Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics

Ognyan Oreshkov

QuIC, Ecole polytechnique de Bruxelles, C.P. 165, Université libre de Bruxelles, 1050 Brussels, Belgium

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It has been shown that it is theoretically possible for there to exist higher-order quantum processes in which the operations performed by separate parties cannot be ascribed a definite causal order. Some of these processes are believed to have a physical realization in standard quantum mechanics via coherent control of the times of the operations. A prominent example is the quantum SWITCH, which was recently demonstrated experimentally. However, the interpretation of such experiments as realizations of a process with indefinite causal structure as opposed to some form of simulation of such a process has remained controversial. Where exactly are the local operations of the parties in such an experiment? On what spaces do they act given that their times are indefinite? Can we probe them directly rather than assume what they ought to be based on heuristic considerations? How can we reconcile the claim that these operations really take place, each once as required, with the fact that the structure of the presumed process implies that they cannot be part of any acyclic circuit? Here, I offer a precise answer to these questions: the input and output systems of the operations in such a process are generally nontrivial subsystems of Hilbert spaces that are tensor products of Hilbert spaces associated with systems at different times---a fact that is directly experimentally verifiable. With respect to these time-delocalized subsystems, the structure of the process is one of a circuit with a causal cycle. This provides a rigorous sense in which processes with indefinite causal structure can be said to exist within the known quantum mechanics. I also identify a whole class of isometric processes, of which the quantum SWITCH is a special case, that admit a physical realization on time-delocalized subsystems. These results unveil a novel structure within quantum mechanics, which may have important implications for physics and information processing.

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[1] L. Hardy, Probability Theories with Dynamic Causal Structure: A New Framework for Quantum Gravity, (2005).

[2] L. Hardy, Quantum Gravity Computers: On the Theory of Computation with Indefinite Causal Structure, in Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle, The Western Ontario Series in Philosophy of Science, vol. 73 (Springer, Dordrecht, 2009); DOI: https:/​/​doi.org/​10.1007/​978-1-4020-9107-0_21; (2007).

[3] G. Chiribella, G. M. D'Ariano, P. Perinotti, and B. Valiron, Quantum computations without definite causal structure, Phys. Rev. A 88, 022318 (2013); DOI: https:/​/​doi.org/​10.1103/​PhysRevA.88.022318; (2009).

[4] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Transforming quantum operations: quantum supermaps, Europhys. Lett. 83, 30004 (2008); DOI: https:/​/​doi.org/​10.1209/​0295-5075/​83/​30004; (2008).

[5] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Theoretical framework for quantum networks, Phys. Rev. A 80, 022339 (2009); DOI: https:/​/​doi.org/​10.1103/​PhysRevA.80.022339; (2009).

[6] The idea of quantum computation beyond causal circuits was notably first considered by Deutsch Deutsch who studied modifications of quantum theory in the vicinity of closed timelike curves, which led to the study of the computational power of such models (see Ref. Aaronson). Although linked to time travel in a different sense Chiribella12, the quantum SWITCH is motivated by the idea of `quantum superpositions of different causal structures' as opposed to classically definite backgrounds with timelike cycles.

[7] D. Deutsch, Quantum mechanics near closed timelike lines, Phys. Rev. D 44, 3197 (1991); DOI: https:/​/​doi.org/​10.1103/​PhysRevD.44.3197.

[8] S. Aaaronson and J. Watrous, Closed timelike curves make quantum and classical computing equivalent, Proc. R. Soc. A 465, 631-647 (2009); DOI: https:/​/​doi.org/​10.1098/​rspa.2008.0350; (2008).

[9] O. Oreshkov, F. Costa, and Č. Brukner, Quantum correlations with no causal order, Nat. Commun. 3, 1092 (2012); DOI: https:/​/​doi.org/​10.1038/​ncomms2076; (2011).

[10] O. Oreshkov and C. Giarmatzi, Causal and causally separable processes, New J. Phys. 18, 093020 (2016); DOI: https:/​/​doi.org/​10.1088/​1367-2630/​18/​9/​093020; (2015).

[11] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, Witnessing causal nonseparability, New J. Phys. 17, 102001 (2015); DOI: https:/​/​doi.org/​10.1088/​1367-2630/​17/​10/​102001; (2015).

[12] J. Wechs, A. A. Abbott, and C. Branciard, On the definition and characterisation of multipartite causal (non)separability, New J. Phys. 21, 013027 (2019); DOI: https:/​/​doi.org/​10.1088/​1367-2630/​aaf352; (2018).

[13] Ä. Baumeler and S. Wolf, Perfect signaling among three parties violating predefined causal order, Proceedings of International Symposium on Information Theory (ISIT) 2014, 526-530, 2014; DOI: https:/​/​doi.org/​10.1109/​ISIT.2014.6874888; (2013).

