Local Measurement and
Quantum Teleportation
in a Relativistic Detector-Field System
Shih-Yuin Lin
林
世
昀
Department of Physics
National Changhua University of Education
國立彰化師範大學物理系
In collaboration with K. Shiokawa, C.-H. Chou, and B. L. Hu
Jun 8, 2012 @ IPAS
Outline
I. Introduction
II. Issues of Q Teleportation in Relativistic QFT
III. Alice-Rob Problem
IV. The Model
V. Physical Fidelity of Quantum Teleportation
VI. Summary
I. Introduction
Beginning of Relativistic Quantum Information: 1935
„
Every single localized quantum object, no matter pure or mixed, can be
described by a local hidden variable theory, which is essentially classical.
.
„
Quantum entanglement of two or more quantum objects
"I would not call that one but rather the characteristic trait of quantum
mechanics, the one that enforces its entire departure from classical line
of thought."
- Erwin Schrödinger (1935), in response to...
When relativity meets quantum theory: EPR "paradox".
I. Introduction
„
Entanglement, Measurement, and Quantum Nonlocality
“Quantum nonlocality”
The quantum state of an entangled pair A and B in Hilbert space
HA x HB will be changed simultaneously by a local measurement in a
subspace (e.g. HB.) at some moment t . [The state of A is affected by
a local operation on B.]
“Spooky action-at-a-distance”
If, in addition, A and B are localized objects and separated at a
distance, the quantum state of A will be changed instantaneously
(wave-function collapse) by a local measurement at another point
(position of B) on the same time-slice.
No violation of causality.
Quantum Teleportation: 2-Level System
Ψ
[Jozsa, Wootters, Bennett,
Brassard, Crepeau, Peres 1993]
Φ 1 =α H 1+β V
Φ−
1
Ψ+
Shared entangled pair
23
=
= Φ 1 ⊗ Φ+
= Φ+
In-state
Φ+
123
(
1
H
2
2
H
3
+ V
2
V
3
)
Ψ−
12
12
12
12
23
(
⊗ (α H
⊗ (α V
⊗ (α V
⊗ α H
Local Operation and Classical Communication (LOCC)
[Bouwmeeter, Pan, Mattle, Eibl, Weinfurter, Zeilinger, 1997]
3
+β V
3
3
−β V
3
3
+β H
3
3
−β H
3
)+
)+
)+
)
Quantum Teleportation: Continuous Variables
[Vaidman 1994]
In-state
|q1 + i p1>
p1
Shared entangled pair:
EPR state with
q2 + q3 = 0, p2 − p3 = 0
q1
[Braunstein, Kimble 1998]
EPR state
Two-mode squeeze state
[Furusawa, Sorensen,
Braustein, Fuchs, Kimble,
Polzik 1998]
Quantum Teleportation: Continuous Variables
[Vaidman 1994]
In-state
|q1 + i p1>
Joint
measurement
p'3 = p3 + b = p1
q'3 = q3 + a = q1
p3 = p1− b
q3 = q1 − a
Shared entangled pair:
EPR state with
q2 + q3 = 0, p2 − p3 = 0
[Braunstein, Kimble 1998]
EPR state
Two-mode squeeze state
[Furusawa, Sorensen,
Braustein, Fuchs, Kimble,
Polzik 1998]
I. Introduction
„
“Natural” assumptions in quantum teleportation
- The projective joint measurement by Alice and the operation on
Bob’s qubit/HO are done locally not only in
Hilbert space (each on a specific dimension), but also in
position space (each at a point-like event in spacetime).
- Alice and Bob must have the full knowledge about the state of the
entangled pair at every moment to make the teleportation successful.
World record: Ursin et al., Nature Physics 3, 481 - 486 (2007)
II. Issues of Quantum Teleportation
in Relativistic Quantum Field Theory
II. Issue 1: Localities
„
Quantum field theory is "local",
meaning that no coupling between fields at
different spacetime points,
which implies causality:
[φ(A), φ(B)] = 0 if B is outside the lightcone of A
Indeed, for a particle moving in a quantum field,
dissipation is related to the retarded Green f 'n (explicitly causal)
and the stochastic (quantum noise) field ξ has correlation
-- Hadamard function, nonzero outside the lightcone.
