Quantum state estimation and large deviations

Abstract

In this paper we propose a method to estimate the density matrix ρof a d-level quantum system by measurements on the N-fold system. The scheme is based on covariant observables and representation theory of unitary groups and it extends previous results concerning the estimation of the spectrum of ρ. We show that it is consistent (i.e. the original input state ρis recovered with certainty if N \to \infty), analyze its large deviation behavior, and calculate explicitly the corresponding rate function which describes the exponential decrease of error probabilities in the limit N \to \infty. Finally we discuss the question whether the proposed scheme provides the fastest possible decay of error probabilities.

LaTex2e, 40 pages, 2 figures. Substantial changes in Section 4: one new subsection (4.1) and another (4.2 was 4.1 in the previous version) completely rewritten. Minor changes in Sect. 2 and 3. Typos corrected. References added. Accepted for publication in Rev. Math. Phys

References in corpus (5)

• Estimating the spectrum of a density operator
• Quantum universal variable-length source coding
• Two quantum analogues of Fisher information from a large deviation viewpoint of quantum estimation
• Collective vs local measurements in qubit mixed state estimation
• Relative Entropy and Single Qubit Holevo-Schumacher-Westmoreland Channel Capacity

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• Spectrum estimation of density operators with alkaline-earth atoms
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• Sample efficient tomography via Pauli Measurements
• Linear programming with unitary-equivariant constraints
• A General Quantum Duality for Representations of Groups with Applications to Quantum Money, Lightning, and Fire
• Large deviation principle for moment map estimation
• Comparison of some methods of quantum state estimation
• Estimating the eigenstates of an unknown density operator without damaging it
• Testing identity of collections of quantum states: sample complexity analysis