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arXiv:quant-ph/0106035

The Geometry of Single-Qubit Maps

Daniel K. L. Oi

Published Jun 6, 2001• 7 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0106035

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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.
Using the known possibility to associate the completely positive maps with density matrices and recent results on expressing the density matrices with sets of classical probability distributions of dichotomic random variables we construct the probability representation of the completely positive maps. In this representation, any completely positive map of qubit state density matrix is identified with the set of classical coin probability distributions. Examples of the maps of qubit states are studied in detail. The evolution equation of quantum states is written in the form of the classical-like kinetic equation for probability distributions identified with qubit state.
In their seminal 1961 paper, Sudarshan, Mathews and Rau investigated properties of the dynamical A and B maps acting on n-dimensional quantum systems. The nature of dynamical maps in open quantum system evolutions has attracted great deal of attention in the later years. However, the novel paper on the A and B dynamical maps has not received its due attention. In this tutorial article, we review the properties of A and B forms associated with the dynamics of finite dimensional quantum systems. In particular, we investigate a canonical structure associated with the A form and establish its equivalence with the associated B form. We show that the canonical structure of the A form captures the completely positive (not completely positive) nature of the dynamics in a succinct manner. This feature is illustrated through physical examples of qubit channels. Quanta 2021; 10: 34–41.
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.
Linear maps of matrices describing the evolution of density matrices for a quantum system initially entangled with another are identified and found to be not always completely positive. They can even map a positive matrix to a matrix that is not positive, unless we restrict the domain on which the map acts. Nevertheless, their form is similar to that of completely positive maps. Only some minus signs are inserted in the operator-sum representation. Each map is the difference of two completely positive maps. The maps are first obtained as maps of mean values and then as maps of basis matrices. These forms also prove to be useful. An example for two entangled qubits is worked out in detail. The relation to earlier work is discussed.