Locality of QFT
Quantum nonlocality
B'
II. Issue 2: Local Objects
„
In theoretical particle physics, formally,
- an elementary particle is local in k-space, and so non-local in position space;
- a spatially localized state (e.g. Newton-Wigner state) or a wave packet
of a free elementary particle will spread in time.
In QFT:
What is a quantum object always local in space?
What is a spatially local measurement?
Out-Particles
interaction region
In-Particles
(something localized in k-space)
II. Issue 2: Local Objects
Moving cavities?
II. Issue 2: Local Objects
Atoms, mesons, or the Unruh-DeWitt detectors? – effective theories
massless scalar field
internal: HO x 3
Entangled pair
QA , QC
could have a
switching function
for QC
t=0
prescribed
trajectories
QB
point-like in space,
bilinear interaction
[ DeWitt 1979 ]
II. Issue 3: Field States and Reference Frames
In quantum field theory, a pure field-state at some moment is a
functional of field-strength at every (real or momentum) space point
Quantization
η ~ time
[SYL, Chou, Hu, 2010]
in M 4
Wave functional of the vacuum state (no-particle state in M 4)
which is defined over the whole time-slice at that moment η.
II. Issue 3: Field States and Reference Frames
Quantum states of fields do not make sense without a reference frame!
Quantum states in the non-asymptotic region defined on two
different time-slices intersecting at some spacetime point P could be
very different even for the sector defined at P.
[Aharonov, Albert, 1984]
II. Issue 3: Field States and Reference Frames
„
In the interaction region, dynamical variables in different coordinates
are incomparable unless they are living on the same time-slice at
some moments of each coordinate, where transformation between
dynamical variables is clear.
Ex: An alternative metric in
(1+1)D Minkowski space
comparable
Interaction region
(A<1)
Not comparable
η = constant
start
ξ=
constant
„
t0
II. Issue 4: Spatially Local Measurement
„
Dynamical variables of quantum fields could be essentially nonlocal in
space:
φ k ~ plane waves
Q1: How to perform a spatially local measurement in a quantum field?
A1: One (perhaps the only one) possibility is to measure a point-like
object coupled with the field, e.g. an Unruh detector or an atom.
(Projective measurement in the interaction region!)
Sounds OK. Then
Q2: Are two post-measurement states measured in the same spatially
local event but collapsed on different time-slices in different frames
consistent?
A2: In the Unruh-DeWitt detector theory (a linear system)…
II. Issue 4: Spatially Local Measurement
Assume the initial state of the combined system is Gaussian, and the
detector is measured at t = t1. Started with the post-measurement state
at t = t1 , the quantum state continued to evolve to t = t2 and still in the
form
Suffice to calculate the factors
,
,
,
compare
at t = t2 , or equivalently,
the evolution of
the symmetric two-point correlators
t2
measure
t1
,
,
.
η = constant
start
ξ=
constant
„
t0
II. Issue 4: Spatially Local Measurement
„
Examine the two-point correlators of the field at t = t2 (
can be expressed by…
t2
t2
t1
t0
t2
t0
t2
t1
t0
η1
t0
1. damped HO localized inside the point-like detector(s) and
2. the retarded fields sourced from the detector(s) .
η1
)
II. Issue 3: Field State and Local Measurement
No such kind of terms:
t2
t1
t0
„
Wave functional at t2 is a function of explicitly covariant objects independent
of the data on t1 or τ1 -slice outside the detector.
So the wave functional in the alternative coord. at t2 can be transformed to the
conventional one under a spatial coord. transformation.
Each PMWF evolves to the same wave functional at t = t2 !
It does not matter which time-slice at t1 that the post-measurement wave
functional has collapsed onto.
II. Issue 3: Field State and Local Measurement
„
In short, in our linear system:
Wave functionals collapsed by the same spatially localized
measurement on different time slices in different reference
frames will all evolve to the “same” field states, up to a coordinate
transformation, when comparable.
The locality of measurement event and the covariance of the
mode-functions (or the operators of the dynamical degrees of
freedom) guarantee the consistency, once all the degrees of
freedom are considered. [SYL, arXiv:1104.0772]
„
In a nonlinear system?