[14] Ä. Baumeler, A. Feix, and S. Wolf, Maximal incompatibility of locally classical behavior and global causal order in multi-party scenarios, Phys. Rev. A 90, 042106 (2014); DOI: https:/​/​doi.org/​10.1103/​PhysRevA.90.042106; (2014).

[15] Ä. Baumeler and S. Wolf, The space of logically consistent classical processes without causal order, New J. Phys. 18, 013036 (2016); DOI: https:/​/​doi.org/​10.1088/​1367-2630/​18/​1/​013036; (2015).

[16] C. Branciard, M. Araújo, A. Feix, F. Costa, and Č. Brukner, The simplest causal inequalities and their violation, New J. Phys. 18, 013008 (2016); DOI: https:/​/​doi.org/​10.1088/​1367-2630/​18/​1/​013008; (2015).

[17] S. S. Bhattacharya and M. Banik, Biased Non-Causal Game, (2015).

[18] A. Feix, M. Araújo, and Č. Brukner, Causally nonseparable processes admitting a causal model, New J. Phys. 18, 083040 (2016); DOI: https:/​/​doi.org/​10.1088/​1367-2630/​18/​8/​083040; (2016).

[19] A. A. Abbott, C. Giarmatzi, F. Costa, and C. Branciard, Multipartite Causal Correlations: Polytopes and Inequalities, Phys. Rev. A 94, 032131 (2016); DOI: https:/​/​doi.org/​10.1103/​PhysRevA.94.032131; (2016).

[20] N. Miklin, A. A. Abbott, C. Branciard, R. Chaves, C. Budroni, The entropic approach to causal correlations, New J. Phys. 19, 113041 (2017); DOI: https:/​/​doi.org/​10.1088/​1367-2630/​aa8f9f; (2017).

[21] J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 3, 195-200 (1964); DOI: https:/​/​doi.org/​10.1103/​PhysicsPhysiqueFizika.1.195.

[22] O. Oreshkov and N. J. Cerf, Operational quantum theory without predefined time, New J. Phys. 18, 073037 (2016); DOI: https:/​/​doi.org/​10.1088/​1367-2630/​18/​7/​073037; (2014).

[23] R. Silva, Y. Guryanova, A. J. Short, P. Skrzypczyk, N. Brunner, and S. Popescu, Connecting processes with indefinite causal order and multi-time quantum states, New J. Phys. 19 , 103022 (2017); DOI: https:/​/​doi.org/​10.1088/​1367-2630/​aa84fe; (2017).

[24] M. Araújo, P. A. Guérin, and Ä. Baumeler, Quantum computation with indefinite causal structures, Phys. Rev. A 96, 052315 (2017); DOI: https:/​/​doi.org/​10.1103/​PhysRevA.96.052315; (2017).

[25] S. Milz, F. A. Pollock, T. P. Le, G. Chiribella, and K. Modi, Entanglement, non-Markovianity, and causal non-separability, New J. Phys. 20, 033033 (2018); DOI: https:/​/​doi.org/​10.1088/​1367-2630/​aaafee; (2017).

[26] G. Chiribella, Perfect discrimination of no-signalling channels via quantum superposition of causal structures, Phys. Rev. A 86, 040301 (2012); DOI: https:/​/​doi.org/​10.1103/​PhysRevA.86.040301; (2011).

[27] C. Branciard, Witnesses of causal nonseparability: an introduction and a few case studies, Sci. Rep. 6, 26018 (2016); DOI: https:/​/​doi.org/​10.1038/​srep26018; (2016).

[28] T. Colnaghi, G. M. D'Ariano, P. Perinotti, and S. Facchini, Quantum computation with programmable connections between gates, Phys. Lett. A 376, 2940 - 2943 (2012); DOI: https:/​/​doi.org/​10.1016/​j.physleta.2012.08.028; (2011).

[29] M. Araújo, F. Costa, and Č. Brukner, Computational advantage from quantum-controlled ordering of gates, Phys. Rev. Lett. 113, 250402 (2014); DOI: https:/​/​doi.org/​10.1103/​PhysRevLett.113.250402; (2014).

[30] A. Feix, M. Araújo, and Č. Brukner, Quantum superposition of the order of parties as a communication resource, Phys. Rev. A 92, 052326 (2015); DOI: https:/​/​doi.org/​10.1103/​PhysRevA.92.052326; (2016).

[31] P. A. Guérin, A. Feix, M. Araújo, and Č. Brukner, Exponential Communication Complexity Advantage from Quantum Superposition of the Direction of Communication, Phys. Rev. Lett. 117, 100502 (2016); DOI: https:/​/​doi.org/​10.1103/​PhysRevLett.117.100502; (2016).

