Symmetric logarithmic derivative for general n-level systems and the quantum Fisher information tensor for three-level systems

TL;DR

This paper derives a general formula for the symmetric logarithmic derivative in quantum systems and applies it to three-level systems, enabling the calculation of the quantum Fisher information tensor for q-trits.

Contribution

It provides an explicit, geometrically derived formula for the symmetric logarithmic derivative applicable to arbitrary mixed quantum systems, including three-level systems.

Findings

Derived explicit formula for symmetric logarithmic derivative in quantum systems.

Applied the formula to three-level systems (q-trits) for the first time.

Calculated the quantum Fisher information tensor for q-trits, including degenerate cases.

Abstract

Within a geometrical context, we derive an explicit formula for the computation of the symmetric logarithmic derivative for arbitrarily mixed quantum systems, provided that the structure constants of the associated unitary Lie algebra are known. To give examples of this procedure, we first recover the known formulae for two-level mixed and three-level pure state systems and then apply it to the novel case of U(3), that is for arbitrarily mixed three-level systems (q-trits). Exploiting the latter result, we finally calculate an expression for the Fisher tensor for a q-trit considering also all possible degenerate subcases.