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Journal of Modern Optics

Optimizations of quantum measurement processes for signal detection and information transmission in quantum systems

Masashi Ban, Masao Osaki, Osamu Hirota

Published Nov 1, 1996• 13 citations
Publisher

Abstract

The calculation method proposed here obtains an optimum detection process for an M-ary signal whose quantum states are linearly independent and nonorthogonal. Optimizations are carried out by minimizing the average probability of error or by maximizing the mutual information in the signal detection process. Quantum detection operators, in terms of which the signal detection process is mathematically described, are assumed to be projection operators defined on a Hilbert space H s spanned by the quantum states of the signal; that is, the best projection operators that describe the signal detection process are obtained. The optimization procedure is equivalent to finding an appropriate unitary transformation in the Hilbert space H s. For a binary quantum state signal, the optimum detection processes in which the average probability of error is minimized and in which the mutual information is maximized are obtained. It is found that the optimum detection process minimizing the average probability of error is the same as that maximizing the mutual information.

DOI
10.1080/09500349608232890

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The calculation method proposed here obtains an optimum detection process for an M-ary signal whose quantum states are linearly independent and nonorthogonal. Optimizations are carried out by minimizing the average probability of error or by maximizing the mutual information in the signal detection process. Quantum detection operators, in terms of which the signal detection process is mathematically described, are assumed to be projection operators defined on a Hilbert space H s spanned by the quantum states of the signal; that is, the best projection operators that describe the signal detection process are obtained. The optimization procedure is equivalent to finding an appropriate unitary transformation in the Hilbert space H s. For a binary quantum state signal, the optimum detection processes in which the average probability of error is minimized and in which the mutual information is maximized are obtained. It is found that the optimum detection process minimizing the average probability of error is the same as that maximizing the mutual information.
Abstract A significant issue at present in the field of quantum information theory is the implementation of an optimum quantum receiver in which the error probability is minimized. Because a quantum receiver is represented by detection operators, in order to implement an optimum quantum receiver, it is important to derive optimum detection operators that minimize the error probability. Although it is generally difficult to analytically derive such a detection operator, optimum detection operators have been obtained on several specific signal systems in which signals are in the pure state. On the other hand, because mixed‐state signals are associated with more general signals, including classical noise, it is more important to address mixed‐state signals. However, for these signals, only basic optimum detection operators, such as binary signals in a two‐dimensional system and M‐ary symmetric signals, have been obtained. Regarding these optimum detection operators, it has been suggested that optimum detection operators for pure‐state signals can also be a reasonably good signal detection process for mixed‐state signals. Based on this suggestion, this paper examines the extent to which the optimum detection operators for pure‐state signals can be applied to mixed‐state signals. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 86(10): 8–18, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/ecjc.1153
Three equally separated nonorthogonal quantum bits that lie on the same plane are prepared for signal processing. One of the qubits is randomly picked as the single-qubit signal. The remaining two qubits then form the two-qubit signal. We find that the optimal probability operator measure for the two-qubit signals can be interpreted as the optimal orthogonal measurement in the four-dimensional Hilbert space of the signal states. We also find that the two-qubit signals have a better detection probability than the single-qubit. Possible generalization to the case of N nonorthogonal qubits is also discussed.
We consider the communication systems sending the classical (Shannon) information through the quantum channel. The classical information source X is represented with symbols and their a priori probabilities At a transmitter, each symbol is modulated to a quantum state and detected at a receiver by the quantum mechanically generalized measurement, so-called the probability operator-valued measure Here we call a detection operator since it represents a detection process deciding a received signal to be ‘j’. The detection operator is no-negative, and resolution of identity, where is the identity operator on the signal Hilbert space spanned by signal quantum states. This detection process can be regarded as a quantum channel characterized by the conditional probability defined as follows:
Consideration of a binary quantum signal detection problem in a two-dimensional Hilbert space. The optimum detection problem is reduced to the problem of finding the locus of a vector which has a maximum projection along the fixed a priori probability vector of hypotheses. It is shown that the desired locus can be determined by a geometrical method involving the use of a randomized decision strategy. It is further shown that this geometrical method can be applied to the optimum solution of a binary detection problem described in a product Hilbert space.