The von Neumann Model of
Measurement in Quantum Mechanics
Pier A. Mello
Instituto de Fı́sica, UNAM, México
Collaborators:
L. M. Johansen, Buskerud University College, Kongsberg, Norway
A. Kalev, University of Singapore
M. Revzen, Technion, Haifa, Israel
Applications of Quantum Mechanics 2013, Guadalajara, July 2013
Version: Mon., Jul. 15/13, 8pm
1
INTRODUCTION
In a general quantum measurement
one obtains information on the system of interest
by coupling it to an
auxiliary degree of freedom, or probe,
and then detecting some property of the probe
using a measuring device.
This procedure was described by von Neumann in his classic book
We refer to it as von Neumannn’s model (vNM)
a
Within the vNM, the combined system
– system proper plus probe –
is given a dynamical description.
a J.
von Neumann, Mathematical Foundations of Quantum Mechanics. Princeton Univ. Press, Princeton, N. J., 1955
2
Example 1: Stern Gerlach experiment
z
i) Observable we want to get information about:
the z-component of the spin of a particle
ii) Auxiliary degree of freedom or probe: position r of the electron
–still a microscopic quantity–
after it leaves the magnet, and this is what is
recorded by a detecting device: a position detector.
E.g., for s = 1/2, we may wish information on the two components
hψspin |Pz± |ψspin i = |h±|ψspin i|2
of the original state
3
Example 2: Cavity Quantum Electrodynamics (QED)
experimentsa
i) Observable: the number n of photons in a cavity
ii) Probe: the atoms sent through the cavity:
after they leave the cavity,
they are detected by a detecting device. a
EM cavity
atoms
a Haroche
et al., Phys. Rev. 45, 5193 (1992); 53, 1295 (1996)
4
SUMMARY
1. Single Measurements in Quantum Mechanics
The system is coupled to a probe with arbitrary coupling strength
a
a) Main goal: We’ll study what information can we obtain on the
system proper detecting the probe
b) As a by-product of that analysis, we obtain the
reduced density operator of the system proper
after its interaction with the probe,
and derive the so-called Lüders rule b as the limiting case of strong
coupling.
a L.
M. Johansen and P. A. Mello, Phys. Lett. A 372, 5760 (2008)
b G. Lüders, Ann. Phys. Lpz 8(6), 322 (1951)
5
2. Successive Measurements in Quantum Mechanics.
The von Neumann model is generalized to two probes that interact
successively with the system proper a
a) Main goal: We describe what information can we obtain on the
system proper detecting the probes
Indeed: We describe a state reconstruction scheme based on the
procedure of successive measurements b
b) We obtain the so-called “Wigner’s formula” c as the
strong-coupling limit of the above formalism.
c) We obtain the so-called Kirkwood’s quasi-probability
distribution d as the weak-coupling limit of the above formalism.
a L.
M. Johansen and P. A. Mello, Phys. Lett. A 372, 5760 (2008)
b A. Kalev and P. A. Mello, J. Phys. A 45, 235301 (2012)
c E. P. Wigner, Am. J. Phys. 31, 6 (1963)
d J. G. Kirkwood, Phys. Rev. 44, 31 (1933)
6
d) We find a generalized transform of observables and states, in
terms of complex quasi-probabilities
e) We describe Wigner transform in terms of
successive measurements of position and momentum
e P.
e
A. Mello and M. Revzen, ArXiv: Quant-Phys 1307.2877, Ann. Phys.
(NY), submitted.
7
The Stern-Gerlach experiment
z
y
x
The Stern-Gerlach (SE) experiment has become a paradigm for
models of measurement in QM.
The experiment was first performed as early as 1922.
It is explained in every textbook on QM (e.g., Ballentine, Peres)
and various refinements have been presented in the literature a .
Surprisingly, its complete (non-relativistic) solution has been given
only recently b
a A.
Peres, PRD 22, 879 (1980), P. Alstrøm et al., Am. J. Phys. 50, 697
(1982), D. E. Platt, Am. J. Phys. 60, 306 (1992)
b M. Scully et al., Found. of Phys. 17, 575 (1987)
8
By complete solution we mean one that takes into account:
i) the translational and transverse motions of the atom, and
ii) a confined magnetic field B(r) that satisfies
∇ · B = 0,
∇ × B = 0.
Here we shall work with a model of the complete problem
that has a simple exact solution
and still shows the physical characteristics we want to exhibit,
i.e., the bending of the trajectory depending on the z-projection of
the spin.
9
z
y
x
Assume:
i) B ≡ 0 outside the gaps
ii) Only Bz is significant
iii) Inside the gaps: Bz (z) = B ′ (z)z
iv) Simulate the translational motion using a
t-dependent interaction, which lasts for the time the particle is
inside the gap of the magnet.
This could be achieved by adopting a
frame of reference moving with the particle
10
We use the model Hamiltonian
p̂2z
H(t) =
− µB θt1 ,τ (t)Bz′ (0)ẑ σ̂z ,
2m
θt1 ,τ (t)

 1, t ∈ (t − τ /2, t + τ /2)
1
1
=
 0 t∈
/ (t1 − τ /2, t1 + τ /2)
·
¸
1
θt1 ,τ (t) = τ
θt1 ,τ (t) = τ g(t) ;
τ
11
Z
∞
−∞
g(t)dt = 1
We use the simplified model Hamiltonian:
p̂2z
H(t) =
− ǫδ(t − t1 )σ̂z ẑ = Kz + V ,
2m
ǫ = µB Bz′ (0)τ . Simplification : g(t) ≈ δ(t − t1 )
Schrödinger eqn. and initial condition :
ih̄
∂|ψ(t)i
= H(t)|ψ(t)i
∂t
(0)
|ψ(0)i = |ψspin i|χ(0) i
where:
σ̂z
ẑ, p̂z
(0)
|ψspin i
|χ(0) i
= the observable for the system proper
:
the probe canonical variables
= initial state of the system proper
= initial state of the probe
12
We find the final (after interaction is over) evolution operator:
i
i
i
Uf ≡ U (t > t1 ) = e− h̄ K̂z (t−t1 ) e h̄ ǫσ̂z ẑ e− h̄ K̂z t1
=
free evolution from t1 to t
×interaction at t1
×free evolution from t = 0 to t1
13
and the final (after interaction is over) state:
X
i
i
(0)
|Ψ(t)if =
Pσ |ψspin ie− h̄ K̂z (t−t1 ) e h̄ ǫσẑ |χ(t1 )i
σ=±1
The component
(0)
Pσ |ψspin i
of the original spin state gets correlated with the probe state
i
h̄
e ǫσẑ |χ(t1 )i
which, in the z-representation, is
i
e h̄ ǫσz χ(z, t1 )
i.e., it gets a boost pσ = ǫσ in the z direction
14
THE PROBABILITY OF A PROBE POSITION z
pf (z, t) =
=
f hΨ|Pz |Ψif
X
σ=±1
=
X
σ=±1
¯2
¯
¯
¯
(0)
(0)
− h̄i K̂z (t−t1 ) h̄i ǫσ ẑ
|χ(t1 )i¯
e
hψspin |Pσ |ψspin i ¯hz|e
Wσ(σ̂z ) |χf (z, σ; t)|2
χf (z, σ; t) ≡
Z
⇓ plane wave : mom. = ǫσ
i
′
U0 (z, z ′ ; t − t1 ) e h̄ ǫσz χ(z ′ , t1 )dz ′
z
W | χ ( z, σ =1, t) |
1
f
(0)
| χ ( z) |
2
2
t1
t
2
| χ ( z, t −) |
1
W | χ ( z, σ =1, t) |
−1
f
15
2
CONDITIONS GENERALLY REQUIRED TO A MEASUREMENT
OF Âs OBSERVING Âprobe (a)
Here, Âs = σ̂z and Âprobe = ẑ
a) V̂ = f (Âs ) Here, indeed: V̂ = −ǫδ(t − t1 )σ̂z ẑ
b) [V̂ , Âprobe ] 6= 0, so that hÂprobe i changes,
enabling one to get information on Âs by observing Âprobe .
Although here [V̂ , ẑ] = 0, we have [K̂z , ẑ] 6= 0;
K̂z causes a displacement of the two wave packets: wait long
enough: the 2 packets separate & we can measure σ = ±1.
AND FOR A Q.M. NON-DEMOLITION MEASUREMENT a :
c) [Ĥ, Âs ] = 0. Here, indeed: [Ĥ, σ̂z ] = 0.