III. Alice-Rob Problem
III. Alice-Rob Problem
Rob: Bob at rest in Rindler frame, i.e. uniformly linearly accelerated
[Alsing, Milburn 2003]
field + cavities in relativistic motion
III. Alice-Rob Problem
Vacuum is not vacuous: Unruh effect [Unruh 1976]
A detector uniformly accelerated in Minkowski vacuum will experience
a thermal bath at the Unruh temperature:
T=0
TU = 1K for
a = 2.4 x 1020 m/s2
* If
is initially in its ground state, in weak-coupling limit,
the transition probability to its 1st excited state is
still early time
- uniform acceleration : aμ aμ = a2 = constant (a: proper acceleration)
- Minkowski vacuum: No particle (field quanta) state of the field
for Minkowski observer
III. Alice-Rob Problem
- Alsing, Milburn 2003 (field + cavities in relativistic motion)
[Schützhold, Unruh 2005]
- Fuentes, Mann 2005 (field modes + observer in relativity)
Quantum Nonlocality
= Entanglement
+ Local Meaurement
[SYL, Chou, Hu, PRD 2008: also on entanglement only]
III. Alice-Rob Problem
- Alsing, Milburn 2003 (QT, field + cavities in relativistic motion)
- Fuentes, Mann 2005 (QE, field modes + observer in relativity)
- SYL, Chou, Hu 2008 (QE, Unruh-DeWitt HO detectors)
- Landulfo, Matsas 2009
(QT, 2-level detectors,
finite duration of coupling/acceleration,
teleportation in future asymptotic region)
- Tom Shiokawa 2009 (QT, HO detectors,
teleportation in interaction region,
“pseudo-fidelity”)
Interaction
region
Our Setup [Ver.1] (Continuous Variables)
Teleport an unknown coherent state of QC from Alice to Rob, who
share an entangled pair of the Unruh-DeWitt detectors QA , QB
coupled with a common quantum field.
QA , QC
QB
Alice
Rob
Alice performs a
joint measurement
on QA and QC ,
then send the
outcome to Rob.
[SYL 2011]
Rob receives the
classical signal from
Alice, then performs
a displacement on QB
t=0
1/b
1/a
QA and QB are initially in a
two-mode squeezed state,
Φ(x) in the Minkowski vacuum
[SYL, Chou, Hu, 2008]
III. Alice-Rob Problem
Issues of QT from Alice to Rob in a relativistic field:
„
Causality and event horizon:
Classical signals of the outcome from Alice have to be able to reach
Rob to complete the teleportation of quantum information.
Our Setup [Ver.2]
Teleport an unknown coherent state of QC from Alice to Rob (or Rob’),
who share an entangled pair of the Unruh-DeWitt detectors QA , QB
in a common quantum field.
QA , QC
QB
Rob’
Alice
Rob
Alice performs a
joint measurement
on QA and QC
[SYL 2011]
Rob or Rob’ receives the
classical signal from
Alice, then performs a
displacement on QB
Rob’ becomes inertial after
this moment. Acceleration
drops from a to 0 here
[Ostapchuk, SYL, Mann,
Hu, 2011]
t=0
1/b
1/a
QA and QB are initially in a
two-mode squeezed state,
Φ(x) in the Minkowski vacuum
[SYL, Chou, Hu, 2008]
III. Alice-Rob Problem
Issues of QT from Alice to Rob in a relativistic field:
„
Causality and event horizon:
Classical signals of the outcome from Alice have to be able to reach
Rob to complete the teleportation of quantum information.
„
Non-Uniformly accelerated detectors [Ostapchuk, SYL, Mann, Hu, 2011]
III. Alice-Rob Problem
Issues of QT from Alice to Rob in a relativistic field:
„
Causality and event horizon:
Classical signals of the outcome from Alice have to be able to reach
Rob to complete the teleportation of quantum information.
„
Non-Uniformly accelerated detectors [Ostapchuk, SYL, Mann, Hu, 2011]
„
Field as environment:
Decoherence, Unruh effect and (dis-)entanglement dynamics
Dynamics of the entangled pair depend on the motions of Alice and Rob
III. Alice-Rob Problem
Issues of QT from Alice to Rob in a relativistic field:
„
Causality and event horizon:
Classical signals of the outcome from Alice have to be able to reach
Rob to complete the teleportation of quantum information.
„
Non-Uniformly accelerated detectors [Ostapchuk, SYL, Mann, Hu, 2011]
„
Field as environment:
Decoherence, Unruh effect and (dis-)entanglement dynamics
Dynamics of the entangled pair depend on the motions of Alice and Rob
„
Frame dependence of entanglement (spacelike correlations) and
measurement (wave functional collapse) [SYL 2011]
III. Alice-Rob Problem: Frame Dependence
„
In particle physics, it is sufficient to calculate the S-matrix in the
“in-out” formalism.
