Quantum mechanics with indefinite causal order Flaminia Giacomini, Esteban Castro- Ruiz, Časlav rukner University of Vienna Ins@tute for Quantum Op@cs and Quantum Informa@on, Vienna Vienna Theory Lunch Seminar 5th pril 2016
Contents (1) Introduc8on: Opera@onal meaning of causality What is indefinite causality and why do we care? (2) The W- matrix framework: ssump@ons Characterisa@on: allowed and forbidden terms Infinite dimensions: problems and solu@ons (3) The quantum switch: a causally nonseparable system (4) Summary and Conclusions
Causally- ordered framework Future Opera8onally Causality as possibility of signaling C Elsewhere O Signaling as sta@s@cal correla@ons between two variables of two par@es Past
Causally- ordered framework Elsewhere Future O Past C Opera8onally Causality as possibility of signaling One direc@onal signaling: from O to from to O > @melike separated No signaling from O to C > spacelike separated N..: always verified in typical laboratory situa@ons
Indefiniteness of causality No global space@me structure Quantum Informa@on tools to formulate physical theories on indefinite causal structures g µ? C We want to preserve: O Linearity No logical paradoxes
Superposi8on of backgrounds Lower is slower =2 =2
Superposi8on of backgrounds =2 =2
Superposi8on of backgrounds? M j M i M. Zych, F. Costa, I. Pikovski and Č. rukner, in prepara(on
The causal inequality a t b y 1 2 X ab X y We want to maximise p(b, y a, b)+ X p(x, a a, b) x! p(a, b) a In a causally ordered scenario x 1 8 X (p (b a, b)+p (a a, b)) apple 3 4 ab Can we violate the causal inequality? Yes re these processes physical? (Open ques@on) O. Oreshkov, F. Costa,and Č. rukner, Nat.Comm. 3, (2012)
Fixed background assump8ons (1) Closed laboratories: and can receive and send a signal, but their laboratories are isolated from the rest of the world between these two events; (2) Free choice: the secngs of the local observers are only correlated with events in their causal future; (3) Causal structure: the main events are located in a causal structure, i.e. a par@al order which defined the possible direc@ons of the signaling.
Fixed background assump8ons (1) Closed laboratories: and can receive and send a signal, but their laboratories are isolated from the rest of the world between these two events; (2) Free choice: the secngs of the local observers are only correlated with events in their causal future; (3) Causal structure: the main events are located in a causal structure, i.e. a par@al order which defined the possible direc@ons of the signaling. It is possible to violate the causal inequality!
The W- matrix formalism M j Definite causal order on a local level (laboratories)? No assump@ons on the global causal background structure M i Local opera@ons: CP maps. Sum over outcomes is also trace- preserving (CPTP) X Tr 2 Mi = 1 1 i O. Oreshkov, F. Costa,and Č. rukner, Nat.Comm. 3, (2012)
The W- matrix formalism M j Definite causal order on a local level (laboratories)? No assump@ons on the global causal background structure M i Local opera@ons: CP maps. Sum over outcomes is also trace- preserving (CPTP) X Tr 2 Mi = 1 1 i O. Oreshkov, F. Costa,and Č. rukner, Nat.Comm. 3, (2012) The process matrix W is defined as the most general operator which sa@sfies p(mi,mj )=Tr W (Mi Mj ) X ij hi = Tr( ) p(m i,m j )=1
Possible correla8ons ll causally ordered situa@ons are included in the formalism: shared states, channels, and their probabilis@c mixtures. M i M j + M j M i + M j M i Signaling from to Signaling from to Shared state between and The set of possible W strictly contains the set of correla@ons allowed by QM. QM W
