Structure of minimum error discrimination for linearly independent states

Abstract

In this paper we study the Minimum Error Discrimination problem (MED) for ensembles of linearly independent (LI) states. We define a bijective map from the set of those ensembles to itself and we show that the Pretty Good Measurement (PGM) and the optimal measurement for the MED are related by the map. In particular, the fixed points of the map are those ensembles for which the PGM is the optimal measurement. Also, we simplify the optimality conditions for the measurement of an ensemble of LI states.

Certified answers for ordered quantum discrimination problems

“…The QCP problem is depicted in figure (6). A source prepares systems in a default state |0 for some time and suddenly it changes and prepares systems in a mutated state |φ .…”

“…Despite being such a fundamental task, analytical solutions for optimal discrimination schemes in the multihypothesis case remains a challenge (see [6] for recent developments). Essentially only the two state [1] and symmetric states cases [7][8][9] have been solved (see [10][11][12] for reviews on state discrimination).…”

Locally distinguishing quantum states with limited classical communication

“…One is to find an optimal strategy to approximately distinguish those states under LOCC (in this regard, see Refs. [27][28][29][30][31][32][33][34][35] for the minimal-error discrimination of nonorthogonal states). Otherwise, those states can be distinguished perfectly under LOCC using suitable resources such as pure state entanglement shared among the parties [36][37][38][39][40][41][42][43][44][45][46][47][48], or multiple identical copies of the states to be distinguished [12,22,49,50].…”

Quantum process discrimination with restricted strategies

“…It is known that if all quantum testers are allowed, then the problem of finding the maximum success probability of guessing which process was applied can be formalized as a semidefinite programming problem, and its Lagrange dual problem has zero duality gap [20]. Many discrimination problems of quantum states, measurements, and channels have been addressed through the analysis of their dual problems [7,16,[19][20][21][22][23][24][25][26][27][28][29][30][31]. However, in a general case where the allowed testers are restricted, the problem cannot be formalized as a semidefinite programming problem.…”