(0)
Starting with |ψspin i = |σ = 1i, state won’t get a component
|σ = −1i
a N.
Imoto et al., PRA 32, 2287 (1985); S. Haroche, op. cit.
16
THE PROBABILITY OF A PROBE MOMENTUM pz
f hΨ|Ppz |Ψif
pf (pz , t) =
X
=
σ=±1
X
=
¯2
¯
i
i
¯
¯
(0)
(0)
hψspin |Pσ |ψspin i ¯hpz |e− h̄ K̂z (t−t1 ) e h̄ ǫσẑ |χ(t1 )i¯
Wσ(σ̂z )
σ=±1
¯
¯2
¯ (0)
¯
¯χ̃ (pz − ǫσ)¯
p
z
ε
| χ~ ( p ) |
(0)
z
~(0)
W + | χ ( pz− ε) |
2
2
~(0)
|χ (p )|
2
z
_
0
t1
t+
1
−ε
17
t
~(0)
W − | χ ( p z+ ε ) |
2
CONDITIONS GENERALLY REQUIRED TO A
MEASUREMENT OF Âs OBSERVING Âprobe
(a)
Here, Âs = σ̂z and Âprobe = p̂z
a) V̂ = f (Âs ) Here, indeed: V̂ = −ǫδ(t − t1 )σ̂z ẑ
b) [V̂ , Âprobe ] 6= 0, to enable measuring Âs by observing Âprobe .
Here, [V̂ , p̂z ] ∝ [ẑ, p̂z ] 6= 0.
AND FOR A Q.M. NON-DEMOLITION MEASUREMENT
a
c) [Ĥ, Âs ] = 0. Here, indeed: [Ĥ, σ̂z ] = 0.
(0)
Consequence: Starting with |ψspin i = |σ = 1i, state won’t get a
component |σ = −1i
a N.
Imoto et al., PRA 32, 2287 (1985); S. Haroche, op. cit.
18
SINGLE MEASUREMENTS
system s : Â
V
<=======>
probe π : Q̂, P̂
Single “measurement” (detecting the probe) of system observable:
 =
P
n
an Pan
having eigenvalues an and eigenprojectors Pan .
Q̂, P̂ : Hermitean operators for probe position and momentum.
von Neumann’s model (vNM):
system coupled to probe via the interaction:
Ĥ(t) = V̂ (t) = ǫ δ(t − t1 )ÂP̂ ,
t1 > 0.
We disregard here (it can be fully taken into account) the intrinsic
evolution of the system and the probe.
Recall Stern-Gerlach: V̂ (t) ≈ −ǫg(t − t1 )ŝz ẑ
19
Initial condition at t = 0:
(0)
ρ(0) = ρ(0)
⊗
ρ
s
π
Evolution operator:
Rt ′ ′
i
i
−
V̂ (t )dt
Û (t) = e h̄ 0
= e− h̄ ǫ θ(t−t1 )ÂP̂ .
After the system-probe interaction is over (i.e., for t > t1 ):
(Â)
ρf
=
(0)
P
ρ
′
an s Pan′
nn
P
h
− h̄i ǫan P̂
e
(0) i
ρπ e h̄ ǫan′ P̂
i
.
Notice:
i
1) The occurrence of the displacement operator e− h̄ ǫan P̂
2) System and probe are now correlated
20
{ Recall:
(Â)
ρf
After the system-probe interaction:
h i
i
P
i
(0)
(0)
= nn′ Pan ρs Pan′ e− h̄ ǫan P̂ ρπ e h̄ ǫan′ P̂ }
For t > t1 we detect the
probe position Q̂ to obtain information on the system.
From Born’s rule, probability density for Q for t > t1 is
(Â)
pf (Q)
=
(Â)
T r(ρf PQ )
=
P
(Â)
n Wan p0 (Q − ǫan )
(0)
(Â)
Wan = Tr(ρs Pan ) ,
is the “Born probability” for the result an , and
(0)
p0 (Q − ǫan ) = hQ − ǫan |ρπ |Q − ǫan i
is the original p0 (Q) (having width = σQ ) displaced by ǫan .
21
COMMENT
(0)
1) Knowing the system state ρs
(Â)
and thus Wan ,
(Â)
we can predict the detectable quantity pf (Q).
(Â)
2) More interestingly, detecting pf (Q)
we can retrieve information on the system state
This we now examine
22
TRANSLATION TO STERN-GERLACH (2 levels)
H(t) = ǫδ(t − t1 )ÂP̂
 ⇒ σ̂z
P̂ ⇒ ẑ
Q̂ ⇒ −p̂z
Q(=−pz )
Q(=−pz )
εa
p (Q)
o
0
Wa p (Q − ε a )
_
1
o
1
1
p (Q)
o
t>t
t1
t
1
Wa
εa
23
_
2
p (Q − ε a )
2
o
2
(Â)
Wan = 0.1, 0.2, 0.2, 0.15, 0.2, 0.05, 0.1
pf (Q) for ǫ = 1, σQ = 0.05: Strong Coupling
(0)
pf (Q) mirrors Born p.d. of e-values of  in state ρs
P
1.5
1.25
1
0.75
0.5
0.25
2
4
6
8
10
Q
pf (Q) for ǫ = 1, σQ = 1: Weak Coupling
pf (Q) does not mirror Born p.d. of e-values of Â
But see comments below !
P
0.175
0.15
0.125
0.1
0.075
0.05
0.025
2
4
6
8
10
Q
24
COMMENTS
1. What we detect
(Â)
is the probe-position probability pf (Q)
(Â)
2. pf (Q) “mirrors” the e-values an
only in the idealized limit of very strong coupling, ǫ/σQ ≫ 1.
(Â)
3. In this limit, pf (Q) integrated around an gives
(Â)
the so-called Born’s probability Wan of an .
This is the high-resolution limit.
This is the information we can find on the system proper
We’ll extend it to any resolution!
Indeed, we’ll find cases where it is
advantageous to use low resolution
25
AVERAGE OF Q̂ AFTER INTERACTION
From:
(Â)
pf (Q) =
P
(Â)
n Wan po (Q − ǫan )
and assuming the original Q distribution centered at Q = 0:
(Â)
1
h
Q̂i
f
ǫ
=
(Â)
a
W
n n an
P
=
P
(0)
(0)
n an Tr(ρs Pan ) = Tr(ρs Â) = hÂi0 , ∀ǫ
(Â)
Result: Detecting hQ̂if
allows extracting hÂi0 , i.e.,
the Born average of the observable  in the original state of the
system.
It is remarkable that result is independent of ǫ a
(See next figure)
a Aharonov,
Albert and Vaidman, PRL 60, 1351 (1988)
26
hQif
pf (Q) for ǫ = 1, σQ = 0.05: Strong Coupling
(0)
Wan = 0.1, 0.2, 0.2, 0.15, 0.2, 0.05, 0.1
(Within e-value range!)
↓ pf (Q) mirrors Born p.d. of e-values of  in state ρ0s
P
1.5
1.25
1
0.75
0.5
0.25
2
4
6
hQif
8
pf (Q) for ǫ = 1, σQ = 1: Weak Coupling
(Within e-value range!)
↓
P
Q
10
(0)
Wan = 0.1, 0.2, 0.2, 0.15, 0.2, 0.05, 0.1
pf (Q) does not mirror Born p.d. of e-values of Â
0.175
0.15
0.125
0.1
0.075
0.05
0.025
2
4
6
8
10
Q
27
SECOND AND HIGHER MOMENTS OF Q
1
ǫ2
1.
h
(Â)
hQ̂2 if
(Â)
Measuring hQ̂2 if
−
i
(0)
= Tr(ρs Â2 ) = hÂ2 i0 ,
∀ǫ
2
allows extracting hA2 i0
and knowing σQ
(Â)
pf (Q)
2.
2
σQ
a
=
(Â)
(0)
P
n (ρs )nn p0 (Q − ǫan )
Measuring pf (Q) gives information on the diagonal elements
(0)
(ρs )nn of the original density operator, (not the off-diagonal ones)
i
D
E
hP
(0)
(Â)
ikǫÂ
ikǫan
p̃
(k)
=
e
)
e
p̃0 (k)
(ρ
3. Char. fctn : p̃f (k) =
s
0
nn
n
0
Measuring
a Kalev
(Â)
pf (Q)
and inferring
(Â)
p̃f (k)
and Mello, unpublished
28
D
allows extracting eikǫÂ
E
0
MEASURING THE OBSERVABLE: Â ≡ PROJECTOR :
P̂aν
Eigenvalues of P̂aν : τ = 1, 0
Eigenprojectors of P̂aν : (P̂aν )τ
(P̂aν )1 = P̂aν .
