Optimum unambiguous discrimination of two mixed quantum states

Abstract

We investigate generalized measurements, based on positive-operator-valued measures, and von Neumann measurements for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. In particular, we derive the conditions under which the failure probability of the measurement can reach its absolute lower bound, proportional to the fidelity of the states. The optimum measurement strategy yielding the fidelity bound of the failure probability is explicitly determined for a number of cases. One example involves two density operators of rank d that jointly span a 2d-dimensional Hilbert space and are related in a special way. We also present an application of the results to the problem of unambiguous quantum state comparison, generalizing the optimum strategy for arbitrary prior probabilities of the states.Comment: final versio

Minimum-error discrimination between mixed quantum states

“…For distinguishing between two mixed quantum states, general upper and lower bounds have been derived for the optimal failure probability by Rudolph et al [27], and, furthermore, for distinguishing between m mixed states, Feng et al [28] obtained a general lower bound on the minimum failure probability. The analytical results for the optimal unambiguous discrimination between two mixed quantum states have been derived by Raynal et al [29], by Herzog and Bergou [30], and by Zhou et al [31]. More references regarding unambiguous discrimination of mixed quantum states may be referred to [32].…”

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

“…Some special cases have been given for which the corresponding problem can be reduced to USD of pure state case, such as in state filtering [9,10,11] or state comparison [1,11]. The underlying reduction theorems have been stated in [12].…”

Practical private database queries based on a quantum-key-distribution protocol

“…Using the criteria of Refs. [20,21] one can show that these two density matrices cannot be discriminated unambiguously for the single-qubit case. The best chance to guess the state correctly is 85.36%, as for the previous attack.…”

“…In general one must assume that Alice disposes of a quantum memory and is hence not forced to measure directly as in step 2. Instead she can keep the photon and, once Bob has announced the state pair, apply the optimal unambiguous state discrimination (USD) measurement [20,21] that will correctly tell her which of the two announced states has actually been sent. The success probability of the USD measurement is, for the case of two equally likely states, bounded by 1 − F (ρ 0 ,ρ 1 ), where F (ρ 0 ,ρ 1 ) is the fidelity between the two quantum states one seeks to discriminate.…”