Quantum State Estimation and Large Deviations

Abstract

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

Sample-optimal tomography of quantum states

“…qubits), the optimal infidelity was shown in [9][10][11][12][13] to scale as 1/n. This scaling was generalized to qudits in [14] (see also Section 6.4 of [15]), but with an uncontrolled dependence on d (i.e.n scales as f (d)/δ for unknown f (·)); see also [16]. In many settings (e.g.…”

“…An independent and concurrent work [39] analyzes Keyl's measurement strategy [16], and proves that it only requires n = O(dr/ 2 ) copies to achieve accuracy in trace distance. This improves on our corollary for trace distance by removing the logarithmic factor, but does not imply our fidelity bound, which is incomparable to theirs.…”

“…When k = 1 this is the PGM, but the cases k = 2 and k = 3 have also been found useful in specific settings; see [43] for a review. If we take k → ∞ here then this corresponds precisely to the Keyl "rotated-highest-weight" strategy [16]. It is possible that this framework could be used to formally compare the performance of these different strategies.…”

The large deviation approach to statistical mechanics

“…As for classical version of that theorem, the quantum version plays an important role in the theory of estimation and in information theory, as generalized to the quantum world [1,5,153]. Finally, a quantum adaptation of the Freidlin-Wentzell theory of dynamical systems perturbed by noise can be found in [19].…”

Comparison of Some Methods of Quantum State Estimation

“…Large deviations can also be used to analyze the performance of state estimation schemes [11], when the qubit is in a mixed state. An optimal estimation scheme is also proposed based on covariant observables.…”