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arXiv:0708.2323 · Physical Review A

Optimal unambiguous discrimination of quantum states

M. A. Jafarizadeh, Mahdi Rezaei, Naser Karimi, A. R. Amiri

Published Apr 16, 2008• 40 citations
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Abstract

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 $N$ 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.

DOI
10.1103/physreva.77.042314

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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.
In the present paper, an exact analytic solution for the optimal unambiguous state discrimination (OPUSD) problem involving an arbitrary number of pure linearly independent quantum states with real and complex inner product is presented. Using semidefinite programming and Karush-Kuhn-Tucker convex optimization method, we derive an analytical formula which shows the relation between optimal solution of unambiguous state discrimination problem and an arbitrary number of pure linearly independent quantum 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.
An exact analytical solution to the optimal unambiguous state discrimination involving two finite-dimensional coherent states that occur with given prior probabilities is presented using the Lewenstein–Sanpera decomposition method. Furthermore, a numerical method is advised for efficient solving of the unambiguous state discrimination of the two finite-dimensional coherent states with the same dimensions. In this manner, it is shown that the maximum success rate for the unambiguous states discrimination of the two finite-dimensional coherent states with the arbitrary prior probability is decreased by increasing the dimensionality of the finite-dimensional coherent states. Also, the success rate for the unambiguous states discrimination of the two coherent states satisfies the upper bound proportional to the fidelity of the states for a given prior probability.
We consider the problem of determining the state of an unknown quantum sequence without error. The elements of the given sequence are drawn with equal probability from a known set of linearly independent pure quantum states with the property that their mutual inner products are all real and equal. This problem can be posed as an instance of unambiguous state discrimination where the states correspond to that of all possible sequences having the same length as the given one. We calculate the optimum probability by solving the optimality conditions of a semidefinite program. The optimum value is achievable by measuring individual members of the sequence, and no collective measurement is necessary.
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.
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.