Indefinite causal order in quantum mechanics

Faculty of Physics, University of Vienna &Institute for Quantum Optics and Quantum Information, Vienna

Mateus Araujo, Cyril Branciard, Fabio Costa, Adrian Feix, Christina Giarmatzi, Ognyan Oreshkov, Magdalena Zych

The 11th “Vienna Central European Seminar on Particle Physics and Quantum Field Theory”

Časlav Brukner

Possible causal influences

𝑏 𝑎 𝑏𝑎

𝑎𝑎 𝑏 𝑐 causes causes and have a

common cause

Question: Can one have situations in which the causal order is not fixed, but rather is subject to quantum indefiniteness?

Dynamical causal structure of general relativity

Superposition principle of quantum mechanics+

Question: Can one have situations in which the causal order is not fixed, but rather is subject to quantum indefiniteness?

Dynamical causal structure of general relativity

Superposition principle of quantum mechanics+

Question: Can one have situations in which the causal order is not fixed, but rather is subject to quantum indefiniteness?

Dynamical causal structure of general relativity

Superposition principle of quantum mechanics+

Question: Can one have situations in which the causal order is not fixed, but rather is subject to quantum indefiniteness?

Outline

“Causal inequalities“ Device-independent approach to causality

Framework for quantum mechanics with no assumed global causal structure:Device-dependent approach to causalityCausally non-separable processes

Quantum computation with indefinite order of gatesComputational task that cannot be accomplished on a

computer with fixed order of gates

Physical realization of causally non-separable processes

Linear optical schemes & via superposition of large masses

“Correlation does not imply causation”

and are correlated, but …does cause ,

or cause , or both have a common cause ?

𝑏 𝑎 𝑏𝑎

𝑎𝑎 𝑏 𝑐

The notion of “causation”

𝑎 ,𝑥

Necessity of interventions, or free variables, statistically independent of “the rest of the

experiment”

𝑏 , 𝑦

),,( yxbap

yxbpyxbapa

,),,( xapyxbap

b

),,(One-directional

signalling

Cause

Effect

The notion of “causation”

𝑎 ,𝑥

Necessity of interventions, or free variables, statistically independent of “the rest of the

experiment”

𝑏 , 𝑦

),,( yxbap

One-directional signalling

𝑥𝑎

𝑦𝑏

yxbpyxbapa

,),,( xapyxbap

b

),,(

Definite causal order

No-signalling One-directional

signalling

Space-like separated

Time-like separated Alice before Bob

Causal inequalities(Device-independent approach to causality)

𝑦𝑏𝑥

𝑎

Causal correlations: either A signals to B or B signals to A, or no-signalling or a convex combination of these situations.

System enters each laboratory only once. The laboratories are shielded and interact with “the

outside world” only through the system entering and exiting it.

Assumptions:

Causal inequalities(Device-independent approach to causality)

Causal correlations: either A signals to B or B signals to A, or no-signalling or a convex combination of these situations.

),|,()1(),|,(),,( yxbapyxbapyxbap ABBA

ybpyxbap BA

a

BA ),,(

xapyxbap BA

b

BA ),,(

ybpyxbap BA

a

BA ),,(

yxapyxbap BA

b

BA ,),,(

No-signalling One-directional signalling: Alice to Bob

The simplest causal inequality

𝑦𝑏𝑥

𝑎

Causal correlations satisfy causal inequalities, which are facets of the causal polytope

1 bit input, 1 bit output

1 bit input, 1 bit output

Guess Your Neighbour’s Input (GYNI) game:

21),( xbyap

C. Branciard, M. Araújo, A. Feix, F. Costa, Č. B., to appear in New J. Phys. (2015)

𝑦𝑏𝑥

𝑎

“Non-causal correlations”(violating causal inequalities)

Interpretation: Both A signals to B and B signals to A, although the system enters only once the laboratory and the laboratories are shielded.

yxbpyxbapa

,),,(

yxapyxbapb

,),,( Two-directional signalling

Past space-like surface

Future space-like surface Output Hilbert spaceℋ2

Input Hilbert spaceℋ1

Local quantum laboratory(Device-dependent approach to causality)

Past space-like surface

Future space-like surface Output Hilbert spaceℋ2

Input Hilbert spaceℋ1

Local quantum laboratory(Device-dependent approach to causality)

𝑀 𝑎∈ℒ (ℋ 1⊗ℋ2)

Transformations = completely positive (CP) trace-nonincreasing maps

𝑎

𝑎 cb

Many local quantum laboratories

Process matrix:

𝑝 (𝑎 ,𝑏 ,𝑐 ,… )=Tr ¿

𝐴1

𝐴2

𝐵1

𝐵2

𝐶1

𝐶2

)

O. Oreshkov, F. Costa, Č.B., Nature Communication 3: 1092 (2012)

Probabilities:

Quantum mechanics locally valid.(no globar causal order fixed)

𝑎 cb

Conditions on W

Normalization:

𝑝 (𝑎 ,𝑏 ,𝑐 ,… )≥0

𝐴1

𝐴2

𝐵1

𝐵2

𝐶1

𝐶2

O. Oreshkov, F. Costa, Č.B., Nature Communication 3: 1092 (2012)

Positivity:

Quantum mechanics locally valid.(no global causal order fixed)

∑𝑎 ,𝑏 ,𝑐

𝑝 (𝑎 ,𝑏 ,𝑐 ,… )=1 and has a specific form

Forbidden processes

Single Loops

𝑊 𝐴1 𝐴2⊗𝑊 𝐵1𝐵 2

Double Loops

𝑊 𝐴1𝐵 2⊗𝑊 𝐴2 𝐵1

𝐴1

𝐴2 𝐵2

𝐵1 𝐵1𝐴1

𝐴2 𝐵2

O. Oreshkov, F. Costa, Č.B., Nature Communication 3: 1092 (2012)

No “grandfather paradox”