(P̂aν )0 = I − P̂aν ,
Ĥ(t) = V̂ (t) = ǫ δ(t − t1 )P̂aν P̂ ,
(P̂aν )
pf
t1 > 0.
(0)
(Q) = Tr[ρ(0)
s (P̂aν )0 ] p0 (Q) + Tr[ρs (P̂aν )1 ] p0 (Q − ǫ)
|
0
|
1
Q/ ε
hProbe positioni gives directly the
probability of aν in the original state:

(Â)
1
 Tr(ρ(0)
X
P̂
)
=
W
s
a
a
ν
ν
τ Tr[ρ(0)
=
)
]
=
(
P̂
aτ τ
s
(P̂aν )
 Tr[ρ(0)
(
P̂
)
]
=
W
s
a τ =1
0 < <Q>/ ε < 1
1
(P̂ )
hQ̂if aν
ǫ
τ =0
ν
29
τ =1
We can rephrase this result as:
(P̂
)
Inferring the probability for τ = 1, i.e., Wτ =1aν ,
(P̂
)
from the measurement of 1ǫ hQ̂if aν ,
is equivalent to the retrieving of the diagonal elements
(0)
haν |ρs |aν i
of the density operator
The extension of this idea to successive measurements will allow
retrieving the full density operator
30
THE REDUCED DENSITY OPERATOR FOR THE SYSTEM
(Â)
over the probe:
h i
´
i
X³
i
− h̄ ǫan P̂ (0) h̄ ǫan′ P̂
Pan ρ(0)
=
.
ρπ e
s Pan′ T rπ e
Trace ρf
(Â)
ρs,f
nn′
(Â)
ρs,f
=
P
nn′
(0)
g(ǫ(an − an′ )) Pan ρs Pan′
Characteristic function of the probe momentum distribution:
D i E(0)
h
i
i
− h̄ β P̂
g(β) = e− h̄ β P̂
= Tr ρ(0)
; β = ǫ(an − an′ )
π e
π
Particular case: pure Gaussian state for the probe:
2
2 −1/4 −Q2 /4σQ
χ(Q) = (2πσQ )
:
e
g(ǫ(an − an′ )) =
Z
−
χ∗ (Q − ǫan′ )χ(Q − ǫan )dQ = e
31
ǫ2
8σ 2
Q
(an −an′ )2
The strong-coupling limit:
(Â)
ρs,f
=
(0)
P
n Pan ρs Pan
This is called the von Neumann-Lüders rule:
We have thus reproduced the result of a
non-selective projective measurement of the observable  !!
(f )
ρs
(0)
and ρs
are not connected by a unitary transformation
(0)
(0)
(0)
E.g.: i) for ρs = |ψs ihψs |,
(Â)
initial e-values: 1, 0, · · · , 0
ii) for ρs,f , e-values have changed:
³
´
³
´
(Â)
ρs,f Paν |ψs(0) i = hψs(0) |Paν |ψs(0) i Paν |ψs(0) i
No contradiction, because the whole ρ̂ evolves unitaritly,
while here we are dealing with the
reduced density operator, i.e., traced over the probe
32
EFFECT OF THE PROBE DETECTOR
(Â)
Above we computed the reduced ρs,f
after the system-probe interaction
Then the probe is detected (D)
Make a model for the π − D interaction,
taking place at t2 > t1 ,
provided this interaction does not involve the system proper s
(after detection)
the final density operator after detection
Call ρf
(after detection)
Trace ρf
over the probe and the detector:
³
´
(after detection)
(after detection)
ρs,f
= T rπD ρf
i
´
h
X³
− h̄i ǫan P̂ (0) h̄i ǫan′ P̂ (0) †
(0)
ρ̂D UπD
ρπ e
Pan ρs Pan′ T rπD UπD e
=
nn′
=
X
nn
(0)
′ )) Pa ρ
g(ǫ(a
−
a
n
n
s Pan′
n
′
Same as before. If π − D interaction involves s, above does not hold.
33
SUCCESSIVE MEASUREMENTS
“Measure” the system observable
 =
P
n
an Pan
and later
P
B̂ = m bm Pbm .
Employ two probes, which we detect.
Q̂i , P̂i : Hermitian operators for probes position and momentum,
which we detect
System coupled to the probes via the interaction:
Ĥ(t) = V̂ (t) = ǫ1 δ(t − t1 )ÂP̂1 + ǫ2 δ(t − t2 )B̂ P̂2 ,
0 < t1 < t2 .
Disregard the intrinsic evolution of the system and the probes and
assume that V̂ (t) represents the full Hamiltonian.
34
Initial condition.
At t = 0:
(0) (0) (0)
ρ(0) = ρs ρπ1 ρπ2
Then, for t > t2 :
(B̂←Â)
ρf
=
X
(Pbm Pan ρ(0)
s Pan′ Pbm′ )
nn′ mm′
³ i
´³ i
´
i
i
ˆ
ˆ
ˆ
ˆ
h̄ ǫ1 an′ P1
h̄ ǫ2 bm′ P2
· e− h̄ ǫ1 an P1 ρ(0)
e− h̄ ǫ2 bm P2 ρ(0)
π1 e
π2 e
At t > t2 we detect the two probe positions and momenta
to obtain information about the system
Two examples are considered below
35
MEASURING THE CORRELATION OF THE
TWO PROBES POSITIONS
(B̂←Â)
hQ̂1 Q̂2 if
ǫ1 ǫ2
=ℜ
(B̂←Â)
P
nm an bm Wbm an (ǫ1 ) ,
a
where
(B̂←Â)
Wbm an (ǫ1 )
=
P
n′
λ(ǫ1 (an − an′ ))Tr
and
h
i
(0)
ρs (Pan′ Pbm Pan )
λ(β) = g(β) + 2h(β)
E(0)
E(0)
D i
i
1 D − i β P̂1
β
P̂
−
− h̄ β P̂1
e 2h̄
,
h(β) =
Q̂1 e 2h̄ 1
g(β) = e
β
π
π
a hQ̂ Q̂ i(B̂←Â)
1 2 f
is a complicated function of ǫ1 ; but it’s linear in ǫ2 , associated
(Â)
with the last measurement, just as, for a single measurement, hQ̂if
36
∝ǫ
COMMENT
(0)
1) Kwowing the original system state ρs ,
the auxiliary function
i
h
P
(B̂←Â)
(0)
ℜWbm an (ǫ1 ) = n′ λ(ǫ1 (an − an′ ))Tr ρs (Pan′ Pbm Pan )
allows predicting
(B̂←Â)
the detectable quantity hQ̂1 Q̂2 if
It extends to 2 measurements the probability
(Â)
(0)
Wan = Tr(ρs Pan ) which, for single measurements,
P
(Â)
1
allows predicting the detectable quantity ǫ hQ̂if = n an Wan
(B̂←Â)
2) More interestingly, detecting hQ̂1 Q̂2 if
, we investigate
what information can we retrieve on the system state
This we examine in what follows
37
MEASURING THE MOMENTUM-POSITION
PROBES CORRELATION:
1
(B̂←Â)
h
P̂
Q̂
i
1
2
ǫ1 ǫ 2
(B̂←Â)
W̃bm an (ǫ1 )
=
P
n′
λ̃(β) =
=
1
2
2σQ
1
ℑ
nm an bm W̃bm an (ǫ1 )
λ̃(ǫ1 (an − an′ ))Tr
λ̄(β)
,
λ̄(0)
(B̂←Â)
P
h
λ̄(β) =
i
(0)
ρs (Pan′ Pbm Pan )
1 ∂g(β)
β ∂β
(B̂←Â)
1) Auxiliary function ℑW̃bm an (ǫ1 ) allows predicting the
(B̂←Â)
detectable quantity hP̂1 Q̂2 if
(B̂←Â)
2) Again, detecting hP̂1 Q̂2 if
, we investigate
what information can we retrieve on the system state
38
TRANSLATION TO STERN-GERLACH (2 levels)
Ĥ(t) = ǫ1 δ(t − t1 )ÂP̂1 + ǫ2 δ(t − t2 )B̂ P̂2 ,
0 < t1 < t2 .
