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Physical Review A

Quantum-state estimation by self-learning measurements

Dietmar G. Fischer, Stefan H. Kienle, Matthias Freyberger

Published Feb 11, 2000• 69 citations
Publisher

Abstract

We describe strategies for quantum-state estimation based on self-learning algorithms. In contrast to the optimal estimation procedures proposed earlier, our schemes rely on measurements performed separately on each quantum system in a finite ensemble. We numerically simulate our strategies in the case of finite ensembles of qubits and compare the resulting average fidelities to the fidelity of optimal quantum-state estimation.

DOI
10.1103/physreva.61.032306

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We describe strategies for quantum-state estimation based on self-learning algorithms. In contrast to the optimal estimation procedures proposed earlier, our schemes rely on measurements performed separately on each quantum system in a finite ensemble. We numerically simulate our strategies in the case of finite ensembles of qubits and compare the resulting average fidelities to the fidelity of optimal quantum-state estimation.
The optimal estimation of a quantum mechanical 2-state system (qubit) - with N identically prepared qubits available - is obtained by measuring all qubits simultaneously in an entangled basis. We report the experimental estimation of qubits using a succession of N measurements on individual qubits where the measurement basis is changed during the estimation procedure conditioned on the outcome of previous measurements (self-learning estimation). The performance of this adaptive algorithm is compared with other algorithms using measurements in a factorizing basis.
We describe an optimized, self-correcting procedure for the Bayesian inference of pure quantum states. By analyzing the history of measurement outcomes at each step, the procedure returns the most likely pure state, as well as the optimal basis for the measurement that is to follow. The latter is chosen to maximize, on average, the fidelity of the most likely state after the measurement. We also consider a practical variant of this protocol, where the available measurement bases are restricted to certain limited sets of bases. We demonstrate the success of our method by considering in detail the single-qubit and two-qubit cases, and comparing the performance of our method against other existing methods.
We present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system. The algorithm specifies a physically realizable (i.e., finite) positive operator valued measurement on a finite number of identically prepared systems. We illustrate the general formalism by applying it to different scenarios of the state estimation of NNN independent and identically prepared two-level systems (qubits).
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.
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.