The Multi-round Process Matrix

Timothée Hoffreumon and Ognyan Oreshkov

Centre for Quantum Information and Communication (QuIC), École polytechnique de Bruxelles, CP 165, Université libre de Bruxelles, 1050 Brussels, Belgium.

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We develop an extension of the process matrix (PM) framework for correlations between quantum operations with no causal order that allows multiple rounds of information exchange for each party compatibly with the assumption of well-defined causal order of events locally. We characterise the higher-order process describing such correlations, which we name the multi-round process matrix (MPM), and formulate a notion of causal nonseparability for it that extends the one for standard PMs. We show that in the multi-round case there are novel manifestations of causal nonseparability that are not captured by a naive application of the standard PM formalism: we exhibit an instance of an operator that is both a valid PM and a valid MPM, but is causally separable in the first case and can violate causal inequalities in the second case due to the possibility of using a side channel.

Quantum communication protocols involve several parties that send and receive quantum systems on which they perform operations. These protocols usually assume a well-defined causal ordering of the operations of the parties. The process matrix (PM) framework relaxes the assumption that separate operations must always occur in a definite order. In this framework, there can be situations in which the correlations established between separate parties cannot be understood as arising from a probabilistic mixture of different orderings of their operations. For example, in a communication between two parties, where the parties apply one operation each, there can be more exotic scenarios than Alice acting either before or after Bob. Such scenarios in which the operations of the parties do not have definite causal order are called causally nonseparable processes. Processes of this kind have been conjectured to be relevant at the intersection of quantum mechanics and gravity, but some are also known to admit experimental realizations with known physics in terms of operations delocalized in time.

In its original formulation, the PM framework assumes that each party is restricted to a single round of information exchange, where the party receives a quantum system in, applies an operation on it, and sends a quantum system out. Practical communication protocols, however, generally involve multiple rounds of information exchange between the parties, who can use local memory and condition the operations they apply at a given time on information obtained in the past.

In this article, we extend the PM framework by relaxing the assumption of each party having only one round of information exchange with the others. In the multi-round process matrix (MPM) framework, the parties can receive and send multiple times while storing information in a local memory under the assumption of well-defined causal order of the events in each laboratory. We characterize the set of MPMs via handy mathematical conditions expressed in terms of superoperator projectors, and show that any MPM is an affine (but not necessarily convex) combination of MPMs describing fixed causal order between the operations of the parties. We also formulate a notion of causal nonseparability for the MPM building upon the one for the PM. Even though every MPM turns out to be formally equivalent to a PM on more parties, we find that there are multi-round manifestations of causal nonseparability that are not captured by a naïve application of the standard PM formalism. Specifically, we provide an example of an operator that is causally separable if considered as a PM but causally nonseparable if considered as an MPM. This demonstrates how the ability of the parties to use local memory between their successive operations, which underlies the MPM framework, can activate nontrivial effects.

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