†
′
†
⟨ψ|E|φ⟩
⟨φ|E|ψ⟩
U (t, t ) =⟨φ|U
U (t−t(t) )=UU†(t)U
(t ) A(t) := U (t) ⟨φ
A⟨φ|ψ⟩
U2(t)
† (t )|ψ⟩
|ψ⟩
+(1−α)
⟨φ|U
(t
)
U
(t
)|ψ⟩
f
i
i
f
+ (1 − α)
= tr(W E)†= α
†
|U (tf ) U (ti )|ψ⟩
⟨φ|ψ⟩
′
†
′
2
⟨φ|U
(ti ) U (tf )|ψ⟩
⟨ψ|φ⟩
†
−t ) = U (t)U (t ⟨x|A|ψ⟩
) A(t) := U (t) A U (t)
A(λ) =
⟨x|ψ⟩
|φk ⟩
"
2
|ψ⟩
⟨φn |A|ψ⟩
⟨x|A|
A(λ) ⟨φ
= n |ψ⟩
⟨x|ψ
K(α) = ⟨φ|uA⟨(
AB
below.
all
briefly
describe
below.
tion
theory).
Notes
of mathematical
the
Mathematical
Notations
Used.
Throughout
this
paper
istribution
theory).
formal”
computation
and
state
our
results
in
a
laxer
way.
C
ctively
called
the
Fourier
transform
and
the
inverse
Fourier
transform
of
f
.
The
α
ic properties of the distribution wAB [φ](a, b) regarding integrations are× mentioned
theInterest.
real field
R
or
the
complex
field
C
, and
define
Ka normalised
:= K For
\ {0}. The
c
ˆ
Definition
of
the
Distribution
of
Let
|φ⟩
∈
H,
∥φ∥
=
1
be
ve
p
F
that
maps
f
to
its
Fourier
transform
f
is
called
the
Fourier
transformation.
or has confirmed that most of the arguments can be elaborated in the
C isMarginals
denoted by c. In order
to avoid confusion, w
complex
number
c ∈ its
α [φ](a,
n
∗
Quasiprobability
Distribution
w
b)
and
Total
Integration.
Observing
that
the
value
of
the
Fourier
transform
a Hilbert
space
H, and
let
A,AB
B be
self-adjoint
operators
on
H.
For
a×normalised
real
number
αvev∈
(nd
Rinof
),
the
properties
under
the
convolution
(4),
involution
f
(x)
:=
f
(−x)
integration
theory
and
the
theory
of
generalised
functions
(Schwartz’s
on
the following
Distribution
of
Interest.
Let
|φ⟩
∈
H,
∥φ∥
=
1
be
a
efinition
of
the
Distribution
of
Interest.
Let
|φ⟩
∈
H,
∥φ∥
=
1
be
a
normalised
J.
Lee
and
I.
T.,
in
preparation
of
all
natural
numbers
including
0
by
N
,
and
N
:=
N
\
{0}.
Sinc
family
of
joint
quasiprobabilities
0
0
α
coincides
with
the
total
integration
of
the
original
function
as
asic
properties
of
the
distribution
w
[φ](a,
b)
regarding
integrations
are
mentioned
we
define
a
distribution
f
)(x)
:=
f
(x
+
a)
are
basic:
slation
(τ
a
AB
is
on
quantum
mechanics,
Hilbert
spaces
are
always
assumed
to be
).
n a Hilbert
space
H,let
and
be self-adjoint
operators
a real
number
bert
space H,
and
A,let
B A,
beBself-adjoint
operators
on on
H. H.
ForFor
a real
number
αα
tegration.
Observing that the value of the Fourier transform at the point 0
otherwise.
Conforming
to the convention in physical literature, an inne
"
ˆ
!
e
define
a
distribution
a distribution
(ffor
∗ g)
=the
f
·
ĝ,
(8)
s with
the total
integration
of
original
function
as
distribution
two
(non-commuting)
observables
A,
B
α
i(1−α)sA
itB
iαsA
on a complex
linear
in its first argument and linea
t) :=fˆ(0)
⟨φ,
e= spaceeis−i⟨0,x⟩
eanti-linear
e f (x)
φ⟩,
uAB [φ](s,
dm
n (x)
self-adjoint
operator
A
on
a
Hilbert
space
H,
we
denote
its domain by
Distribution
of
Interest.
Let
|φ⟩
∈
H,
∥φ∥
=
1
be
a
normalised
vector
∗!
ˆ value of the Fourier
n
f
(9)
f%
=
Integration.
Observing that
the
transform
at
the
point
0
R
α
i(1−α)sA
itB
iαsA
define
its:=
expectation
value
α
i(1−α)sA
itBebya
iαsA
!
[φ](s,
t)
⟨φ,
e
e φ⟩,
φ⟩,
u
−i⟨0,x⟩
AB
H,
and
let
A,
B
be
self-adjoint
operators
on
H.
For
real
number
α ∈ R,
[φ](s,
t)
:=
⟨φ,
e
e
e
u
ˆ
des
with
the
total
integration
of
the
original
function
as
e
f
(x)
dm
(x)
AB
f
(0)
=
n
and its Fourier transform
i⟨a,x⟩ ˆ
#
⟨φ, Aφ⟩
n
f
(t),
a
∈
(τ
f
)(t)
=
e
R
,
(10)
=
f
(x)
dm
(x),
a
n
R
[A;
φ]
:=
,
|φ⟩
∈
dom(A),
ution
!!
∥φ∥2
Rn
n ) ⊂ L1 (Rn ),
nd
its transormation
Fourier transform
Fourier
transformation
−i⟨0,x⟩
∧
α
α
urier
F
plays
particularly
well
on
the
subspace
S
(
R
ˆ
=
(x)
(x),
Fourier transform wfAB
For
simplicity
of dm
notation,
we have
ef:=
f (x)
(0)[φ](a,
= b)
n
(udm
[φ])
(a,
b), we adopt the natural units, where(20)
n (x)
AB
n
n!
n ), of which inverse is given by
αa linear
i(1−α)sA
itB
iαsA
R
R
)
onto
S
(
R
becomesuone
bijection
of
S
(
has
⟨φ, e !
e e
φ⟩,
(18)
AB [φ](s, t) :=
∧
α
α −i(as+bt)
α
Fourier
Transformation.
It proves
useful
tot).
first renormalise the n
w
[φ](a,
b)
:=
(u
[φ])
(a,
b),
e
u
[φ](s,
t)
dm
(s,
=
2
∧
AB
AB
α
α
AB
!
3
f (x)[φ])
dm
(20)
n (x),
n
n ), n ∈ N × by
wAB [φ](a, b) =:= R(u
(a,
b),
2
on
(
R
,
B
measure
β
R
! αn
n AB
α
w
[φ](a,
b)
dm
(a,
b)
=
ŵ
0)
!
