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arXiv:quant-ph/0602093 · Physical Review A

Optimal unambiguous discrimination of two subspaces as a case in mixed-state discrimination

János A. Bergou, Edgar Feldman, Mark S. Hillery

Published Mar 21, 2006• 51 citations
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Abstract

We show how to optimally unambiguously discriminate between two subspaces of a Hilbert space. In particular we suppose that we are given a quantum system in either the state $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$, where $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ can be any state in the subspace ${S}_{1}$, or $\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩$, where $\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩$ can be any state in the subspace ${S}_{2}$, and our task is to determine in which of the subspaces the state of our quantum system lies. We do not want to make any error, which means that our procedure will sometimes fail if the subspaces are not orthogonal. This is a special case of the unambiguous discrimination of mixed states. We present the positive operator valued measures that solve this problem and several applications of this procedure, including the discrimination of multipartite states without classical communication.

DOI
10.1103/physreva.73.032107

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We show how to optimally unambiguously discriminate between two subspaces of a Hilbert space. In particular we suppose that we are given a quantum system in either the state \ensuremath∣\ensuremathψ1⟩\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩\ensuremath∣\ensuremathψ1​⟩, where \ensuremath∣\ensuremathψ1⟩\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩\ensuremath∣\ensuremathψ1​⟩ can be any state in the subspace S1{S}_{1}S1​, or \ensuremath∣\ensuremathψ2⟩\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩\ensuremath∣\ensuremathψ2​⟩, where \ensuremath∣\ensuremathψ2⟩\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩\ensuremath∣\ensuremathψ2​⟩ can be any state in the subspace S2{S}_{2}S2​, and our task is to determine in which of the subspaces the state of our quantum system lies. We do not want to make any error, which means that our procedure will sometimes fail if the subspaces are not orthogonal. This is a special case of the unambiguous discrimination of mixed states. We present the positive operator valued measures that solve this problem and several applications of this procedure, including the discrimination of multipartite states without classical communication.
We study the measurement for the unambiguous discrimination of two mixed quantum states that are described by density operators ${\ensuremath{\rho}}_{1}$ and ${\ensuremath{\rho}}_{2}$ of rank ddd, the supports of which jointly span a 2d2d2d-dimensional Hilbert space. Based on two conditions for the optimum measurement operators and on a canonical representation for the density operators of the states, two equations are derived that allow the explicit construction of the optimum measurement, provided that the expression for the fidelity of the states has a specific simple form. For this case, the problem is mathematically equivalent to distinguishing pairs of pure states, even when the density operators are not diagonal in the canonical representation. The equations are applied to the optimum unambiguous discrimination of two mixed states that are similar states, given by ${\ensuremath{\rho}}_{2}=U{\ensuremath{\rho}}_{1}{U}^{\ifmmode\dagger\else\textdagger\fi{}}$, and that belong to the class where the unitary operator UUU can be decomposed into multiple rotations in the ddd mutually orthogonal two-dimensional subspaces determined by the canonical representation.
Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e., allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the NNN given nonorthogonal states ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ are divided into two sets and the first set consists of one state only while the second consists of all of the remaining states. We present the derivation of the optimal measurement strategy, in terms of a generalized measurement (positive-operator-valued measure), to distinguish $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [Phys. Rev. Lett. 90, 25901 (2003)] as the emphasis there was on appplication of the results to probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.
We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.
We present a generic study of the unambiguous discrimination between two mixed quantum states. We derive operational optimality conditions and show that the optimal measurements can be classified according to their rank. In Hilbert space dimensions less than or equal to 555, this leads to the complete optimal solution. We demonstrate our method with a physical example, namely, the unambiguous comparison of nnn quantum states, and find the optimal success probability.
We study the determination of an unknown mixed state of a ddd-dimensional quantum system by means of unambiguous state discrimination. We show that optimal and nonoptimal unambiguous state discrimination can be used to reconstruct unknown states of a qubit. This result is extended to the case of a qudit by a sequence of reconstructions in two-dimensional subspaces. The total number of projections scales approximately as 2d22{d}^{2}2d2 for ddd large; this is twice as much as in the case of tomography based on mutually unbiased bases or symmetric informationally complete positive-operator-valued measures, and less than the d3{d}^{3}d3 projections required by standard tomography.