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arXiv:quant-ph/0605172 · Journal of Physics A Mathematical and Theoretical

Trace distance from the viewpoint of quantum operation techniques

A E Rastegin

Published Jul 19, 2007• 16 citations
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Abstract

In the present paper, the trace distance is exposed within the quantum operations formalism. The definition of the trace distance in terms of a maximum over all quantum operations is given. It is shown that for any pair of different states there are an uncountably infinite number of maximizing quantum operations. Conversely, for any operation of the described type, there are an uncountably infinite number of those pairs of states that the maximum is reached by the operation. The behaviour of the trace distance under considered operations is studied. Relations and distinctions between the trace distance and the sine distance are discussed.

DOI
10.1088/1751-8113/40/31/026

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In the present paper, the trace distance is exposed within the quantum operations formalism. The definition of the trace distance in terms of a maximum over all quantum operations is given. It is shown that for any pair of different states there are an uncountably infinite number of maximizing quantum operations. Conversely, for any operation of the described type, there are an uncountably infinite number of those pairs of states that the maximum is reached by the operation. The behaviour of the trace distance under considered operations is studied. Relations and distinctions between the trace distance and the sine distance are discussed.
The distinguishability between two quantum states can be defined in terms of their trace distance. The operational meaning of this definition involves a maximization over measurement projectors. Here we introduce an alternative definition of distinguishability which, instead of projectors, is based on maximization over normalized states (density matrices). It is shown that this procedure leads to a distance (between two states) that in contrast to the usual approach based on a 1-norm, is based on an infinite-norm. Properties such as convexity, monotonicity, and invariance under unitary transformations are fulfilled. Equivalent operational implementations based on maximization over classical probabilities and hypothesis-testing scenarios are also established. When considering the action of completely positive transformations, contractivity is only granted for unital maps. This feature allows us to introduce a measure of the quantumness of nonunital maps that can be written in terms of the proposed distinguishability measure and corresponds to the maximal possible deviation from contractivity. Particular examples sustain the main results and conclusions.
Introducing contravariant trace-densities for quantum states, we restore one to one correspondence between quantum operations described by normal CP maps and their trace densities as Hermitian positive operator-valued contravariant kernels. The CB-norm distance between two quantum operations with type one input algebras is explicitly expressed in terms of these densities, and this formula is also extended to a generalized CB-distances between quantum operations with type two inputs. A larger C-distance is given as the natural norm-distance for the channel densities, and another, Helinger type distance, related to minimax mean square optimization problem for purification of quantum channels, is also introduced and evaluated in terms of their contravariant trace-densities. It is proved that the Helinger type complete fidelity distance between two channels is equivalent to the CB distance at least for type one inputs, and this equivalence is also extended to type two for the generalized CB distance. An operational meaning for these distances and relative complete fidelity for quantum channels is given in terms of quantum encodings as generalized entanglements of quantum states opposite to the inputs and the output states.
In recent years, several measures have been proposed for characterizing the coherence of a given quantum state. We derive several results that illuminate how these measures behave when restricted to pure states. Notably, we present an explicit characterization of the closest incoherent state to a given pure state under the trace distance measure of coherence. We then use this result to show that the states maximizing the trace distance of coherence are exactly the maximally coherent states. We define the trace distance of entanglement and show that it coincides with the trace distance of coherence for pure states. Finally, we give an alternate proof to a recent result that the ${\ensuremath{\ell}}_{1}$ measure of coherence of a pure state is never smaller than its relative entropy of coherence.
We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well defined for channels between finite-dimensional algebras, but it also applies to a certain class of channels between infinite-dimensional algebras (explicitly, those channels that possess an operator-valued Radon-Nikodym density with respect to the trace in the sense of Belavkin-Staszewski) and induces a metric on the set of quantum channels that is topologically equivalent to the CB-norm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fidelity (“generalized transition probability”) of Uhlmann, is topologically equivalent to the trace-norm distance.