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Europhysics Letters (EPL)

The universal unambiguous discriminator between two unknown qudit states

Wenshi Wang, Tao Zhou

Published Jul 29, 2021• 1 citation
Publisher

Abstract

Abstract In this work, the unambiguous discrimination between two unknown states in d -dimensional Hilbert space is investigated. To achieve this goal, the Hilbert space of the states is divided into a series of subspaces, where the components of the two states projected in some of these subspaces can be unambiguously distinguished, and then the unambiguous discriminator between two unknown qudit states can be constructed. Moreover, the optimal success probability of the discrimination between two unknown states is obtained and the form of detection operators is given.

DOI
10.1209/0295-5075/ac18fa

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Abstract In this work, the unambiguous discrimination between two unknown states in d -dimensional Hilbert space is investigated. To achieve this goal, the Hilbert space of the states is divided into a series of subspaces, where the components of the two states projected in some of these subspaces can be unambiguously distinguished, and then the unambiguous discriminator between two unknown qudit states can be constructed. Moreover, the optimal success probability of the discrimination between two unknown states is obtained and the form of detection operators is given.
We discuss the problem of designing unambiguous programmable discriminators for any nnn unknown quantum states in an mmm-dimensional Hilbert space. The discriminator is a fixed measurement that has two kinds of input registers: the program registers and the data register. The quantum state in the data register is what users want to identify, which is confirmed to be among the nnn states in program registers. The task of the discriminator is to tell the users which state stored in the program registers is equivalent to that in the data register. First, we give a necessary and sufficient condition for judging an unambiguous programmable discriminator. Then, if m=nm=nm=n, we present an optimal unambiguous programmable discriminator for them, in the sense of maximizing the worst-case probability of success. Finally, we propose a universal unambiguous programmable discriminator for arbitrary nnn quantum states.
We discuss the problem of designing unambiguous programmable discriminators for any nnn unknown quantum states in an mmm-dimensional Hilbert space. The discriminator is a fixed measurement which has two kinds of input registers: the program registers and the data register. The program registers consist of the nnn states, while the data register is prepared among them. The task of the discriminator is to tell us which state stored in the program registers is equivalent to that in the data register. First, we give a necessary and sufficient condition for judging an unambiguous programmable discriminator. Then, if m=nm=nm=n, we present an optimal unambiguous programmable discriminator for them, in the sense of maximizing the worst-case probability of success. Finally, we propose a universal unambiguous programmable discriminator for arbitrary nnn quantum states. We also show how to use this universal discriminator to unambiguously discriminate mixed states.
We construct a device that can unambiguously discriminate between two unknown quantum states. The unknown states are provided as inputs, or programs, for the program registers, and a third system, which is guaranteed to be prepared in one of the states stored in the program registers, is fed into the data register of the device. The device will then, with some probability of success, tell us whether the unknown state in the data register matches the state stored in the first or the second program register. We show that the optimal device, i.e., the one that maximizes the probability of success, is universal. It does not depend on the actual unknown states that we wish to discriminate.
We study the determination of an unknown mixed state of a ddd-dimensional quantum system by means of unambiguous state discrimination. We show that optimal and nonoptimal unambiguous state discrimination can be used to reconstruct unknown states of a qubit. This result is extended to the case of a qudit by a sequence of reconstructions in two-dimensional subspaces. The total number of projections scales approximately as 2d22{d}^{2}2d2 for ddd large; this is twice as much as in the case of tomography based on mutually unbiased bases or symmetric informationally complete positive-operator-valued measures, and less than the d3{d}^{3}d3 projections required by standard tomography.
We show how to optimally unambiguously discriminate between two subspaces of a Hilbert space. In particular we suppose that we are given a quantum system in either the state $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$, where $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ can be any state in the subspace S1{S}_{1}S1​, or $\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩$, where $\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩$ can be any state in the subspace S2{S}_{2}S2​, and our task is to determine in which of the subspaces the state of our quantum system lies. We do not want to make any error, which means that our procedure will sometimes fail if the subspaces are not orthogonal. This is a special case of the unambiguous discrimination of mixed states. We present the positive operator valued measures that solve this problem and several applications of this procedure, including the discrimination of multipartite states without classical communication.
The universal programmable discriminator is a device for discrimination between unknown quantum states. It has two kinds of registers: the program register and the data register. The states that are stored in registers are all unknown. It is assured that the data state is identical with one of the program states with the certain probability. The aim is to optimally identify the state stored in the data register with one of the program states that it is done by discriminating between unknown states of the combined system contains program and data registers. We design an optimal universal unambiguous programmable discriminator that has NNN qubit systems in the program register, to unambiguously identify the qubit state of data with one of NNN program qubit states. All unknown qubit states are pure and ND{N}_{D}ND​ copies of the data state are available. We determine the optimal positive-operator valued measure (POVM) elements in two different ways. First, we find optimal POVMs that maximize the average of the overall success probability of discrimination between NNN unknown pure states of the combined system. In another way, the problem of discrimination between NNN unknown pure states of the combined system is reduced to discrimination between NNN known average mixed states. It is shown that theses two different ways lead to the same results. We show that at least $N\ensuremath{-}1$ copies of the data state are necessary for discriminating between NNN unknown states of the combined system.