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arXiv:0807.2668 · Quantum Information and Computation

Mixing doubly stochastic quantum channels with the completely depolarizing channel

John Watrous

Published May 1, 2009• 18 citations
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Abstract

It is proved that every doubly stochastic quantum channel that is properly averaged with the completely depolarizing channel can be written as a convex combination of unitary channels. It follows that within the space of doubly stochastic quantum channels, there is a ball with positive radius around the completely depolarizing channel within which all channels are convex combinations of unitary channels.

DOI
10.26421/qic9.5-6-4

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It is proved that every doubly stochastic quantum channel that is properly averaged with the completely depolarizing channel can be written as a convex combination of unitary channels. It follows that within the space of doubly stochastic quantum channels, there is a ball with positive radius around the completely depolarizing channel within which all channels are convex combinations of unitary channels.
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.
The set of doubly-stochastic quantum channels and its subset of mixtures of unitaries are investigated. We provide a detailed analysis of their structure together with computable criteria for the separation of the two sets. When applied to O ( d )-covariant channels this leads to a complete characterization and reveals a remarkable feature: instances of channels which are not in the convex hull of unitaries can become elements of this set by either taking two copies of them or supplementing with a completely depolarizing channel. These scenarios imply that a channel whose noise initially resists any environment-assisted attempt of correction can become perfectly correctable.
We investigate the classical capacity of two quantum channels with memory: a periodic channel with depolarizing channel branches and a convex combination of depolarizing channels. We prove that the capacity is additive in both cases. As a result, the channel capacity is achieved without the use of entangled input states. In the case of a convex combination of depolarizing channels the proof provided can be extended to other quantum channels whose classical capacity has been proven to be additive in the memoryless case.
A canonical form for unital qubit channels under local unitary transforms is obtained. In particular, it is shown that the eigenvalues of the Choi matrix of a unital quantum channel form a complete set of invariants of the canonical form. It follows immediately that every unital qubit channel is the average of four unitary channels. More generally, a unital qubit channel can be expressed as the convex combination of unitary channels with convex coefficients p1,…,pmp_1, \dots, p_mp1​,…,pm​ as long as 2(p1,…,pm)2(p_1, \dots, p_m)2(p1​,…,pm​) is majorized by the vector of eigenvalues of the Choi matrix of the channel. A unital qubit channel in the canonical form will transform the Bloch sphere onto an ellipsoid. We look into the detailed structure of the natural linear maps sending the Bloch sphere onto a corresponding ellipsoid.
A canonical form for unital qubit channels under local unitary transforms is obtained. In particular, it is shown that the eigenvalues of the Choi matrix of a unital quantum channel form a complete set of invariants of the canonical form. It follows immediately that every unital qubit channel is the average of four unitary channels. More generally, a unital qubit channel can be expressed as the convex combination of unitary channels with convex coefficients p1,…,pmp_1, \dots, p_mp1​,…,pm​ as long as 2(p1,…,pm)2(p_1, \dots, p_m)2(p1​,…,pm​) is majorized by the vector of eigenvalues of the Choi matrix of the channel. A unital qubit channel in the canonical form will transform the Bloch sphere onto an ellipsoid. We look into the detailed structure of the natural linear maps sending the Bloch sphere onto a corresponding ellipsoid.
Depolarizing maps acting on an NNN-dimensional system are completely positive maps resulting into compression of the Bloch ``ball'' along ${N}^{2}\ensuremath{-}1$ polarization directions. In the qubit case these maps are a convex sum of four extremal maps and form a simplex in the space of compression coefficients along the three polarization directions. We calculate the compression domain for three- and four-level systems. For a three-level system the region has curved surfaces, but it is a simplex for a four-level system. We conjecture that it is a simplex in the case of 2n{2}^{n}2n-level systems.