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arXiv:1308.1378 · Physical Review A

Programmable discrimination with an error margin

Gael Sentís, E. Bagán, John Calsamiglia, R. Muñoz-Tapia

Published Nov 7, 2013• 14 citations
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Abstract

The problem of optimally discriminating between two completely unknown qubit states is generalized by allowing an error margin. It is visualized as a device---the programmable discriminator---with one data and two program ports, each fed with a number of identically prepared qubits---the data and the programs. The device aims at correctly identifying the data state with one of the two program states. This scheme has the unambiguous and the minimum-error schemes as extremal cases, when the error margin is set to zero or it is sufficiently large, respectively. Analytical results are given in the two situations where the margin is imposed on the average error probability---weak condition---or it is imposed separately on the two probabilities of assigning the state of the data to the wrong program---strong condition. It is a general feature of our scheme that the success probability rises sharply as soon as a small error margin is allowed, thus providing a significant gain over the unambiguous scheme while still having high confidence results.

DOI
10.1103/physreva.88.052304

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The problem of optimally discriminating between two completely unknown qubit states is generalized by allowing an error margin. It is visualized as a device---the programmable discriminator---with one data and two program ports, each fed with a number of identically prepared qubits---the data and the programs. The device aims at correctly identifying the data state with one of the two program states. This scheme has the unambiguous and the minimum-error schemes as extremal cases, when the error margin is set to zero or it is sufficiently large, respectively. Analytical results are given in the two situations where the margin is imposed on the average error probability---weak condition---or it is imposed separately on the two probabilities of assigning the state of the data to the wrong program---strong condition. It is a general feature of our scheme that the success probability rises sharply as soon as a small error margin is allowed, thus providing a significant gain over the unambiguous scheme while still having high confidence results.
We describe a class of programmable devices that can discriminate between two quantum states. We consider two cases. In the first, both states are unknown. One copy of each of the unknown states is provided as an input, or program, for the two program registers, and the data state, which is guaranteed to be prepared in one of the program states, is fed into the data register of the device. This device will then tell us, in an optimal way, which of the templates stored in the program registers the data state matches. In the second case, we know one of the states while the other is unknown. One copy of the unknown state is fed into the single program register, and the data state which is guaranteed to be prepared in either the program state or the known state, is fed into the data register. The device will then tell us, again optimally, whether the data state matches the template or is the known state. We determine two types of optimal devices. The first performs discrimination with minimum error, and the second performs optimum unambiguous discrimination. In all cases we first treat the simpler problem of only one copy of the data state and then generalize the treatment to nnn copies. In comparison to other works we find that providing n>1n>1n>1 copies of the data state yields higher success probabilities than providing n>1n>1n>1 copies of the program states.
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.
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.
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.
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.
There are two common settings in a quantum-state discrimination problem. One is minimum-error discrimination where a wrong guess (error) is allowed and the discrimination success probability is maximized. The other is unambiguous discrimination where errors are not allowed but the inconclusive result ``I do not know'' is possible. We investigate a discrimination problem with a finite margin imposed on the error probability. The two common settings correspond to the error margins 1 and 0. For arbitrary error margin, we determine the optimal discrimination probability for two pure states with equal occurrence probabilities. We also consider the case where the states to be discriminated are multipartite and show that the optimal discrimination probability can be achieved by local operations and classical communication.