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Quantum Cramér-Rao bound on quantum metric as a multi-observable uncertainty relation

Wei Chen

Abstract

A version of quantum Cramér-Rao bound dictates that the covariance of any set of operators is bounded by a product of the derivatives of expectation values and the inverse of quantum metric. We elaborate that because quantum metric itself is the covariance of the generators of translation in the parameter space, quantum metric in any dimension is bounded by a product of itself and Berry curvature. The generator formalism further indicates that the bound is equivalent to a multi-observable uncertainty relation, which in the two-operator case recovers the Robertson-Schrödinger uncertainty relation. The momentum space quantum metric and spin operators of three-dimensional topological insulators under magnetic field are used to demonstrate the validity of the three-operator version of these bounds.

Quantum Cramér-Rao bound on quantum metric as a multi-observable uncertainty relation

Abstract

A version of quantum Cramér-Rao bound dictates that the covariance of any set of operators is bounded by a product of the derivatives of expectation values and the inverse of quantum metric. We elaborate that because quantum metric itself is the covariance of the generators of translation in the parameter space, quantum metric in any dimension is bounded by a product of itself and Berry curvature. The generator formalism further indicates that the bound is equivalent to a multi-observable uncertainty relation, which in the two-operator case recovers the Robertson-Schrödinger uncertainty relation. The momentum space quantum metric and spin operators of three-dimensional topological insulators under magnetic field are used to demonstrate the validity of the three-operator version of these bounds.
Paper Structure (10 sections, 52 equations, 1 figure)

This paper contains 10 sections, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quantum Cramér-Rao bound on variance of operators
  3. Geometric interpretation
  4. Uncertainty interpretation
  5. Applications to 3D topological insulators
  6. Geometric interpretation in 3D class AII topological insulators
  7. Uncertainty relation for spin operators in 3D class AII topological insulators
  8. Conclusions
  9. Proof of the operator version of quantum Cramér-Rao bound
  10. Index-explicit inequalities in 3D

Figures (1)

  • Figure 1: (a) The bound on quantum metric represented by $(V_{xx}^{g},V_{yy}^{g},V_{zz}^{g})$ and (b) the uncertainty relation for spin operators represented by $(V_{xx}^{\Lambda},V_{yy}^{\Lambda},V_{zz}^{\Lambda})$ in the lattice model of 3D class AII TI in the presence of a magnetic field $(B_{x},B_{y},B_{z})=(0.1,0.2,0.3)$, plotted along the high-symmetry line $\Gamma-X-M-R-\Gamma$. (c) and (d) show the same quantities but in an extremely large magnetic field $(B_{x},B_{y},B_{z})=(0.5,1,2)$. One sees that all these quantities remain positive everywhere, indicating that the bounds and uncertainty relations are always satisfied.