Error trade-off relations for two-parameter unitary model with commuting generators

TL;DR

This paper explores the existence and characteristics of error trade-off relations in two-parameter unitary models with commuting generators, demonstrating their generic presence in finite-dimensional quantum systems through analytical and numerical analysis.

Contribution

It establishes the generic occurrence of error trade-off relations in finite-dimensional quantum models with commuting generators, supported by analytical and numerical examples.

Findings

Error trade-off relations exist in finite-dimensional systems with commuting generators.

Trade-off relations can be characterized by SLD and RLD Cramer-Rao bounds intersecting.

Range of reference states where trade-offs occur is analytically determined to be up to half of the possible range.

Abstract

We investigate whether a trade-off relation between the diagonal elements of the mean square error matrix exists for the two-parameter unitary models with mutually commuting generators. We show that the error trade-off relation which exists in our models of a finite dimension system is a generic phenomenon in the sense that it occurs with a finite volume in the spate space. We analyze a qutrit system to show that there can be an error trade-off relation given by the SLD and RLD Cramer-Rao bounds that intersect each other. First, we analyze an example of the reference state showing the non-trivial trade-off relation numerically, and find that its eigenvalues must be in a certain range to exhibit the trade-off relation. For another example, one-parameter family of reference states, we analytically show that the non-trivial relation always exists and that the range where the trade-off relation exists is up to about a half of the possible range.