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arXiv:0806.1172 · Physical Review Letters

Optimal Quantum Tomography of States, Measurements, and Transformations

Alessandro Bisio, Giulio Chiribella, Giacomo Mauro D’Ariano, S. Facchini, Paolo Perinotti

Published Jan 9, 2009• 46 citations
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Abstract

We present the first complete optimization of quantum tomography, for states, positive operator value measures, and various classes of transformations, for arbitrary prior ensemble and arbitrary representation, giving corresponding feasible experimental schemes in terms of random Bell measurements.
DOI
10.1103/physrevlett.102.010404

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Quantum tomography is the process of reconstructing the ensemble average of an arbitrary operator (observable or not, including the density matrix), which may not be directly accessible by feasible detection schemes, starting from the measurement of a complete set of observables i.e. a quorum . The measurement of a quorum thus represents a complete characterization of the quantum state. The operator expression in terms of a quorum corresponds to an expansion on an irreducible set of operators in the Liouville space. We give two general characterizations of these sets, and show that all the known quantum tomographies can be described in this framework. New operatorial resolutions are also given that may be used in novel reconstruction schemes.
. The paper is devoted to the mathematical foundation of the quantum tomography using the theory of square-integrable representations of unimodular Lie groups. 1. introduction In Quantum Mechanics a physical system is associated with a Hilbert space H: the states are described by positive trace-class trace-one operators T on H, the physical quantities by self-adjoint operators A on H and the physical content of the theory is given by the expectation values Tr (AT ). The state T is completely determined by Tr (Q n T ) for Q n running on a suitable set fQ n g of observables and, for arbitrary operator A, Tr (AT ) can be computed in terms of Tr (Q n T ). In order to implement this scheme one has to estimate Tr (Q n T ) experimentally, facing the problems arising from statistical errors and instrumental noise. Moreover, the number of experimental data is clearly nite, while A and T are operators on an innite dimensional Hilbert space and the set fQ n g is innite. The problem of determ...
We examine a variety of strategies for numerical quantum-state estimation from data of the sort commonly measured in experiments involving quantum-state tomography. We find that, in some important circumstances, an elaborate and time-consuming numerical optimization to obtain the optimum density matrix corresponding to a given data set is not necessary and that cruder, faster numerical techniques may well be sufficient; in other words, ``the best'' is the enemy of ``good enough.''