2
nsform
−i(as+bt) α
AB
AB [φ](0,
−n/2
!
e
uAB [φ](s,
dm:=2 (s,
=R−i(as+bt)
2
dmt)n (x)
(2π)t). dβn (x),
then
α
α
2
are
interested
in
the
latter
wAB
[φ](a,
b), of which prope
α
αdistribution
e
u
[φ](s,
t)
dm
(s,
t).
=
R
sFor our purpose, we
2
AB
wAB [φ](a, b) and
dm
(a,
b)
=
ŵ
[φ](0,
0)
p
2
αthe convolution
renormalized
n-dimensional
Lebesgue measure
AB
-norm
and
by
the
L
2 accordingly redefine
=
u
[φ](−0,
−0)
R
2
∧
AB
αshall briefly !
R
we
describe
below.
!"
#1/p
wAB
[φ](a, b) :=
(uαAB [φ])
(a, b),
α
α interested in the latter
ααdistribution wAB [φ](a,
2 (x)
or our purpose, we!are
b),
of ,which
prop
p
p
n
∥f
∥
:=
|f
(x)|
dm
f
∈
L
(
R
)
wAB
[φ](a, b) dm2 (a, b) =
[φ](0,
0)
=ŵuAB
[φ](−0,
−0)
=
∥φ∥
=
1,
p
n
AB
α
purpose,
we are
interested
in the
distribution w"RAB [φ](a, b), of which prope
R2
−i(as+bt)
α latter
e2.1.
shall briefly
describe
below.
α
e Distribution
uAB [φ](s,
t)[φ](a,
dm22 (s,
t). its Marginals (19) (21)
=
Quasiprobability
wAB
b)
and
=
∥φ∥
=
1,
α
(f[φ](−0,
∗joint
g)(x)quasiprobability
:=
= αuAB
−0) fα(x − y)g(y) dmn (y), f, g ∈ L1 (
briefly describe below.
R2
Two basic
properties
of the implies
distribution
[φ](a, b) regarding
integrations
are mentio
which
in particular
that w
the
[φ](a,
b) in question
is a
Rw
ABdistribution
AB
2
α
=
∥φ∥
=b)1,we
(21) is no ris
respectively.
For
brevity,
write
dm
.1.
Quasiprobability
Distribution
w
[φ](a,
and
its
Marginals
1 = dm whenever there
α
below.
distribution.
α
AB
implies
the distribution
w
[φ](a,
b)
in
is
a
quasiprobability
eparticular
are interested
in that
the latter
distribution
w
[φ](a,
b),
which
properties
1 (question
nof
ˆ, fˇ : Rn
AB
AB
R
),
recall
that
the
functions
f
Now,
for
a
function
f
∈
L
α
α b)
uasiprobability
Distribution
w
[φ](a,
and its5 Marginals
wo
basic
properties
of
the
distribution
w
" integrations are ment
AB
AB [φ](a, b) regarding
ion.
cribe below.
n
n
α
R (29) on the marginal
on
as
yifdirectly
demonstrate
utilising
the
self-adjoint
(9)), i.e.,results (28),
the distribution
uABitisby
α(cf. previous
!
α
αI denotes the identity
w
[φ]
=
w
[φ],
(30)ope
is
the
special
case
for
T
=
tI,
where
α
AB
=
(F
⊗
I)u
[φ](a,
0)
α
AB
α
α
AB
[φ](a,
however for later
convenience,
we take
a
the quasiprobability
w
=
(F
⊗
I)u
[φ](a,
0)
∗ [φ](a,
wAB
dm(b)
(22)
P[A = a; distribution
wAB [φ]] :=
α
α b)b),
AB
AB
=convenient
uA(t)
[φ],
(u=AB0,[φ])
#
† " to define
† (31)
AB
det(T
)
Rit is
α
)
U
(t
)|ψ⟩
)
A(t)
U
(tfthat
)|φ⟩
⟨ψ|U
(t
⟨φ|U
(t
f (cf.
ii.e.,
i#
"
φ instead.
ernative
approach based
on
the
relations
(11)
and
(12)
Now,
observing
is
self-adjoint
(9)),
the distribution
uAB
!
alif Distribution.
We
define
the
marginal
distributions
of
the
quasiprobability
λ(A) = α
(a)$!
= F µ̌A+(1−α)
φ
†
† (t )|ψ⟩
%
marginals
α
α
⟨φ|U
(t
)
U
(t
)|ψ⟩
⟨φ|U
(t
)
U
F
µ̌
(a)
=
f
i
i
f
P[B
=
a;
w
[φ]]
:=
w
[φ](a,
b)
dm(a)
(23)
if
α
=
1/2
(one
may
check
the
last
statement
directly
from
definition).
A
∧
AB
AB
αα
ution as
∗
α α
aw(u
[φ](a,
b)=!=ua
(uAB [φ])φf (a,
b):=
(31)
R
AB
AB [φ])
AB [φ],
(x)
f
(x)
dm
(x)
δ0 (x)
0
n
=
µ
(a),
φ†n
α coincide
′α
†α A′ with
below that the marginal
distributions
respectively
the
probability
∧
−1
R
:= ′=
w
[φ](a,
b)
dm(b)
(22)
P[A =
a; w
)AB
=[φ]]
U (t−t
)=
U
(t)U
(t
)
A(t)
:=
U
(t) A U (t)
|φk ⟩ (34
|ψ
U1.
(t,
tPreliminaries
=
µ
(a),
AB(∂1 u
α
i
[φ])
(a,
b)
A
AB
α = 1/2 the
(one
may and
check
the
last statement
directly
from
definition).
asiexpectation
Value
Quasicovariance
of
w
[φ](a,
b)
R the ideal
onifdescribing
behaviour
of
the
outcome
of
quantum
measurement
of
AB
and analogously,
!
†
where
δ
denotes
the
delta
distribution
centred
at
0.
0
Notes
of
the
Mathematical
Notations
Used.
Throughout
this p
)|ψ⟩
U
(t,
t
)|φ⟩
⟨φ|U
(t,
t
)
U
(t,
t
α
i
f
f
and analogously,
φ
has A and B.
cevables
properties
of the
quasiprobability
distribution
w
[φ](a,
b)
regarding
its
“quasiα
α
α
AB
P[B In
= a;
w
[φ]]
:=
w
[φ](a,
b)
dm(a)
(23)
[φ]]
=and
µBdefine
(b).oneKhas
P[B
=
b;
w
!
× := K
AB
AB
AB
α
relation
to
the
Fourier
transformation,
the
real
field
R
or
the
complex
field
C
,
\ {0}. T
φ
αand
we make
a small
mathematical
preparation.
For aof
self-adjoint
operator
X
asiexpectation
Value
and
Quasicovariance
w
[φ](a,
b)
1
R
nhis,value”
and
“quasicovariance”
is
presented,
and
its
relation
to
the
(quantum)
P[B
=!|3⟩)
b; wABα[φ]] = µB (b).
α
α
AB
√
(|1⟩
+
|2⟩
+
|ψ⟩
=
[φ]]
=
a
w
[φ](a,
b)dm
(a,
b)
E[A;
w
2 distribution
AB
AB
The
results
show
that
the
quasiprobability
w
[φ](a,
b)confusio
has som
sed
vector
|φ⟩
∈
H,
∥φ∥
=
1,
let
C
is
denoted
by
c.
