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arXiv:1112.0931

Group-theoretic approach for multi-copy programmable discriminators between two unknown qudit states

Tao Zhou, Xiaohua Wu, Long, Gui Lu

Published Dec 5, 2011• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1112.0931

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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.
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The discrimination of any pair of unknown quantum states is performed by devices processing three parts of inputs: copies of the pair of unknown states we want to discriminate are, respectively, stored in two program systems and copies of data, which is guaranteed to be one of the unknown states, in a third system. We study the efficiency of such programmable devices with the inputs prepared with nnn and mmm copies of unknown qubits used as programs and data, respectively. By finding a symmetry in the average inputs, we apply the Jordan-basis method to derive their optimal unambiguous discrimination and minimum-error discrimination schemes. The dependence of the optimal solutions on the a priori probabilities of the mean input states is also demonstrated.
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.
Quantum state discrimination is a fundamental primitive in quantum statistics where one has to correctly identify the state of a system that is in one of two possible known states. A programmable discrimination machine performs this task when the pair of possible states is not a priori known but instead the two possible states are provided through two respective program ports. We study optimal programmable discrimination machines for general qubit states when several copies of states are available in the data or program ports. Two scenarios are considered: One in which the purity of the possible states is a priori known, and the fully universal one where the machine operates over generic mixed states of unknown purity. We find analytical results for both the unambiguous and minimum error discrimination strategies. This allows us to calculate the asymptotic performance of programmable discrimination machines when a large number of copies are provided and to recover the standard state discrimination and state comparison values as different limiting cases.
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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.