Several quantities of interest in quantum information, including entanglement and purity, are nonlinear functions of the density matrix and cannot, even in principle, correspond to proper quantum o...

Quantum estimation for quantum technology

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.

Optimum testing of multiple hypotheses in quantum detection theory

Minimum mean-squared error of estimates in quantum statistics

Given a finite number of quantum states with {\em a priori} probabilities, the positive operator-valued measure that maximizes the Shannon mutual information is investigated The group covariant case is examined in detail

Information and quantum measurement

A quantum-mechanical form of the Cramer-Rao inequality is derived, setting a lower bound to the variance of an unbiased estimate of a parameter of a density operator. It is applied to the estimates of parameters such as amplitude, arrival time, and carrier frequency of a coherent signal as picked up by an ideal receiver in the presence of thermal noise. The estimation of parameters of a noise-like signal is also treated.

The minimum variance of estimates in quantum signal detection

Basing decisions and estimates on simultaneous approximate measurements of noncommuting observables in a quantum receiver is shown to be equivalent to measuring commuting projection operators on a larger Hilbert space than that of the receiver itself. The quantum-mechanical Cramer-Rao inequalities derived from right logarithmic derivatives and symmetrized logarithmic derivatives of the density operator are compared, and it is shown that the latter give superior lower bounds on the error variances of individual unbiased estimates of arrival time and carrier frequency of a coherent signal. For a suitably weighted sum of the error variances of simultaneous estimates of these, the former yield the superior lower bound under some conditions.

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Noncommuting observables in quantum detection and estimation theory

An iterative procedure is described for reducing the Bayes cost in decisions among M>2 quantum hypotheses by minimizing the average cost in binary decisions between all possible pairs of hypotheses: the resulting decision strategy is a projection-valued measure and yields an upper bound to the minimum attainable Bayes cost. From it is derived an algorithm for finding the optimum measurement states for choosing among M linearly independent pure states with minimum probability of error. The method is also applied to decisions among M unimodal coherent quantum signals in thermal noise.

Bayes-cost reduction algorithm in quantum hypothesis testing (Corresp.)

The cumulative distribution of the envelope of the sum of two sine waves and narrowband Gaussian noise, the difference of the phases of the sine waves being uniformly distributed over (0, 2 pi ), is expressed as an integral that is evaluated by numerical quadrature. Approximation for the distribution valid in the limit of large signal-to-noise ratio are derived. The probability density function of the envelope is similarly treated. >

Distribution of the sum of two sine waves and Gaussian noise

The complex amplitude of a coherent quantum signal in the presence of thermal noise is to be estimated when its real and imaginary parts have a Gaussian prior distribution. Cost functions of Gaussian, power-law, and delta-function forms are shown all to lead to the same optimum estimator.

C. Helstrom

Papers

The minimum variance of estimates in quantum signal detection

Noncommuting observables in quantum detection and estimation theory

Bayes-cost reduction algorithm in quantum hypothesis testing (Corresp.)

Distribution of the sum of two sine waves and Gaussian noise

Quantum Bayes estimation of the amplitude of a coherent signal (Corresp.)