Out-Particles
interaction region
In-Particles
(something localized in k-space)
III. Alice-Rob Problem: Frame Dependence
„
In particle physics, it is sufficient to calculate the S-matrix in the
“in-out” formalism. But we are looking at the measurement
process in the interaction region (the “in-in” formalism).
Future asymptotic region
Out-Particles
Caution:
1. Particle concept is well defined;
interaction region
2. Vacuum state is Lorentz invariant;
3. One/multi-particle state is “covariant”,
so is entanglement between particles;
- only in the asymptotic regions in an
asymptotic Minkowski spacetime.
In-Particles
(something localized in k-space)
Past asymptotic region
III. Alice-Rob Problem
Issues of QT from Alice to Rob in a relativistic field:
„
Causality and event horizon:
Classical signals of the outcome from Alice have to be able to reach
Rob to complete the teleportation of quantum information.
„
Non-Uniformly accelerated detectors [Ostapchuk, SYL, Mann, Hu, 2011]
„
Field as environment:
Decoherence, Unruh effect and (dis-)entanglement dynamics
Dynamics of the entangled pair depend on the motions of Alice and Rob
„
Frame dependence of entanglement (spacelike correlations) and
measurement (wave functional collapse) [SYL 2011]
„
Fiducial time-slice in non-equilibrium field theory:
Choice of coordinate is not arbitrary.
Our Setup [Ver.2]
„
Moreover, the concepts of
(1) the physical fidelity of quantum teleportation (timelike) and
(2) quantum entanglement between Alice and Rob (spacelike)
are incommensurable in general.
QA , QC
QB
Rob’
Alice
Rob
Alice performs a
joint measurement
on QA and QC
[SYL 2011]
Rob or Rob’ receives the
classical signal from
Alice, then performs a
displacement on QB
Rob’ becomes inertial after
this moment. Acceleration
drops from a to 0 here
[Ostapchuk, SYL, Mann,
Hu, 2011]
t=0
1/b
1/a
QA and QB are initially in a
two-mode squeezed state,
Φ(x) in the Minkowski vacuum
[SYL, Chou, Hu, 2008]
IV. The Model
IV. The Model: 3 identical Unruh-DeWitt detectors in a
scalar field in (3+1)D Minkowski space
massless scalar field
internal: HO x 3
Entangled pair
QA , QC
prescribed
trajectories
point-like in space,
bilinear interaction
[ DeWitt 1979 ]
QB
could have a
switching function
for QC
t=0
Initial state is defined here.
IV. The Model
„
Initial state of the combined system at t = 0
where -
Or, created right before
the joint measurement by Alice
- Minkowski vacuum
-
- Two-mode squeeze state in (K, Δ) representation [Unruh, Zurek 1989]
EPR state with
is a Gaussian state.
- Coherent state
w/o uncertainty
IV. The Model
„
Schrödinger Picture (
;
)
(K-Δ) representation of Wigner functional [Unruh, Zurek (1989)]
(Characteristic functional of Wigner functional [e.g. Mandel, Wolf (1995)])
where
Right before the joint measurement by Alice
At some moment
, Alice perform a Gaussian measurement
locally in space on A and C. Right before the measurement,
Coordinate time
Here x0 = t = t1
QA , QC
Alice performs a
joint measurement
on QA and QC
t=0
1/b
1/a
QB
Right after the joint measurement by Alice
Right after the joint measurement, the quantum state collapsed to
where ( m, n = A, C , the outcome
)
Two-mode
squeeze state
Here
such that
: probability of finding A and C in
Quantum Teleportation: Continuous Variables
[Vaidman 1994]
In-state
|q1 + i p1>
Joint
measurement
p'3 = p3 + b = p1
q'3 = q3 + a = q1
p3 = p1− b
q3 = q1 − a
Shared entangled pair:
EPR state with
q2 + q3 = 0, p2 − p3 = 0
[Braunstein, Kimble 1998]
EPR state
Two-mode squeeze state
[Furusawa, Sorensen,
Braustein, Fuchs, Kimble,
Polzik 1998]
Pseudo-Fidelities
Imagine Rob gets the outcome β instantaneously (in some observer’s
frame) at
right after measurement and performs a displacement
accordingly on B, then B becomes
Def: “Pseudo-fidelity”
and averaged pseudo-fidelity
(initially
),
Our Setup
Teleport an unknown coherent state of QC from Alice to Rob
QA , QC
Alice performs a
joint measurement
on QA and QC
QB
Imagine Rob performs
a displacement on QB
here according to
Alice’s outcome
(Averaged) pseudo-fidelity is
evaluated on this time-slice
for observer at rest in M4
t=0
1/b
1/a
QA and QB are initially in a
two-mode squeezed state,
Φ (x) in the Minkowski vacuum
Pseudo-Fidelities
Averaged pseudo-fidelity on t1-slice
Logarithmic negativity
of A and B evaluated on
t1-slice
1/2
Fidelity of classical teleportation
(using a coherent state instead of
the EPR state)
