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arXiv:quant-ph/0505183 · Physical Review A

Optimal discrimination of quantum operations

Massimiliano Federico Sacchi

Published Jun 30, 2005• 185 citations
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Abstract

We address the problem of discriminating with minimal error probability two given quantum operations. We show that the use of entangled input states generally improves the discrimination. For Pauli channels we provide a complete comparison of the optimal strategies where either entangled or unentangled input states are used.
DOI
10.1103/physreva.71.062340

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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.
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.
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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 address the problem of unambiguous discrimination between a given set of quantum operations. The necessary and sufficient condition for them to be unambiguously distinguishable is derived in the cases of single use and multiple uses respectively. For the latter case we explicitly construct the input states and corresponding measurements that accomplish the task. It is found that the introduction of entanglement can improve the discrimination. The advantage of our strategy over minimum-error discrimination is also discussed.