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Optical Fiber Communication Conference

Quantum State Transformations, Majorization and Distinguishability

Anthony Chefles

Published Jun 13, 2001• 0 citations
OpenAlex

Abstract

We give a unified treatment of quantum operations which transform one set of pure states into a another pure set. Extremal cases are deterministic transformations, which have unit success probability, and unambiguous state discrimination, which reveals the state and has the lowest success probability for a fixed initial ensemble. We obtain necessary and, for linearly independent initial states, sufficient conditions for the realisability of a given transformation with a fixed probability of success for each state. We then examine the relationship between deterministic transformations, majorization and distinguishability.

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We give a unified treatment of quantum operations which transform one set of pure states into a another pure set. Extremal cases are deterministic transformations, which have unit success probability, and unambiguous state discrimination, which reveals the state and has the lowest success probability for a fixed initial ensemble. We obtain necessary and, for linearly independent initial states, sufficient conditions for the realisability of a given transformation with a fixed probability of success for each state. We then examine the relationship between deterministic transformations, majorization and distinguishability.
In quantum operations, probabilities characterize both the degree of the success of a state transformation and, as density operator eigenvalues, the degree of mixedness of the final state. We give a unified treatment of $\mathrm{pure}\ensuremath{\rightarrow}\mathrm{pure}$ state transformations, covering both probabilistic and deterministic cases. We then discuss the role of majorization in describing the dynamics of mixing in quantum operations. The conditions for mixing enhancement for all initial states are derived. We show that mixing is monotonically decreasing for deterministic $\mathrm{pure}\ensuremath{\rightarrow}\mathrm{pure}$ transformations, and discuss the relationship between these transformations and deterministic local operations with classical communication entanglement transformations.
We derive a necessary condition for the existence of a completely-positive, linear, trace-preserving map which deterministically transforms one finite set of pure quantum states into another. This condition is also sufficient for linearly-independent initial states. We also examine the issue of quantum coherence, that is, when such operations maintain the purity of superpositions. If, in any deterministic transformation from one linearly-independent set to another, even a single, complete superposition of the initial states maintains its purity, the initial and final states are related by a unitary transformation.
It is shown that different distinguishability measures impose different orderings on ensembles of N pure quantum states. This is demonstrated using ensembles of equally probable, linearly independent, symmetrical pure states, with the maximum probabilities of correct hypothesis testing and unambiguous state discrimination being the distinguishability measures. This finding implies that there is no absolute scale for comparing the distinguishability of any two ensembles of N quantum states, and that distinguishability comparison is necessarily relative to a particular discrimination strategy.
Abstract It is well known that nonorthogonal quantum states cannot be reliably distinguished; however, for a number of sets of quantum states, the operation of unambiguous discrimination is possible, which either provides full information or yields an inconclusive result. In this paper, a generalization of such a transformation is constructed that has an increased success probability and makes the states more distinguishable. It is shown that after this transformation the states can be reliably distinguished without loss of the total success probability.
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.