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Optimal generalized variance and quantum Fisher information

Géza Tóth, Dėnes Petz

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

We define the generalized variance based on requiring that (i) it equals the usual variance for pure states and (ii) it is concave. For a quantum system of any size, we show that the usual variance is the smallest generalized variance, which makes it optimal for using it in entanglement criteria based on uncertainty relations. Similarly, we define the generalized quantum Fisher information, replacing the requirement of concavity by convexity. For rank-2 density matrices, we show that the quantum Fisher information is the largest among generalized quantum Fisher informations. We relate our findings to the results of [D. Petz, J. Phys. A: Math. Gen. 35, 79 (2003); P. Gibilisco, F. Hiai and D. Petz, IEEE Trans. Inform. Theory 55, 439 (2009)].

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We define the generalized variance based on requiring that (i) it equals the usual variance for pure states and (ii) it is concave. For a quantum system of any size, we show that the usual variance is the smallest generalized variance, which makes it optimal for using it in entanglement criteria based on uncertainty relations. Similarly, we define the generalized quantum Fisher information, replacing the requirement of concavity by convexity. For rank-2 density matrices, we show that the quantum Fisher information is the largest among generalized quantum Fisher informations. We relate our findings to the results of [D. Petz, J. Phys. A: Math. Gen. 35, 79 (2003); P. Gibilisco, F. Hiai and D. Petz, IEEE Trans. Inform. Theory 55, 439 (2009)].
We show that the variance is its own concave roof. For rank-2 density matrices and operators with zero diagonal elements in the eigenbasis of the density matrix, we prove analytically that the quantum Fisher information is four times the convex roof of the variance. Strong numerical evidence suggests that this statement is true even for operators with nonzero diagonal elements or density matrices with a rank larger than 2.2.2. We also find that within the different types of generalized quantum Fisher information considered in Petz [J. Phys. A 35, 929 (2002)] and Gibilisco, Hiai, and Petz [IEEE Trans. Inf. Theory 55, 439 (2009)], after appropriate normalization, the quantum Fisher information is the largest. Hence, we conjecture that the quantum Fisher information is four times the convex roof of the variance even for the general case.
We examine important properties of the difference between the variance and the quantum Fisher information over four, i.e., (ΔA)2−FQ[ϱ,A]/4.(ΔA)^2-F_{\rm Q}[\varrho,A]/4.(ΔA)2−FQ​[ϱ,A]/4. We find that it is equal to a generalized variance defined in Petz [J. Phys. A 35, 929 (2002)] and Gibilisco, Hiai, and Petz [IEEE Trans. Inf. Theory 55, 439 (2009)]. We present an upper bound on this quantity that is proportional to the linear entropy. As expected, our relation shows that for states that are close to being pure, the quantum Fisher information over four is close to the variance. We also obtain the variance and the quantum Fisher information averaged over all Hermitian operators, and examine its relation to the von Neumann entropy. Apart from the usual quantum Fisher information, we also consider the Kubo-Mori-Bogoliubov quantum Fisher information.
We discuss recent findings relating the quantum Fisher information to convex roofs of variances. We present several improvements on the Robertson-Schrodinger uncertainty relation. In all these improvements, we consider a decomposition of the density matrix into a mixture of pure states, and use the fact that the Robertson-Schrodinger uncertainty relation is valid for all these components. By considering a convex roof of the bound, we obtain an alternative derivation of the relation in F. Frowis, R. Schmied, and N. Gisin [Phys. Rev. A 92, 012102 (2015)], and we can also list a number of conditions that are needed to saturate the relations. We also gain new insights about the Cramer-Rao bound. By considering a concave roof of the bound in the Robertson-Schrodinger uncertainty relation, we obtain other type of improvements. We consider similar techniques for uncertainty relations with three variances. Finally, we present further uncertainty relations that provide lower bounds on the metrological usefulness of bipartite quantum states based on the variances of canonical position and momentum operators for two-mode continuous variable systems and angular momentum operators for spin systems.
We present several inequalities related to the Robertson-Schr\"odinger uncertainty relation. In all these inequalities, we consider a decomposition of the density matrix into a mixture of states, and use the fact that the Robertson-Schr\"odinger uncertainty relation is valid for all these components. By considering a convex roof of the bound, we obtain an alternative derivation of the relation in Fr\"owis et al. [Phys. Rev. A 92, 012102 (2015)], and we can also list a number of conditions that are needed to saturate the relation. We present a formulation of the Cram\'er-Rao bound involving the convex roof of the variance. By considering a concave roof of the bound in the Robertson-Schr\"odinger uncertainty relation over decompositions to mixed states, we obtain an improvement of the Robertson-Schr\"odinger uncertainty relation. We consider similar techniques for uncertainty relations with three variances. Finally, we present further uncertainty relations that provide lower bounds on the metrological usefulness of bipartite quantum states based on the variances of the canonical position and momentum operators for two-mode continuous variable systems. We show that the violation of well-known entanglement conditions in these systems discussed in Duan et al. [Phys. Rev. Lett. 84, 2722 (2000)] and Simon [Phys. Rev. Lett. 84, 2726 (2000)] implies that the state is more useful metrologically than certain relevant subsets of separable states. We present similar results concerning entanglement conditions with angular momentum operators for spin systems.
We propose an entanglement criterion in terms of quantum Fisher information, which complements the criterion based on variance and local uncertainty relations [H. F. Hofmann and S. Takeuchi, Phys. Rev. A 68, 032103 (2003)]. We illustrate the significance of this criterion by showing that it can reveal entanglement undetectable by the variance method. The dual relation between the criterion based on quantum Fisher information and that based on variance are highlighted. The combination of these two criteria leads to a refined method for entanglement detection, which is stronger than when either one is used alone.
Quantum Fisher information places the fundamental limit to the accuracy of estimating an unknown parameter. Here we shall provide the quantum Fisher information an operational meaning: a mixed state can be so prepared that a given observable has the minimal averaged variance, which equals exactly to the quantum Fisher information for estimating an unknown parameter generated by the unitary dynamics with the given observable as Hamiltonian. In particular we shall prove that the quantum Fisher information is the convex roof of the variance, as conjectured by Toth and Petz based on numerical and analytical evidences, by constructing explicitly a pure-state ensemble of the given mixed state in which the averaged variance of a given observable equals to the quantum Fisher information.