Post on 14-Sep-2020
Quantum processes on time-delocalized systems
Ognyan Oreshkov
Centre for Quantum Information and Communication, Université libre de Bruxelles
Causality in the quantum world, Anacapri, 2019
Outline
0) Background
1) The quantum SWITCH on time delocalized systems
2) Realization of a class of isometric extensions of bipartite processes (based on arXiv:1801.07594)
3) Recent (with R. Lorenz and J. Barret): all unitarily extensible bipartite processes are causally separable
4) New (with J. Wechs, Cyril Branciard, …): realization of processes violating causal inequalities
p(i, j, k, l, m)Joint probabilities
Hardy, PIRSA:09060015; Chiribella, D’Ariano, Perinotti, PRA 81, 062348 (2010) [arXiv 2009]
Quantum systems
{Pm}
C D
I
F
Time
G
E
H
I
U {Fl}{Nk}
t1
t2
t3
{Si}
{Ej}M
A Bt0
System A à Hilbert space
From a global perspective, a system is generally a subsystem (tensor factor of a subspace) of a larger Hilbert space (P. Zanardi, PRL (2001); Zanardi, Lidar, Lloyd, PRL (2004)):
The most general faithful encoding of quantum information
Quantum operation from A to B à quantum instrument (collection of CP maps):
, where is CPTP.
(usually at a given time)
Viola, Knill, Laflamme, JPA (2001); E. Knill, PRA (2006); Kribs and Spekkens PRA (2006)
Quantum systems
Quantum systems
A subsystem is defined by the algebra of operators acting (locally) on that systems:
Example:
U
A B
C D
A B
A’ B’
U Id Id=
p(i, j, k, l, m)Joint probabilities
Quantum systems
{Pm}
C D
I
F
Time
G
E
H
I
U {Fl}{Nk}
t1
t2
t3
{Si}
{Ej}M
A Bt0
{Pm}
C D
I
F
Time
G
E
H
I
U {Fl}{Nk}
t1
t2
t3
p(i, j, k, l, m)Joint probabilities
{Si}
{Ej}M
A Bt0
Quantum systems
{Pm}
C D
I
F
Time
G
E
H
I
U {Fl}{Nk}
t1
t2
t3
p(i, j, k, l, m)Joint probabilities
{Si}
{Ej}M
A Bt0
Quantum systems
Chiribella, D’Ariano, Perinotti, PRA (2009)
Quantum combs:
{Ck} {Djl}
C D
I
F
Time
H
U
{Nk}
t1
t2
t3
M
At0
Quantum systems
Chiribella, D’Ariano, Perinotti, PRA (2009)
Quantum combs:
{Ck} An operation from AF to DHI
C D
I
F
Time
H
U
{Nk}
t1
t2
t3
M
At0
Quantum systems
{Ck}
Can do tomography on it.
preparation
measurement
Time-delocalized systems: subsystems of Hilbert spaces that are tensor products of systems at different times
D
C
B
Time
t1
t2
t3
At0
Time-delocalized systems: subsystems of Hilbert spaces that are tensor products of systems at different times
U
Example
≡
A B
A’ B’
Id IdId
No assumption of global causal order
AliceBob
O. O., F. Costa, and Č. Brukner, Nat. Commun. 3, 1092 (2012).
The quantum process matrix framework
Joint probabilities
Process matrix CJ operators
The quantum process matrix framework
1. Non-negative probabilities:
2. Probabilities sum up to 1:
on all CPTP , , ...
The quantum process matrix framework
Joint probabilities
Process matrix CJ operators
An equivalent formulation as a second-order transformation:[Quantum supermaps, Chiribella, D’Ariano, and Perinotti, EPL 83, 30004 (2008)]
CPTP CPTP = CPTP
AO
AI BI
BO
aI
aO
bI
bO bOaO
aI bI
The quantum process matrix framework
a channel from AOBO to AIBI
W
The process matrix framework
W
Wcausal
Wcausally separable
Wnc
can violate causal inequalities
cannot violate causal inequalities
Do such processes have a physical realization?
