Designing optimal quantum detectors via semidefinite programming

Abstract

We consider the problem of designing an optimal quantum detector to minimize the probability of a detection error when distinguishing between a collection of quantum states, represented by a set of density operators. We show that the design of the optimal detector can be formulated as a semidefinite programming problem. Based on this formulation, we derive a set of necessary and sufficient conditions for an optimal quantum measurement. We then show that the optimal measurement can be found by solving a standard (convex) semidefinite program followed by the solution of a set of linear equations or, at worst, a standard linear programming problem. By exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum, the optimal measurement can be computed very efficiently in polynomial time.Using the semidefinite programming formulation, we also show that the rank of each optimal measurement operator is no larger than the rank of the corresponding density operator. In particular, if the quantum state ensemble is a pure-state ensemble consisting of (not necessarily independent) rank-one density operators, then we show that the optimal measurement is a pure-state measurement consisting of rank-one measurement operators.

Upper and lower bounds on optimal success probability of quantum state discrimination with and without inconclusive results

“…with a positive semidefinite operatorX ′ m+1 , whereX 0 = ρ 0 , andX m+1 (m ∈ I M−1 ) is an optimal solution to problem (19). We derive a new upper bound on Q, namely, Q = TrX M−1 .…”

“…We derive a new upper bound on Q, namely, Q = TrX M−1 . According to Lemma 1, the optimal solution to problem (19) is expressed aŝ X m+1 =X m + (ρ m+1 −X m ) + . The proposed upper bound Q can thus be expressed as…”

“…Instead of analytical approaches, we can use numerical methods. It is known that the design of an optimal success probabilities can be treated as a positive semidefinite programming problems [19]. In many cases, an optimal value can be computed in polynomial time by well known algorithms for solving semidefinite programs such with interior point methods.…”

Orthogonal and projected orthogonal matched filter detection

“…This problem has also been solved in the context of quantum detection [9] and in the context of general inner product shaping [6]; the solution in [6,9] is equal to the OMF signals corresponding to the set transformation given by (11). We may then interpret the OMF demodulator as a correlation demodulator matched to a set of orthonormal signals that are closest in the least-squares sense to the signals s k (t).…”

“…Then s k (t) = X s k for some s k ∈ R m , and S = X S where S is the m × m matrix of columns s k . We may then expressĜ of (11) in terms of X and S aŝ…”