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Journal of Modern Optics

Discrimination of quantum states

János A. Bergou

Published Jan 7, 2010• 218 citations
Publisher

Abstract

In quantum information processing and quantum computing protocols the carrier of information is a quantum system and information is encoded in the state of a quantum system. After processing the information it has to be read out what is equivalent to determining the final state of the system. When the possible final states are not orthogonal this is a highly nontrivial task that constitutes the general area of what is known as quantum state discrimination. It consists in finding measurement schemes that, according to some figure of merit, will determine the state of the system. Optimized measurement schemes often lead to generalized measurements (Positive Operator Valued Measures [POVMs]). In this tutorial review we illustrate the power of the POVM concept on examples relevant to applications in quantum cryptography. In order to keep the flow of the presentation we give a brief introduction to the quantum theory of measurements, including generalized measurements (POVMs), in the Appendices.

DOI
10.1080/09500340903477756

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In quantum information processing and quantum computing protocols the carrier of information is a quantum system and information is encoded in the state of a quantum system. After processing the information it has to be read out what is equivalent to determining the final state of the system. When the possible final states are not orthogonal this is a highly nontrivial task that constitutes the general area of what is known as quantum state discrimination. It consists in finding measurement schemes that, according to some figure of merit, will determine the state of the system. Optimized measurement schemes often lead to generalized measurements (Positive Operator Valued Measures [POVMs]). In this tutorial review we illustrate the power of the POVM concept on examples relevant to applications in quantum cryptography. In order to keep the flow of the presentation we give a brief introduction to the quantum theory of measurements, including generalized measurements (POVMs), in the Appendices.
In quantum information processing and quantum computing protocols information is encoded in the state of a quantum system. After processing the information it has to be read out which is equivalent to determining the final state of the system. When the possible final states are not orthogonal this is a highly nontrivial task that constitutes the general area of what is known as quantum state discrimination. It consists in finding measurement schemes that, according to some figure of merit, will determine the state of the system. Optimized measurement schemes often lead to generalized measurements (Positive Operator Valued Measures, POVMs). In this tutorial review we begin with a brief introduction to the theory of generalized measurements and illustrate the power of the concept on examples relevant to applications in quantum cryptography.
In quantum information and quantum computing, the carrier of information is some quantum system and information is encoded in its state. The state, however, is not an observable in quantum mechanics, thus a fundamental problem arises: after processing the information it has to be read out or, in other words, the state of the system must be determined. When the possible target states are orthogonal, this is a relatively simple task, provided the set of possible target states is known. But when the possible target states are not orthogonal their discrimination is far from being trivial even if their set is known. Thus the problem of discriminating among non-orthogonal states is ubiquitous in quantum information and quantum computing, underlying all communication and computing schemes. It is the subject of this talk to review recent theoretical and experimental advances in this field.
Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.
The positive operator valued measure (POVM), also known as the probability operator valued measure, is useful in quantum information processing. The POVM consists of a set of non-negative quantum-mechanical Hermitian operators that add up to the identity. The probability that a quantum system is in a particular state is given by the expectation value of the POVM operator corresponding to that state. Following a brief review of the mathematics and history of POVMs in quantum theory, and an expository discussion of the quantum mechanics of photonic qubits, a particular implementation of a POVM for use in the measurement of photonic qubits is reviewed.
In quantum information and quantum computing, the carrier of information is some quantum system and information is encoded in its state. After processing the state in the quantum processor, the information has to be read out. Clearly, this task is equivalent to determining the final state of the system. We begin by briefly reviewing various possible state discrimination strategies that are optimal with respect to some reasonable criteria and report on recent advances in the unambiguous discrimination of mixed quantum states. This strategy has been successfully applied to devise a class of novel probabilistic quantum algorithms and has been demonstrated experimentally, using a linear optical implementation via generalized interferometers. In the second part we present a scheme for communication via completely unknown quantum states. In this context we discuss programmable quantum state discriminators that are universal, i.e. perform optimally on average, independently of the actual states used for the communication scheme. We conclude with a discussion of possible experimental implementations of the proposed device.