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!