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MAXIMUM FISHER INFORMATION IN MIXED STATE QUANTUM SYSTEMS

Alessandra Luati

Published Jan 1, 2004• 28 citations
OpenAlex

Abstract

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

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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 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.
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.
Braunstein and Caves (Braunstein S L and Caves C M 1994 Phys. Rev. Lett. 72 3439-43) proposed to use Helstrom's quantum information number to define, meaningfully, a metric on the set of all possible states of a given quantum system. They showed that the quantum information is nothing other than the maximal Fisher information in a measurement of the quantum system, maximized over all possible measurements. Combining this fact with classical statistical results, they argued that the quantum information determines the asymptotically optimal rate at which neighbouring states on some smooth curve can be distinguished, based on arbitrary measurements on n identical copies of the given quantum system. We show that the measurement which maximizes the Fisher information typically depends on the true, unknown, state of the quantum system. We close the resulting loophole in the argument by showing that one can still achieve the same, optimal, rate of distinguishability, by a two-stage adaptive measurement procedure. When we consider states lying not on a smooth curve, but on a manifold of higher dimension, the situation becomes much more complex. We show that the notion of `distinguishability of close-by states' depends strongly on the measurement resources one allows oneself, and on a further specification of the task at hand. The quantum information matrix no longer seems to play a central role.
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.