arXiv:1001.1385v1 [quant-ph] 9 Jan 2010 Efficient Optimal Minimum Error Discrimination of Symmetric Quantum States Antonio Assalini, Gianfranco Cariolaro, and Gianfranco Pierobon Department of Information Engineering (DEI), University of Padua, Via Gradenigo 6/B, 35131, Padova, Italy (Dated: January, 2010) This paper deals with the quantum optimal discrimination among mixed quantum states enjoying geometrical uniform symmetry with respect to a reference density operator ρ 0 . It is well-known that the minimal error probability is given by the positive operator-valued measure (POVM) obtained as a solution of a convex optimization problem, namely a set of operators satisfying geometrical symmetry, with respect to a reference operator Π 0 , and maximizing Tr ( ρ 0 Π 0 ) . In this paper, by resolving the dual problem, we show that the same result is obtained by minimizing the trace of a semidefinite positive operator X commuting with the symmetry operator and such that X ≥ ρ 0 . The new formulation gives a deeper insight into the optimization problem and allows to obtain closed-form analytical solutions, as shown by a simple but not trivial explanatory example. Besides the theoretical interest, the result leads to semidefinite programming solutions of reduced complex- ity, allowing to extend the numerical performance evaluation to quantum communication systems modeled in Hilbert spaces of large dimension. set of smaller dimension. The simplified formulation of the dual problem is illustrated with an example where a closed-form solution is easily found. For problems of large dimension, that cannot be solved analytically, the reduced number of variables in the simplified dual state- ment becomes useful to the numerical solution by means of semidefinite programming (SDP) tools. In a quantum system Alice prepares the quantum chan- nel into one of several quantum states. Bob measures the quantum channel by a set of measurement operators and, on the basis of the result, it guesses the choice made by the transmitter. These actions lead to a classical chan- nel, and the problem arises of finding the measurement operators that provide optimal performance according to a predefined criterion, in this paper the minimum error probability. However, to solve the problem, except for some particular cases, has appeared to be a very diffi- cult task since the pioneering contributions in the seven- ties [1–3]. In recent years, particular attention has been paid to quantum states satisfying geometrical symmetry [4–7], in view of applications to optical communication systems. In some specific cases, including symmetric pure quan- tum states [4, 5] and symmetric mixed quantum states with a characteristic structure [6], the solution named square root measurement (SRM) proves optimal. Never- theless, in general, SRM represents a suboptimal strat- egy, although it provides pretty good performance in many scenarios. In this paper, we are concerned with the construction of optimal POVM for the discrimination of symmetric mixed quantum states. We present some results that provide intuition into the problem and offer perspectives on its solution. Optimal quantum state discrimination represents a convex optimization problem, and, as such, it can be formulated in a primal and in a dual form, with the lat- ter having a reduced number of variables and constraints [3, 6]. In this paper we investigate how primal and dual problems simplify with symmetric quantum states. For the primal problem this study was already considered in [6]. Herein, we extend the analysis to the dual prob- lem, where the optimal solution can be searched in a The analytical solution of PP1 is in general difficult since P$_{c}$ has to be maximized over the whole M -tuple of measure operators Π$_{i}$ . As a matter of fact, closed-form results to the primal problem are available only for some Primal problem ( PP1 ). Find the maximum of the probability of correct detection P$_{c}$ = ∑ M − 1 i =0 q$_{i}$ Tr ( ρ$_{i}$ Π$_{i}$) over the class of the POVM on H . II. GENERAL FORMULATION OF MINIMUM ERROR DISCRIMINATION I. INTRODUCTION