Optimal discrimination of quantum states with a fixed rate of inconclusive outcomes

Abstract

We present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between unambiguous and minimum error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function of the rate of inconclusive answers. A critical value of this rate is identified that coincides with the minimum failure probability in the cases where unambiguous discrimination is possible and provides a natural generalization of it when states cannot be unambiguously discriminated. The method is illustrated on two explicit examples: discrimination of two pure states with arbitrary prior probabilities and discrimination of trine states. State discrimination has long been recognized to play a central role in quantum information and quantum computing. In these fields the information is encoded in the state of quantum systems; thus, one often needs to identify in which of N known states {ρ i } N i=1 one such system was prepared. If the possible states are mutually orthogonal this is an easy task: we just set up detectors along these orthogonal directions and determine which one clicks (assuming perfect detectors). However, if the states are not mutually orthogonal the problem is highly nontrivial and optimization with respect to some reasonable criteria leads to complex strategies often involving generalized measurements. Finding such optimal strategies is the subject of state discrimination.The two fundamental state discrimination strategies are discrimination with minimum error (ME) and unambiguous discrimination (UD). In ME, every time a system is given and a measurement is performed on it a conclusion must be drawn about its state. Accordingly, the measurement is described by an N -element positive operator valued measure (POVM), where each element represents a conclusive outcome. Errors are permitted and in the optimal strategy the probability of making an error is minimized [1]. In UD, no errors are tolerated but at the expense of permitting an inconclusive measurement outcome, represented by the positive operator 0 . Hence, the corresponding POVM is = { i } N i=0 . When 0 clicks we do not learn anything about the state of the system and in the optimal strategy the probability of the inconclusive outcome is minimized [2]. It has been recognized that states can be discriminated unambiguously only if they are linearly independent [3]. Discrimination with maximum confidence (MC) can be applied to states that are not necessarily independent and for linearly independent states it reduces to unambiguous discrimination [4,5], so the MC scheme can be regarded as a generalized UD strategy.It is clear that UD (or MC for linearly dependent stat...

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

“…(15). Obviously, the equality in (15) holds if and only if {Φ,1 −Φ} is a minimum-error measurement, i.e., (16) holds [29].…”

“…Obtaining an optimal inconclusive measurement is generally a more difficult task than obtaining a minimum-error measurement. In fact, closed-form analytical expressions for optimal inconclusive measurements are only known for very special cases (e.g., [14][15][16][17][18]). Instead of analytical approaches, we can use numerical methods.…”

Programmable discrimination with an error margin

“…This result was derived in [13] and its generalization to arbitrary prior probabilities in [14] (also in [15], by fixing an inconclusive rate Q instead of an error margin). Note that the POVM E is fully determined by the angle φ, which in turn is fully determined by the margin r through Eq.…”

“…In particular, we use the basis |(j A j B )j AB j C ;jm to diagonalize σ 1 and |j A (j B j C )j BC ;jm to diagonalize σ 2 , where j A = j C = n/2, j B = n /2 and j AB = j BC = (n + n )/2. The diagonal form of σ 1 is (15) and the analogous form of σ 2 is obtained by coupling j B and j C instead of j A and j B . The key property of the angular momentum basis is that it satisfies the orthogonality relation…”

“…Hence by introducing an error margin we can unify minimum-error and unambiguous discrimination. Both become extremal points of the unified discrimination with error margin scheme [13][14][15]. Interpolating between these two extemal cases may have practical interest in some situations.…”