 ⇒ σ̂z
B̂ ⇒ σ̂x
P̂1 ⇒ ẑ
P̂2 ⇒ x̂
Q̂2 ⇒ −p̂x
Q̂1 ⇒ −p̂z
Q (=−p z )
1
Q (=−p )
L
R
1
z
SG oriented
SG oriented
0
t
1
in z−direction
in x−direction
Q (=−p x )
2
t1
t
Q (=−p )
2
x
2
1) send 1 atom from L to R
Measure Q1 = −pz , Q2 = −px ([Q̂1 , Q̂2 ] = 0) and construct Q1 Q2
PN (i) (i)
Q1 Q2
Pi=1
N
(i) (i)
1
Within an ensemble, construct hQ1 Q2 i = N
1
2) In another ensemble, construct hP1 Q2 i = N
39
i=1
P1 Q2
WE FIRST EXAMINE SOME PROPERTIES OF
THE AUXILIARY FUNCTIONS
h
i
P
(B̂←Â)
(0)
Wbm an (ǫ1 ) = n′ λ(ǫ1 (an − an′ ))Tr ρs (Pan′ Pbm Pan )
h
i
P
(0)
(B̂←Â)
W̃bm an (ǫ1 ) = n′ λ̃(ǫ1 (an − an′ ))Tr ρs (Pan′ Pbm Pan )
1) The strong-coupling limit: ǫ1 → ∞:
(B̂←Â)
Wbm an (ǫ1 )
(B̂←Â)
W̃bm an (ǫ1 )
→
(B̂←Â)
W
Wbm an
= Tr
h
i
(0)
ρs (Pan Pbm Pan )
We recover the jpd given by Wigner’s rule (real and non-negative),
obtained as follows, e.g., for projective measurements on a pure
state and no degeneracy:
P (bm , an )
= P (bm |an )P (an ) = |hbm |an i|2 |han |ψi|2
= hψ|Pan Pbm Pan |ψi
40
(B̂←Â)
Wbm an (ǫ1 )
h
P
i
(0)
ρs (Pan′ Pbm Pan )
λ(ǫ1 (an − an′ ))Tr
h
i
P
(0)
(B̂←Â)
W̃bm an (ǫ1 ) = n′ λ̃(ǫ1 (an − an′ ))Tr ρs (Pan′ Pbm Pan )
=
n′
2) Weak-coupling limit: ǫ1 → 0:
(B̂←Â)
Wbm an (ǫ1 )
(B̂←Â)
W̃bm an (ǫ1 )
→
(B̂←Â)
Kbm an
= Tr
h
i
(0)
ρs (Pbm Pan )
We recover the Kirkwood’s-Dirac joint quasi-probability,
complex, in general a
a R.
Feynmann, Negative Probability, Essays in honour of D. Bohm, 1987
41
COMMENTS
(B̂←Â)
(B̂←Â)
a) For intermediate, arbitrary ǫ1 , Wbm an (ǫ1 ) and W̃bm an (ǫ1 )
can be regarded as two auxiliary functions,
(B̂←Â)
ℜWbm an (ǫ1 ) being taylored for predicting hQ̂1 Q̂2 i, and
(B̂←Â)
ℑWbm an (ǫ1 ) for hP̂1 Q̂2 i
We’ll see that for some special Â’s and B̂’s
we can have the inverse case: from the measurable quantities
(0)
hQ̂1 Q̂2 i and hP̂1 Q̂2 i we’ll be able to reconstruct ρs
********
b) Particular case: π1 described by a pure Gaussian state:
−
λ(β) = λ̃(β) = g(β) = e
(B̂←Â)
(B̂←Â)
β2
8σ 2
Q
Wbm an (ǫ1 ) = W̃bm an (ǫ1 )
42
Recall:
hQ̂1 Q̂2 i (B̂←Â)
ǫ1 ǫ 2
=ℜ
P
(B̂←Â)
nm an bm Wbm an (ǫ1 )
3) For ǫ1 → 0 the probes correlation reduces to:
h
i
X
1
1
hQ̂1 Q̂2 i(B̂←Â) =
an bm Tr ρ(0)
s (Pbm Pan + Pan Pbm )
ǫ1 ǫ2
2
nm
h
i
X
1
H
=
an bm WbM
= Tr ρ(0)
s (B̂ Â + ÂB̂)
m an
2
nm
h
i
1
H
(0)
WbM
Tr
ρ
=
a
s (Pbm Pan + Pan Pbm )
m n
2
=Margenau-Hill distribution
= real part of the Kirkwood quasi-probability distribution:
it may take negative values (and cannot be regarded as a joint
probability in the classical sense)
We now explain
43
If [Pbm , Pan ] 6= 0, the quantity
1
(Pbm Pan + Pan Pbm )
2
has at least one negative e-value
Ŝmn ≡
For this pair of variables, i.e., an , bm , and for the state = e-vector
that gives rise to a negative e-value, the MH pd
H
WbM
= hŜmn i0 < 0, and hQ̂1 Q̂2 i/ǫ1 ǫ2
m an
may lie outside the range [(an bm )min , (an bm )max ].
bm
Example: N = 2:
 
 
1
0



,
an = 1, 0; |1i =
, |0i =
0
1
 


1
1
1
1
.
= 1, 0; |1) = √   , |0) = √ 
2 1
2 −1
44
1
X
1
H
MH
an bm WbM
hQ̂1 Q̂2 i(B̂←Â) =
=
W
= hS11 i
11
m an
ǫ1 ǫ2
n,m=0
S11


1 2 1 
=
4 1 0
λ+ =
λ− =
1
4 (1
1
4 (1
+
−
√
√
2) > 0 ⇒ |ψ+ i
2) < 0 ⇒ |ψ− i
√
1
1
(B̂←Â)
hQ̂1 Q̂2 i
= hψ− | S11 |ψ− i , = (1 − 2) < 0
ǫ1 ǫ2
4
which lies outside the interval defined by the possible values of
s · s′ = 0, 1
45
Recall:
(B̂←Â)
Wbm an (ǫ1 )
=
P
n′
(0)
λ(ǫ1 (an − an′ ))Tr[ρs (Pan′ Pbm Pan )]
4) The commutative case. For
[Pan , Pbm ] = 0, ∀n, m ,
(B̂←Â)
Wbm an (ǫ1 )
= Tr
h
i
(0)
ρs (Pbm Pan )
,
∀ǫ1
reduces to the standard, real and non-negative,
quantum-mechanical definition of the joint probability of an and
bm for commuting observables. We also have the standard result:
i
h
1
hQ1 Q2 i = Tr ρ(0)
s (ÂB̂)
ǫ1 ǫ2
46
5) Measure subsequently the same observable (Particular case of 4)
i
h
X
(0)
(Â←Â)
λ(ǫ1 (an − an′ ))Trs ρs (Pan′ Pn̄ Pan )
Wan̄ an (ǫ1 ) =
n′
= Trs
hQ1 Q2 i
ǫ1 ǫ2
C(Q1 , Q2 )
=
(Gauss)
=
=
X
n,n̄
p
r
³
ρ(0)
s Pan
´
δan̄ ,an
Â)
an an̄ ℜWa(n̄Â←
an (ǫ1 )
=
X
n
Wa(nÂ) a2n = hÂ2 i0
hQ1 Q2 i − hQ1 ihQ2 i
[hQ21 i − hQ1 i2 ][hQ22 i − hQ2 i2 ]
σQi
varÂ
r
→
1,
as
→0
´2
´2
³
³
ǫi
σQ1
σQ2
var + ǫ1
var + ǫ2
47
Q2 / ε
2
a
N
a2
a
1
a1
a
a
2
N
Q1 / ε
σ
1
p(Q1 , Q2 ) in the strong-coupling limit, ǫQi i → 0:
correlation=1 between outcome of probes 1 and 2
48
6) Successive measurements and weak values
a
WV of Â, with preselection |ψi and post-selection |φi:
(Â)W
=
=
hφ|Â|ψi
hφ|ψi
³
(0)
b
´
Tr ρ̂s P̂φ Â
hψ|φihφ|Â|ψi
hψ|P̂φ Â|ψi
´
³
=
⇒
(0)
hψ|φihφ|ψi
hψ|P̂φ |ψi
Tr ρ̂s P̂φ
= correlation fctn between observable  and projector P̂φ
X hP̂φ P̂a i
n
=
an
hP̂φ i
n
X
=
an × “complex” probability of an , conditioned by φ
n
a L.
