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arXiv:quant-ph/0611133

Unambiguous State Discrimination of two density matrices in Quantum Information Theory

Philippe Raynal

Published Nov 12, 2006• 20 citations
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Abstract

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 $r$ in a 2$r$-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 $n$ 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.

DOI
10.48550/arxiv.quant-ph/0611133

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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.
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 ${\ensuremath{\rho}}_{1}=U{\ensuremath{\rho}}_{0}{U}^{\ifmmode\dagger\else\textdagger\fi{}}$, 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.
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 investigate generalized measurements, based on positive-operator-valued measures, and von Neumann measurements for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. In particular, we derive the conditions under which the failure probability of the measurement can reach its absolute lower bound, proportional to the fidelity of the states. The optimum measurement strategy yielding the fidelity bound of the failure probability is explicitly determined for a number of cases. One example involves two density operators of rank ddd that jointly span a 2d2d2d-dimensional Hilbert space and are related in a special way. We also present an application of the results to the problem of unambiguous quantum state comparison, generalizing the optimum strategy for arbitrary prior probabilities of the states.
We analyze the optimal unambiguous discrimination of two arbitrary mixed quantum states. We show that the optimal measurement is unique and we present this optimal measurement for the case where the rank of the density operator of one of the states is at most 2 (“solution in four dimensions”). The solution is illustrated by some examples. The optimality conditions proven by Eldar et al. [Phys. Rev. A 69, 062318 (2004)] are simplified to an operational form. As an application we present optimality conditions for the measurement, when only one of the two states is detected. The current status of optimal unambiguous state discrimination is summarized via a general strategy.
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.