Pseudo-Fidelities
Averaged pseudo-fidelity on t1-slice
Logarithmic negativity of A and B
1/2
Fidelity of classical teleportation
1. Quantum entanglement of A and B varies much slower than Fav.
2. Fav can be less than ½ while A and B are still entangled.
3. The oscillation of Fav is due to the natural oscillation of the two-mode
squeeze state of A and B (i.e. the distortion from the initial state).
4. The change of the oscillating period of Fav is due to time-dilation of B
observed in Minkowski frame.
Pseudo-Fidelities
Averaged pseudo-fidelity on t1-slice
Logarithmic negativity of A and B
Fidelity of classical teleportation
1/2
QB
QA
γt1 = 0.2
γt1 = 2
Pseudo-Fidelities
Averaged pseudo-fidelity on t1-slice
Logarithmic negativity of A and B
1/2
Fidelity of classical teleportation
1. Quantum entanglement of A and B varies much slower than Fav.
2. Fav can be less than ½ while A and B are still entangled.
3. The oscillation of Fav is due to the natural oscillation of the two-mode
squeeze state of A and B (i.e. the distortion from the initial state).
4. The change of the oscillating period of Fav is due to time-dilation of B
observed in Minkowski frame.
Our Setup
Teleport an unknown coherent state of QC from Alice to Rob
QA , QC
QB
Rob’
Rob
Alice performs a
joint measurement
on QA and QC
Or evaluated on this time-slice
for observer at rest in
quasi-Rindler frame
Imagine Rob performs
a displacement on QB
here according to
Alice’s outcome
(Averaged) pseudo-fidelity
is evaluated on this time-slice
for observer at rest in M4
t=0
1/b
1/a
QA and QB are initially in a
two-mode squeezed state,
Φ (x) in the Minkowski vacuum
Pseudo-Fidelities
(Averaged) pseudo-fidelities (as well as quantum entanglement) are
frame-dependent !
: best averaged fidelity of QT
in Minkowski frame
in quasi-Rindler frame
To achieve the best fidelity, both Rob and Alice must have the full knowledge
on the state of AB-pair at every moment, which depends on their motion.
Pseudo-Fidelities: Frame-Dependence
Entanglement is a necessary condition for the best averaged
pseudo-fidelities of quantum teleportation better than the classical one.
[Can be shown explicitly in weak coupling limit.]
in Minkowski frame
in quasi-Rindler frame
Neither entanglement nor
the best averaged fidelity
is sensitive to a here
The larger a ,
the lower entanglement, and
the lower the best averaged fidelity
V. Physical Fidelity of Quantum Teleportation
V. Physical Fidelity
Teleport an unknown coherent state of QC from Alice to Rob’
QA , QC
Rob’
Alice
Alice performs a
joint measurement
on QA and QC
QB
Rob’ receives the
classical signal from
Alice, then performs a
displacement on QB
Rob’ becomes inertial after
this moment. Acceleration
drops from a to 0 here
t=0
1/b
1/a
QA and QB are initially in a
two-mode squeezed state,
Φ (x) in the Minkowski vacuum
V. Physical Fidelity
Teleport an unknown coherent state of QC from Alice to Rob’
QA , QC
QB
Rob’
Alice
Q: Wave functional of the
combined system
collapses on this time-slice?
Alice performs a
joint measurement
on QA and QC
Or this time-slice?
t=0
1/b
1/a
QA and QB are initially in a
two-mode squeezed state,
Φ (x) in the Minkowski vacuum
V. Physical Fidelity
Teleport an unknown coherent state of QC from Alice to Rob’
QA , QC
QB
Rob’
Alice
Q: Wave functional of the
combined system
collapses on this time-slice?
