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IEEE Transactions on Information Theory

Time Reversal and Exchange Symmetries of Unitary Gate Capacities

Aram W. Harrow, Peter W. Shor

Published Jan 1, 2010• 12 citations
Publisher

Abstract

Unitary gates are interesting resources for quantum communication in part because they are always invertible and are intrinsically bidirectional. This paper explores these two symmetries: time-reversal and exchange of Alice and Bob. We present examples of unitary gates that exhibit dramatic separations between forward and backward capacities (even when the back communication is assisted by free entanglement) and between entanglement-assisted and unassisted capacities, among many others. Along the way, we will give a general time-reversal rule for relating the capacities of a unitary gate and its inverse that will explain why previous attempts at finding asymmetric capacities failed. Finally, we will see how the ability to erase quantum information and destroy entanglement can be a valuable resource for quantum communication.

DOI
10.1109/tit.2009.2034899

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Unitary gates are interesting resources for quantum communication in part because they are always invertible and are intrinsically bidirectional. This paper explores these two symmetries: time-reversal and exchange of Alice and Bob. We present examples of unitary gates that exhibit dramatic separations between forward and backward capacities (even when the back communication is assisted by free entanglement) and between entanglement-assisted and unassisted capacities, among many others. Along the way, we will give a general time-reversal rule for relating the capacities of a unitary gate and its inverse that will explain why previous attempts at finding asymmetric capacities failed. Finally, we will see how the ability to erase quantum information and destroy entanglement can be a valuable resource for quantum communication.
Unitary gates are an interesting resource for quantum communication in part because they are always invertible and are intrinsically bidirectional. This paper explores these two symmetries: time-reversal and exchange of Alice and Bob. We will present examples of unitary gates that exhibit dramatic separations between forward and backward capacities (even when the back communication is assisted by free entanglement) and between entanglement-assisted and unassisted capacities, among many others. Along the way, we will give a general time-reversal rule for relating the capacities of a unitary gate and its inverse that will explain why previous attempts at finding asymmetric capacities failed. Finally, we will see how the ability to erase quantum information and destroy entanglement can be a valuable resource for quantum communication.
A unitary interaction coupling two parties enables quantum or classical communication in both the forward and backward directions. Each communication capacity can be thought of as a tradeoff between the achievable rates of specific types of forward and backward communication. Our first result shows that for any bipartite unitary gate, bidirectional coherent classical communication is no more difficult than bidirectional classical communication --- they have the same achievable rate regions. Previously this result was known only for the unidirectional capacities (i.e., the boundaries of the tradeoff). We then relate the tradeoff for two-way coherent communication to the tradeoff for two-way quantum communication and the tradeoff for coherent communication in one direction and quantum communication in the other.
We consider interactions as bidirectional channels. We investigate the capacities for interaction Hamiltonians and nonlocal unitary gates to generate entanglement and transmit classical information. We give analytic expressions for the entanglement generating capacity and entanglement-assisted one-way classical communication capacity of interactions, and show that these quantities are additive, so that the asymptotic capacities equal the corresponding 1-shot capacities. We give general bounds on other capacities, discuss some examples, and conclude with some open questions.
Two-qubit operations may be characterized by their capacities for communication, both with and without free entanglement, and their capacity for creating entanglement. We establish a set of inequalities that give an ordering to the capacities of two-qubit unitary operations. Specifically, we show that the capacities for entanglement creation and bidirectional communication without entanglement assistance are at least as great as half the bidirectional communication capacity with entanglement assistance. In addition, we show that the bidirectional communication that can be performed using an ensemble may be increased via a two-qubit unitary operation by twice the operation's capacity for entanglement.
We consider interactions as bidirectional channels, and investigate the capacities for interaction Hamiltonians and nonlocal unitary gates to generate entanglement and transmit classical information. We give analytic expressions for the entanglement generating capacity and entanglement-assisted 1-way classical communication capacity of interactions, and show that these quantities are additive, so that the asymptotic capacities equal the corresponding 1-shot capacities. We give bounds on other capacities and discuss their additivity, along with other open questions.
Time-reversal symmetry is a fundamental principle of physics describing the invariance of physical laws under reversal of the direction of time. We formulate a Bell-inequality-like test of this antiunitary symmetry using only forward access and trusted quantum operations: entanglement converts temporal input--output relations into measurable spatial exchange symmetry. For nnn-qubit unitary dynamics, we prove that reliably distinguishing the time-reversal-symmetric circular ensembles from Haar-random dynamics requires Ω(min⁡{2n/2,2n−e})Ω(\min\{2^{n/2},2^{n-e}\})Ω(min{2n/2,2n−e}) queries for any classically adaptive protocol. Here, e=min⁡{es,em}e=\min\{e_{\mathrm s},e_{\mathrm m}\}e=min{es​,em​} with ese_{\mathrm s}es​ and eme_{\mathrm m}em​ representing the probe and measurement logarithmic entanglement negativities, respectively. Maximally entangled probes and SWAP measurements reduce this cost to a constant number of queries. Furthermore, we develop a time-reversal symmetry test for arbitrary fixed, compatible probes and measurements, relate its query complexity to their logarithmic negativities, and match the lower-bound scaling in the high-entanglement regime by optimizing the probe and measurement. Our results establish a quantitative connection between entanglement and time-reversal symmetry, bridging two central concepts in quantum information science and fundamental physics.