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A state estimation scheme for finite quantum systems

Thomas Baier

Published Jan 1, 2009• 0 citations
Publisher

Abstract

The estimation of the state of a finite quantum system is studied. It is assumed that N identical copies of the unknown state are at hand and statistical data is obtained from different separate measurements. Complementary (or quasi-orthogonal) measurements, defined by an orthogonality relation, play a special role among separate measurements. Motivated by recent results on the existence of certain complementary subalgebras, the effect of complementarity of the measurements on the state estimation procedure is examined. For an unconstrained estimate optimality is shown for finite sample sizes, and for a constrained estimate, that always gives approvable results, optimality is shown in the limit of large N.

DOI
10.60644/3288

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The estimation of the state of a finite quantum system is studied. It is assumed that N identical copies of the unknown state are at hand and statistical data is obtained from different separate measurements. Complementary (or quasi-orthogonal) measurements, defined by an orthogonality relation, play a special role among separate measurements. Motivated by recent results on the existence of certain complementary subalgebras, the effect of complementarity of the measurements on the state estimation procedure is examined. For an unconstrained estimate optimality is shown for finite sample sizes, and for a constrained estimate, that always gives approvable results, optimality is shown in the limit of large N.
Abstract: The estimation of the density matrix of a k-level quantum system is studied when the parametrization is given by the real and imaginary part of the entries and they are estimated by independent measurements. It is established that the properties of the estimation procedure depend very much on the invertibility of the true state. In particular, in case of a pure state, the estimation should be constrained to ensure the positive definiteness of the estimate. An efficient constraining algorithm is proposed and it yields an asymptotically unbiased estimate. Moreover, several estimation schemes are compared for the unknown state of a qubit when one copy is measured at a time. It is shown that the average mean quadratic error matrix is the smallest if the applied observables are complementary. All the results are illustrated by computer simulations. PACS numbers: 03.67.-a, 03.65.Wj, 03.65.Fd Key words: State determination, density matrix, unconstrained estimate,
The estimation of the density matrix of a k -level quantum system is studied when the parametrization is given by the real and imaginary part of the entries, and they are estimated by independent measurements. It is established that the properties of the estimation procedure depend very much on the invertibility of the true state. In particular, in the case of a pure state, the estimation should be constrained to ensure the positive definiteness of the estimate. An efficient constraining algorithm is proposed and it yields an asymptotically unbiased estimate. Moreover, several estimation schemes are compared for the unknown state of a qubit when one copy is measured at a time. It is shown that the average mean quadratic error matrix is the smallest if the applied observables are complementary. All the results are illustrated by computer simulations.
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.
Abstract: Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. Two subalgebras A1 and A2 of B(H) are called complementary if the traceless subspaces of A1 and A2 are orthogonal (with respect to the Hilbert-Schmidt inner product). When both subalgebras are maximal Abelian, then the concept reduces to complementary observables or mutually unbiased bases. In the paper several characterizations of complementary subalgebras are given in the general case and several examples are presented. For a 4-level quantum system, the structure of complementary subalgebras can be described very well, the Cartan decomposition of unitaries plays a role. It turns out that a measurement corresponding to the Bell basis is complementary to any local measurement of the two-qubit-system.
We consider the problem of determining the weights of a quantum ensemble. That is to say, given a quantum system that is in a set of possible known states according to an unknown probability law, we give strategies to estimate the individual probabilities, weights, or mixing proportions. Such strategies can be used to estimate the frequencies at which different independent signals are emitted by a source. They can also be used to estimate the weights of particular terms in a canonical decomposition of a quantum channel. The quality of these strategies is quantified by a covariance-type error matrix. According with this cost function, we give optimal strategies in both the single-shot and multiple-copy scenarios. The latter is also analyzed in the asymptotic limit of large number of copies. We give closed expressions of the error matrix for two-component quantum mixtures of qubit systems. The Fisher information plays an unusual role in the problem at hand, providing exact expressions of the minimum covariance matrix for any number of copies.