M. Johansen, unpublished
b Aharonov, Albert and Vaidman, PRL 60, 1351 (1988)
49
Consider a successive measurement experiment governed by
Ĥ(t) = V̂ (t) = ǫ1 δ(t − t1 )ÂP̂1 + ǫ2 δ(t − t2 )P̂φ P̂2 ,
0 < t1 < t2 .
One can show
lim
hQ̂1 Q̂2 i(P̂φ ⇐Â)
lim
hP̂1 Q̂2 i(P̂φ ⇐Â)
ǫ1 →0
ǫ1 →0
ǫ1 hQ̂2
i(P̂φ ⇐Â)
ǫ1 hQ̂2
i(P̂φ ⇐Â)
=
i
h
1 hÂP̂φ + P̂φ Âi
= ℜ (Â)W
2
hP̂φ i
=
i
h
1 hP̂φ Â − ÂP̂φ i
2
= 2σP1 ℑ (Â)W
2
2σQ1
2ihP̂φ i
50
STATE RECONSTRUCTION SCHEME
BASED ON SUCCESSIVE MEASUREMENTS
Consider two orthonormal bases |ki, |µ),
Given k there is only one vector
Given µ there is only one vector (no degeneracy)
The two bases are mutually non-orthogonal: hk|µ) 6= 0, ∀k, µ
Then the two bases have no common eigenvectors: complementary.
v^
^j
^
u
o.
^
k = w^
^i
1st basis: î, ĵ, k̂
2nd basis: û, v̂, ŵ
Assume they have 1 common e-vector: k̂ = ŵ
Then û, v̂ ⊥ k̂ and î, ĵ ⊥ ŵ,
contradicting the assumption of no mutually orthogonal e-vectors
51
Consider the system-probes interaction:
V̂ (t) = ǫ1 δ(t − t1 )Pk P̂1 + ǫ2 δ(t − t2 )Pµ P̂2
i.e., the system observables are the rank-one projectors
Pk = |kihk| and Pµ = |µ)(µ|
onto the k- and µ-state of the first and second basis, respectively.
52
The position-position correlations
hQ̂1 Q̂2 i(Pµ ←Pk )
ǫ 1 ǫ2
=ℜ
(P ←P )
Wστ µ k (ǫ1 )
(P ←P )
W11 µ k (ǫ1 )
=
P1
P1
= Tr
(P ←Pk )
µ
στ
W
στ
τ,σ=0
τ ′ =0
³
′
λ(ǫ1 (τ − τ ))Tr
(0)
ρs P k P µ P k
´
(P ←Pk )
(ǫ1 ) = ℜW11 µ
³
(0)
ρs (Pk )τ ′ (Pµ )σ (Pk )τ
+ λ(ǫ1 )
P
k′ (6=k)
Tr
³
´
(0)
ρs P k ′ P µ P k
(0)
This last eqn. can be inverted to obtain ρ̂s :
P
(Pµ ←Pk )
(0) ′
(µ|k′ i
1
(ǫ1 ) (µ|ki ,
hk|ρs |k i = G ′ (ǫ1 ) µ W11
k k
where: Gkk′ (ǫ1 ) = δk,k′ + λ(ǫ1 )(1 − δk,k′ )
Require (µ|ki =
6 0, ∀µ, k
The two basis must be mutually non-orthogonal
53
(ǫ1 ),
´
,
CONCLUSION
(P ←Pk )
The set of complex quantities W11 µ
(ǫ1 ) , ∀k, µ,
(0)
contain all the information about the state of the system ρs .
(P ←P )
If we could infer W11 µ k (ǫ1 ), ∀k, µ, from measurement
(0)
outcomes, we could reconstruct the full ρs .
(P ←Pk )
Notice: the full complex W11 µ
(ǫ1 ) is needed for tomography
However, from the detected position-position correlation
(P ←Pk )
hQ̂1 Q̂2 i(Pµ ←Pk ) /ǫ1 ǫ2 , we directly extract only ℜW11 µ
(P ←Pk )
To find ℑW11 µ
(ǫ1 )
(ǫ1 ), we use the momentum-position correlation
—————
(P̂ )
Recall: In the single-measurement case, the detected hQ̂if aν allows
(0)
reconstructing the diagonal m. el. of ρs :
(P̂aν )
1
h
Q̂i
f
ǫ
(P̂
= Wπ=1aν
54
)
(0)
= haν |ρs |aν i
The momentum-position correlations
hP̂1 Q̂2 i(Pµ ←Pk )
ǫ1 ǫ2
(P ←P )
W̃11 µ k (ǫ1 )
= Tr
³
=
2
2σP
1
h̄
(0)
ρs P k P µ P k
(P ←Pk )
ℑ W̃11 µ
´
+ λ̃(ǫ1 )
P
(ǫ1 ),
k′ (6=k)
Tr
³
(0)
ρs P k ′ P µ P k
One can show that from the
(P ←P )
measurable ℑW̃11 µ k (ǫ1 )
(P ←P )
one can infer ℑW11 µ k (ǫ1 )
and this enables the complete state reconstruction
This completes our procedure
55
´
,
(P ←Pk )
ℑW11 µ
(P ←Pk )
(ǫ1 ) FROM THE MEASURED ℑW̃11 µ
³
´
X
(0)
Let :
Tr ρs Pk′ Pµ Pk
= rµk + isµk
(ǫ1 ),
k′ (6=k)
(P ←Pk )
(ǫ1 )
= xµk + iyµk
(P ←Pk )
(ǫ1 )
= x̃µk + iỹµk
W11 µ
W̃11 µ


xµk



yµk



 ỹµk
2
= |hk|µ)|
P
µ′
xµ′ k + λr rµk − λi sµk
= λi rµk + λr sµk
= λ̃i rµk + λ̃r sµk
For given k, µ: 3 Eqs. in the 3 unknowns: rµk , sµk , yµk
Measured quantities: xµk , ỹµk
yµk
´
2
X
|λ(ǫ
)|
ℑ{λ(ǫ1 )λ̃∗ (ǫ1 )} ³
1
=
xµ′ k +
xµk −|hk|µi|2
ỹµk
∗
∗
ℜ{λ(ǫ1 )λ̃ (ǫ1 )}
ℜ{λ(ǫ1 )λ̃ (ǫ1 )}
µ′
56
REMARK
For an N -dimensional Hilbert space,
(0)
hk|ρs |k ′ i is constructed in terms of the 2N 2 correlations
hQ̂1 Q̂2 i(Pµ ←Pk ) , and hP̂1 Q̂2 i(Pµ ←Pk )
These are not all independent, because
(0)
(0)
(0)
ρ̂s = [ρ̂s ]† and Trρ̂s = 1 impose N 2 + 1 restrictions,
leaving N 2 − 1 independent correlations to be measured
E.g., for N = 2 it’s enough to measure:
hQ̂1 Q̂2 i(P+ ←P0 ) ,
hQ̂1 Q̂2 i(P− ←P0 ) ,
57
hP̂1 Q̂2 i(P− ←P0 )
Example: N = 2,
k = 0, 1, µ = +, −
ǫ2
− 21
8σ
Q1
Probes in a pure Gaussian state: g(ǫ1 ) = e
Result:
(0)
= x+,0 + x−,0
(0)
= 1 − x+,0 − x−,0
x+,0 − x−,0 − 2iy−,0
=
g(ǫ1 )
x+,0 − x−,0 + 2iy−,0
=
g(ǫ1 )
ρ00
ρ11
(0)
ρ01
(0)
ρ10
58
The strong-coupling limit ǫ1 → ∞ (when we get Wigner’s rule)
´
³
P
(Pµ ←Pk )
(0)
(ǫ1 ) = k′ Gkk′ (ǫ1 )Tr ρs Pk′ Pµ Pk
Pure
Recall: W11
−
Gaussian probe state: Gkk′ = δkk′ + (1 − δkk′ )e
ǫ2
1
8σ 2
Q1
As ǫ1 → ∞,
´
³
(Pµ ←Pk )
2
(ǫ1 ) → Tr ρ(0)
W11
s Pk Pµ Pk = |hk|µ)| ρkk
(P ←Pk )
and W11 µ
(ǫ1 ) only contains information on the diagonal
(0)
elements of ρs .
