PHYSICAL REVIEW A 87, 042329 (2013) Comparison of error probability bounds in quantum state discrimination Roberto Corvaja * Department of Information Engineering, University of Padova, Via G. Gradenigo 6/B - 35131 Padova, Italy (Received 2 January 2013; revised manuscript received 19 February 2013; published 23 April 2013) In quantum discrimination, the value of the minimum error probability and the set of measurement operators which achieve this minimum are often difficult to derive. Here we present a comparison of the performance obtained by the optimal solution and by the available bounds, namely the square root measurement (SRM) and the Chernoff bound. Applied to some Gaussian states, namely to coherent states with thermal noise, it is shown that the SRM provides a much tighter bound with respect to the Chernoff bound, with a comparable numerical complexity. DOI: 10.1103/PhysRevA.87.042329 PACS number(s): 03.67.Hk I. INTRODUCTION In the discrimination of quantum states, not only is it difficult to derive the set of measurement operators achieving the minimum error probability, but in many cases it is difficult to derive also the actual value of the error probability. Closed- form expressions can be obtained only in the cases of pure states with high-symmetry properties. Also, Helstrom’s bound for the binary case, in the general case of mixed quantum states, requires a singular value decomposition for the determination of the positive eigenvalues of the difference matrix (Helstrom matrix) needed for the evaluation of the error probability. In many other cases, the optimal measurement set is not known and only numerical solutions are available, requiring one to resort to a heavy convex optimization problem [1] to determine the measurement operators. However, suboptimal bounds can be derived, in particular the square root measurement (SRM) [2,3], which obtains the set of measurement operators from the Gram operator or from the Gram matrix, and the quantum Chernoff bound, which recently received a great deal of attention, especially for Gaussian quantum states [4], as a simple way to estimate the performance of quantum discrimination [5–7]. Applied to Gaussian states, other bounds are derived in [8] for binary hypothesis testing by fixing one of the conditional error probabilities and minimizing the other conditional error probability. The Chernoff quantum is limited in that it can only be applied to binary quantum systems, and in this work we compare the SRM and the Chernoff bounds with the optimum error probability in terms of both performance gap and complexity. In fact, we notice that several studies in the literature considered the bounds separately and when possible evaluated their relation to the optimum value. In this paper, we will make a systematic comparison of the two bounds. It can be seen that, applied to coherent states with thermal noise, the SRM provides a tighter bound to the error probability than the Chernoff bound with comparable computational complexity. Also, the SRM is available easily also for the M-ary case and has the additional advantage of providing the optimal solution when the states are pure and exhibit geometrical uniform symmetry [9]. * corvaja@dei.unipd.it II. QUANTUM DISCRIMINATION A general M-ary quantum system with mixed states is described by M density operators {ρ 0 ,...,ρ M−1 } in an N - dimensional Hilbert space H, where N may possibly be infinite. The eigenvalues of the operators span a subspace U ⊆ H. Quantum discrimination is the operation of choosing among the possible density operatorsρ i , i = 0,1,...,M − 1, performed by applying a positive operator valued measurement (POVM) set, that is, a set of M positive semidefinite operators  0 ,..., M−1 with the condition M−1  i =0  i = P U , (1) where P U is the projector operator onto U . In other words,  i must give a resolution of the identity in the subspace U . The probability that the detection outcome is j , provided that the density operator is ρ i , i.e., the transition probability p(j |i ), is given by p(j |i ) = Tr(ρ i  j ), i,j = 0,1,...,M − 1, (2) and the corresponding error probability in the detection becomes P e = 1 − M−1  i =0 q i p(i |i ) = 1 − M−1  i =0 q i Tr(ρ i  i ), (3) where q i ,i = 0,...,M − 1, are the a priori probabilities. For the binary case, the optimum solution is available by the decomposition of the difference operator D = q 1 ρ 1 − q 0 ρ 0 [10], obtaining the Helstrom bound [11] for the minimum error probability achievable, P e = q 1 −  η k >0 η k , (4) where η k are the eigenvalues of D and the sum extends over the positive ones. In the following, for simplicity, we assume that all the states are equiprobable, that is, q i = 1/M, i = 0,...,M − 1. Apart for the binary case, the optimal detection set of POVM can be obtained by convex semidefinite programming (CSP) [9], while suboptimal solutions are achievable by square root measurement (SRM) [3] or by the Chernoff bound, both giving an upper bound to the error probability. The conditions for the optimum POVM set in [12,13] lead to a convex 042329-1 1050-2947/2013/87(4)/042329(4) ©2013 American Physical Society