In
order
to
avoid
complex
number
c
∈
α
AB
es
below
that
the
marginal
distributions
respectively
coincide
with
the
probability
2
3
cn properties
of
the quasiprobability
distribution
b) regarding
its “quasi- α
value andwith
covariance
of the observables
A andw&
B
is[φ](a,
revealed
below.
R
−i⟨q,x⟩
AB
×
! show
The
results
that including
the
quasiprobability
distribution
wnAB
(f
)(q)
=
e
f
(x)
dm
(x)
φ all
T
T
useful
properties
that
are
also
shared
with
the
familiar
Wigner
quasiprobabilit
of
natural
numbers
0
by
N
,
and
N
:=
N
\
{0}.
0
0
(B)
:=
⟨φ,
E
(B)φ⟩
(24)
µ
ution
describing
the behaviour
of the
outcome
ofprobability
the
quantum
measurement of
measure
n value”
and “quasicovariance”
isX−1
presented,
its relation
X
1and
n to the (quantum)
∧ ideal
α
R
= see
i below
(∂|φ⟩
[φ])
(a,
b) |2⟩
dm
(a,
b)are
√
=
(|1⟩
+
−
|3⟩)
useful
properties
that
are
also
shared
thealways
familiar
Wignerto
1 uAB
2with
In
fact,
we
shall
shortly
that
the
Wigner
quasiprobability
distribut
!
is
on
quantum
mechanics,
Hilbert
spaces
assumed
ervables
A
and
B.
2
X. quasiprobability
Defining
the
defined
spectralA
measure
value andmeasure
covariance
ofthrough
the observables
and
is ofrevealed
below.
ne probability
Value.
Defining
the
“quasiexpectation
value”
of
the
distriRthe
3 BEX
α
−i⟨q,T
x⟩
In
fact,
we
shall
shortly
see
below
that
the
Wigner
quasiproba
[φ](a,
b)
o
a
special
case
to
the
family
of
quasiprobability
distributions
w
otherwise.
Conforming
to
the
convention
in
physical
literature,
an
=
e
f
(x)
dm
(x
e[φ](a,
this, we
make
a
small
mathematical
preparation.
For
a
self-adjoint
operator
X
and
n
AB
−1
α
ansform
!
value = i P(∂
b)expectation
as
[φ]) i(0,
0)
=
1, 2, 3
Pi R
=n |ci ⟩⟨ci |
1 u|i⟩⟨i|
B
AB
i =
−itx
interest.
on=a=
complex
linear
space
is anti-linear
in its (25)
first argument
and
aν)(t)
special
case
to
the
family
of quasiprobability
distributions
ailsed
vector
|φ⟩
∈
H,
∥φ∥
1,
let
(F
ν̂(t)
:=
e
dν(x)
!
"
!
n Value.
Defining the “quasiexpectation
value” of the quasiprobability
distri"
R
∂
∗
⟨b|P
α self-adjoint
α
i(1−α)sA
itB(a,
iαsA
i |a⟩
−i⟨T
q,x⟩
A
on
a
Hilbert
space
H,
we
denote
its
doma
"
φ= i−1operator
interest.
[φ]]
:=
a
w
[φ](a,
b)dm
b),
(32)
E[A;
w
6
⟨φ,
e
e
e
φ⟩
2
=
e
f
(x)
dm
(x
(P
)
=
ABr. h.µs. is
ABEX
n
(B)
:=
⟨φ,
(B)φ⟩
(24)
[φ](a,
b)
as
lex
measure
ν,
where
the
understood
as
the
Lebesgue
integration
with
i
w
"
X
B
⟨b|a⟩ (s,t)=(0,0)
R ∂s
define
its
expectation
value
by
Rn
!
ν, one has
6
X. Defining the
the probability measure
α defined through
α the spectral measure E
X of
∗
α
ˆ
aφ⟩w
wAB
[φ](a,
b)dm
(a,
b),= f⟨φ,
(32)
E[A;
= (1
−
α)⟨φ,
Aφ⟩ +
α⟨φ,22(a,
Aφ⟩b),
(TAφ⟩
q),
α wAB [φ]] :=
itB
E[B;
b
b)dm
(33)
AB[φ](a,
[φ](0,
t)
=
⟨φ,
e
u
[A;A|φ⟩
φ] :== a|φ⟩2 , |φ⟩ ∈ dom(A),
AB
transform
!
R
A|ψ⟩
=
a|ψ⟩
! !
∥φ∥
=
[A;
φ].
(35
−itx
∗
itb
(F
ν)(t)
=
ν̂(t)
:=
e
dν(x)
(25)
where
T denotes
the
adjoint
(in
this
case,
the outcomes
transpose)ofof t
e with
d⟨φ,
E
=
that they respectively
expectation
values
of
the
α coincide
α the
X (b)φ⟩
⟨A⟩
:=
⟨ψ|A|ψ⟩
⟨B⟩
:=
⟨φ,
Bφ⟩
For
we2adopt
E[B; wAB
[φ]]simplicity
:=R b wofABnotation,
[φ](a,
b)dm
(a, b), the natural units, where
(33) we
Rn
analogous
argument
holds
for
the
B,
which
leads
to
RA case
measurement
of
the
observables
and
B
on
the
quantum
state integration
|φ⟩. Indeed,with
one
φ
mplex measure ν, where the
r.
h.
s.
is
understood
as
the
Lebesgue
(26)
= µ̌B (t),
that
respectively
coincide
with
values
ofon
the
outcomes
of (36t
α the expectation
tly
it byFourier
utilising
the
results
(28),
(29)
the
marginals
to demonstrate
ν,they
one has
Transformation.
It
proves
useful
to
first
renormalise
[φ]]
=
[B;
φ].
E[B;
wprevious
AB
ise
α
n
n ),the
× byconvenience,
measurement
of the observables
B
on
quantum
state |φ⟩. Indeed,
one
b),
however
for
later
we
take
an
siprobability distribution
wAB [φ](a,
α
(RitB
,B
n
∈
N
measure
βnAonand
−
x
)t
(x
f
i
n
n
ccasionally called the (symmetric) quantum covariance of the observable A and B.
occasionally
called the is
(symmetric)
quantum covariance
of the observable
A and
B.
The
demonstration
quite straightforward.
Utilising
the same
technique
introduc
The demonstration is quite straightforward. Utilising the same technique introduced in
he Quantum
previous
paragraph,
first
observe
that
Covariance.
Defining
the
“quasicovariance” of the quasiprobability distributhe previous
paragraph,
first
observe
that
covariance
α
!
wAB [φ](a, b) asDefining the
mtion
Covariance.
of the quasiprobability distribu! “quasicovariance”
α
αα
α
[φ]]
=
ab
w
[φ](a,
b)dm
E[AB;
w
[φ]]
=
ab
w
[φ](a,
b)dm
b) 2 (a, b)
E[AB;
w
2 (a,
AB AB
AB
αAB
α
α
α
B [φ](a, b) as
2 E[A; w
R
[φ]]
:=
E[(A
−
[φ]])(B
−
E[B;
w
[φ]]);
w
CV[A,
B;
w
R
nce.
