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arXiv:quant-ph/9707028 · Physical Review Letters

Universal Algorithm for Optimal Estimation of Quantum States from Finite Ensembles via Realizable Generalized Measurement

R. Derka, Vladimír Bužek, ARTUR K. EKERT

Published Feb 23, 1998• 251 citations
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Abstract

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 $N$ independent and identically prepared two-level systems (qubits).

DOI
10.1103/physrevlett.80.1571

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•Oct 26, 2004•0 citations
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).
Optimal and finite positive operator valued measurements on a finite number NNN of identically prepared systems have recently been presented. With physical realization in mind, we propose here optimal and minimal generalized quantum measurements for two-level systems. We explicitly construct them up to N=7N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}7N=7 and verify that they are minimal up to N=5N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5N=5.
Positive-operator-valued measurements on a finite number of N identically prepared systems of arbitrary spin J are discussed. Pure states are characterized in terms of Bloch-like vectors restricted by a SU(2J+1)\mathrm{SU}(2J+1)SU(2J+1) covariant constraint. This representation allows for a simple description of the equations to be fulfilled by optimal measurements. We explicitly find the minimal positive-operator-valued measurement for the N=2N=2N=2 case, a rigorous bound for N=3,N=3,N=3, and set up the analysis for arbitrary 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.
A direct derivation is given for the optimal mean fidelity of quantum state estimation of a ddd-dimensional unknown pure state with its NNN copies given as input, which was first obtained by Hayashi in terms of an infinite set of covariant positive operator valued measures (POVM's) and by Bru\ss{} and Macchiavello establishing a connection to optimal quantum cloning. An explicit condition for POVM measurement operators for optimal estimators is obtained, by which we construct optimal estimators with finite POVMs using exact quadratures on a hypersphere. These finite optimal estimators are not generally universal, where universality means the fidelity is independent of input states. However, any optimal estimator with finite POVM for M(>N)M(>N)M(>N) copies is universal if it is used for NNN copies as input.
A direct derivation is given for the optimal mean fidelity of quantum state estimation of a d-dimensional unknown pure state with its N copies given as input, which was first obtained by Bruss and Macchiavello by establishing a connection to optimal quantum cloning. An explicit condition for POVM measurement operators for optimal estimators is obtained, by which we show optimal estimators with finite POVM exist but not generally universal, where universality means the fidelity is independent of input states. However any optimal estimator for M(>N) copies is universal if it is used for N copies as input.