Alice performs a
joint measurement
on QA and QC
Or this time-slice?
t=0
1/b
A: Doesn’t matter!!
1/a
QA and QB are initially in a
two-mode squeezed state,
Φ (x) in the Minkowski vacuum
V. Physical Fidelity
Teleport an unknown coherent state of QC from Alice to Rob’
QA , QC
QB
Rob’
Alice
The reduced state of QB
collapsed at τ1 then
evolved to
is consistent with
the reduced state of QB
collapsed at τ1’ then
evolved to
.
t=0
1/b
1/a
A: Doesn’t matter!!
V. Physical Fidelity
Teleport an unknown coherent state of QC from Alice to Rob’
QA , QC
QB
Rob’
Alice
The reduced state of QB
collapsed at τ1 then
evolved to
is consistent with
the reduced state of QB
collapsed at τ1’ then
evolved to
.
t=0
1/b
1/a
Actually ---
The reduced state of QB collapsed in all frames will become consistent
when Rob’ is entering the future lightcone of the measurement event x!
V. Physical Fidelity
Teleport an unknown coherent state of QC from Alice to Rob’
QA , QC
QB
Rob’
Alice
Since the collapsed state is
complicated, it is convenient
to collapse the reduced state
of QB directly at
,
namely, (almost) on the future
lightcone of the measurement
event by Alice.
t=0
1/b
1/a
The reduced state of QB collapsed in all frames will become consistent
when Rob’ is entering the future lightcone of the measurement event x!
V. Physical Fidelity
Teleport an unknown coherent state of QC from Alice to Rob’
QA , QC
QB
Rob’
Alice
Since the collapsed state is
complicated, it is convenient
to collapse the reduced state
of QB directly at
,
namely, (almost) on the future
lightcone of the measurement
event by Alice.
t=0
1/b
1/a
Assume Rob’ performs the displacement on QB right after receiving
the classical signal from Alice, namely, at
.
V. Physical Fidelity
Assume Rob’ performs the displacement on QB right after receiving
the classical signal from Alice, namely, at
.
QA , QC
QB
Rob’
Alice
Def: Averaged physical
fidelity of quantum telportation
evaluated on the future
lightcone of the measurement
event by Alice.
t=0
1/b
1/a
(The best possible fidelity of
QT one can have in this open
quantum system.)
V. Physical Fidelity
Averaged physical fidelity of quantum telportation
Pseudo-Fav
in M4
The larger a,
the earlier
becomes
less than ½.
EN/10 evaluated almost
on the future lightcone of
the measurement event
by Alice.
Physical
Pseudo-Fav in M4
Pseudo-Fav in M4
Physical
Physical
Still, entanglement evaluated almost on the future lightcone of
the measurement event is a necessary condition for
> ½.
V. Physical Fidelity
Averaged physical fidelity of quantum telportation drops to ½ much
earlier than any off-shell fidelity.
Pseudo-F+av in t
1/b (c = 1)
The larger aτ2,
the later
Rob’ has, so
the lower value of the best
averaged physical fidelity
at that
.
Unruh effect is not manifest here, though
VI. Summary
VI. Summary [arXiv:1204.1525, 1104.0772]
„
A consistent Alice-Rob’ CV quantum teleportation scenario, with
quantum nonlocality, relativistic locality, entanglement, and fidelities
naturally integrated, has been illustrated in a Unruh-DeWitt detector theory.
„
Spatially local measurement in QFT can be realized by measuring
an object localized in space and coupling to the field.
Consistency is guaranteed by the covariance of mode functions and
the locality of the measurement.
VI. Summary [arXiv:1204.1525, 1104.0772]
„
Entanglement between two spatially local objects is frame-dependent. The
most relevant quantity to physical processes is the entanglement evaluated
almost on the future lightcone of the measurement event, which is a
necessary condition for the best averaged physical fidelities of QT beating
the classical one. (Quantum discord is not so relevant to LOCC.)
„
Full knowledge about the entangled pair is necessary for a 100% successful
QT.
„
The larger a, the earlier (in the moment when the measurement by Alice
occurs in Minkowski time) the best averaged physical fidelity becomes less
than ½. This is mainly due to the longer time needed for classical signal to
catch Rob’, not dominated by the Unruh effect.
Thank you!