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arXiv:0804.3100 · Physical Review A

Geometry of depolarizing channels

Kuldeep Dixit, E. C. G. Sudarshan

Published Sep 4, 2008• 9 citations
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Abstract

Depolarizing maps acting on an $N$-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 ${2}^{n}$-level systems.

DOI
10.1103/physreva.78.032308

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Depolarizing maps acting on an NNN-dimensional system are completely positive maps resulting into compression of the Bloch ``ball'' along N2\ensuremath−1{N}^{2}\ensuremath{-}1N2\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.
A generalized depolarizing channel acts on an NNN-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=2dN={2}^{d}N=2d (i.e., the system consists of ddd 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.
Quantum channels can be mathematically represented as completely positive trace-preserving maps that act on a density matrix. A general quantum channel can be written as a convex sum of `extremal' channels. We show that for an NNN-level system, the extremal channel can be characterized in terms of N2N^2N2-NNN real parameters coupled with rotations. We give a representation for NNN= 2, 3, 4.
We introduce a 3-parameter class of maps (1) acting on a bipartite system which are a natural generalisation of the depolarizing channel (and include it as a special case). Then, we find the exact regions of the parameter space that alternatively determine a positive, completely positive, entanglement-breaking, or entanglement-annihilating map. This model displays a much richer behaviour than the one shown by a simple depolarizing channel, yet it stays exactly solvable. As an example of this richness, positive partial transposition but not entanglement-breaking maps is found in Theorem 2. A simple example of a positive yet indecomposable map is provided (see the Remark at the end of Section IV). The study of the entanglement-annihilating property is fully addressed by Theorem 7. Finally, we apply our results to solve the problem of the entanglement annihilation caused in a bipartite system by a tensor product of local depolarizing channels. In this context, a conjecture posed in the work of Filippov [J. Russ. Laser Res. 35, 484 (2014)] is affirmatively answered, and the gaps that the imperfect bounds of Filippov and Ziman [Phys. Rev. A 88, 032316 (2013)] left open are closed. To arrive at this result, we furthermore show how the Hadamard product between quantum states can be implemented via local operations.
The physically allowed quantum evolutions on a single qubit can be described in terms of their geometry. From a simple parameterisation of unital single-qubit channels, the canonical form of all such channels can be given. The related geometry can be used to understand how to approximate positive maps by completely-positive maps, such as in the case of optimal eavesdropping strategies. These quantum channels can be generated by the appropriate network or through dynamical means. The Strømer-Woronowisc result can also be understood in terms of this geometry.