Defining theAB“quasicovariance”
of
distribuABthe quasiprobability
AB
AB [φ]]
! !
∧
α
α
α
α
αα∂ u
α dm
α
=−
i−2 w
(∂
[φ])
(a,
b)
(a,
b)
∧
[φ]] := E[(A=
E[A;
w
[φ]])(B
−
E[B;
w
[φ]]);
w
[φ]]
−2
α
s CV[A, B; wAB
[φ]]
−
E[A;
w
[φ]]
·
E[B;
w
[φ]],
(37)
E[AB;
1
2
2
AB(∂AB∂ u
AB
AB
AB
AB
=R
iAB
[φ])
(a,
b)
dm
(a,
b)
1 2 AB
2
2
α R
−2
α E[A; w α [φ]] · E[B; w α [φ]],
[φ]]
−
(37)
=
E[AB;
w
=αi AB
(∂1 ∂2 uAB [φ]) (0, 0)
α
α AB
ABα
where
[φ]])(B
wAB
[φ]]);
wAB [φ]]
; wAB [φ]] := E[(A − E[A; wAB
−2 ! − E[B;
α
"
=
i
(∂
∂
u
[φ])
(0,
0)
1 2 AB α
α
" (a, b),
∂
∂
with
[φ]]
:=
ab
w
[φ](a,
b)dm
(38)
E[AB;
w
−2
i(1−α)sA
itB
iαsA
"2
AB
ABe
"
α
α
α
⟨φ,
e
e
φ⟩
=
i
2 [φ]] · E[B; w
"
!∂s ∂tw∂AB
[φ]],
(37)
= E[AB; wAB [φ]] − E[A;
R∂
"
AB
(s,t)=(0,0)
−2
α
α ⟨φ, ei(1−α)sA eitB eiαsA φ⟩"
=
i
[φ]]case
:= α = ab
w∂t
[φ](a, b)dm
b),quantity
(38)
E[AB;
"
one finds below
that, w
for
the
1/2,
it coincides
with
the
2 (a,
AB
AB
∂s
= (1 − Rα)⟨φ,
ABφ⟩ + α⟨φ, BAφ⟩,
2
(s,t)=(0,0)(40)
!
B;
φ]
[(AB
+ BA)/2;
φ]+−α⟨φ,
φ] · [B; φ]
(39)
ds which,
belowfor
that,
for theα case
αreduces
= :=
1/2,
it α)⟨φ,
coincides
with
the[A;
quantity
α(1to
=
−
ABφ⟩
BAφ⟩,
theα choice
=[A,
1/2,
ab wAB [φ](a, b)dm2 (a, b),
(38)
E[AB; wAB [φ]] :=
2
2
2
R
1/2
(symmetric)
occasionally called
the
quantum
covariance
of
the
observable
A
and (41)
B. (39)
[φ]]
=
[(AB
+
BA)/2;
φ].
E[AB;
w
[A,
B;
φ]
:=
[(AB
+
BA)/2;
φ]
−
[A;
φ]
·
[B;
φ]
mixture
of
ordering
AB
hich, for the choice α = 1/2, reduces to
at α = 1/2,
at, for
case
it
with the
quantity
Thethe
demonstration
is quite
straightforward.
Utilising
the same technique introduced in
it coincides
reduces
to
Combining the results (35), (36) and (41),
one then finally has
1/2
the
previous
paragraph,
first
observe
that
nally
called
the
(symmetric)
quantum
covariance
of the
observable
+ BA)/2;
φ]. A and B.
E[AB; wAB [φ]] = [(AB
1/2BA)/2;
[A, B; φ] :=
[(AB
+
φ] −+Utilising
[A; φ]φ]·the
φ]
(39) in
demonstration
isCV[A,
quite
straightforward.
same
introduced
BA)/2;
− [B;
[A;
φ] · technique
[B; φ]
B; wAB [φ]] = ! [(AB
α
α
[φ]]
=
ab
w
b)dm2finally
(a, b) has
E[AB;
w
AB (36)
AB [φ](a,
ombining
the results
(35),
and
(41),
one
then
vious
paragraph,
first observe
that
= R2 [A, B; φ],
(42)
the (symmetric) quantum
of the observable
A and B.
!
! covariance
quantum
covariance
∧
1/2
−2
α
=
i
(∂
∂
u
[φ])
(a,φ]b)−dm2[A;
(a,introduced
b)
α
α
which
was
to
be
demonstrated.
1the
2 +AB
on is quite
straightforward.
Utilising
same
technique
[φ]]
[(AB
BA)/2;
φ] · [B; φ]in
CV[A,
=w ab
w= [φ](a,
b)dm
(a, b)
E[AB;
w [φ]]B;
AB
AB
2
2
RAB
2 next subsection, the two-state value emerges as the condiAs we shall shortly see in the
R
raph, first observe that !
α B; φ],
α 0)
=(∂distribution
[A,
= i−2
∂
u
tional expectation of the quasiprobability
1 2 AB [φ])w(0,
AB [φ](a, b). It occurred to the author
!
−2
α
∧
special
case shall
of the mention.
distribution defined
h
two
R wellearlier
rves
as awe
generalisation
of the other
known proposals
ofψ quasiprobabilty
distributions,
=
W
(x,
p),
1/2
1/
Various
authors
fromp̂ the (special case) relatio
1/2
1/2
i1/2y
∗infer
i1/2y
Quasiprobability
Distributions
Proposed
1/2 p̂
ψ
ψ
w
[ψ](p,ψ⟩
x) = (F ⊗ I)(I ⊗
F)up̂x̂
which two we shall mention.
⟨x,
= (F
I)⟨x,
wp̂x̂ [ψ](p,
x)⊗(x,
=
(F
⊗ew
I)(I
⊗ψ⟩
F )u
(x,
p) =
x),
(58)
p̂x̂
=
WW
p),
p̂x̂ [ψ](p,
p̂x̂ e[ψ](p, x)
formwαof [φ](a,
the conditional
quasiexpectation of the q
!
y
of
quasiprobabilty
distributions
b)
of
our
interest
r for
Quasiprobability
Distribution.
Wigner
Quasiprobability
Distribut
where
we
used
(56)
inThe
the
second
i1/2y p̂
AB
i1/2y
p̂
∗equality.
i1/2y
p̂
2 ), A =
the relation
choice
H =to
L2previously
(Rhave
p̂, known
B
=
x̂
and
α
=
1/2.
Indeed,
one
has
ψ
=
(F
⊗
I)⟨x,
e
ψ⟩
⟨x,
e
ψ⟩
=
(F
⊗
I)⟨x,
e
−ixy
quasiprobability
distributions
Π
=
1.
