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Physical Review Letters

Optimal Unambiguous Discrimination of Pure Quantum States

János A. Bergou, Ulrike Futschik, Edgar Feldman

Published Jun 21, 2012• 77 citations
Publisher

Abstract

A complete geometric view is presented for the optimal unambiguous discrimination among N > 2 pure states. A single intuitive picture contains all aspects of the problem: linear independence of the states, positivity of the detection operators, and a graphic method for finding and classifying the optimal solutions. The method is illustrated on the example of three states. We show that the problem depends on the phases of the complex inner products only through an invariant combination, the Berry phase φ, and present complete analytical results for φ = 0 and φ = π. The optimal solution exhibits full permutational symmetry and is single valued for a large range of parameters. However, for φ = 0 it can be bivalued: beyond a critical value of the parameters a second, less symmetric solution becomes optimal. The bifurcation is analogous to a second-order symmetry-breaking phase transition. We conclude with a discussion of the unambiguous discrimination of N > 3 states.

DOI
10.1103/physrevlett.108.250502

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A complete geometric view is presented for the optimal unambiguous discrimination among N > 2 pure states. A single intuitive picture contains all aspects of the problem: linear independence of the states, positivity of the detection operators, and a graphic method for finding and classifying the optimal solutions. The method is illustrated on the example of three states. We show that the problem depends on the phases of the complex inner products only through an invariant combination, the Berry phase φ, and present complete analytical results for φ = 0 and φ = π. The optimal solution exhibits full permutational symmetry and is single valued for a large range of parameters. However, for φ = 0 it can be bivalued: beyond a critical value of the parameters a second, less symmetric solution becomes optimal. The bifurcation is analogous to a second-order symmetry-breaking phase transition. We conclude with a discussion of the unambiguous discrimination of N > 3 states.
We consider unambiguous discrimination of three pure quantum states. Necessary and sufficient conditions to decide which states should be detected for optimal measurement of unambiguous discrimination are provided in terms of inner products and the geometric phase $\mathrm{\ensuremath{\Phi}}$. We get the optimal measurement and the optimal failure probability when the optimal unambiguous discrimination does not require the detection of every given quantum state. When at least two quantum states are orthogonal to each other, we supply the optimal measurement and optimal failure probability in analytic form. When all three quantum states are not orthogonal to each other and $\mathrm{\ensuremath{\Phi}}\ensuremath{\ne}0$, we find an analytic condition to determine the zero and nonzero elements for an optimal positive operator valued measure. We explain how to determine the solution in a geometric manner. Using the known solution of a case where the mutual inner products are real, we check the necessary and sufficient conditions when $\mathrm{\ensuremath{\Phi}}=\ensuremath{\pi}$ and analyze the property of the singular point when $\mathrm{\ensuremath{\Phi}}=0$ and the relation between the optimal points.
Complete solutions are given in a closed analytic form for unambiguous discrimination of three general pure states with real mutual inner products. For this purpose, we first establish some general results on unambiguous discrimination of nnn linearly independent pure states. The uniqueness of solution is proved. The condition under which the problem is reduced to an ($n\ensuremath{-}1$)-state problem is clarified. After giving the solution for three pure states with real mutual inner products, we examine some difficulties in extending our method to the case of complex inner products. There is a class of set of three pure states with complex inner products for which we obtain an analytical solution.
Given nnn linearly independent pure states and their prior probabilities, we study the optimum unambiguous state discrimination problem. We derive the conditions for the optimum measurement strategy to achieve the maximum average success probability and establish two sets of equations that must be satisfied by the optimum solution in different situations. We also provide the detailed steps to find the optimum measurement strategy. The method and results we obtain are given a geometrical illustration with a numerical example. Furthermore, using these equations, we derive a formula which shows a clear analytical relation between the optimum solution and the nnn states to be discriminated. We also solve a generalized equal-probability measurement problem analytically. Finally, as another application of our result, the unambiguous discrimination problem of three pure states is studied in detail and analytical solutions are obtained for some interesting cases.
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.
Unambiguous discrimination between nonorthogonal but linearly independent quantum states is a challenging problem in quantum information processing. In this study, using the connection between Lewenstein-Sanpera decomposition (LSD) and optimal unambiguous state discrimination (OPUSD), an analytic relation for the feasible region of NNN linearly independent quantum states is presented in terms of inner product of reciprocal states. Then, for the real inner product of states, an exact analytic solution for the OPUSD problem involving an arbitrary number of pure linearly independent quantum states is presented using the Karush-Kuhn-Tucker convex optimization method. In another approach, an analytic relation for the feasible region for an arbitrary number of pure linearly independent quantum states is presented in terms of the inner product of states. To this end, the relevant semidefinite programming task is reduced to a linear programming (LP) one with a feasible region of polygon type which can be solved via simplex method. Moreover, using the obtained feasible region, an exact analytic solution to an OPUSD problem involving two and three pure linearly independent quantum states is provided.
We present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between unambiguous and minimum error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function of the rate of inconclusive answers. A critical value of this rate is identified that coincides with the minimum failure probability in the cases where unambiguous discrimination is possible and provides a natural generalization of it when states cannot be unambiguously discriminated. The method is illustrated on two explicit examples: discrimination of two pure states with arbitrary prior probabilities and discrimination of trine states.