Example of forbidden terms Terms which cause probability to be not normalised Local loops Logical paradoxes (i.e. signaling to the past) Global loops M j M i M i
Infinite dimensional systems Paper at arxiv:1510.06345
Infinite dimensional systems Causal order of events Global @me and metric proper@es of the space@me Infinite dimensional Hilbert space External degrees of freedom (posi@on and momentum) Wave func@on propagators as channels Space@me proper@es g µ
The problem of singulari8es In finite dimensions the characterisa@on of the W heavily relies on the dimension of the Hilbert spaces. W = In infinite dimensions d!1 1 d 1 d 1 1 + + + // Signaling from to It is important that Signaling from to No signaling 0 apple Tr W M i M j apple 1 W is trace class in the input Hilbert spaces The local opera@ons are trace class in the output Hilbert spaces Probabili@es are well defined
The problem of singulari8es Wigner func@ons Equivalent expression for the W- matrix with no singulari@es W ( 1,..., 2 )= Z dz 1...dz 2 e i ~ p i 1 z 1...e ~ p 2 z 2 z 1 Dx 1 2... x z 2 2 2 W x 1 z 1 2... x z E 2 2 2 i =(x i,p i ) point in the phase space The basis of complex exponen@als allows the explicit characterisa@on of the W- matrix Fourier transform of the Wigner func@on conjugate to in the phase space e i ~ i i i i
Normalisa@on of probability: 1= 1 (2 ) 4 Z Characterisa8on of W d 1 d 2 d 1 d 2 W ( 1, 2, 1, 2 ) M ( 1, 2 ) M ( 1, 2 ) CPTP condi@on on local opera@ons: M ( 1, 0) =2 ( 1 ) M ( 1, 0) =2 ( 1 ) Consistency of W when tracing over one observer: Tr (WM )=W for any allowed M The characterisa@on is the same as in finite dimensions. The formalism can be fully generalised to infinite- dimensional Hilbert spaces.
Example: a quantum system with indefinite causal structure It can not be wriien as a convex mixture of ordered processes: W 6= W +(1 )W Experimental implementa@on in finite dimensions: L. M. Procopio et al, Nat. Comms. 6, 7913 (2015)
Example: a quantum system with indefinite causal structure It can not be wriien as a convex mixture of ordered processes: W 6= W +(1 )W Superposi8on of causal orders Experimental implementa@on in finite dimensions: L. M. Procopio et al, Nat. Comms. 6, 7913 (2015)
t O t The quantum switch C M C Coherent superposi@on of the order of quantum opera@ons. t 2 M i M j Green func@on of the free par@cle G(x x,t 2 t 1 ) t 1 M i M j t I x x x Chiribella Giulio, et al., Phys. Rev. 88.2 (2013): 022318
t O t The quantum switch C M C Coherent superposi@on of the order of quantum opera@ons. t 2 M i M j Green func@on of the free par@cle G(x x,t 2 t 1 ) W- matrix is pure t 1 t I M i x M j x x W = wihw Û: free- par@cle propagator + = Z dx xxi + + + + + + wi Û 2 C 1 Û2 2 1 ÛI1 2 I 0c i+û 2 C 1 Û2 2 1 ÛI1 2 I 1c i Chiribella Giulio, et al., Phys. Rev. 88.2 (2013): 022318 Control qubit
The quantum switch Measurement + reprepara@on in the local laboratories Z Mi = dx x ihx O i R i Interference term due to superposi@on of causal orders. G(x 0 x, t 0 t) M j M i M C C Signaling in both direc@ons Pure W- matrix The process is causally nonseparable t t
Open ques8ons Conclusions (1) The W- matrix framework can describe more general situa@ons than those allowed by standard Quantum Mechanics and avoids logical paradoxes (2) The set of allowed W contains: processes which violate the causal inequality: physical? causally nonseparable processes which can be experimentally implemented (the quantum switch) causally separable and causally ordered processes (3) The framework can be fully generalised to infinite- dimensional Hilbert spaces. (1) Which W- matrices describe processes in which the indefiniteness of causal structure is due to gravita@onal interac@on? (2) Can we formulate Quantum Fields on indefinite causal structure? (Long term goal)