Increased Efficiency of Quantum State Estimation Using Non-Separable Measurements
Abstract
We address the "major open problem" of evaluating how much increased efficiency in estimation is possible using non-separable, as opposed to separable, measurements of N copies of m-level quantum systems. First, we study the six cases m = 2, N = 2,...,7 by computing the the 3 x 3 Fisher information matrices for the corresponding optimal measurements recently devised by Vidal et al (quant-ph/9812068) for N = 2,...,7. We obtain simple polynomial expressions for the ("Gill-Massar") traces of the products of the inverse of the quantum Helstrom information matrix and these Fisher information matrices. The six traces all have minima of 2 N -1 in the pure state limit, while for separable measurements (quant-ph/9902063), the traces can equal N, but not exceed it. Then, the result of an analysis for m = 3, N = 2 leads us to conjecture that for optimal measurements for all m and N, the "Gill-Massar trace" achieves a minimum of (2N-1)(m-1) in the pure state limit.
Cite
@article{arxiv.quant-ph/0006009,
title = {Increased Efficiency of Quantum State Estimation Using Non-Separable Measurements},
author = {Paul B. Slater},
journal= {arXiv preprint arXiv:quant-ph/0006009},
year = {2009}
}
Comments
Sixteen pages, six postscript figures, we include an additional analysis (in sec. III D 3) for N = 2 copies of the 3-level quantum systems, leading us to conjecture that for (non-separable) optimal measurements of N copies of m-level quantum systems, the "Gill-Massar" trace converges downward in the pure state limit to (2N-1)(m-1), for all m and N, while (as Gill and Massar established) it can not exceed N(m-1) for separable measurements. This conjecture conforms with our previously obtained result of 2N -1, for m = 2 and N = 2,...7