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arXiv:2012.02965 · Journal of Physics A Mathematical and Theoretical

Higher-order uncertainty bounds for mixed states

Alex J Belfield, Dorje C Brody

Published Sep 15, 2021• 3 citations
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Abstract

Uncertainty lower bounds for parameter estimations associated with a unitary family of mixed-state density matrices are obtained by embedding the space of density matrices in the Hilbert space of square-root density matrices. In the Hilbert-space setup the measure of uncertainty is given by the skew information of the second kind, while the uncertainty lower bound is given by the Wigner–Yanase skew information associated with the conjugate observable. Higher-order corrections to the uncertainty lower bound are determined by higher-order quantum skew moments; expressions for these moments are worked out in closed form.

DOI
10.1088/1751-8121/ac2717

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Uncertainty lower bounds for parameter estimations associated with a unitary family of mixed-state density matrices are obtained by embedding the space of density matrices in the Hilbert space of square-root density matrices. In the Hilbert-space setup the measure of uncertainty is given by the skew information of the second kind, while the uncertainty lower bound is given by the Wigner–Yanase skew information associated with the conjugate observable. Higher-order corrections to the uncertainty lower bound are determined by higher-order quantum skew moments; expressions for these moments are worked out in closed form.
Given a pure state vector | x ⟩ and a density matrix , the function defines a probability density on the space of pure states parameterised by density matrices. The associated Fisher–Rao information measure is used to define a unitary invariant Riemannian metric on the space of density matrices. An alternative derivation of the metric, based on square-root density matrices and trace norms, is provided. This is applied to the problem of quantum-state estimation. In the simplest case of unitary parameter estimation, new higher-order corrections to the uncertainty relations, applicable to general mixed states, are derived.
The variance of an observable in a quantum state is usually used to describe Heisenberg uncertainty relation. For mixed states, the variance includes quantum and classical uncertainties. By means of the skew information and the decomposition of the variance, a stronger uncertainty relation was presented by Luo [Phys. Rev. A 72, 042110 (2005)]. In this paper, by using Wigner-Yanase-Dyson information which is a generalization of the skew information, we propose a general uncertainty relation of mixed states.
Abstract In this paper, we build some uncertainty relations for arbitrary N quantum channels based on the Wigner–Yanase skew information. It is shown that our lower bounds are stronger than the ones proposed by Zhang et al (2021 Phys. Lett. A 387 127029; 2021 Laser Phys. Lett. 18 095204) respectively in all intervals. In particular, we derive several uncertainty relations for N unitary channels. Moreover, we establish several descending sequences of lower bounds for the uncertainty relations of N quantum channels. Detailed examples are provided.
A closed-form expression for Wigner–Yanase skew information in mixed-state quantum systems is derived. It is shown that limit values of the mixing coefficients exist such that Wigner–Yanase information is equal to Helstrom information. The latter constitutes an upper bound for the classical expected Fisher information, hence the inverse Wigner–Yanase information provides an approximate lower bound to the variance of an unbiased estimator of the parameter of interest. The advantage of approximating Helstrom’s sharp bound lies in the fact that Wigner–Yanase information is straightforward to compute, while it is often very difficult to obtain a feasible expression for Helstrom information. In fact, the latter requires the solution of an implicit second order matrix differential equation, while the former requires just scalar differentiation.
Abstract In this work, we derive state-dependent uncertainty relations (uncertainty equalities) in which commutators of incompatible operators (not necessarily Hermitian) are explicitly present and state-independent uncertainty relations based on the Wigner-Yanase (-Dyson) skew information. We derive uncertainty equality based on standard deviation for incompatible operators with mixed states, a generalization of previous works in which only pure states were considered. We show that for pure states, the Wigner-Yanase skew information based state-independent uncertainty relations become standard deviation based state-independent uncertainty relations which turn out to be tighter uncertainty relations for some cases than the ones given in previous works, and we generalize the previous works for arbitrary operators. As the Wigner-Yanase skew information of a quantum channel can be considered as a measure of quantum coherence of a density operator with respect to that channel, we show that there exists a state-independent uncertainty relation for the coherence measures of the density operator with respect to a collection of different channels. We show that state-dependent and state-independent uncertainty relations based on a more general version of skew information called generalized skew information which includes the Wigner-Yanase (-Dyson) skew information and the Fisher information as special cases hold. In qubits, we derive tighter state-independent uncertainty inequalities for different form of generalized skew informations and standard deviations, and state-independent uncertainty equalities involving generalized skew informations and standard deviations of spin operators along three orthogonal directions. Finally, we provide a scheme to determine the Wigner-Yanase (-Dyson) skew information of an unknown observable which can be performed in experiment using the notion of weak values.
Tighter sum uncertainty relations via (α,β,γ)(α,β,γ)(α,β,γ) weighted Wigner-Yanase-Dyson skew information
We establish tighter uncertainty relations for arbitrary finite observables via (α,β,γ)(α,β,γ)(α,β,γ) weighted Wigner-Yanase-Dyson ((α,β,γ)(α,β,γ)(α,β,γ)WWYD) skew information. The results are also applicable to the (α,γ)(α,γ)(α,γ) weighted Wigner-Yanase-Dyson ((α,γ)(α,γ)(α,γ)WWYD) skew information and the weighted Wigner-Yanase-Dyson (WWYD) skew information. We also present tighter lower bounds of quantum channels and unitary channels via (α,β,γ)(α,β,γ)(α,β,γ) modified weighted Wigner-Yanase-Dyson ((α,β,γ)(α,β,γ)(α,β,γ)MWWYD) skew information. Detailed examples are provided to illustrate tightness of our uncertainty relations.