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arXiv:quant-ph/0702022 · Physical Review A

Optimal unambiguous state discrimination of two density matrices: A second class of exact solutions

Philippe Raynal, Norbert Lütkenhaus

Published Nov 21, 2007• 22 citations
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Abstract

We consider the unambiguous state discrimination (USD) of two mixed quantum states. We study the rank and the spectrum of the elements of an optimal USD measurement. This naturally leads to a partial fourth reduction theorem. This theorem shows that either the failure probability equals its overall lower bound given in terms of the fidelity or a two-dimensional subspace can be split off from the original Hilbert space. We then use this partial reduction theorem to derive the optimal solution for any two equally probable geometrically uniform states ${\ensuremath{\rho}}_{0}$ and ${\ensuremath{\rho}}_{1}=U{\ensuremath{\rho}}_{0}{U}^{\ifmmode\dagger\else\textdagger\fi{}}$, ${U}^{2}=\mathbb{1}$, in a four-dimensional Hilbert space. This represents a second class of analytical solutions for USD problems that cannot be reduced to some pure state cases. We apply our result to answer two questions that are relevant in implementations of the Bennett and Brassard 1984 quantum key distribution protocol using weak coherent states.

DOI
10.1103/physreva.76.052322

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•Jan 1, 2008•0 citations
We consider the unambiguous state discrimination (USD) of two mixed quantum states. We study the rank and the spectrum of the elements of an optimal USD measurement. This naturally leads to a partial fourth reduction theorem. This theorem shows that either the failure probability equals its overall lower bound given in terms of the fidelity or a two-dimensional subspace can be split off from the original Hilbert space. We then use this partial reduction theorem to derive the optimal solution for any two equally probable geometrically uniform states \ensuremathρ0{\ensuremath{\rho}}_{0}\ensuremathρ0​ and \ensuremathρ1=U\ensuremathρ0U\ifmmode†\else\textdagger\fi{\ensuremath{\rho}}_{1}=U{\ensuremath{\rho}}_{0}{U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremathρ1​=U\ensuremathρ0​U\ifmmode†\else\textdagger\fi, U2=1{U}^{2}=\mathbb{1}U2=1, in a four-dimensional Hilbert space. This represents a second class of analytical solutions for USD problems that cannot be reduced to some pure state cases. We apply our result to answer two questions that are relevant in implementations of the Bennett and Brassard 1984 quantum key distribution protocol using weak coherent states.
In this thesis we study the problem of unambiguously discriminating two mixed quantum states. We first present reduction theorems for optimal unambiguous discrimination of two generic density matrices. We show that this problem can be reduced to that of two density matrices that have the same rank rrr in a 2rrr-dimensional Hilbert space. These reduction theorems also allow us to reduce USD problems to simpler ones for which the solution might be known. As an application, we consider the unambiguous comparison of nnn linearly independent pure states with a simple symmetry. Moreover, lower bounds on the optimal failure probability have been derived. For two mixed states they are given in terms of the fidelity. Here we give tighter bounds as well as necessary and sufficient conditions for two mixed states to reach these bounds. We also construct the corresponding optimal measurement. With this result, we provide analytical solutions for unambiguously discriminating a class of generic mixed states. This goes beyond known results which are all reducible to some pure state case. We however show that examples exist where the bounds cannot be reached. Next, we derive properties on the rank and the spectrum of an optimal USD measurement. This finally leads to a second class of exact solutions. Indeed we present the optimal failure probability as well as the optimal measurement for unambiguously discriminating any pair of geometrically uniform mixed states in four dimensions. This class of problems includes for example the discrimination of both the basis and the bit value mixed states in the BB84 QKD protocol with coherent states.
Recently the problem of unambiguous state discrimination of mixed quantum states has attracted much attention. So far, bounds on the optimum success probability have been derived [T. Rudolph, R. W. Spekkens, and P. S. Turner, Phys. Rev. A 68, 010301(R) (2003)]. For two mixed states they are given in terms of the fidelity. Here we give tighter bounds as well as necessary and sufficient conditions for two mixed states to reach these bounds. Moreover we construct the corresponding optimal measurement strategies. With this result, we provide analytical solutions for unambiguous discrimination of a class of generic mixed states. This goes beyond known results which are all reducible to some pure state case. Additionally, we show that examples exist where the bounds cannot be reached.
We present reduction theorems for the problem of optimal unambiguous state discrimination of two general density matrices. We show that this problem can be reduced to that of two density matrices that have the same rank n and are described in a Hilbert space of dimensions 2n.2n.2n. We also show how to use the reduction theorems to discriminate unambiguously between N mixed states (N> 2).(N>~2).(N> 2).
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 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.
The discrimination of two mixed quantum states is a fundamental task in quantum state estimation and quantum information theory. In quantum state discrimination a quantum system is assumed to be in one of two possible – in general mixed – non-orthogonal quantum states. The discrimination then consists of a measurement strategy that allows to decide in which state the system was before the measurement. In unambiguous state discrimination the aim is to make this decision without errors, but it is allowed to give an inconclusive answer. Especially interesting are measurement strategies that minimize the probability of an inconclusive answer. A starting point for the analysis of this optimization problem was a result by Eldar et al. [Phys. Rev. A 69, 062318 (2004)], which provides non-operational necessary and sufficient conditions for a given measurement strategy to be optimal. These conditions are reconsidered and simplified in such a way that they become operational. The simplified conditions are the basis for further central results: It is shown that the optimal measurement strategy is unique, a statement that is e.g. of importance for the complexity analysis of optimal measurement devices. The optimal measurement strategy is derived for the case, where one of the possible input states has at most rank two, which was an open problem for many years. Furthermore, using the optimality criterion it is shown that there always exists a threshold probability for each state, such that below this probability it is optimal to exclude this state from the discrimination strategy. If the two states subject to discrimination can be brought to a diagonal structure with (2×2)-dimensional blocks, then the unambiguous discrimination of these states can be reduced to the unambiguous discrimination of pure states. A criterion is presented that allows to identify the presence of such a structure for two self-adjoint operators. This criterion consists of the evaluation of three commutators and allows an explicit construction of the (2×2)-dimensional blocks. As an important application of unambiguous state discrimination, unambiguous state comparison, i.e., the question whether two states are identical or not, is generalized and optimal measurements for this problem are constructed. If for a certain family of states, a physical device maps the input state to an output state, such that a second device can be built that yields back the original input state, such a map is called reversible on this family. With respect to state discrimination, such reversible maps are particularly interesting, if the output states are pure. A complete characterization of all families that allow such a reversible and purifying map is provided. If the states are mapped to pure states, but the map itself is not reversible, upper and lower bounds are analyzed for the “deviation from perfect faithfulness”, a quantity which measures the deviation from a reversible mapping.