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Journal of Mathematics

Local Channels under Decomposition of Quantum Gates in Bipartite Systems

Yihui Lao, Pei Wang, Shanli Liao

Published Mar 14, 2023• 0 citations
Publisher

Abstract

For a bipartite quantum system with Hilbert space H m ⊗ H n , this paper considers a quantum channel ϕ from the system to itself which preserves the set of separable states. Within this class of quantum channels, the paper characterizes those channels which preserve the Tsallis entropy of combinations of separable states through decomposing quantum gates, where the characterization is given in terms of the action of the channel on separable states.

DOI
10.1155/2023/2130374

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For a bipartite quantum system with Hilbert space H m ⊗ H n , this paper considers a quantum channel ϕ from the system to itself which preserves the set of separable states. Within this class of quantum channels, the paper characterizes those channels which preserve the Tsallis entropy of combinations of separable states through decomposing quantum gates, where the characterization is given in terms of the action of the channel on separable states.
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We extend the definition of the conditional min-entropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an operational interpretation as a guessing probability when one of the subsystems is classical. We then show that the extended conditional min-entropy can be used to fully characterize when two bipartite quantum channels are related to each other via a superchannel (also known as supermap or a comb) that is acting on one of the subsystems. This relation is a pre-order that extends the definition of “quantum majorization” from bipartite states to bipartite channels, and can also be characterized with semidefinite programming. As a special case, our characterization provides necessary and sufficient conditions for when a set of quantum channels is related to another set of channels via a single superchannel. We discuss the applications of our results to channel discrimination, and to resource theories of quantum processes. Along the way we study channel divergences, entropy functions of quantum channels, and noise models of superchannels, including random unitary superchannels and doubly-stochastic superchannels. For the latter we give a physical meaning as being completely-uniformity preserving.
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