N = 2:
1 (0)
1 (0)
(−←0)
(+←0)
(ǫ1 ) → ρ00
(ǫ1 ) → ρ00 , W11
W11
2
2
1 (0)
1 (0)
(−←1)
(+←1)
(ǫ1 ) → ρ11
(ǫ1 ) → ρ11 , W11
W11
2
2
(0)
(0)
and only ρ00 and ρ11 can be retrieved
59
The weak-coupling limit ǫ1 → 0
´
³
P
(Pµ ←Pk )
(0)
Recall: W11
(ǫ1 ) = k′ Gkk′ (ǫ1 )Tr ρs Pk′ Pµ Pk
2
−ǫ21 /8σQ
1
Gkk′ (ǫ1 ) = δkk′ + (1 − δkk′ )e
;
As ǫ1 → 0:
³
´
(Pµ ←Pk )
(0)
W11
(ǫ1 ) → Tr ρ(0)
s Pµ Pk = Kµk = hk|ρs |µ)(µ|ki .
N = 2:
(0)
(0)
ρ00 + ρ01
(+←0)
W11
(ǫ1 ) →
,
2
(0)
(0)
ρ10 + ρ11
(+←1)
W11
(ǫ1 ) →
,
2
(0)
(0)
ρ00 − ρ01
(−←0)
W11
(ǫ1 ) →
2
(0)
(0)
ρ10 − ρ11
(−←1)
W11
(ǫ1 ) →
2
(0)
and the four ρs m. els can be retrieved
To reconstruct a QM state using the successive-measurement
scheme, it’s better to perform measurements with weak coupling ǫ1 ,
rather than with strong coupling
60
The intermediate case: ǫ1 arbitrary
Recall:
ǫ2
− 21
8σ
Q1
(P ←P )
W11 µ k (ǫ1 ) = xµ,k + iyµ,k , and g(ǫ1 ) = e
and our result for N = 2:
(0)
= x+,0 + x−,0
(0)
= 1 − x+,0 − x−,0
x+,0 − x−,0 − 2iy−,0
=
g(ǫ1 )
x+,0 − x−,0 + 2iy−,0
=
g(ǫ1 )
ρ00
ρ11
(0)
ρ01
(0)
ρ10
Even if g 6= 0, but g ≪ 1, a small expt. uncertainty in extracting
hQ1 Q2 i(µ←k) and hP1 Q2 i(µ←k) , which give
(µ←k)
W11
(ǫ1 ) = xµ,k + iyµ,k , is divided by a small number g ≪ 1
(0)
when k 6= k ′ , and this makes the error in extracting ρkk′ large
61
P
(Pµ ←Pk )
(µ|k′ i
(ǫ1 ) (µ|ki
=
Summarizing:
µ W11
´
³
P
(Pµ ←Pak )
(0)
W11
(ǫ1 ) = k′ Gkk′ (ǫ1 )Tr ρs Pk′ Pµ Pk
(0)
hk|ρs |k ′ i
1
Gk′ k (ǫ1 )
Gkk′ (ǫ1 ) = δkk′ + (1 − δkk′ )λ(ǫ1 )
========================
i) The strong-coupling limit (ǫ1 → ∞, Gk′ k (ǫ1 ) → δk′ k ):
³
´ X
X
W
hk|ρ(0)
Wµk = Trs (ρs Pan ) ,
Tr ρ(0)
s |ki →
s Pk Pµ Pk =
µ
µ
in terms of Wigner’s joint probability.
Only ρs diagonal elements can be retrieved.
This is precisely the limit in which Wigner’s formula is obtained.
————————–
ii) The weak-coupling limit (ǫ1 → 0, Gk′ k (ǫ1 ) → 1):
X
(µ|k ′ i X
(µ|k ′ i
(0) ′
(0)
hk|ρs |k i →
Trs (ρs Pµ Pk )
=
Kµk
(µ|ki
(µ|ki
µ
µ
in terms of Kirkwood’s joint quasi-probability.
62
THE QUANTUM-MECHANICAL EXPECTATION VALUE OF
AN OBSERVABLE IN TERMS OF ITS “TRANSFORM” AND
THE QUASI-PROBABILITY
The above tomographic approach is particularly attractive, because
(P ←Pk )
W11 µ
(ǫ1 ) can be interpreted
as a quasi-probability
in the following sense
63
Consider the Hermitean operator Ô for the system.
From the above inversion formula we find its expectation value:
(0)
T rs (ρ̂s Ô)
=
(P ←Pk )
µ
W
11
kµ
P
(ǫ1 ) O(µ, k)
• The “transform” of the operator Ô as a function of µ, k:
O(µ, k) =
X (µ|k ′ i hk ′ |Ô|ki
k′
(µ|ki Gk′ k (ǫ1 )
• The “quasi-probability” (in general complex)
a function of µ, k:
(P ←P )
W11 µ k (ǫ1 )
a R.
.
a
(0)
for state ρs
Feynmann, Negative Probability, Essays in honour of D. Bohm, 1987
64
as
Structure similar to that of a number of transforms that express
the QM expectation value of an observable in terms of its
transform and a quasi-probability distribution:
• Wigner’s transform of the observable and Wigner function
(joint quasi-probability): defined in phase space (q, p). The
pair of variables (q, p) label the states of coordinate and
momentum bases.
Wigner function: a representation in phase space of the
quantum state.
• Present transform and joint quasi-probability: defined for the
pair of variables (µ, k), that label the states |µ), |ki of the two
bases.
Joint quasi-probability: a representation of the quantum state.
65
Wigner Function
and the Successive Measurement of
Position and Momentum a
Pier A. Mello, Instituto de Fı́sica, and
Michael Revzen, Technion - Israel Institute of Technology, Haifa
32000, Israel
Wigner function is a “quasi-distribution” that provides a
representation of the state of a quantum mechanical system in the
phase space of position and momentum.
We find a relation between Wigner function
and the successive measurement of position and momentum,
as described by von Neumann’s model of measurement.
We show that one can relate Wigner function to Kirkwood joint
quasi-distribution of position and momentum,
a ArXiv:
Quant-Phys 1307.2877; submitted to Annals of Physics (NY)
66
the latter, in turn, being a particular case of successive
measurements.
We first consider the case of a quantum mechanical system
described in a continuous Hilbert space,
and then turn to the case of a discrete, finite-dimensional Hilbert
space.
67
Introduction
Wigner function was originally introduced to provide a
phase-space representation of the state of a
Quantum-Mechanical system described in a continuous
Hilbert space.
Quantum Mechanics precludes a proper joint probability
distribution of position q and momentum p.
Indeed, the Wigner function is termed a
“quasi-distribution”,
as it may become negative in some portions of phase
space.
However, in many respects it plays a role similar to
the phase space distribution function in classical
statistical mechanics.
68
Therefore, we find it natural to inquire whether one can
relate Wigner function to appropriate measurements of
position and momentum.
69
Here we show that one can “measure” the Wigner function of a
system by detecting the correlation functions of two “probes”, in
an experiment where the observables are, successively, projectors
associated with the position and the momentum of the system.
This is demonstrated by first relating the Wigner function to the
so-called Kirkwood joint quasi-distribution of position and
momentum and, second, relating Kirkwood distribution to probe
correlations in successive measurements which, in turn, are
experimentally accessible.
The relation between Kirkwood distribution and probe correlations,
is understood within the spirit of von Neumann’s model of
measurement.
70
Wigner function and Kirkwood quasi
distribution: continuous Hilbert space
The Wigner transform of an operator defined in a
continuous Hilbert space
The Wigner transform (WT) of an operator  is a mapping from
Hilbert space to phase space
It can be expressed as the inverse Fourier transform of the
characteristic function of the operator.
In units with q and p are dimensionless, and h̄ = 1, we define
Z ∞Z ∞
1
W̃Â (u, v)ei(uq+vp) dudv
(1)
WÂ (q, p) =
2π −∞ −∞
i
h
−i(uq̂+v p̂)
.
(2)
W̃Â (u, v) = Tr Âe
71
When the operator  is the density operator ρ̂,
we speak of its WT as the Wigner function (WF) of the state.
The WT of an operator  can also be expressed as
WÂ (q, p) = Tr[ÂP̂ (q, p)],
P̂ (q, p) being thr Hermitean operator
Z ∞
¯
¯
ED
y
y
¯
¯
P̂ (q, p) =
e−ipy ¯q −
q + ¯ dy .
