# 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

### Abstract

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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[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.

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[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).

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[2] Giulio Chiribella and Hlér Kristjánsson, "Quantum Shannon theory with superpositions of trajectories", Proceedings of the Royal Society of London Series A 475 2225, 20180903 (2019).

[3] Philippe Allard Guérin and Časlav Brukner, "Observer-dependent locality of quantum events", New Journal of Physics 20 10, 103031 (2018).

[4] Daniel Ebler, Sina Salek, and Giulio Chiribella, "Enhanced Communication with the Assistance of Indefinite Causal Order", Physical Review Letters 120 12, 120502 (2018).

[5] Julian Wechs, Alastair A. Abbott, and Cyril Branciard, "On the definition and characterisation of multipartite causal (non)separability", New Journal of Physics 21 1, 013027 (2019).

[6] Sina Salek, Daniel Ebler, and Giulio Chiribella, "Quantum communication in a superposition of causal orders", arXiv:1809.06655.

[7] Lucien Hardy, "Implementation of the Quantum Equivalence Principle", arXiv:1903.01289.

[8] Philippe Allard Guérin, Marius Krumm, Costantino Budroni, and Časlav Brukner, "Composition rules for quantum processes: a no-go theorem", New Journal of Physics 21 1, 012001 (2019).

[9] Giulio Chiribella, Manik Banik, Some Sankar Bhattacharya, Tamal Guha, Mir Alimuddin, Arup Roy, Sutapa Saha, Sristy Agrawal, and Guruprasad Kar, "Indefinite causal order enables perfect quantum communication with zero capacity channel", arXiv:1810.10457.

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[11] Lucien Hardy, "The Construction Interpretation: Conceptual Roads to Quantum Gravity", arXiv:1807.10980.

[12] Alastair A. Abbott, Julian Wechs, Dominic Horsman, Mehdi Mhalla, and Cyril Branciard, "Communication through coherent control of quantum channels", arXiv:1810.09826.

[13] Yu Guo, Xiao-Min Hu, Zhi-Bo Hou, Huan Cao, Jin-Ming Cui, Bi-Heng Liu, Yun-Feng Huang, Chuan-Feng Li, and Guang-Can Guo, "Experimental investigating communication in a superposition of causal orders", arXiv:1811.07526.

[14] C. T. Marco Ho, Fabio Costa, Christina Giarmatzi, and Timothy C. Ralph, "Violation of a causal inequality in a spacetime with definite causal order", arXiv:1804.05498.

[15] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, "Quantum Causal Models", arXiv:1906.10726.

[16] Philippe Allard Guérin, Giulia Rubino, and Časlav Brukner, "Communication through quantum-controlled noise", Physical Review A 99 6, 062317 (2019).

[17] Nicolas Loizeau and Alexei Grinbaum, "Channel capacity enhancement with indefinite causal order", arXiv:1906.08505.

[18] Esteban Castro-Ruiz, Flaminia Giacomini, Alessio Belenchia, and Časlav Brukner, "Time reference frames and gravitating quantum clocks", arXiv:1908.10165.

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