Quantum theory without predefined causal structure

Transcription

1 Quantum theory without predefined causal structure Ognyan Oreshkov Centre for Quantum Information and Communication, niversité Libre de Bruxelles Based on work with Caslav Brukner, Nicolas Cerf, Fabio Costa, Chris<na Giarmatzi Quantum Nonlocality, Causal Structures, and Device-independent Quantum Information, Tainan, 2015

2 Opera<onal Approach from Hardy (2001) Significant progress in understanding QM from operational perspective. Hardy (2001), Barrett (2005), Dakic and Brukner (2009), Massanes and Mülelr (2010), Chiribella, D Ariano, and Perinotti (2010).

3 Opera<onal Approach from Hardy (2001) A temporal sequence is assumed

4 Correla<ons between experiments in space- <me time space A causal structure is assumed

5 Ques<ons 1. Could we understand causal structure from more primitive concepts (e.g., signaling from A to B à A is in the past of B)? 2. Why does signalling always go forward in time? 3. Can we generalize quantum theory so that time and causal structure are not predefined? (Motivation: quantum gravity) 4. What new physical possibilities would this imply?

6 Outline The process framework for operations with no causal order A time-symmetric operational approach to quantum theory Quantum theory without any prior notion of time

7 The process framework Alice Bob No assumption of pre-existing causal order. O. O., F. Costa, and C. Brukner, Nat. Commun. 3, 1092 (2012).

8 The process framework Alice Bob Process (catalogue of probabilities)

9 Quantum processes Local descrip-ons agree with quantum mechanics Transforma<ons = completely posi<ve (CP) maps Kraus representation: Completeness relation:

10 Quantum processes Assumption 1: The probabilities are functions of the local CP maps, Local validity of QM is linear in,,...

11 Choi- Jamiołkowski isomorphism CP maps Posi<ve semidefinite matrices

12 The process matrix Representation Process matrix

13 The process matrix Representation Process matrix Similar to Born s rule but can describe signalling!

14 The process matrix Conditions on W (assuming the parties can share entanglement): 1. Non-negative probabilities: 2. Probabilities sum up to 1: on all CPTP maps,,... Simple characterization via the allowed terms in a Hilbert-Schmidt basis

15 Example: bipar<te state

16 Example: channel B A

17 Example: channel with memory Aà B (The most general possibility compatible with no signalling from B to A)

18 Bipar<te processes with causal realiza<on no signalling from A to B (ch. with memory from A to B) no signalling from B to A (ch. with memory from B to A)

19 Bipar<te processes with causal realiza<on no signalling from A to B (ch. with memory from A to B) no signalling from B to A (ch. with memory from B to A) More generally, we may conceive causally separable processes (probabilistic mixtures of fixed-order processes):

20 Bipar<te processes with causal realiza<on no signalling from A to B (ch. with memory from A to B) no signalling from B to A (ch. with memory from B to A) More generally, we may conceive causally separable processes (probabilistic mixtures of fixed-order processes): Are all possible W causally separable?

21 Causal game y x a b b Alice is given bit a and Bob bit b. Bob is given an additional bit b that tells him whether he should guess her bit (b =1) or she should guess his bit (b =0). Alice produces x and Bob y, which are their best guesses for the value of the bit given to the other. The goal is to maximize the probability for correct guess:

22 Causal game y x a b b Definite causal order between the events in the experiment à

23 A non- causal process Can achieve probability of success. two-level systems The operations of Alice and Bob do not occur in a definite order! More info: O. O., F. Costa, and C. Brukner, Nat. Commun. 3, 1092 (2012).

24 A non- causal process Can achieve probability of success. Can such process be realized in practice? We don t know.

25 But it is not a priori impossible From the outside the experiment may still agree with standard unitary evolution in time. time unitary transformation E.g., quantum switch (Chiribella, D Ariano, Perinotti and Valiron, arxiv: , PRA 2013)

26 Other causal inequali<es and viola<ons Simplest bipartite inequalities: Branciard, Araujo, Feix, Costa, Brukner, arxiv: (2015) Multiparite inequalities: Baumeler and Wolf - violation with perfect signaling: Proc. ISIT 2014, (2014) - violation by classical local operations: Phys. Rev. A 90, (2014) arxiv: (2015) Biased version of the original inequality: Bhattacharya and Banik, arxiv: (2015)

27 Formal theory of causality and causal separability See O. O. and C. Giarmatzi, arxiv: Captures the possibility for dynamical causal relations:

28 Formal theory of causality and causal separability See O. O. and C. Giarmatzi, arxiv: Captures the possibility for dynamical causal relations:

29 This framework s<ll assumes <me locally, and it is <me- asymmetric. What is the origin of this time asymmetry? Could we relax the assumption of time also locally?

