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arXiv:1201.1172

Bounds on the distance between a unital quantum channel and the convex hull of unitary channels, with applications to the asymptotic quantum Birkhoff conjecture

Nengkun Yu, Runyao Duan, Quanhua Xu

Published Jan 5, 2012• 3 citations
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Abstract

Motivated by the recent resolution of Asymptotic Quantum Birkhoff Conjecture (AQBC), we attempt to estimate the distance between a given unital quantum channel and the convex hull of unitary channels. We provide two lower bounds on this distance by employing techniques from quantum information and operator algebras, respectively. We then show how to apply these results to construct some explicit counterexamples to AQBC. We also point out an interesting connection between the Grothendieck's inequality and AQBC.

DOI
10.48550/arxiv.1201.1172

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Optimal convex approximation of quantum channels based on ααα-affinity
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The diamond distance between two quantum channels determines the optimal probability of distinguishing them from a single use. We give a quantum algorithm that estimates the diamond distance between two unknown ddd-dimensional unitary channels to within additive error ε\varepsilonε, using O(d/ε)O(\sqrt{d}/\varepsilon)O(d​/ε) queries to the unitary channels and their inverses. The query complexity is optimal, as it matches the lower bound due to Jeon and Oh (npj Quantum Inf. 2026).
Birkhoff's Theorem states that doubly stochastic matrices are convex combinations of permutation matrices. Quantum mechanically these matrices are doubly stochastic channels, i.e. they are completely positive maps preserving both the trace and the identity. We expect these channels to be convex combinations of unitary channels and yet it is known that some channels cannot be written that way. Recent work has suggested that nnn copies of a single channel might approximate a mixture (convex combination) of unitaries. In this paper we show that n(n+1)/2n(n+1)/2n(n+1)/2 copies of a symmetric unital quantum channel may be arbitrarily-well approximated by a mixture (convex combination) of unitarily implemented channels. In addition, we prove that any extremal properties of a channel are preserved over nnn (and thus n(n+1)/2n(n+1)/2n(n+1)/2) copies. The result has the potential to be completely generalized to include non-symmetric channels.