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IEEE International Symposium on Information Theory (ISIT)

Unambiguous discrimination between two rank-2 mixed quantum states

Mihailo Stojnic, Babak Hassibi

Published Jun 1, 2007• 1 citation
Publisher

Abstract

In this paper we consider the problem of the optimal quantum unambiguous detection between two mixed quantum states. More specifically, we consider two mixed quantum states of rank 2 which lie in a Hilbert space of dimension 4. Using duality theory we explicitly characterize the optimal measurement operators. Furthermore, as a by-product of our framework we obtain a closed form solution of unambiguous discrimination between a pure and a mixed quantum state.

DOI
10.1109/isit.2007.4557236

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In this paper we consider the problem of the optimal quantum unambiguous detection between two mixed quantum states. More specifically, we consider two mixed quantum states of rank 2 which lie in a Hilbert space of dimension 4. Using duality theory we explicitly characterize the optimal measurement operators. Furthermore, as a by-product of our framework we obtain a closed form solution of unambiguous discrimination between a pure and a mixed quantum state.
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 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.
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 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.
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 S1{S}_{1}S1​, or $\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩$, where $\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩$ 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 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).