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arXiv:0909.1973 · Physical Review A

Geometry of generalized depolarizing channels

Christian K. Burrell

Published Oct 27, 2009• 9 citations
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Abstract

A generalized depolarizing channel acts on an $N$-dimensional quantum system to compress the ``Bloch ball'' in ${N}^{2}\ensuremath{-}1$ directions; it has a corresponding compression vector. We investigate the geometry of these compression vectors and prove a conjecture of Dixit and Sudarshan [Phys. Rev. A 78, 032308 (2008)], namely, that when $N={2}^{d}$ (i.e., the system consists of $d$ qubits), and we work in the Pauli basis then the set of all compression vectors forms a simplex. We extend this result by investigating the geometry in other bases; in particular we find precisely when the set of all compression vectors forms a simplex.

DOI
10.1103/physreva.80.042330

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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.
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