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Physical Review A

Designing an optimal, universal, programmable, and unambiguous discriminator for N unknown qubits

M. A. Jafarizadeh, Payman Mahmoudi, Derakhshan Akhgar, E. Faizi

Published Nov 9, 2017• 9 citations
Publisher

Abstract

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 $N$ qubit systems in the program register, to unambiguously identify the qubit state of data with one of $N$ program qubit states. All unknown qubit states are pure and ${N}_{D}$ 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 $N$ unknown pure states of the combined system. In another way, the problem of discrimination between $N$ unknown pure states of the combined system is reduced to discrimination between $N$ 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 $N$ unknown states of the combined system.

DOI
10.1103/physreva.96.052111

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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−1N\ensuremath{-}1N\ensuremath−1 copies of the data state are necessary for discriminating between NNN unknown states of the combined system.
The discrimination between two unknown states can be performed by a universal programmable discriminator, where the copies of the two possible states are stored in two program systems respectively and the copies of data, which we want to confirm, are provided in the data system. In the present paper, we propose a group-theretic approach to the multi-copy programmable state discrimination problem. By equivalence of unknown pure states to known mixed states and with the representation theory of U(n)U(n)U(n) group, we construct the Jordan basis to derive the analytical results for both the optimal unambiguous discrimination and minimum-error discrimination. The POVM operators for unambiguous discrimination and orthogonal measurement operators for minimum-error discrimination are obtained. We find that the optimal failure probability and minimum-error probability for the discrimination between the mean input mixd states are dependent on the dimension of the unknown qudit states. We applied the approach to generalize the results of He and Bergou (2007) from qubit to qudit case, and we further solve the problem of programmable dicriminators with arbitrary copies of unknown states in both program and data systems.
A universal programmable discriminator can perform discrimination between two unknown states, and the optimal solution can be approached via discrimination between the two averages over the uniformly distributed unknown input pure states, as has been widely discussed in previous works. In this paper, we consider the success probabilities of the optimal universal programmable unambiguous discriminators when applied to the pure input states. More precisely, the analytic results of the success probabilities are derived with the expressions of the optimal measurement operators for the universal discriminators, and we find that the success probabilities are independent of the dimension ddd while the amount of copies in the two program registers is equal. The success probability of the programmable unambiguous discriminator can asymptotically approach that of the usual unambiguous discrimination (state comparison) as the number of copies in the program registers (data register) goes to infinity.
The discrimination between two unknown states can be performed by a universal programmable discriminator, where the copies of the two possible states are stored in two program systems respectively and the copies of data, which we want to confirm, are provided in the data system. In the present paper, we propose a group-theretic approach to the multi-copy programmable state discrimination problem. By equivalence of unknown pure states to known mixed states and with the representation theory of U(n) group, we construct the Jordan basis to derive the analytical results for both the optimal unambiguous discrimination and minimum-error discrimination. The POVM operators for unambiguous discrimination and orthogonal measurement operators for minimum-error discrimination are obtained. We find that the optimal failure probability and minimum-error probability for the discrimination between the mean input mixd states are dependent on the dimension of the unknown qudit states. We applied the approach to generalize the results of He and Bergou (Phys. Rev. A {\bf 75}, 032316 (2007)) from qubit to qudit case, and we further solve the problem of programmable dicriminators with arbitrary copies of unknown states in both program and data systems.
We investigate the universal programmable unambiguous discrimination of two unknown quantum states with two copies. We first present the optimal positive-operator valued measure (POVM) elements that determine the individual success probabilities. In terms of the positivity of an inconclusive operator, we derive the average success probability, which is higher than the average success probability of discriminating two unknown states with one copy given in the reference [Bergou and Hillery, Phys. Rev. Lett. 94, 160501 (2005)].
This work investigates which sets of quantum states give rise to the highest achievable success probability in minimum-error state discrimination if multiple copies of the unknown state are given. Specifically, we consider uniformly distributed ensembles of the form {1N,ρi⊗k}i=1N\left\{\frac{1}{N},ρ_i^{\otimes k}\right\}_{i=1}^N{N1​,ρi⊗k​}i=1N​, where NNN states in dimension ddd are provided in kkk identical copies, and derive universal limits in this scenario. For pure state ensembles, we prove that whenever NNN is large enough to support a state kkk-design, these designs will exactly give rise to the maximally discriminable sets. We further show that when NNN exceeds the size required for a kkk-design, mixed states can outperform all pure state ensembles. We then recognise that the problem of most discriminable classical states in the multi-copy regime is in one-to-one correspondence to the concept of the multiplicative Bayes capacity of independent uses of classical channels, a concept that emerges naturally in the context of classical information leakage. This connection allows us to completely solve the classical analogue of our problem when N≥(d+k−1k)N\geq \binom{d + k - 1}{k}N≥(kd+k−1​), and to prove that quantum systems offer a quadratic advantage (in number of copies kkk) over classical ones. Then, we prove that this quantum over classical advantage is strongly reduced when one is restricted to real quantum states, more precisely, when N≥k+1N \geq k + 1N≥k+1, pure real qubits only offer a constant advantage over classical bits. Finally, we introduce computational techniques to find sets of most discriminable ensembles and to obtain rigorous universal upper bounds on the maximal success probability for multi-copy state discrimination in cases that are analytically intractable.