Admissible processes

𝑊 𝐴1 𝐴2 𝐵1⊗ Ι 𝐵2

Channels from Alice to Bob

𝐵1

𝐴1

𝐴2

Time-like separated

𝑊 𝐴1𝐵1⊗ Ι 𝐴2 𝐵2

States

𝐴1 𝐵1

Space-like separated

O. Oreshkov, F. Costa, Č.B., Nature Communication 3: 1092 (2012)

There are causally non-separable processes

Causally separable processes

𝑊 𝐴1 𝐴2 𝐵1𝐵 2=𝜆𝑊 𝐴≼𝐵+(1− 𝜆 )𝑊 𝐵≼𝐴

𝑊 𝐴≼𝐵 A signals to B, or no signaling

Most general processes compatible with definite causal structure (convex mixtures of ordered processes):

𝑊 𝐵≼𝐴 B signals to A, or no signaling

Ordered processes:

Alice

Bob

¿0 ⟩∨𝜓 ⟩

Channel from Alice to Bob

Alice

Bob

¿1⟩∨𝜓 ⟩

Channel from Bob to Alice

Quantum switch –quantum control of causal order

Path degree of freedom (quantum control)

Internal degree of freedom

Quantum switch –quantum control of causal order

|0 ⟩∨𝜓 ⟩❑

|1 ⟩∨𝜓 ⟩❑

𝐵1 𝐵2

𝐴2𝐴1

𝐶1

Alice

Bob

Cleve

Quantum switch is a causally nonseparable process𝑊 ≠ 𝜆𝑊 𝐴≼𝐵≼𝐶+(1− 𝜆 )𝑊 𝐵≼𝐴≼𝐶

Idea of the proof: The switch allows two-way signalling (contains both and terms)

Quantum switch –quantum control of causal order

|0 ⟩∨𝜓 ⟩❑

|1 ⟩∨𝜓 ⟩❑

𝐵1 𝐵2

𝐴2𝐴1

𝐶1

Alice

Bob

Cleve

Quantum switch is a causally nonseparable process𝑊 ≠ 𝜆𝑊 𝐴≼𝐵≼𝐶+(1− 𝜆 )𝑊 𝐵≼𝐴≼𝐶

However, the switch cannot violate a „causal inequality“

„Superposition of unitarities“

|0 ⟩∨𝜓 ⟩❑

|1 ⟩∨𝜓 ⟩❑

𝐵1 𝐵2

𝐴2𝐴1

𝐶1

Alice

Bob

Cleve

|𝜓 𝑓𝑖𝑛𝑎𝑙 ⟩=1√2

(¿0 ⟩¿¿1𝑈 2𝑈 1|𝜓 ⟩2+¿1⟩1𝑈 1𝑈 2|𝜓 ⟩2)¿

Computational advantages

1C. Branciard, M. Araújo, A. Feix, F. Costa, and C.B., arXiv:1508.01704 (2015)2T. Colnaghi, G. M. D’Ariano, S. Facchini, and P. Perinotti, Phys. Lett. A 376, 2940-2943 (2012)

3G. Chiribella, Phys. Rev. A 86, 040301(R) (2012)4M. Araujo, F. Costa, and C.B., Phys. Rev. Lett. 113, 250402 (2015)

5A. Feix, M. Araujo, and C.B. Phys. Rev. A 92, 052326 (2015)

“Guess Your Neighbour's Input” game1

Controlled permutations of unitary gates2

Discrimination between pairs of commuting or anti-commuting unitaries with a single use3

Reduction of the query complexity from to 4

Reduction of communication complexity5

Experimental Demonstration

L. M. Procopio et. al, Nature Communication 6, 7913 (2015).

Order of events determined by their position in space-time

A

B

General relativity: space-time is dynamical

Dynamics of the clock described in:

Proper times of clocks a & b

General relativity: space-time is dynamical

M. Zych, F. Costa, I. Pikovski, Č. B., Nature Communication 2:505 (2011)I. Pikovski, M. Zych, F. Costa, Č. B., Nature Physics 11, 668 (2015)

M. Zych, F. Costa, I. Pikovski, Č. B., Nature Communication 2:505 (2011)I. Pikovski, M. Zych, F. Costa, Č. B., Nature Physics 11, 668 (2015)

Dynamics of the clock described in:

Proper times of clocks a & b

General relativity: space-time is dynamical

Quantum control of temporal order

Gravitational quantum switch

Summary and Outlook

Global causal order need not be a necessary element of quantum theory.

There exist physical processes that are causally nonseparable.

Causally nonseparable processes is a new resource for quantum information processing

The framework also includes processes with no (known) physical interpretation

Does a theory of quantum gravity provide a physical interpretation to such processes?

Thank you!

quantumfoundations.org

BA

Rr rB

Potential

r

Force-free time dilation

Relative degrees of freedom

Clock in a superposition Fixed backgroud

Local clockSuperposition of backgrouds

=

running clock in a superposition

QM+GR: interference cannot be observed since the which-path information is stored in the clock time

Clock in a superposition

GR: time shown by the clock depends on the path taken

QM: either path or interference

Visibility modulation

𝒱=|cos( ∆𝐸𝑔∆ 𝑥𝑡2ℏ𝑐2 )|=|cos (∆𝜏𝑡⊥ 𝜋2 )|

Difference in proper time

Orthogonalization time

For one needs

𝑡⊥=ℏ𝜋Δ𝐸

M. Zych, F. Costa, I. Pikovski, Č. B., Nature Communication 2:505 (2011)I. Pikovski, M. Zych, F. Costa, Č. B., Nature Physics 11, 668 (2015)

∆ 𝑥