This
view
is
confirmed
in
the
general
fram
f
Wigner
Quasiprobability
Distribution.
The
Wigner
Quasiprobability
Distribution
is
a
e equality.
ψ(x
− y/2)ψ(x
+ y/2) dm(y)
= defined
other
known
proposals
of
quasiprobabilty
distributions,
ave
used
(56)
in the
second
!
case
ofwell
the
distribution
earlier
!
1/2
1/2
R= (F
pecial case of the distribution
it earlier
has
been
to be nothing
but the con
wp̂x̂ [ψ](p,defined
x)
⊗ I)(I
⊗ Fdemonstrated
)up̂x̂ [ψ](p, x)
−ixy
−ixy
!
e
ψ(x − y/2)ψ(x
+ y/2)=is
dm(y)
= The
especial
ψ(xcase
− y/2)
Kirkwood
Function.
Kirkwood
function
a
of
1/2
α
ψ
Wigner function WW
1/2
i1/2y
p̂ [ψ](p,
∗
i1/2y
p̂
ixy
R
[φ](a
provided
(!)
quasiprobability
distribution
w
(x,
p)
w
x),
R
ψ⟩
⟨x,
e
ψ⟩
=ψψ(x
(F
⊗!
I)⟨x,
e
AB
(x,
p)
=
w=
[ψ](p,
x),
(58)
=
+
y/2)ψ(x
−
y/2)e
dm(y)
p̂x̂ isp̂x̂a special case of !
unction. earlier
The Kirkwood
function
the
distribution
d
!
R
ixy
=
ψ(x
+
y/2)ψ(x
−
y/2)e
2
2
−ixy
ibution.
The
Wigner
Quasiprobability
Distribution
is =
a1dm(y)ψ(x + y/2)ψ(x
2
2
rchoice
the choice
H
=
L
(
R
),
A
=
p̂,
B
=
x̂
and
α
=
1/2.
Indeed,
one
has
e
ψ(x
−
y/2)ψ(x
+
y/2)
dm(y)
=
H = L (R ), A==Wp̂,ψB
=p),
x̂ Direct
and α =
1/2.b;Indeed,
one
hasQuasiprobab
[φ](a,
b)
K(a,
φ)
=
w
R
AB
2.4.
Representation
of
the
(x,
(5
R
R
defined earlier 1/2
!
1/2
1
ψ
w
= (Fb;=
⊗φ)
I)(I
⊗w
F)u
[ψ](p,
x)
W
(x,
p),
1/2
1/2
[φ](a,
b)
K(a,
=
p̂x̂ [ψ](p, x)
p̂x̂
ixyan explicit computation
ψ
AB
For
later
use,
we
give
of in
w
=
ψ(x
+
y/2)ψ(x
−
y/2)e
dm(y)
for
the
case
α
=
1.
The
demonstration
is
quite
straightforward,
for
w
[ψ](p,
x)
=
(F
⊗
I)(I
⊗
F
)u
[ψ](p,
x)
=
W
(x,
p),
1/2
ψ
p̂x̂
p̂x̂
have
the
second
equality.
i1/2y p̂
∗
i1/2y p̂
R
W used
(x, p)(56)
= win
[ψ](p,
x),
(58)
ψ⟩
⟨x,
e
ψ⟩
=
(F
⊗
I)⟨x,
e
p̂x̂
"
#
below
that
have
used (56) in the
second
equality.
αwhere
= 1.weThe
demonstration
is
quite
straightforward,
for
indeed
one
has
from
ψ
i1/2y p̂
∗
i1/2y p̂
#
!
1
=
W
(x,
p),
(59)
ψ⟩
⟨x,
e
ψ⟩
=
(F
⊗
I)⟨x,
e
where
we
have
used
(56)
in
the
second
equality.
[φ](a,
b)
=
⟨b,
E
(·)φ⟩
∗
⟨b,
E
(·)φ⟩
w
α
A # b)0= ⟨b, E
A (·)φ⟩1 (
−ixy AB
"
[φ](a,
w
gives
the
unique
case
when
becomesAreal defin
e one
ψ(xhas
−is
y/2)ψ(x
+ y/2)
= !
=Function.
p̂, B = x̂ andThe
α = Kirkwood
1/2. Indeed,
ABdm(y)
(1−
function
a
special
case
of
the
distribution
1
where
we
have
used
(56)
in
the
second
equality.
R
Kirkwood Function.
The
Kirkwood
function
is aE
special
case
of the distribution defi
b)
=
⟨b,
E
(·)φ⟩
∗
⟨b,
(·)φ⟩
(a)
wAB [φ](a,
!
−ixy
1
A
A
0
1/2
e
ψ(x
−
y/2)ψ(x
+
y/2)
dm(y)
=
∗
⟨b,
E
(·)φ⟩)
(a)
=
(⟨φ,
b⟩δ
0
A
where
|b⟩
denotes
the
eigenvector
associated
tois the
ixy
Kirkwood
Function.
The
Kirkwood
function
as
)earlier
= (F ⊗ I)(I ⊗ F )up̂x̂ [ψ](p,
= x)
ψ(x
+
y/2)ψ(x
−
y/2)e
dm(y)
R function
1 is a special case of the distribution defined
Kirkwood
Function.
The
Kirkwood
R
!
R C R
I
CI
|φk ⟩ |ψ⟩
ator
B,
and
the
s.b)
of
equality is und
1 r.
[φ](a,
b)h.⟨φ,
(6
K(a,
b;
φ)
=
w
earlier
⟨b,
EAB
(a)theEabove
(⟨φ,
b⟩δ
AB
(a)φ⟩
=
b⟩⟨b,
0∗
A (·)φ⟩)
[φ](a,
K(a,
b;
φ)
=
w
i1/2y p̂function
∗ =
i1/2y
p̂
Kirkwood
A
ψ
earlier
ψ⟩ ⟨x,
e
ψ⟩
= (F ⊗ I)⟨x, e
ixy
==W a (x,
p),
(59)1 α ∈
ψ(x
+
y/2)ψ(x
−
y/2)e
dm(y)
(a)φ⟩
scaled
by
the
real
parameter
→
%
⟨b,
E
A
[φ](
K(a,
b; φ)
=has
w
1
!= 1.
se
The
demonstration
is
quite
straightforward,
for
indeed
one
has
from
(5
AB
⟩ forα
|ψ⟩
|φ
⟩
|φ
⟩
|φ
⟩
α
=
0
(or
)
[φ](a,
b)
(60)
K(a,
b;
φ)
=
w
the
case
α
=
1.
The
demonstration
is
quite
straightforward,
for
indeed
one
from
1
2
3
(a)φ⟩
=
⟨φ,
AB
=
⟨φ,
b⟩⟨b,
a⟩⟨a,
φ⟩
Rb⟩⟨b, EA
−ixyused (56) in the second
1−
αdm(y)
(recall the definition# of #the scaling (15) and
here
we have
equality.
"
"
e
ψ(x
−
y/2)ψ(x
+
y/2)
=
ψ
the
case
α = 1. The demonstration
is from
quite(57):
straight
the case
α
= 1.