[32] N. Friis, V. Dunjko, W. Dür, and H. J. Briegel, Implementing quantum control for unknown subroutines, Phys. Rev. A 89, 030303(R) (2014); DOI: https:/​/​doi.org/​10.1103/​PhysRevA.89.030303; (2014).

[33] L. M. Procopio et al., Experimental Superposition of Orders of Quantum Gates, Nat. Commun. 6, 7913 (2015); DOI: https:/​/​doi.org/​10.1038/​ncomms8913; (2014).

[34] N. Friis, A. A. Melnikov, G. Kirchmair, and H. J. Briegel, Coherent controlization using superconducting qubits, Sci. Rep. 5, 18036 (2015); DOI: https:/​/​doi.org/​10.1038/​srep18036; (2015).

[35] G. Rubino, L. A. Rozema, A. Feix, M. Araújo, J. M. Zeuner, L. M. Procopio, Č. Brukner, and P. Walther, Experimental Verification of an Indefinite Causal Order, Sci. Adv. 3, e1602589 (2017); DOI: https:/​/​doi.org/​10.1126/​sciadv.1602589; (2016).

[36] G. Rubino, L. A. Rozema, F. Massa, M. Araújo, M. Zych, Č. Brukner, and P. Walther, Experimental Entanglement of Temporal Orders, (2017).

[37] K. Goswami, C. Giarmatzi, M. Kewming, F. Costa, C. Branciard, J. Romero, and A. G. White, Indefinite Causal Order in a Quantum Switch, Phys. Rev. Lett. 121, 090503 (2018); DOI: https:/​/​doi.org/​10.1103/​PhysRevLett.121.090503; (2018).

[38] L. Viola, E. Knill, and R. Laflamme, Constructing Qubits in Physical Systems, J. Phys. A 34, 7067 (2001); DOI https:/​/​doi.org/​10.1088/​0305-4470/​34/​35/​331; (2001).

[39] E. Knill, Protected realizations of quantum information, Phys. Rev. A 74, 042301 (2006); DOI: https:/​/​doi.org/​10.1103/​PhysRevA.74.042301; (2006).

[40] D. W. Kribs and R. W. Spekkens, Quantum Error Correcting Subsystems are Unitarily Recoverable Subsystems, Phys. Rev. A 74, 042329 (2006); DOI: https:/​/​doi.org/​10.1103/​PhysRevA.74.042329; (2006).

[41] P. Zanardi, Virtual Quantum Subsystems, Phys. Rev. Lett. 87, 077901 (2001); DOI: https:/​/​doi.org/​10.1103/​PhysRevLett.87.077901; (2001).

[42] P. Zanardi, D. Lidar, and S. Lloyd, Quantum Tensor Product Structures are Observable Induced, Phys. Rev. Lett. 92, 060402 (2004); DOI: https:/​/​doi.org/​10.1103/​PhysRevLett.92.060402; (2003).

[43] M. Araújo, Adrien Feix, Miguel Navascués, and Časlav Brukner, A purification postulate for quantum mechanics with indefinite causal order, Quantum 1, 10 (2017); DOI: https:/​/​doi.org/​10.22331/​q-2017-04-26-10; (2016).

[44] A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Math. Phys. 3, 4, 275-278 (1972); DOI https:/​/​doi.org/​10.1016/​0034-4877(72)90011-0.

[45] M.-D. Choi, Completely positive linear maps on complex matrices, Lin. Alg. Appl. 10, 285-290 (1975); DOI: https:/​/​doi.org/​10.1016/​0024-3795(75)90075-0.

[46] O. Oreshkov and N. J. Cerf, Operational formulation of time reversal in quantum theory, Nature Phys. 11, 853-858 (2015); DOI: https:/​/​doi.org/​10.1038/​nphys3414; (2015).

[47] P. Perinotti, Causal Structures and the Classification of Higher Order Quantum Computations, in Time in physics, R. Renner and S. Stupar (eds), Tutorials, Schools, and Workshops in the Mathematical Sciences, (Birkhäuser, Cham, 2017); DOI: https:/​/​doi.org/​10.1007/​978-3-319-68655-4_7;.

[48] A. Kissinger and S. Uijlen, A categorical semantics for causal structure, Logical Methods in Computer Science, Volume 15, Issue 3 (August 9, 2019), lmcs:5681; DOI: https:/​/​doi.org/​10.23638/​LMCS-15(3:15)2019; (2017).

[49] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, (Cambridge University Press, Cambridge, 2000); DOI https:/​/​doi.org/​10.1017/​CBO9780511976667.