30 The circuit framework for opera<onal probabilis<c theories Circuit (an acyclic composition of operations with no open wires): {L k } {P l } Probabilistic structure A C {M i } {N j } B D Joint probabilities p(i, j, k, l) p(i, j, k, l) 0, ijkl p (i, j, k, l)=1 Hardy, PIRSA: ; Chiribella, D Ariano, Perinotti, PRA 81, (2010) [arxiv 2009]

31 Time- asymmetry of standard quantum theory Causality axiom [Chiribella, D Ariano, Perinotti, PRA 81, (2010), PRA 84, (2011)]: Also Pegg, PLA 349, 411 (2006), ( weak causality ). {E j } A p(i, j) p(½ i, E j ) {½ i } In quantum theory, p(ρ i, E j ) = Tr( ˆρ i Ê j ). The marginal probabilities of the preparation events are independent of the measurement: p(½ i {E j } ) = p(½ i ) No signalling from the future

32 What do we call opera<on?

33 What do we call opera<on? Two ideas: O.O. and N. Cerf, Nature Phys. 11, 853 (2015)

34 What do we call opera<on? Two ideas: O.O. and N. Cerf, Nature Phys. 11, 853 (2015) Idea 1. The closed-box assumption {L k } {P l } The events in a box are correlated with other events only as a result of information exchange through the wires A C {N j } D B {M i }

35 What do we call opera<on? Two ideas: O.O. and N. Cerf, Nature Phys. 11, 853 (2015) Idea 1. The closed-box assumption B {M i } à An operation can be realized inside an isolated box. A

36 What do we call opera<on? Two ideas: O.O. and N. Cerf, Nature Phys. 11, 853 (2015) Idea 2. No post-selection B The choice of operation can be known before the operation is applied {M i } (Gives the idea that an operation can be chosen.) A à The causality axiom describes a constraint on pre-selected operation.

37 What do we call opera<on? Two ideas: O.O. and N. Cerf, Nature Phys. 11, 853 (2015) Idea 2. No post-selection B The choice of operation can be known before the operation is applied {M i } (Gives the idea that an operation can be chosen.) A The very concept of operations is time-asymmetric!

38 O.O. and N. Cerf, Nature Phys. 11, 853 (2015) Proposal: drop the no post-selection criterion B {M i } A Operation = description of the possible events in a box conditional on local information

39 Time- symmetric quantum theory O.O. and N. Cerf, Nature Phys. 11, 853 (2015) Joint probabilities: The basic probability rule. {E j } A {ρ i } p(i, j) = Tr(ρ i E j ) Tr(ρE) where ρ = ρ i, E = E j, i j Tr(ρ) =1 Tr(E) = d [Also Pegg, Barnett, Jeffers, J. Mod. Opt. 49, 913 (2002).]

40 New states and effects States (equivalent preparation events):, where. (ρ, ρ) 0 ρ ρ, Tr(ρ) =1 (E, E) 0 E E, Tr(E) = d Effects (equivalent measurement events):, where. Joint probabilities: (E, E) A (ρ, ρ) p[(ρ, ρ),(e, E)] = Tr(ρE) Tr(ρE), Tr(ρE) 0 = 0, Tr(ρE) = 0 States can be thought of as functions on effects and vice versa.

41 New states and effects States (equivalent preparation events):, where. (ρ, ρ) 0 ρ ρ, Tr(ρ) =1 (E, E) 0 E E, Tr(E) = d Effects (equivalent measurement events):, where. Joint probabilities: (E, E) A (ρ, ρ) The set of states (effects) is not closed under convex combinations! p[(ρ, ρ),(e, E)] = Tr(ρE) Tr(ρE), Tr(ρE) 0 = 0, Tr(ρE) = 0 States can be thought of as functions on effects and vice versa.