Physical realization: there can be an experiment in which every element of the mathematical description of the process (Hilbertspaces, operations on them, …) has a physical counterpart that can be considered a realization of that element.
The quantum SWITCH
W
AO
AI BI
BO
aI
aO
bI
bO
SQ
S’Q’
UA UB
A supermap:
Chiribella, D’Ariano, Perinotti, and Valiron, PRA 88, 022318 (2013), arXiv:0912.0195 (2009)
The quantum SWITCH
AO
AI BI
BO
aI
aO
bI
bO
SQ
S’Q’
UA UB
A supermap:
The quantum SWITCH
AO
AI BI
BO
aI
aO
bI
bO
SQ
S’Q’
UA UB
A supermap:
The quantum SWITCH
AO
AI BI
BO
aI
aO
bI
bO
SQ
S’Q’
UA UB
A supermap:
The quantum SWITCH
WAlice Bob
AO
AI BI
BO
SQ
S’Q’
A process matrix:
Charlie
David
The quantum SWITCH
WAlice Bob
AO
AI BI
BO
SQ
S’Q’
A process matrix:
Charlie
David
The process matrix is not causally separable.
W= |W><W|
Oreshkov and Giarmatzi, NJP 18, 093020 (2016)
Araujo et al., NJP 17, 102001 (2015)
However, it cannot violate causal inequalities.
Experimental realizations of the quantum SWITCH
Procopio et al., Nat. Commun. (2015) Rubio et al., Sci. Adv. (2017) Goswami et al., PRL (2018) ...
Where are the operations of Alice and Bob?
Temporal description
A
CharlieTime
BI
S
S’
Q
Q’
UA
David
aI bI
bOaO
B UB
UA
B UB
bI
bO
UA
A BI
BO
S
S’
Q
Q’
UA
BI
aI
aO
Time
B UBBob
Charlie
DavidO. O., arXiv:1801.07594
Temporal description(simple version)
Bob at a fixed time
Where are the operations of Alice and Bob?
O. O., arXiv:1801.07594
Identifying Alice’s operation
UA
A BI
BO
S
S’
Q
Q’
UA
BI
aI
aO
Time
Claim:
where AI is a nontrivial subsystem of QSBO, defined by the algebra of operators
,
and AO is a nontrivial subsystem of Q’S’BI, defined by the algebra of operators
.
UA
A BI
BO
S
S’
Q
Q’
UA
BI
aI
aO
Time
O. O., arXiv:1801.07594
Identifying Alice’s operation
UA
A BI
BOS
S’
Q
Q’
UA
BI
aI
aO
Time
O. O., arXiv:1801.07594
Identifying Alice’s operation
≡
aI
aO
AI
AO
UA Id
AI
AO
With respect to AI and AO, the experiment has the structure of a circuit with a cycle.
WAlice Bob
AO
AI BI
BO
SQ
S’Q’
Charlie
David
Are there more general processes that exist on time-delocalized systems?
A
Time
BI
S1
S2N
Q
Q’
aI
aO
UA
UA1
N
UA2
S2
S3
S4
S2N-1
Z1
ZN-1
Circuits with coherent control of the times of the operations
(includes the symmetricimplementation of the SWITCH)
Unitarily extendible processes
Araujo, Feix, Navascues, Brukner, Quantum 1, 10 (2017)
W
AO
AI BI
BO
=U
AO
AI BI
BO
DO
CI
~
Claim:
All unitary extensions of bipartite processes have realizations on time-delocalized systems
O. O., arXiv:1801.07594
U1(UA)
BI
BO
aI
aO
U2(UA)
DO
CI
Time
Bob
David
Charlie
EU
BI
BO
DO
CI
Bob
Charlie
≡
AO
AI
aI
aO
UA
David
Proof:
U1(UA)
BI
BO
aI
aO
U2(UA)
DO
CI
EU
BI
BO
DO
CI
AO
AI
aI
aO
UA ≡
Chiribella, D’Ariano, and Perinotti, EPL 83, 30004 (2008)
U
BI
BODO
CI
AO
AI
aI
aO
UA ≡
BI
BODO
CI
AO
AI
aI
aO
UA E
U1
U2
Proof:
Proof:
U
BI
BODO
CI
AO
AI
aI
aO
UA ≡
U1
BI
U2
CI
aI
aO
UA
BODO
Id
~AI~AI
AO~ ~AO
AI – a subsystem of DOBO.
