Unambiguous discrimination between mixed quantum states

Abstract

We prove that the states secretly chosen from a mixed state set can be perfectly discriminated if and only if they are orthogonal. The sufficient and necessary condition under which nonorthogonal mixed quantum states can be unambiguously discriminated is also presented. Furthermore, we derive a series of lower bounds on the inconclusive probability of unambiguous discrimination of states from a mixed state set with a priori probabilities.

Violating the modified Helstrom bound with nonprojective measurements

“…This advantage of nonprojective measurements in state discrimination is so surprising that it has become a featured example in modern quantum information textbooks (e.g., [9,10]), and has led to considerable research, both in theory [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] and experiment [28][29][30][31][32][33][34][35][36][37] (reviewed, e.g., in [38,39]). Most of this work has focused on the extreme cases of zero declining (as with the HB) or zero error (as with USD), with fewer papers considering intermediate cases that minimize the declining probability given a fixed nonzero error rate [23][24][25][26][27].…”

Optimal unambiguous state discrimination of two density matrices: Lower bound and class of exact solutions

“…For the unambiguous discrimination of a pair of mixed states, lower bounds on the failure probability have been found [1,15] and reveal three regimes, depending on the ratio between the two a priori probabilities of the two mixed states. The boundaries of the middle regime were recently refined in [11] but the consequences for the two remaining outer regimes were not addressed.…”

Quantum communications with an anonymous receiver

“…where Tr r is a map of operators known as the partial trace over system r. After simple computation, we can get that ρ i ρ i = δ ii ρ 2 i if and only if a = 0 or b = 0, which is the sufficient and necessary condition for state perfect discrimination [29], that means no measurement can perfectly discriminate ρ i and ρ i , in fact, probabilistic unambiguous discrimination between the two quantum states is also impossible to A since nobody knows the phase θ i except P i in this protocol. Therefore, A cannot distinguish whether P i entangles a qubit or not, i.e., he cannot judge whether P i is P r or not by the attack; moreover, the probability that the attack makes the protocol abort in…”