2
2
−∞
We can also use the mutually unbiased bases (MUB) states
|x′ , θi, eigenstates of the operator X̂θ = q̂ cos θ + p̂ sin θ
–and hence eigenstates of the exponential operator in Eq. (2)–
X̂θ |x′ , θi = x′ |x′ , θi,
to express the operator P̂ (q, p) as
Z π Z ∞
Z ∞
1
−it(x′ −q cos θ−p sin θ) ′
′
dθ
dt|t|e
|x ; θihx′ ; θ|
dx
P̂ (q, p) =
2π 0
0
−∞
72
P̂ (q, p) and the WT of an operator  possess the attributes:
1) Matrix elements of P̂ (q, p) in the coordinate basis:
′
hq|P̂ (q ′ , p′ )|q̄i = eip (q−q̄) δ(q + q̄ − 2q ′ ) .
2) If  = † , its WT is real, since P̂ (q, p) = P̂ † (q, p)
3) The operators P̂ (q, p) fulfill orthogonality and closure
h
i
1
′ ′
Tr P̂ (q, p)P̂ (q , p ) = δ(q − q ′ )δ(p − p′ ) ,
2π
Z ∞Z ∞
1
P̂ (q, p)dqdp = I
2π −∞ −∞
4) The WT of the operators  and B̂ satisfy the
“product formula”, or “overlap formula”
Z ∞Z ∞
dqdp
= Tr(ÂB̂) .
WÂ (q, p)WB̂ (q, p)
2π
−∞ −∞
73
(3)
(4)
(5)
5) The WF for the state ρ̂ satisfies the marginality relation
Tr(ρ̂P̂θx′ )
= hx′ , θ|ρ̂|x′ , θi
Z ∞Z ∞
dqdp
′
(6)
=
Wρ̂ (q, p)δ(x − (q cos θ + p sin θ))
2π
−∞ −∞
I.e., if the system is in state ρ̂,
probability to find it in the pure state |x′ , θi
= integral of WF along the line q cos θ + p sin θ = x′ in phase space.
The marginal probability of q and of p take the standard form.
(6) is called the Radon transform of the Wigner function Wρ̂ (q, p).
6) The WF is normalized as
Z ∞Z ∞
dqdp
=1.
Wρ̂ (q, p)
2π
−∞ −∞
74
(7)
Relation between Wigner function and Kirkwood
quasi-distribution for a continuous Hilbert space
Wigner function in terms of Kirkwood’s quasi-distribution
Z Z
′
′
Wρ̂ (q, p) = 2
dq ′ dp′ e2i(q−q )(p−p ) K(p′ , q ′ ).
(8)
K(p, q) =
Tr(ρ̂ P̂p P̂q ),
(9)
with the projectors
P̂q = |qihq|
P̂p = |qihp|,
(10)
is Kirkwood’s joint quasi-distribution of q and p, which is, in
general, complex.
The operators P̂q and P̂p are not proper position and momentum
projectors, since they are not idempotent.
75
To use the formalism developed in in the previous lectures,
we use the operators P̂qn and P̂pm defined as
Z qn +δ/2
Z pm +δ/2
Pqn =
|qidqhq|,
Ppm =
|pidphp|
(11)
qn −δ/2
Wρ̂ (q, p) = 2
pm −δ/2
∞
X
n,m=−∞
≈ 2
= 2
∞
X
Z
qn +δ/2
dq ′
pm +δ/2
e2i(q−qn )(p−pm ) Trs
n,m=−∞
∞
X
∞
X
′
³
′
dp′ e2i(q−q )(p−p ) Trs ρ̂s P̂p′ P̂q′
pm −δ/2
qn −δ/2
2i(q−qn )(p−pm )
e
n,m=−∞
= 2
Z
Ã
ρ̂s
Z
pm −δ/2
³
Trs ρ̂s P̂pm P̂qn
e2i(q−qn )(p−pm ) K(pm , qn )
n,m=−∞
76
pm +δ/2
´
dp′ P̂p′
Z
qn +δ/2
qn −δ/2
´
dq ′ P̂
From previous lectures:
¸
·
ih̄
1
hQ̂1 Q̂2 i(P̂pm ←P̂qn ) + 2 hP̂1 Q̂2 i(P̂pm ←P̂qn )
limǫ1 →0
ǫ1 ǫ2
2σP1
(P̂
= ℜW11 pm
←P̂qn )
(P̂
(0) + iℑW11 pm
←P̂qn )
(0)
and also
(P̂
←P̂qn )
(0)
W11 pm
³
´
= Trs ρ̂s P̂pm P̂qn ≡ K(pm , qn ),
I.e., Kirkwood’s distribution can be expressed in terms of the
position-position and momentum-position correlation of two probes,
detected in a measurement described by von-Neumman’s model
with very weak coupling, in which one
“measures” in succession the projectors for position and momentum
of the system proper.
77
Thus:
Wρ̂ (q, p) = 2
∞
X
e2i(q−qn )(p−pm )
n,m=−∞
·
¸¾
½
i
1
hQ̂1 Q̂2 i(P̂pm ←P̂qn ) + 2 hP̂1 Q̂2 i(P̂pm ←P̂qn )
× limǫ1 →0
ǫ1 ǫ2
2σP1
Thus Wigner function, which is defined in the system phase space,
can be related to a set of measurable quantities,
consisting of the two-probe correlations detected in the
experimental setup described above,
and thereby reconstructed therefrom.
We recall an analogous result, used in Quantum Optics,
which relates Wigner function to other measurable quantities:
those obtained in a computer-aided tomography scan
78
Wigner function and Kirkwood quasi distribution:
discrete, finite-dimensional Hilbert space
The Wigner transform of an operator, for a discrete,
finite-dimensional Hilbert space
WT of  for a Hilbert space of finite dimensionality N
has been studied extensively in the literature. Here, we propose
(N −1 N −1
)
N
−1
X
1 X X
i 2π
i 2π
k(−p+bq)
N
W̃Â (k, b)e
WÂ (q, p) =
W̃Â (l)e N lq
+
N
b=0 k=1
l=0
)
( ·
´k ¸†
³
W̃ (k, b) = Tr  X̂ Ẑ b
W̃Â (l)
· ³ ´ ¸
†
l
= Tr  Ẑ
q, p = 0, 1, · · · , N − 1: coordinate and momentum
in discrete phase space: N × N set of points.
79
Dimensionality N = prime number larger than 2:
the integers 0, 1, · · · , N − 1 constitute a mathematical field, with
addition, subtraction, multiplication and division defined ModN
This field plays a role similar to that of the real numbers in the
continuous case studied earlier
When N = prime, the problem admits exactly N + 1
mutually unbiased bases (MUB)
The quantities Ẑ and X̂ are the Schwinger operators
80
Schwinger operators and MUB
N -dimensional Hilbert space spanned by N distinct states
|qi, with q = 0, 1, · · · , (N − 1),
subject to the periodic condition
|q + N i = |qi
They form the “reference basis”, or “computational basis” of the
space. Define, with Schwinger
Ẑ|qi
X̂|qi
X̂ N
Ẑ X̂
= ω q |qi,
ω = e2πi/N ,
= |q + 1i.
= Ẑ N = Î
= ω X̂ Ẑ
The two operators Ẑ and X̂ form a complete algebraic set:
only a multiple of the identity commutes with both:
any operator in the N -dim. Hilbert space is a function of Ẑ and X̂
81
We introduce the Hermitean operators p̂ and q̂, which play the role
of “momentum” and “position” as
X̂
= ω −p̂ = e−
Ẑ
= ω q̂ = e
2πi
N p̂
2πi
N q̂
,
.
(12)
(13)
The reference basis is thus the “position basis”
In the continuous limit, [q̂, p̂] = i
The “momentum basis”:
|pi
X̂|pi
=
N
−1
X
q=0
= e−
2πi
e N pq
√
|qi ,
N
2πi
N p
82
|pi
(14)
(15)
The N 2 -dimensional matrix space is spanned by the complete
orthonormal N 2 operators X̂ m Ẑ l , with m, l = 0, 1, ..(N − 1), so
that any N × N matrix can be written as a linear combination of
these N 2 operators.
A familiar example is a 2-dimensional Hilbert space, where any
2 × 2 matrix can be written as a linear combination of the three
Pauli matrices plus the unit matrix, which can also be written as
σx , σz , σx σz and I.
Our complete orthonormal set of N 2 operators can be taken as
(X̂ Ẑ b )k ,
Ẑ l ,
b = 0, 1, · · · , N − 1,
(16)
k = 1, · · · , N − 1,
l = 0, 1, · · · , N − 1 .
83
(17)
The operator X̂ Ẑ b possesses N eigenvectors
X̂ Ẑ b |m, bi = ω m |m; bi,
|m; bi =
N −1
1 X b n(n−1)−nm
√
ω2
|ni,
N n=0
b, m = 0, 1, · · · , N − 1.
|ni, n = 0, · · · , N − 1: the N states of the reference basis.
