Analysis of a convenient information bound for general quantum channels

TL;DR

This paper investigates the bounds of Fisher information in quantum channels, clarifying when these bounds are attainable and providing conditions for optimal measurements, thus advancing quantum parameter estimation theory.

Contribution

It establishes the relationship between the SM bound, SLD quantum information, and Fisher information, and extends attainability conditions to multi-parameter quantum channels.

Findings

The SLD quantum information is bounded above by the SM bound.

Equality conditions for Fisher information and bounds are identified.

Attainability conditions are shown not to hold universally for all channels.

Abstract

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.\ $H(\theta) \leq C_{\Upsilon}(\theta)$ and we find conditions for equality. As the Fisher information is less than or equal to the SLD quantum information, i.e. $F_M(\theta) \leq H(\theta)$, we can deduce when equality holds in $F_M(\theta) \leq C_{\Upsilon}(\theta)$. 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.