[50] E. Castro-Ruiz, F. Giacomini, and Časlav Brukner, Dynamics of Quantum Causal Structures, Phys. Rev. X 8, 011047 (2018); DOI: https:/​/​doi.org/​10.1103/​PhysRevX.8.011047; (2017).

[51] Strictly speaking, in Ref. Chiribella12 it was shown that if there exists a realization of the quantum SWITCH such that Alice's operation is in the past of Bob's operation so that the output ancilla of Alice could be connected to the input ancilla of Bob, this would allow deterministic transmission of information back in time. In the realization discussed here, this condition is not satisfied—the ancillary systems of Alice and Bob cannot be connected to each other as they occupy space-like separated regions. Nevertheless, the full experiment still has the structure of a circuit with a `timelike' cycle, albeit not permitting deterministic time travel, as any quantum process matrix is equivalent to a channel from the output systems of all parties to their input systems OCB.

[52] More precisely, C-SWAP$^{XYZ} = |0\rangle\langle 0|^{X}\otimes \mathbb{I}^{YZ} + |1\rangle\langle 1|^{X}\otimes$SWAP$^{YZ}$, where SWAP$^{YZ}$ is the SWAP operator on $Y$ and $Z$, which can be defined as follows. Consider two systems $Y$ and $Z$ with Hilbert spaces of the same dimension and a linear isomorphism between the states in these Hilbert spaces. An arbitrary vector in the joint system $YZ$ can be written in the form $|\psi\rangle^{YZ} = \sum_{i,j} \psi_{ij}|i\rangle^Y |j\rangle^Z$, where $\{|i\rangle^Y\}$ are orthonormal bases for $Y$ and $Z$, respectively. The action of the operator SWAP$^{YZ}$ on the vector $|\psi\rangle^{YZ}$ is then given by SWAP$^{YZ} |\psi\rangle^{YZ} = \sum_{i,j} \psi_{ij}|j\rangle^Y |i\rangle^Z$.

[53] Of course, if during the working of the device, an adversary turns on unwanted interactions, such as a Hamiltonian on the control qubit that is not diagonal in the logical basis, this could prevent the device from implementing the correct operation on the systems of interest. But this is the case for any physical device implementing an operation, irrespectively of whether the operation is localized or delocalized in time.

[54] Very recently, after the submission of this paper, the author and colleagues J. Barrett and R. Lorenz showed via different methods that all bipartite processes that are unitarily extendible are causally separable, and hence their unitary extensions are variations of the quantum SWITCH (in preparation). Nevertheless, we believe that the proof of realizability presented here has a particular value since it is based on a different idea that could have wider applications. In particular, it provides the basis for the generalization in Sec. 7, and might be useful in the search for realizations of more complicated unitary processes.

[55] W. F. Stinespring, Positive functions on C*-algebras, Proc. Amer. Math. Soc. 6, 211 (1955); DOI: https:/​/​doi.org/​10.2307/​2032342.

[56] Throughout this paper, when we speak about isomorphic mapping between two Hilbert spaces, we understand isometric isomorphism.

[57] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014); DOI: https:/​/​doi.org/​10.1103/​RevModPhys.86.419; (2013).

[58] After this paper appeared, a subsequent paper AllardGuerin claimed to show that all unitary processes admit a representation on time-delocalized subsystems. However, this claim is based on a misunderstanding of the concept of time-delocalized subsystems. The proof claimed in AllardGuerin amounts to the observation (discussed in this paper) that if we have a unitary process, the unitary maps isomorphically the output system of any one party, say Alice, onto a subsystem of the input systems of the rest of the parties, and similarly maps a subsystem of the output systems of the rest of the parties onto the input system of Alice. This by itself does not imply that we can associate the input and output systems of Alice with time-delocalized subsystems (which are subsystems of tensor products of Hilbert spaces associated with concrete physical systems at concrete times).

[59] P. Allard Guérin and Č. Brukner, Observer-dependent locality of quantum events, New J. Phys. 20, 103031 (2018); DOI: https:/​/​doi.org/​10.1088/​1367-2630/​aae742; (2018).

[60] D. Ebler, S. Salek, and G. Chiribella, Enhanced Communication with the Assistance of Indefinite Causal Order, Phys. Rev. Lett. 120, 120502 (2018); DOI: https:/​/​doi.org/​10.1103/​PhysRevLett.120.120502; (2017).

[61] M. Zych, F. Costa, I. Pikovski, and Časlav Brukner, Bell's Theorem for Temporal Order, Nat. Commun. 10, 3772 (2019); DOI: https:/​/​doi.org/​10.1038/​s41467-019-11579-x; (2017).

[62] A. Dimić, M. Milivojević, D. Gočanin, and Časlav Brukner, Simulating spacetime with indefinite causal order via Rindler observers, (2017).

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