Altogether, N + 1 mutually unbiased bases (MUB).
The states with b = 0, i.e.,
N −1
N −1
1 X − 2πi mq
1 X 2πi (N −m)q
e N
eN
|m; 0i = √
|qi, = √
|qi ,
N n=0
N n=0
(18)
are eigenstates of p̂ which, from Eq. (14), can be written as
|m; 0i = |p = −m = (N − m)Mod[N ]i.
84
(19)
The definition
WÂ (q, p) =
(N −1 N −1
)
N
−1
X
2π
1 X X
i 2π
k(−p+bq)
W̃Â (k, b)e N
W̃Â (l)ei N lq
+
N
b=0 k=1
l=0
(20)
W̃Â (k, b)
W̃Â (l)
)
( ·
´k ¸†
³
= Tr  X̂ Ẑ b
· ³ ´ ¸
†
l
= Tr  Ẑ
(21)
(22)
is, for the discrete case, analogous to that for the continuous case.
The operators X̂ Ẑ b , b = 0, · · · N − 1,
define N of a set of N + 1 MUB
The operator Ẑ defines the so-called “reference basis”
85
The definition (20) can be written in terms of MUB as
(0̈: the reference basis)
£
¤D
N −1 N −1 N −1
¯ ¯
E
1 X X X 2πi
¯
¯
k
M
(b)−m
q,p
eN
WÂ (q, p) =
m; b ¯Â¯ m; b − Tr(Â)
N
m=0
b=0̈ k=0

 (−p + bq) Mod[N ], for b = 0, · · · , N − 1 ,
Mq,p (b) =

q,
for b = 0̈ .
Given a pair of variables q, p:
for b = 0̈: Mq,p (0̈) = q
for b = 0: Mq,p (0) = −p Mod[N ] = N − p
for subsequent values of b: Mq,p (b) = (−p + bq) mod[N ]
86
m
N−p
4
3
q
2
1
0
..
0
0
1
2
3
4
b
Figure 1:
Illustration of the function m = Mq,p (b) in the b − m plane, for
N = 5 and the particular pair of “phase-space” values q = 2, p = 1.
Mq,p (b) specifies “points” in an b − m plane
This aggregate of points, for fixed q and p, may be described as a
“line” in the b − m plane. We refer to Mq,p (b) as a line
Further study, based on such a view, is in progress
87
The WT of  can be given the alternative forms
WÂ (q, p)
N
−1 D
X
=
b=0̈
¯ ¯
E
¯ ¯
Mq,p (b); b ¯Â¯ Mq,p (b); b − Tr(Â) ,
= Tr(ÂP̂q,p ) ,
(23)
(24)
where we have defined the Hermitean “line operator”
P̂q,p
=
N
−1
X
b=0̈
=
¯
¯
®­
¯Mq,p (b); b Mq,p (b); b¯ − Î,
(25)
N −1 N −1 N −1
1 X X X 2πi k(−p+bq−m)
eN
|m; bihm; b|
N
m=0
b=0 k=1
N −1 N −1
1 X X 2πi k(q−n)
eN
+
|nihn|.
N
n=0
k=0
88
(26)
The WT and the line operator have the following properties,
analogous to the ones for the continuous case.
1) The matrix elements of the line operator with respect to the
states of the reference basis are given by
hq|P̂q′ p′ |q̄i = δqq′ δq̄q′ − δqq̄ δ2q, 2q′ +1 + δq+q̄, 2q′ +1 e
2πi ′
N p (q−q̄)
.
†
2) Since P̂q,p = P̂q,p
, the WT of a Hermitean operator  is real:
WÂ (q, p) = WÂ⋆ (q, p) , for A† = A.
(27)
3) The N 2 line operators P̂q,p form a complete orthonormal set:
they fulfill the orthogonality and closure relations
h
i
1
Tr P̂q,p P̂q′ ,p′
= δq,q′ δp,p′ ,
N
N −1
1 X
P̂q,p = I
N q,p=0
89
An N × N matrix  can be written as a lin. combin. of the P̂q,p ’s:
Â
=
=
N −1
³
´
1 X
Tr ÂP̂q,p P̂q,p
N q,p=0
N −1
1 X
W (q, p)P̂q,p
N q,p=0 Â
4) The WT’s of the operators  and B̂ fulfill the “product formula”
N −1
1 X
WÂ (q, p)WB̂ (q, p) = Tr(ÂB̂) ,
N q,p=0
which can be proved using the orthogonality of the line operators.
90
5) The WF Wρ̂ (q, p) satisfies the marginality property
³
Tr ρ̂ P̂mb
´
N −1
1 X
Wρ̂ (q, p)δMq,p (b),m
= hm, b|ρ̂|m, bi =
N q,p=0
(28)
I.e., the probability to find the system in the state m
of the basis b (of the set of N + 1 MUBs)
is 1/N times the sum of the WF over the points in the phase-space
plane q, p that satisfy Mq,p (b) = m
The RHS of Eq. (28) defines Radon transform of the WF Wρ̂ (q, p)
1 X
1 X
b = 0̈, T r(ρ̂P̂q0 ,0̈ ) = hq0 |ρ̂|q0 i =
Wρ̂ (q, p)δq,q0 =
Wρ̂ (q0 , p)
N q,p
N p
1 X
b = 0 : T r(ρ̂P̂N −p0 ,0 ) =
Wρ̂ (q, p)δN −p,N −p0
N q,p
1 X
i.e., hp0 |ρ̂|p0 i =
Wρ̂ (q, p0 )
N q
91
,
6) The WF is normalized as
N −1
1 X
Wρ̂ (q, p) = 1 .
N p,q=0
92
(29)
Relation between Wigner function and Kirkwood
quasi-distribution for a discrete,
finite-dimensional Hilbert space
Wigner function in terms of Kirkwwod’s quasi-distribution
Wρ̂ (q, p)
=
N
−1
X
e
N +1
′
′
2πi
N 2(q−q + 2 )(p−p )
Kp′ q′
q ′ ,p′ =0
¯ ¯
E
¯ ¯
+hq|ρ̂|qi − q + (N + 1)/2¯ρ̂¯q + (N + 1)/2
D
The labels occurring in bras and kets must be understood Mod[N ].
Again, Kirkwood’s distribution can be related to the correlations of
two probes, in a very weak-coupling measurement designed to
measure in succession the projectors for position and momentum:
·
¸
i
1
hQ̂1 Q̂2 i(Pp ←Pq ) + 2 hP̂1 Q̂2 i(Pp ←Pq ) .
K(p, q) = limǫ1 →0
ǫ1 ǫ2
2σP1
93
For the WF we thus find
Wρ̂ (q, p) =
N
−1
X
e
′
′
2πi
N 2(q−q +(N +1)/2)(p−p )
q ′ ,p′ =0
¸¾
·
½
i
1
hQ̂1 Q̂2 i(Pp′ ←Pq′ ) + 2 hP̂1 Q̂2 i(Pp′ ←Pq′ )
× limǫ1 →0
ǫ1 ǫ2
2σP1
1
1
(Pq )
+ hQ̂i
− hQ̂i(Pq+(N +1)/2 ) .
ǫ
ǫ
(30)
As a result, Wigner function, which is defined in the system
discrete phase space, can be related to a set of measurable
quantities, consisting of the two-probe and single-probe expectation
values obtained in the experimental setup described above, and
reconstructed therefrom.
94
1
Conclusions
We posed the question whether it is possible to find
appropriate measurements of the system position and momentum
that would allow the reconstruction of Wigner function of the
system state.
We gave an affirmative answer to this question:
the types of measurements needed are
successive measurements of projectors associated with position and
momentum,
of the type envisaged in von Neumann’s model of measurement.
In this model, what one detects are
the correlation functions of the two probes
that are coupled to the system in question in order to allow for the
performance of two successive measurements.
95
We first considered the case in which the system is described in a
continuous Hilbert space,
and then we turned to the study of a description in a
discrete, finite-dimensional Hilbert space.
The Wigner function for the case of a discrete, finite-dimensional
Hilbert space, has been widely studied in the literature.
Here we proposed an alternative version, within a
standard algebraic approach.
It turns out that this version can be re-formulated entirely in terms
of “finite-geometry” concepts, an approach that associates
states and operators in Hilbert space with lines and points of the
geometry
This latter approach is conceptually very attractive, and its
development will be postponed to a future publication.
96