The
demonstration
is(x,
quite
straightforward,
for indeed one has
H for
→ R(A)
⊂
R
I
A
:
H
→
C
I
1⊗ H for
1
w
=
W
p),
R
[φ](a,
b)"b⟩⟨b,
=A (·)φ⟩
⟨b,
E0A (·)φ⟩
∗AK(a,
⟨b,
EA#b;
(·)φ⟩
wAB
[φ](a,
b)
=
⟨b,
E
∗
⟨b,
E
(·)φ⟩
(a)
w
=
φ),
⟨φ,
a⟩⟨a,
φ⟩
1 α(a)
1
0
AB
α
"
!
[φ]
=
(F
⊗
F
)u
w
1 Kirkwood function is a special case
!
AB
A
irkwood Function.
The
the distribution
defined
⟨φ
|A|ψ⟩
1 1of (a)
⟨b, EA (·)φ⟩0 ∗ ⟨b, EA (·)φ⟩
w
k
2
AB [φ](a, b) =
ixy
[φ](a,
b)
=
⟨b,
E
(·)φ⟩
∗
⟨
w
=
⟨ψ|A|ψ⟩
|⟨φ
|ψ⟩|
A
AB
=
ψ(x
+
y/2)ψ(x
−
y/2)e
dm(y)
k
0
we
have
used
(56)
in
the
second
equality.
∗
⟨b,
E
(·)φ⟩)
(a)
=
(⟨φ,
b⟩δ
0 second
= (⟨φ,
K(a,
φ),
A (a)
arlier
∗ ⟨b,
E
b⟩δb;0in
where
(16)
the
equalty.
⟨φk |ψ⟩ we have used
A (·)φ⟩)
= (F ⊗ I)(I ⊗
R
= (⟨φ, b⟩δ0 1∗ ⟨b, EA (·)φ⟩) (a)
b)
(60)
K(a, b; φ)
=
wb⟩⟨b,
(a)φ⟩
=
⟨φ,
E
AB [φ](a,
∗
⟨b,
E
=
(⟨φ,
b⟩δ
A
ψ
0
A
(a)φ⟩
=
⟨φ,
b⟩⟨b,
E
⟨φ
|A|ψ⟩
A
=
W
(x,
p),
(59)
ave used1 (16) in the second=Now,
equalty.
⟨φ, b⟩⟨b, EA (a)φ⟩
⟨A⟩
dλ A(λ)ρ(λ)
= |ψ(x)| λ x
⟨A⟩dBBof= A dλ
A(λ)ρ(λ)
ρ(λ)
|ψ(x)|
dBB==
ability
given
the !outcome
b of
Bλ ofx the ρ(λ)
quasiprobability
distribution
α
α
wAB
[φ](a,
b)w α [φ]].
α
[φ]]
:=
a
dCP[A
=
a|B
=
b;
(45)
CE[A|B
=
b;
w
as
AB
CP[AAB
= a|B = b;⟨φ|A|ψ⟩
wAB [φ]]
:=
|φ1 ⟩We
|ψ⟩
|φ
|φ2 ⟩ |φquasiprobα
1 ⟩⟨ψ|A|ψ⟩
3⟩
Value as Conditional Quasiexpectation.
define
the
conditional
P[B
=
b;
w
[φ]]
⟨A⟩
=
R
⟨A⟩
=
⟨ψ|A|ψ⟩
QM
QM
AB
Aw =
α α[φ](a, b)
w
⟨φ|ψ⟩
α [φ](a,
[φ](a, b)
Aatgivenconditional
the outcomequasiprobability
b of B of
the
quasiprobability
distribution
wmeasurement’
α ‘postselected
AB
AB
w
b)
relevant
to
CP[A
=
a|B
=
b;
w
[φ]]
:=
AB ⊂AB
⟨A⟩ : H →
R(A)
R
I ,
AP[B
⊗
Hα →[φ]]
CI
w :H
(43)
=
=
b;
w
"
" |⟨b, φ⟩|2
AB
!
!
A
A
ai Ei
µ(E) a
=
tr(W
A =α
µ(E) = tr(W E) W −1A
=
iE
i α E) W
α
!
α
⟨x|A|ψ⟩
⟨φ[φ])(a,
wone
wABi[φ](a, b)
[φ](a,
b)k |A|ψ⟩
wAB [φ](a, isb)dm(a)
=
i
F
⊗
F
(∂
u
b)dm(a)
α
2
1
AB
AB
i
the a
denominator
defined
as
in
(23),
which
computes
as
CP[A = a|BA(λ)
= b; w=AB
[φ]] := |⟨φk |ψ⟩| α
,
= = ⟨ψ|A|ψ⟩
R
R
2
⟨x|ψ⟩ P[B
[φ]]
# = b; w$
$
#
|⟨b, φ⟩|
⟨φ
|ψ⟩
AB
k
φ
α
k µ (b)"
"
"
−1
α= |⟨b, φ⟩|2"
[φ]]
=
.
(44)
P[B = =
b; w
α
i
⊗
F
(
∂
u
[φ]|
)(b)
ABIµ(E
B
1
w
[φ](a,
b)
AB
"
s=0
=
µ
E
µ(E
)
{E
}
=
µ
E
)
{E
}
i
i
i
i
i
i in (23),
ABas
⟨ψ|A|φ⟩
⟨φ|A|ψ⟩
preselection
where the denominator
is defined
which
one
computes
as
,
(43)
=
2
(1
− α)i
(A)previous
=
aresult
) =i tr(W
⟨φ
|A|ψ⟩
i+
i= α
i µ(Pi(29).
|⟨b,
φ⟩|
1
he
Now,=A)
we
define
conditional
“quasiexpectation”
of A given the
(1 − α)⟨φ,
AEB (b)φ⟩ + α⟨φ,
E
(b)Aφ⟩
⟨φ|ψ⟩
⟨ψ|φ⟩
postselection
B
|b⟩
φ
α
2
i
α
quasi-expectation
value
[φ]]
=
µ
(b)
=
|⟨b,
φ⟩|
.
P[B
=
b;
w
⟨φ
|ψ⟩
me
b of B of the
quasiprobability
distribution
w
[φ](a,
b)
as B
1
AB
denominator
is defined
as in (23),
which
one
computes
as
AB
⟨φ|A|ψ⟩
⟨φ|A|ψ⟩
b⟩⟨b, Aφ⟩,
(46)
=
α⟨φ,
b⟩⟨b,
Aφ⟩
+
(1
−
α)⟨φ,
!
Aw =
Aw =
⟨φ|ψ⟩
⟨φ|ψ⟩
φ Now, we define
α
2 2 |A|ψ⟩
α
α
⟨φ
fromCE[A|B
the P[B
previous
result
(29).
conditional
“quasiexpectation”
[φ]]
=
µ
(b)
=
|⟨b,
φ⟩|
.
(44)
=
b;
w
[φ]] :=
a
dCP[A
=
a|B
=
b;
w
(45)
= b; wABAB
AB [φ]].
