Quantum superpositions of causal orders as an operational resource

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Quantum superpositions of causal orders as an operational resource

Márcio M. Taddei, Ranieri V. Nery, and Leandro Aolita

Phys. Rev. Research 1, 033174 – Published 13 December 2019

Abstract

Causal nonseparability refers to processes where events take place in a coherent superposition of different causal orders. These may be the key resource for experimental violations of causal inequalities and have been recently identified as resources for concrete information-theoretic tasks. Here, we take a step forward by deriving a complete operational framework for causal nonseparability as a resource. Our first contribution is a formal definition for the specific notion of quantum control of causal orders, a stronger form of causal nonseparability—with the celebrated quantum switch as best-known example—where the causal orders of events for a target system are coherently controlled by a control system. We then build a resource theory—for both generic causal nonseparability as well as quantum control of causal orders—with a physically motivated class of free operations, based on process-matrix concatenations. We present the framework explicitly in the mindset with a control register. However, our machinery is totally versatile, being directly applicable also to scenarios with a target register alone. Moreover, an important subclass of our operations is free not only with respect to causal nonseparability and quantum control of causal orders but it also preserves the very causal structure of causal processes. Hence, our treatment contains, as a built-in feature, the basis of a resource theory of quantum causal networks too. As applications, first, we establish a simple sufficient condition for pure-process free convertibility. This imposes a hierarchy of quantum control of causal orders with the quantum switch at the top. Second, we prove that causal-nonseparability distillation exists. More precisely, we show how to convert multiple copies of a process with arbitrarily little causal nonseparability into fewer copies of a quantum switch. Our findings reveal conceptually new, unexpected phenomena, with both fundamental and practical implications.

Received 25 March 2019
Revised 19 August 2019

DOI:https://doi.org/10.1103/PhysRevResearch.1.033174

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum coherence & coherence measures Quantum correlations in quantum information Quantum entanglement Resource theories

Quantum Information, Science & TechnologyGeneral Physics

Authors & Affiliations

Márcio M. Taddei ^1,*, Ranieri V. Nery^1,2, and Leandro Aolita¹

¹Instituto de Física, Federal University of Rio de Janeiro, 21941-972, P. O. Box 68528, Rio de Janeiro, Brazil
²International Institute of Physics, Federal University of Rio Grande do Norte, 59078-970, P. O. Box 1613, Natal, Brazil

^*marciotaddei@gmail.com

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Vol. 1, Iss. 3 — December - December 2019

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Figure 1
Schematics of a process. (a) Two users, Alice and Bob, in laboratories $A$ and $B$ , respectively, receive a target qudit as local input and subsequently send out an equivalent system as output after some operation. We label Alice's input (output) $A_{I}$ ( $A_{O}$ ), and Bob's $B_{I}$ ( $B_{O}$ ). The events inside each laboratory occur in a definite order: $A_{I}$ ( $B_{I}$ ) happens before $A_{O}$ ( $B_{O}$ ). However, no causal order between $A$ and $B$ is assumed. For practical purposes, but without loss of generality, we do nevertheless assume a past and a future to both $A$ and $B$ . These are realized by laboratories $P$ and $F$ , whose only function is to input into and output from the process the initial and final states of the target qudit, respectively. Finally, there is yet another laboratory, $C$ , operated by a third user, Charlie, who gets as local input a control qubit. This can control the causal order between $A$ and $B$ . The process, denoted by $W$ , then specifies the entire system history—preparations, evolutions, measurements, etc.—outside the laboratories (light gray). The operations inside each laboratory, in turn, are described by instruments (dark-gray). (b) Pictorial representation of processes with definite causal orders $A \to B$ (solid) and $B \to A$ (dashed). Coherent superpositions of the latter give causally nonseparable processes. If, in addition, such superpositions involve entanglement with the control qubit, the process can feature quantum control of causal orders (see Fig. 2 for precise definition).
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Figure 2
Pictorial representation of the internal geometry of the set $P$ of all processes. $CS$ and $S$ are the subsets of causally separable and separable processes, respectively. The processes outside $CS$ are causally nonseparable, whereas those outside $S$ are entangled between control and target. The convex hull of $CS$ and $S$ gives the set $NQC$ (gray) with no quantum control of causal orders. This includes processes (given by convex combinations of elements in $CS$ and $S$ ) that are causally nonseparable and entangled. We define the processes outside $NQC$ to have quantum control of causal orders.
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Figure 3
Schematics of the operational framework of causal nonseparability and quantum control of causal orders. (a) An initial process $W$ (dark gray) is concatenated—i.e., link-product multiplied—with the average process $V$ (light gray) of an instrument at Alice, Bob and Charlie's laboratories, whereas the past and future laboratories are untouched. The final process is thus $V (W) = V * W$ (dashed outline). This has the same configuration of laboratories as the initial process: $P, A^{'}, B^{'}, C^{'}$ , and $F$ , where $A^{'}, B^{'}$ , and $C^{'}$ have the same structure of input and output systems as the initial process' laboratories $A, B$ , and $C$ . The instrument's inputs are $A_{I}, A_{O}^{'}, B_{I}, B_{O}^{'}$ , and $C$ ; and its outputs are $A_{I}^{'}, A_{O}, B_{I}^{'}, B_{O}$ , and $C^{'}$ . All the correlations in the control-versus-target bipartition that $V$ produces are due to 1-way classical communication (double line) of the local instrument outcomes from the control laboratory of Charlie to the target laboratories of Alice and Bob. That is, $V$ is separable in the bipartition and, therefore, creates no entanglement between control and target systems. Furthermore, for each outcome $j$ at $C$ , the outcome process $V_{A B}^{(j)}$ on the target is designed to contain no causal nonseparability either. For this reason, whenever the initial process is causally separable, so is the final one. All this, together with linearity of the link product, implies also that whenever the initial process is not quantum-control causally ordered, neither is the final one. [(b) and (c)] Structure of the instruments on the target register. Each process $V_{A B}^{(j)}$ has six laboratories. Four of them correspond to the inputs $A_{I}$ and $B_{I}$ and outputs $A_{O}$ and $B_{O}$ of the initial process. While the other two correspond to Alice and Bob's final laboratories, $A^{'}$ and $B^{'}$ , both equipped with input and output systems ( $A_{I}^{'}$ and $B_{I}^{'}$ and $A_{O}^{'}$ and $B_{O}^{'}$ , respectively). We consider two elementary types of instruments. (b) The first one gives rise to the class of local operations and ancillary entanglement (LOAE). There, each $V_{A B}^{(j)}$ describes a local (in the $A | B$ bipartition) unitary evolution on the instrument inputs $A_{I}, A_{O}^{'}, B_{I}$ , and $B_{O}^{'}$ together with (arbitrary-dimensional) ancillary registers $\tilde{A}$ and $\tilde{B}$ in a (possibly entangled) state $| {Ψ ぷさい}^{(j)} 〉_{\tilde{A} \tilde{B}}$ , followed by the disposal of the ancilas. (c) In the second one, called probabilistic lab swaps (PLS), each $V_{A B}^{(j)}$ is a pure process where the same unitary operator (either the swap $S$ or the identity $1$ gate) is applied to $A_{I}$ together with $B_{I}$ and to $A_{I}^{'}$ together with $B_{I}^{'}$ . That is, conditioned on Charlie's outcome, Alice and Bob either exchange their systems (through swap gates on their inputs and outputs) or leave them untouched. This probabilistically exchanges the causal orders $A \to B$ and $B \to A$ , but it never introduces causal nonseparability or quantum control of causal orders.
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