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A note on Fisher-Helstrom information inequality in pure state models

Alessandra Luati

Published Jan 1, 2008• 2 citations
OpenAlex

Abstract

This paper concerns the design problem of choosing the measurement that provides the maximum Fisher information for the unknown parameter of a quantum system. We show that when the system under investigation is described by a one-parameter n-dimensional pure state model an optimal measurement exists, such that Fisher information attains the upper bound constituted by Hel-strom information. A characterisation theorem and two strategies of im-plementations are derived and discussed. These results generalise to n-dimensional spaces those obtained for n = 2 by Barndorff-Nielsen and Gill (2000). AMS (2000) subject classification. Primary 62B05; secondary 62F10.

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This paper concerns the design problem of choosing the measurement that provides the maximum Fisher information for the unknown parameter of a quantum system. We show that when the system under investigation is described by a one-parameter n-dimensional pure state model an optimal measurement exists, such that Fisher information attains the upper bound constituted by Hel-strom information. A characterisation theorem and two strategies of im-plementations are derived and discussed. These results generalise to n-dimensional spaces those obtained for n = 2 by Barndorff-Nielsen and Gill (2000). AMS (2000) subject classification. Primary 62B05; secondary 62F10.
We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101–102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439–3443]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481–4490]. In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states.
We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101–102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439–3443]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481–4490]. In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states. 1. Introduction. Quantum
We address the "major open problem" of evaluating how much increased efficiency in estimation is possible using non-separable measurements, rather than separable ones, of N copies of two-level quantum systems. To proceed, we compute the the 3 x 3 Fisher information matrices for the optimal measurements recently devised by Vidal et al (quant-ph/9812068) for N = 2,...,7. We obtain simple polynomial expressions for the traces of the products of the inverse of the quantum Helstrom information matrix and the computed Fisher information matrices. These traces all have minima of 2 N - 1 in the pure state limit, while for separable measurements, as established by Gill and Massar (quant-ph/9902063), they are bound above by N. This indicates both that there is a large discontinuity at the pure states (where the computed Fisher information matrices are indeterminate) and that the gain in using joint measurements over separate measurements is substantial for all impure states. In seeking to extend our results to N > 7, we investigate possible relationships between the Fisher information matrices generated here and the class of monotone Riemannian metrics, studied by Petz and Sudar as well as others.
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 3 × 3 Fisher information matrices for the corresponding optimal measurements recently devised by Vidal et al . 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, 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 (2 N − 1)( m − 1) in the pure state limit.
Open questions from Sarovar and Milburn (2006 J.Phys. A: Math. Gen. 39 8487) are answered. Sarovar and Milburn derived a convenient upper bound for the Fisher information of a one-parameter quantum channel. They showed that for quasi-classical models their bound is achievable and they gave a necessary and sufficient condition for positive operator-valued measures (POVMs) attaining this bound. They asked (i) whether their bound is attainable more generally, (ii) whether explicit expressions for optimal POVMs can be derived from the attainability condition. We show that the symmetric logarithmic derivative (SLD) quantum information is less than or equal to the SM bound, i.e.\ H(θ)≤CΥ(θ)H(θ) \leq C_Υ(θ)H(θ)≤CΥ​(θ) and we find conditions for equality. As the Fisher information is less than or equal to the SLD quantum information, i.e. FM(θ)≤H(θ)F_M(θ) \leq H(θ)FM​(θ)≤H(θ), we can deduce when equality holds in FM(θ)≤CΥ(θ)F_M(θ) \leq C_Υ(θ)FM​(θ)≤CΥ​(θ). Equality does not hold for all channels. As a consequence, the attainability condition cannot be used to test for optimal POVMs for all channels. These results are extended to multi-parameter channels.