B
†
† theαfinal result
ombining
the(tabove
(44)
and
(79),
R results
A(t)
U
A(t)
U
(tfw)|φ⟩
⟨ψ|U
(tdistribution
⟨φ|U
fb) of
i )|ψ⟩
i )⟨φ
outcome
Bintermediate
of (t
the
quasiprobability
[φ](a, b) as
|ψ⟩
2
AB
+(1−α)
(A)
= αresult (29). Now,† we define
⟨ψ|E|φ⟩
⟨φ|E|ψ⟩
⟨ψ|E|φ⟩
⟨φ|E|ψ⟩
evious
conditional
“quasiexpectation”
of
A
given the
†
ving that µ(E)
!
⟨φ|U
(t
)
U
(t
)|ψ⟩
⟨φ|U
(t
)
U
(t
)|ψ⟩
µ(E)
tr(W
+=
(1
−αα) E) = αi ⟨b, Aφ⟩
= tr(W
f E) =i α
f + (1 − α)
⟨b,
Aφ⟩
α distribution
⟨φ|ψ⟩
⟨ψ|φ⟩ α
⟨φ|ψ⟩
⟨ψ|φ⟩
! α
of B of !the
quasiprobability
wαAB+
[φ](a,
b)α)
as
|A|ψ⟩
⟨φ
n
(1
−
[φ]]
=
CE[A|B
= b; wAB
a dCP[A = a|B = b; wAB [φ]].
CE[A|B =−1b; wAB [φ]] :=α
α
⟨b,b)dm(a)
φ⟩
a wAB [φ](a, b)dm(a) =! i ⟨b,
F φ⟩
⊗ F (∂1 uAB [φ])(a,
R
R
CE[A|B
α
[φ]] :=
b; wAB
A(λ) =
⟨φn |ψ⟩
α
R dCP[A
⟨x|A|ψ⟩
a
⟨x|A|ψ⟩
=
a|B
= b; wAB [φ]].
(45) (47)
α
=
[A;
b,
φ].
Observing that †
−1
α A(λ) =
R
†(α)|ψ⟩
=
i
I
⊗
F
(
∂
u
[φ]|
)(b)
⟨x|ψ⟩
⟨x|ψ⟩
1
K(α)
=
⟨φ|u
AB
s=0
)
A(t)
U
(t
)|ψ⟩
)
A(t)
U
(tf )|φ⟩
⟨ψ|U
(t
⟨φ|U
(t
A
f
i
i
in
agreement
with
!
!
+(1−α)
(A)
=α
that
†
† (tE)|ψ⟩
"
"
antity
α
−1
α
=
(1
−
α)⟨φ,
AE
(b)φ⟩
+
α⟨φ,
(b)Aφ⟩
⟨φ|U
(t
)
U
(t
)|ψ⟩
⟨φ|U
(t
)
U
⟨ψ|A|φ⟩
⟨φ|A|ψ⟩
⟨ψ|A|φ⟩
⟨φ|A|ψ⟩
B
B
f
i
i
f
a
w
[φ](a,
b)dm(a)
=
i
F
⊗
F
(∂
u
[φ])(a,
A
A
1
!
!
AB
AB
(1 − α) b)dm(a)
µ(E
=
=α
λ(A)
+) (1
−⟨φ|u
α) A) (α)|χ
a µ(E ) = tr(W
A) = α aK
λ(A) =
(α)
=tr(W
⟩⟨χ +
|ψ⟩
=
k i
A
k
k
R ⟨ψ|φ⟩
⟨φ|ψ⟩
⟨ψ|φ⟩
⟨φ|ψ⟩
α
−1
α
b⟩⟨b,
Aφ⟩,
(46)
=
α⟨φ,
b⟩⟨b,
Aφ⟩
+
(1
−
α)⟨φ,
i
a
w
[φ](a,
b)dm(a)
=
i
F
⊗
F
(∂
u
[φ])(a,
b)dm(a)
i
1 † AB
⟨φ†f , Aφ
⟨φf −1
, Aφi ⟩
′
′
′ i⟩
α AB
α⟩
(t, tR) = U
(t−t
U (t)U
|φ
|ψ⟩s=0 )(b) (48)
+ (1:=−Uα)1(t)
∈
[A;
φf , )φ=
α (tR) A(t)
C,[φ]|
= iA UI2(t)
⊗ ,F (α
∂
u
k
i ] :=
1 AB
3
⟨φ
, φi ⟩
⟨φfand
, φi ⟩(79), the final result
f−1
s, by combining the above =
intermediate
results
α (44)
i I ⊗ F ( ∂1 uAB [φ]|s=0 )(b)
= (1 − α)⟨φ, AE (b)φ⟩ + α⟨φ, E (b)Aφ⟩
i
i
i
R
|χ ⟩
†
|χ ⟩
†
†
|χ ⟩
p
B
† B
AB
⟨φ|χk ⟩
R) be an the
invertible
real matrix.
ransformation
of Distributions.
T ∈
Fourier Transformation.
It provesLet
useful
to GL(n;
first renormalise
n-dimensional
Lebesgue
α
Direct
Representation
ofnkthe
Distribution wAB [φ](a, b)
⟩∈ NQuasiprobability
n ),
× by
k(Rn , ⟨φ|A|χ
on
B
measure
β
ψ|A|ψ⟩
2
n
=
A(λ)ρ(λ) ρ(λ) A
=w|ψ(x)|
λ⟩ x
−1w α [φ](a,
−1
⟨φ|χ
k
ter use, representation
we give an explicit
computation
of
in
terms
of
probability
| det(T
)|
f
(T
(13)(2)
fT (x) :=dm
−n/2x) b) in this subsection. We argue
AB
(x)
:=
(2π)
dβ
(x),
n
n
that
!
A⟩
=
⟨ψ|A|ψ⟩
1 (Rn#
QM
$
tegrable
function
f
∈
L
),
one
has
the
relation
p
2 and the convolution by
"
-norm
accordingly
the
L
α redefineρ(λ)
dλ!
A(λ)ρ(λ)
=
|ψ(x)|
λ x ∗ ⟨b, EA (·)φ⟩α (a),
A⟩and
dBB =
[φ](a,
b)
=
⟨b,
E
(·)φ⟩
(54)
w
A
!
AB
(1−α)
R
I AC
I=
|φ
⟩
|ψ⟩
k
!"
#1/p
ai P
i
"
p g(T x)f (x) dmn (x)p
n
g(x)f
(x)
dm
(x)
=
(14)(3)
n
T
∥f
∥
:=
|f
(x)|
dm
(x)
,
f
∈
L
(
R
),
R
of
the
self-adjoint
oper|b⟩ Pdenotes
associated
to
the
eigenvalue
b
∈
a
P
(W
) W ithe
A⟨A⟩
=eigenvector
p
n
=
⟨ψ|A|ψ⟩
i
i
Rn
with RnQM
Rn
B,⟩ and
the⟩ r. h.
s. ⟩ofi the
above
equality
is understood as the convolution of the function
"
|φ
|φ
|φ
⟩
1
2
3
the
resultbyof
analysis
(change
variables),
whenever
exists.