42 General opera<ons General operations: collections of CP maps {M j }, s.t. Tr( M j ( I )) =1. d A B {M j } A Transformations: (M, M ), where 0 M M, Tr(M( I )) =1. d A j

43 Time reversal symmetry Example: B A {E k } {M j } {ρ i } p(i, j, k) = Tr(E B k M A B j (ρ A i )) Tr(E B M A B (ρ A ))

44 Time reversal symmetry Example: B A {σ k } {N j } {F i } p(i, j, k) = Tr(F i A N j B A (σ k B )) Tr(F A N B A (σ B ))

45 Time reversal symmetry The exact form of time-reversal is determined by physics! {ρ i } {F i } (play the movie backwards)

46 Time reversal symmetry The exact form of time-reversal is determined by physics! {ρ i } {F i } (play the movie backwards) {F i } The time-reversed image of is determined relative to preparations have not been time-reversed. {ρ i } {σ m } that {σ m }

47 Generalized Wigner s theorem O.O. and N. Cerf, Nature Phys. 11, 853 (2015) The time-symmetric theory admits more general symmetry transformations. Time reversal can have this form (Here is an invertible operator and denotes transposition in some basis.)

48 nderstanding the observed asymmetry A toy model of the universe: O.O. and N. Cerf, Nature Phys. 11, 853 (2015) t 1 For an observer at, all future circuits contain standard operations iff. (Implies that we remember the past and not the future!)

49 Note: it is logically possible that non- standard opera<ons were obtainable without post- selec<on

50 A <me- neutral formalism An isomorphism dependent on time reversal TRANSFORMATIONS EFFECTS ON PAIRS OF SYSTEMS O.O. and N. Cerf, arxiv:

51 A <me- neutral formalism An isomorphism dependent on time reversal TRANSFORMATIONS EFFECTS ON PAIRS OF SYSTEMS Joint probabilities: entangled state Φ Φ A 2A 1 (encodes time reversal) measurement i, j,k,l p(i, j, k,l {M i A 2 B 2 },{N j C 2 },,W ) = Tr[W A 1A 2 B 1 B 2 C 1 C 2 D 1 D 2 (M i A 2 B 2 N j C 2 P k B 1 C 1 D 2 Q l A 1 D 1 )] Tr[W A 1A 2 B 1 B 2 C 1 C 2 D 1 D 2 (M i A 2 B 2 N j C 2 P k B 1 C 1 D 2 Q l A 1 D 1 )] process matrix (encodes the connections) = Φ Φ A 1A 2 Φ Φ B 1B 2 Φ Φ C 1C 2 Φ Φ D 1D 2 O.O. and N. Cerf, arxiv:

52 A <me- neutral formalism Can describe circuits with cycles: {L k } H 1 D 2 E 2 B 2 A {N 1 j } C 1 All such circuits can be realized using post-selection. {M i } A 1 (Compatible with closed timelike curves) O.O. and N. Cerf, arxiv:

53 A <me- neutral formalism There exist circuits with cycles that can be obtained without post-selection! the idea of background independence extended to random events (provides a framework to understand experiments realizing the quantum switch)

54 Time- symmetric process matrix formalism Equivalently: external variables W p(i, j, {M i A 1 A 2 },{M j B 1 B 2 },,W ) = Tr[W A 1A 2 B 1 B 2 (M i A 1 A 2 M j B 1 B 2 )] Tr[W A 1A 2 B 1 B 2 (M A 1A 2 M B 1B 2 )] The process matrix : W A 1A 2 B 1 B 2 0, Tr(W A 1A 2 B 1 B 2 ) =1 Note: Any process matrix is allowed. O.O. and N. Cerf, arxiv:

55 Dropping the assump<on of local <me Observation: The predictions are the same whether the systems are of type 1 or type 2. Proposal: There is no a priori distinction between systems of type 1 and 2. The concept of time should come out from properties of the dynamics! O.O. and N. Cerf, arxiv:

56 Dropping the assump<on of local <me Observation: The predictions are the same whether the systems are of type 1 or type 2. Proposal: There is no a priori distinction between systems of type 1 and 2. The concept of time should come out from properties of the dynamics! The general picture: Main probability rule p(i, j, {M i },{N j }, ) = Tr[W wires(m i N j )] (M N )] Tr[W wires O.O. and N. Cerf, arxiv:

57 Limit of quantum field theory R. Oeckl, Phys. Lett. B 575, 318 (2003),..., Found. Phys. 43, 1206 (2013) (the general boundary approach with a few generalization) Time O.O. and N. Cerf, arxiv: Space

58 Proposal: causal structure from correla<ons The causal structure underlying the dynamics in the region is reflected in correlation properties of the state on the boundary. Time O.O. and N. Cerf, arxiv: Space

59 Conclusion It is possible to formulate a QT without any predefined time, which - agrees with experiment - has a physical and informational interpretation - opens up the possibility to understand time and causal structure as dynamical, and explore new forms of dynamics Is the metric/causal structure emergent, or do we need to postulate it as another field? What processes/networks can be realized without post-selection (e.g., can we violate causal inequalities?) How to formulate general covariant laws of dynamics in this framework? What does it imply for the foundations of information processing?

60 Thank you!

61 Formal theory of causality for processes O. O. and C. Giarmatzi, arxiv: A notion of causality should: have a universal expression (implies the multipartite case) allow of dynamical causal order (a given event can influence the order of other events in its future) capture our intuition of causality

62 Formal theory of causality for processes O. O. and C. Giarmatzi, arxiv: General process: Intuition: The choice of setting of a given party can only influence the occurrence of events in the future and the order of such events.

63 Formal theory of causality for processes O. O. and C. Giarmatzi, arxiv: General process: A process is causal iff there exists a random partial order and and a probability distribution such that for every party, e.g., A, and every subset X, Y, of the other parties,

64 Formal theory of causality for processes O. O. and C. Giarmatzi, arxiv: General process: A process is causal iff there exists a random partial order and and a probability distribution such that for every party, e.g., A, and every subset X, Y, of the other parties, (background-independent understanding of causal order)

65 Formal theory of causality for processes O. O. and C. Giarmatzi, arxiv: Consider If no signaling from to exists reduced process conditional process

66 Formal theory of causality for processes O. O. and C. Giarmatzi, arxiv: Theorem (canonical causal decomposition): where (iterative formulation) Describes causal unraveling of the events in the process!

67 Formal theory of causality for processes O. O. and C. Giarmatzi, arxiv: Theorem (canonical causal decomposition): where (iterative formulation) Causal correlations form polytopes! See also Branciard et al., arxiv:

68 Causal separability O. O. and C. Giarmatzi, arxiv: A quantum process is called causally separable iff it can be written in a canonical causal form with every reduced and conditional process being a valid quantum process. (analogy with Bell local and separable quantum states) à Agrees with the bipartite definition

69 Causal separability O. O. and C. Giarmatzi, arxiv: A quantum process is called causally separable iff it can be written in a canonical causal form with every reduced and conditional process being a valid quantum process. (analogy with Bell local and separable quantum states) à Agrees with the bipartite definition In the multipartite case, causality and causal separability are not equivalent! [see also Araujo, Branciard, Costa, Feix, Giarmatzi, Brukner, NJP 17, (2015)]

70 Causal separability O. O. and C. Giarmatzi, arxiv: A quantum process is called causally separable iff it can be written in a canonical causal form with every reduced and conditional process being a valid quantum process. (analogy with Bell local and separable quantum states) à Agrees with the bipartite definition In the multipartite case, causality and causal separability are not equivalent! [see also Araujo, Branciard, Costa, Feix, Giarmatzi, Brukner, NJP 17, (2015)] Is there a simple description of multipartite causally separable processes?

71 Extensive causality and causal separability O. O. and C. Giarmatzi, arxiv: Non-causality can be activated by shared entanglement! à Define extensively causal / extensively causally separable processes as those that remain causal / causally separable under extension with arbitrary input ancilla. There is a simple characterization of multipartite extensively causally separable processes! (see paper)

72 What we know about the classes of quantum processes O. O. and C. Giarmatzi, arxiv:

73 What we know about the classes of quantum processes O. O. and C. Giarmatzi, arxiv: