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Geometry of quantum signal detection.

Robert O. Harger, M. C. Kim

Published Jan 1, 1972• 0 citations
OpenAlex

Abstract

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.

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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

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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.
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.
The problem of specifying the optimum quantum detector in multiple hypotheses testing is considered for application to optical communications. The quantum digital detection problem is formulated as a linear programming problem on an infinite-dimensional space. A necessary and sufficient condition is derived by the application of a general duality theorem specifying the optimum detector in terms of a set of linear operator equations and inequalities. Existence of the optimum quantum detector is also established. The optimality of commuting detection operators is discussed in some examples. The structure and performance of the optimal receiver are derived for the quantum detection of narrow-band coherent orthogonal and simplex signals. It is shown that modal photon counting is asymptotically optimum in the limit of a large signaling alphabet and that the capacity goes to infinity in the absence of a bandwidth limitation.
This paper is concerned with the problem of optimal M-alternative determination of quantum statistical states. A review of newest achievement of solving this problem is given. A notion of an effective decision Hilbert space is introduced and necessary and sufficient condkions for optimality of multiple quantum hypothesis testing in this space are formulated. The general solution is found for the case of a two-dimensional decision space. Another problem solved is that of discrimination of quantum pure non-orthogonal states. The result is represented in explicit analytical form for an "equidiagonal" case, which is quite general. In particular, we find explicit solutions of optimal discrimination problem of homogeneous and equiangle sets of pure states. These results are used for the M-ary detection problem in solving for the quantum coherent non-orthogonal signals. It is proved that the simplex signals are optimal elso in quantum case. The optimal estimatesof phaseandamplitude of quantum coherent signals are found. For decision operators a notion of IT-representation is introduced to get a general quasi-classical (optimal in quasi-classical limit) M-ary detection procedure of stochastic fields and particles, which submits to Bose-Einstein statistics. An optimal solution of problem of non-coherent detection of quantum stochastic (including optical) signals are found in the extreme quantum limit (weaknoise and signals with unknown phase).
A new technique for evaluating the performance of quantum signals observed in the presence noise is described and evaluated. The quantum theory for detecting coherent-state signals has been developed previously, however the quantum "signal plus noise" problem has received little attention due to its complexity. Here we develop a discrete approximation to the coherent-state representation of signal-plus-noise density operators, and present solutions to optimum receiver performance in terms of quantum "measurement states" whose performance is optimized via generalized rotations in Hilbert space. An efficient algorithm for carrying out the required numerical optimization is described and applied to binary signals observed in the presence of noise, for which exact results are available for comparison. The algorithm is then applied to the detection of ternary signals observed in the presence of noise, a previously unsolved problem, and the performance of the optimum receiver characterized.
We demonstrate a class of optimum detection strategies for extracting the maximum information from sets of equiprobable real symmetric qubit states of a single photon. These optimum strategies have been predicted by Sasaki et al. [Phys. Rev. A 59, 3325 (1999)]. The peculiar aspect is that the detections with at least three outputs suffice for optimum extraction of information regardless of the number of signal elements. The cases of ternary (or trine), quinary, and septenary polarization signals are studied where a standard von Neumann detection (a projection onto a binary orthogonal basis) fails to access the maximum information. Our experiments demonstrate that it is possible with present technologies to attain about 96% of the theoretical limit.
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