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arXiv:quant-ph/0506072 · Journal of Optics B Quantum and Semiclassical Optics

Minimum error discrimination of Pauli channels

M F Sacchi

Published Sep 21, 2005• 22 citations
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Abstract

We solve the problem of discriminating with minimum error probability two given Pauli channels. We show that, unlike in the case of discrimination between unitary transformations, the use of entanglement with an ancillary system can strictly improve the discrimination, and any maximally entangled state allows one to achieve the optimal discrimination. We also provide a simple necessary and sufficient condition in terms of the structure of the channels for which the ultimate minimum error probability can be achieved without entanglement assistance. When such a condition is satisfied, the optimal input state is simply an eigenstate of one of the Pauli matrices.

DOI
10.1088/1464-4266/7/10/018

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We solve the problem of discriminating with minimum error probability two given Pauli channels. We show that, unlike in the case of discrimination between unitary transformations, the use of entanglement with an ancillary system can strictly improve the discrimination, and any maximally entangled state allows one to achieve the optimal discrimination. We also provide a simple necessary and sufficient condition in terms of the structure of the channels for which the ultimate minimum error probability can be achieved without entanglement assistance. When such a condition is satisfied, the optimal input state is simply an eigenstate of one of the Pauli matrices.
We solve the problem of discriminating with minimum error probability two given Pauli channels. We show that, differently from the case of discrimination between unitary transformations, the use of entanglement with an ancillary system can strictly improve the discrimination, and any maximally entangled state allows to achieve the optimal discrimination. We also provide a simple necessary and sufficient condition in terms of the structure of the channels for which the ultimate minimum error probability can be achieved without entanglement assistance. When such a condition is satisfied, the optimal input state is simply an eigenstate of one of the Pauli matrices.
Entanglement is known to be a powerful resource that improves performance in various quantum information and computational tasks. A standard example of such a phenomenon is the possibility of perfectly discriminating all four Pauli operations in a single shot via the superdense coding protocol. While entanglement is often a powerful resource for quantum channel discrimination, this is not necessarily the case. In this work, we identify scenarios in which the maximally entangled state is a bad choice of input state and, more generally, show that excessive entanglement can reduce channel discriminability dramatically. To do so, we present an explicit pair of unitary channels which are perfectly discriminable without entanglement, but for which any strategy with maximally entangled input states is εεε-close to a blind uniform guessing strategy. To develop a systematic approach, we introduce the concepts of Maximal Entanglement Worst Case (MEWC) and Maximal Entanglement Best Case (MEBC) pairs of channels, and present conditions for a pair of channels to be MEWC or MEBC. With these conditions, we show that the optimal input states for discriminating MEWC pairs of channels are necessarily separable, and provide non-trivial examples of measurement channels for which entanglement necessarily reduces the maximum probability of discrimination.
I study the problem of optimally discriminating between two open quantum dynamical processes in a single-shot scenario, with the aim of minimizing the error probability of identification. This task requires jointly optimising the input state—potentially entangled with an ancillary system isolated from the dynamics—and the time at which the resulting time-dependent quantum channels, generated by the two dynamical maps, are most distinguishable. As an illustrative case, I focus on Pauli dynamical maps and their associated families of time-dependent Pauli channels. I identify a regime in which separable strategies require waiting indefinitely until the dynamics reaches the stationary state, whereas entangled input states enable optimal discrimination at a finite time, with a strict reduction in error probability. These results highlight the crucial interplay between entanglement and timing in enhancing the distinguishability of quantum dynamical processes.
In this paper, we address the problem of discriminating two given quantum operations. Firstly, based on the Bloch representation of single qubit systems, we give the exact minimum error probability of discriminating two single qubit quantum operations by unentangled input states. In particular, for the Pauli channels discussed in [Phys. Rev. A {\bf 71}, 062340 (2005)], we use a more intuitional and visual method to deal with their discrimination problem. Secondly, we consider the condition for perfect discrimination of two quantum operations. Specially, we get that two generalized Pauli channels are perfectly distinguishable if and only if their characteristic vectors are orthogonal.