AO – a subsystem of CIBI.
~
~
Proof:
U
BI
BODO
CI
AO
AI
aI
aO
UA ≡
U1
BI
U2
CI
aI
aO
UA
BODO
Id
~AI~AI
AO~ ~AO
AI – a subsystem of DOBO.
AO – a subsystem of CIBI.
~
~
Proof:
≡
U1
BI
U2
CI
aI
aO
UA
BODO
Id
~AI~AI
AO~ ~AO
AI – a subsystem of DOBO.
AO – a subsystem of CIBI.
~
~
U
BI
CI
aI
aO
UA
AO~
~AI
BO
DO
Proof:
≡
U1(UA)
aI
aO
U2(UA)
DO
CI
E
BI
BO
U
BI
CI
aI
aO
UA
AO~
~AI
BO
DO
Similarly can prove realizability of the following class:
A2
A1 B1
B2
V is an isometric channel that maps a tensor factor of D2B2 onto A1.
D2
C1
V
Theorem: all bipartite processes that obey the unitary extension postulate are causally separable.
Hence, the unitary extensions are just variations of the quantum SWITCH.
With R. Lorenz and J. Barrett, Cyclic Quantum Causal Models, in preparation.
Idea behind proof
We have the following no-signaling constraints for the unitary channel of the 4-partite process:
AO BODO
AIBI CI
U
(lack of arrow means no signaling)
This follows from the types of terms permitted in a process matrix.
Consider the reduced process obtained by tracing our Charlie:
AO BODO
AIBI CI
U=
AO BODO
AIBI
U
Let U denote the process matrix (the transposed Choi state of the channel U).
Then, there exists a decomposition
such that
[J.-M. Allen et al., PRX 7, 031021 (2017); J. Barrett, R. Lorenz, O. O., in preparation]
Bad news?
Not at all.
Theorem! All tripartite unitarily extendible tripartite processes, including their unitary extensions, have realizations on time-delocalized systems.
With J. Wechs, C. Branciard, … , in preparation.
This includes processes violating causal inequalities!
[E.g., the classical noncausal process by Baumeler and Wolf, NJP (2016)]
Idea of proof
CO
CI
BO
BI
AO
AI
F
P
U UC
cI
cO
variation of the quantum SWITCH on Alice and Bob
There is a universal way of implementing all such processes,such that Alice and Bob act on fixed time-delocalized systems:
A
Time
BIUA
C1
B UB
UA
B UB
(ancillas suppressed)
C2
C3
Charlie’s operation is within the comb around Alice and Bob.
The systems it acts on can be found similarly to the bipartite case.
Example:
XA in
XAout XC
out
XCin
XBout
X Bin
XAin = ¬ X Bout ⋀ X Cout
XBin = ¬ X Cout ⋀ X Aout
XCin = ¬ X Aout ⋀ X Bout
Bameler and Wolf, NJP 18, 013036 (2016)
Violates causal inequalities.
WBW
Example:
XA in
XAout XC
out
XCin
XBout
X Bin
XAin = ¬ X Bout ⋀ X Cout
XBin = ¬ X Cout ⋀ X Aout
XCin = ¬ X Aout ⋀ X Bout
Bameler and Wolf, NJP (2016)
Violates causal inequalities.
Admits unitary extension [Araujo et al., Quantum (2017)]
Y
Z
U
From Julian Wechs:
Closely linked to a circuit previously found by Allard Guérin and Brukner, NJP (2018)
XA in
XAout XC
out
XCin
XBout
X BinxA
in
xAout
…
xCin
xCout
There are (time-nonlocal) classical variables that violate causal inequalities.
1) They form a cyclic causal model as above, which can be tested.
2) Under the assumption of closed laboratories (no extra arrows apart from those shown), the correlations cannot be explained by dynamical causal order.
You can do it on your desk!