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arXiv:0708.0956 · Physical Review A

Quantum estimation via the minimum Kullback entropy principle

Stefano Olivares, Matteo G. A. Paris

Published Oct 22, 2007• 26 citations
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Abstract

We address quantum estimation in situations where one has at disposal data from the measurement of an incomplete set of observables and some a priori information on the state itself. By expressing the a priori information in terms of a bias toward a given state, the problem may be faced by minimizing the quantum relative entropy (Kullback entropy) with the constraint of reproducing the data. We exploit the resulting minimum Kullback entropy principle for the estimation of a quantum state from the measurement of a single observable, either from the sole mean value or from the complete probability distribution, and apply it as a tool for the estimation of weak Hamiltonian processes. Qubit and harmonic oscillator systems are analyzed in some detail.

DOI
10.1103/physreva.76.042120

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We address quantum estimation in situations where one has at disposal data from the measurement of an incomplete set of observables and some a priori information on the state itself. By expressing the a priori information in terms of a bias toward a given state, the problem may be faced by minimizing the quantum relative entropy (Kullback entropy) with the constraint of reproducing the data. We exploit the resulting minimum Kullback entropy principle for the estimation of a quantum state from the measurement of a single observable, either from the sole mean value or from the complete probability distribution, and apply it as a tool for the estimation of weak Hamiltonian processes. Qubit and harmonic oscillator systems are analyzed in some detail.
Knowing and guessing, these are two essential epistemological pillars in the theory of quantum-mechanical measurement. As formulated quantum mechanics is a statistical theory. In general, a priori unknown states can be completely determined only when measurements on infinite ensembles of identically prepared quantum systems are performed. But how one can estimate (guess) quantum state when just incomplete data are available (known)? What is the most reliable estimation based on a given measured data? What is the optimal measurement providing only a finite number of identically prepared quantum objects are available? These are some of the questions we address. We present several schemes for a reconstruction of states of quantum systems from measured data: (1) We show how the maximum entropy (MaxEnt) principle can be efficiently used for an estimation of quantum states on incomplete observation levels. (2) We show how Bayesian inference can be used for reconstruction of quantum states when only a finite number of identically prepared systems are measured. (3) We describe the optimal generalized measurement of a finite number of identically prepared quantum systems which results in the estimation of a quantum state with the highest fidelity.
The minimum-Kullback-entropy principle (mKE) is a useful tool to estimate quantum states and operations from incomplete data and prior information. In general, the solution of an mKE problem is analytically challenging and an approximate solution has been proposed and employed in different contexts. Recently, the form and a way to compute the exact solution for finite dimensional systems has been found, and a question naturally arises on whether the approximate solution could be an effective substitute for the exact solution, and in which regimes this substitution can be performed. Here, we provide a systematic comparison between the exact and the approximate mKE solutions for a qubit system when average data from a single observable are available. We address both mKE estimation of states and weak Hamiltonians, and compare the two solutions in terms of state fidelity and operator distance. We find that the approximate solution is generally close to the exact one unless the initial state is near an eigenstate of the measured observable. Our results provide a rigorous justification for the use of the approximate solution whenever the above condition does not occur, and extend its range of application beyond those situations satisfying the assumptions used for its derivation.
The question of what information about a quantum state may be inferred from a sequence of measurements made on it, is addressed. The main result is that maximum-likelihood estimation gives an arguably natural optimum approach in quantum theory. It singles out a particular choice of basis, in which the state is expressed. Various incompatible and incomplete observations made on the ensemble are treated as a complete generalized measurement. Consequently, the maximum likelihood (relative entropy) is the best measure for relating experimental data with theoretical predictions of quantum theory.
Quantum-state reconstruction on a finite number of copies of a quantum system with informationally incomplete measurements, as a rule, does not yield a unique result. We derive a reconstruction scheme where both the likelihood and the von Neumann entropy functionals are maximized in order to systematically select the most-likely estimator with the largest entropy, that is, the least-bias estimator, consistent with a given set of measurement data. This is equivalent to the joint consideration of our partial knowledge and ignorance about the ensemble to reconstruct its identity. An interesting structure of such estimators will also be explored.
Gibbs states are familiar from statistical mechanics, yet their use is not limited to that domain. For instance, they also feature in the maximum entropy reconstruction of quantum states from incomplete measurement data. Outside the macroscopic realm, however, estimating a Gibbs state is a nontrivial inference task, due to two complicating factors: the proper set of relevant observables might not be evident a priori; and whenever data are gathered from a small sample only, the best estimate for the Lagrange parameters is invariably affected by the experimenter's prior bias. I show how the two issues can be tackled with the help of Bayesian model selection and Bayesian interpolation, respectively, and illustrate the use of these Bayesian techniques with a number of simple examples.