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Electronics and Communications in Japan (Part III Fundamental Electronic Science)

Relationship between optimum quantum detection operators for pure and mixed‐state signals

Yuji Fujihara, Tsuyoshi Sasaki Usuda, Ichi Takumi, Masayasu Hata

Published Apr 14, 2003• 6 citations

Abstract

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

DOI: 10.1002/ecjc.1153

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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

A method is given for calculating the minimum average error probability attainable in deciding between a mixed state and a pure state of a quantum system. It is applied, first, to the detection of a noisy coherent pulse represented by a state with a Gaussian P-representation and, second, to the detection of a signal represented at the transmitter by a two-photon coherent state (TCS). In both problems the field of the receiver is free of background noise and, when no signal was transmitted, is in the ground state.

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.

A general method for solving an important class of quantum detection problems will be presented and evaluated. The quantum theory for detecting pure states for communications purposes has been developed over two decades ago, however the mixed state problem representing signal plus noise states has received little attention due to its great complexity. Here we develop a practical model for solving the mixed-state problem using a discrete approximation to the coherent-state representation of signal plus noise density operators.

For discrimination between two signaling states of a qubit, the optimal detector minimizing the probability of error is applied to the situation where detection has to be performed from a noisy qubit affected by an arbitrary quantum noise separately characterized. With no noise, any pair of orthogonal pure quantum states is optimal for signaling as it enables error-free detection. In the presence of noise, detection errors are in general inevitable, and the pairs of signaling states best resistant to such noise are investigated. With an arbitrary quantum noise, modeled as a channel affecting the qubit, and when minimum-error detection is performed from the output, a characterization of the optimal input signaling pairs and of their best detection performance is obtained through a simple maximization of a quadratic scalar criterion in three constrained real variables. This general characterization enables to establish that such optimal signaling pairs are always made of two orthogonal pure quantum states, but that they must be specifically selected to match the noise properties and prior probabilities. The maximization is explicitly solved for several generic quantum noise processes relevant to the qubit, such as the squeezed generalized amplitude damping noise which describes interaction with a thermal bath representing a decohering environment and which includes as special cases both the generalized and the regular amplitude damping noise processes, and such as general Pauli noise processes which include for instance the bit-flip noise and the depolarizing noise. Also, examined is the situation of one imposed (pure or mixed) signaling state, for which the other associated signaling state optimal for noisy detection is determined as a pure state, yet not necessarily orthogonal.