1 integration
" f (x of
(fthe
∗ g)(x)
−αy)g(y)
f, gconvolution
∈ the
Lconjugate
(Rn ),
(a)φ⟩ scaled
real:=
parameter
∈ R, dm
andn (y),
its
complex
scaled
by (4)
b,
EAclassical
A|ψ⟩
⟨φ|A|ψ⟩
Rnai Pi
µ(P
)
=
tr(W
P
)
W
A
=
t(recall
the
familiar
scaling
=
definition
of ⊗
theH
scaling
(15) and (16)). To see this, first observe that
w R
) A⊂respectively.
I the
A
:
H
→
C
I
w
i dm1"= dm
# whenever there is no risk for confusion.
|ψ⟩ ⟨φ|ψ⟩ For brevity, we write
x
−n
×
1
n
α
α
(x)
:=
|t|
f
,
t
∈
R
f
), recall
fˇ : Rn → C defined by(15)
Now, for a function w
ftAB
∈L
[φ](R
= (F
⊗ Ftthat
)uABthe
[φ] functions fˆ, scaling
"
⟨x|A|ψ⟩ A = ⟨φ|A|ψ⟩
2 ⟨φk |A|ψ⟩
w ⟨ψ|A|ψ⟩
α
A(λ) =
=
|ecial
−i⟨q,x⟩
ˆ
=
(F
⊗
I)(I
⊗
F
)u
[φ].n (x), For the case t = 0,(55)
⟨φ|ψ⟩
case
for
T
=
tI,
where
I
denotes
the
identity
operator.
i.e.,(5)
e
f
(x)
dm
f
(q)
:=
AB
⟨x|ψ⟩
A|ψ⟩
⟨φk |ψ⟩
Rn
0, it is convenient
to define⟨ψ|A|φ⟩ "
⟨φ|A|ψ⟩
x|ψ⟩
i⟨q,x⟩ %
⟨x|A|ψ⟩
+ (1
− α) $fˇ!(q) :=
tr(W A) = α
e
f (x) dmn (x),
(6)
A(λ) =
⟨φ|ψ⟩
⟨ψ|φ⟩
Rn i(1−α)sA
⟨x|ψ⟩
1 |A|ψ⟩
α
iαsA
(x)
:=
f
(x)
dm
(x)
δ
(x),
(16)
f
(I
⊗
F
)u
[φ](s,
b)
=
⟨φ,
e
E
(b)e
φ⟩
0
n
0B
|b⟩
AB
⟨ψ|A|φ⟩
⟨φ|A|ψ⟩
n
n
R
given
with
the
scalar
product
on
R
"
φ1 |ψ⟩
+ (1 − α) ⟨φ|A|ψ⟩ by i(1−α)sA
α
⟨ψ|A|φ⟩
(1 ⟨φ,
−$
α)
= ⟨φ|ψ⟩
ai µ(Pi ) = tr(W A) = α ⟨ψ|φ⟩+ =
b⟩⟨b, eiαsA φ⟩
ne
⟨φ|ψ⟩
⟨ψ|φ⟩
†
denotes
the delta
distribution
centred
at 0.
n
U
U †⟨x,
(t
)|φ⟩
⟨ψ|U
(ti ) A(t)
i(ti )|ψ⟩
fy⟩
:=
x
y
,
x,
y
∈
R
,
(7)
i
i
i(1−α)sA ∗
iαsA
+(1−α)
⟨φ
|A|ψ⟩
φ⟩ ⟨b, e
φ⟩
(56)
⟨b,i=1
e
† (t )|ψ⟩
2
tion
transformation,
has
(ti ) U † (tf )|ψ⟩=one
i to the Fourier ⟨φ|U
!
⟨φ
|ψ⟩
are respectively
called the Fourier transform
and
the inverse Fourier transform of f . The C2
†
†
−i⟨q,x⟩
&
U(x)
(tfdm
)|φ⟩
⟨ψ|U
(ti ) A(t)
⟨φ|U (tf ) A(t) U (ti )|ψ⟩
(f
)(q)
=
e
f
T
T
ˆ isn (x)
F† that maps
f to†its Fourier
transform
f
called the Fourier
+(1−α)
α linear map
#
$ transformation. For
⟨ψ|U (t ) A(t) U (tR )|φ⟩ †
n
R
µ(E) = tr(W E)
W
ai EiA
A=
i
#
"
Concluding
Remarks
$
"
µ
Ei =
µ(Ei ) {Ei }
The weak value
may
be
regarded
as
the
average
of
the
‘classical
weak
i
i
values’ with respect to the (conditional) weak quasiprobability
associated with the given transition processes. The imaginary part
⟨φ|A|ψ⟩
describes the degree
= interference involved in the processes.
Aw of
⟨φ|ψ⟩
The weak quasiprobability (or the weak value) has a natural position
in HVT (ontological model)
when complexity
is allowed. It admits an
⟨ψ|E|φ⟩
⟨φ|E|ψ⟩
(1 − α)
µ(E)
= tr(W
E) = α , which is +
arbitrary
parameter
related
to the ratio of mixture
⟨φ|ψ⟩
⟨ψ|φ⟩
between the forward and
backward
processes.
TEST SPACE
C
which mayRbe C
relevant
to
R
I
C
I ‘postselected
|φk ⟩ |ψ⟩
R
I The
CI joint|φquasiprobability,
|ψ⟩
⟨x|A|ψ⟩
k⟩
=
measurement’ inA(λ)
the conditional
form, admits a family containing the
⟨x|ψ⟩
|φ1 ⟩the|ψ⟩
|φ1 ⟩ |φfunction
Kirkwood
|φ1 ⟩ |φ2Wigner
⟩ |φ3 ⟩ function
α = 0, 1 ( α = 1/2) and
2 ⟩ |φ3 ⟩ ( α = 0, 1 ). α = 1/2
"
⟨ψ|A|φ⟩
⟨φ|A|ψ⟩
A
⟨A⟩
:
H
→
R(A)
⊂
R
I
Aw : H ⊗ H → CI
→
R(A)
⊂
R
I
A
:
H
⊗
H
→
C
I
w ) = tr(W A) = α
+
(1
−
α)
ai µ(E
λ(A) =
i quasiprobability lies at the heart of the weak value
In
all
aspects,
⟨φ|ψ⟩
⟨ψ|φ⟩
!
!
2 ⟨φik |A|ψ⟩
= ⟨ψ|A|ψ⟩
|⟨φk |ψ⟩|
and, possibly,
at the
⟨φk |ψ⟩
⟨φ1 |A|ψ⟩
2 ⟨φk |A|ψ⟩
heart of quantum|⟨φmechanics.
k |ψ⟩|
⟨φ |ψ⟩
†
k
k
⟨φ1 |A|ψ⟩
†
= ⟨ψ|A|ψ⟩
Thank you!