Maximum-confidence discrimination among symmetric qudit states

O. Jiménez, M. A. Solís-Prosser, A. Delgado, and L. Neves

Phys. Rev. A 84, 062315 – Published 19 December 2011

Abstract

We study the maximum-confidence (MC) measurement strategy for discriminating among nonorthogonal symmetric qudit states. Restricting to linearly dependent and equally likely pure states, we find the optimal positive operator valued measure (POVM) that maximizes our confidence in identifying each state in the set and minimizes the probability of obtaining inconclusive results. The physical realization of this POVM is completely determined and it is shown that after an inconclusive outcome, the input states may be mapped into a new set of equiprobable symmetric states, restricted, however, to a subspace of the original qudit Hilbert space. By applying the MC measurement again onto this new set, we can still gain some information about the input states, although with less confidence than before. This leads us to introduce the concept of sequential maximum-confidence (SMC) measurements, where the optimized MC strategy is iterated in as many stages as allowed by the input set, until no further information can be extracted from an inconclusive result. Within each stage of this measurement our confidence in identifying the input states is the highest possible, although it decreases from one stage to the next. In addition, the more stages we accomplish within the maximum allowed, the higher will be the probability of correct identification. We will discuss an explicit example of the optimal SMC measurement applied in the discrimination among four symmetric qutrit states and propose an optical network to implement it.

Received 26 September 2011

DOI:https://doi.org/10.1103/PhysRevA.84.062315

Authors & Affiliations

O. Jiménez^1,2, M. A. Solís-Prosser^2,3,4, A. Delgado^2,3,4, and L. Neves^2,3,4,*

¹Departamento de Física, Facultad de Ciencias Básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile
²Center for Optics and Photonics, Universidad de Concepción, Casilla 4016, Concepción, Chile
³MSI-Nucleus on Advanced Optics, Universidad de Concepción, Casilla 160-C, Concepción, Chile
⁴Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile

^*leonardo.neves@cefop.udec.cl

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Vol. 84, Iss. 6 — December 2011

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Figure 1
Schematic diagram of an $n$ -stage sequential MC measurement for discrimination among $N$ equiprobable symmetric qudit states ${| ψ_{j} 〉}$ . In the first stage, the $D$ -dimensional system is coupled to a two-dimensional ancilla [see Eq. (24)] which is measured in the logical basis ${{| 0 〉}_{a} {, | 1 〉}_{a}}$ . The outcome “0,” obtained with probability $1 - P (?)$ , projects the input states onto ${| u_{j} 〉}$ , which are subjected to the $N$ -outcome POVM (27) and correctly identified with probability $D / N$ . The outcome “1,” obtained with probability $P (?)$ , projects the input states onto the failure states ${| ξ_{j} 〉}$ , which are restricted to a $(D - d)$ -dimensional Hilbert space. The second stage—and each posterior one—starts with the failure states “ $ξ_{j}$ ” of the preceding stage, and the above procedure is repeated within a lower-dimensional space. In each stage, the “ $u_{j}$ ” states are correctly identified with the probability indicated in the dashed box. Due to the one-to-one correspondence between the states “ $u_{j}$ ” and $| ψ_{j} 〉$ , this probability is the confidence that one identifies $| ψ_{j} 〉$ in that stage of the sequential measurement. This sequence ends up in the $n^{th}$ stage when either the magnitude of the failure state coefficients are all equal (a) or the failure space is one dimensional (b). In either case, the total probability of correctly identifying the input states is given by Eq. (34).Reuse & Permissions
Figure 2
(a) Comparison of the maximum-confidence figure of merit for the optimal MC strategy (applied in the first and second stages of the SMC measurement), and the optimal ME strategy, as a function of the magnitude of the input state coefficients (37). (b) Comparison of the maximum-confidence figure of merit for the optimal MC strategy applied in the second stage and the optimal ME strategy, if the latter had been applied in the second stage of the SMC measurement. These probabilities are plotted as a function of the magnitude of either nonvanishing failure state coefficient (40).Reuse & Permissions
Figure 3
Probability of correctly identifying the input states for the first stage (a) and the complete SMC measurement (b). (c) Comparison between the probabilities achieved by the optimal ME and SMC measurements. (d) Probability of gaining no information about the input states after a two-stage SMC measurement. All the probabilities are plotted as a function of the magnitude of the input-state coefficients (37).Reuse & Permissions
Figure 4
Proposed optical network that implements the optimal two-stage SMC measurement for the discrimination among four symmetric qutrit states. Each dashed box from I to VIII corresponds to a step in the process—from state preparation to measurement—which is discussed in detail in the text. The dark (light) gray shaded region represents the first (second) stage of the SMC measurement. The inset shows the symbols for the single photon counting modules (SPCM) as well as the optical elements used in the scheme. Abbreviations: HWP, half-wave plate; PBS, polarizing beam splitter; PS, phase-shifter; BS, 50:50 beam splitter; mirror.Reuse & Permissions

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