# Polynomial measure of coherence

## TL;DR

This paper introduces a polynomial-based measure of quantum coherence, explores its properties, and provides analytical formulas and bounds for symmetric and mixed states, advancing the quantification of coherence in quantum systems.

## Contribution

It defines a polynomial coherence measure, investigates its properties, and derives analytical formulas and bounds, especially for symmetric and mixed states.

## Key findings

- No polynomial coherence measure satisfies the zero criterion for all incoherent states except in the qubit case.
- A necessary condition for polynomial coherence measures is established.
- Analytical formulas and lower bounds are derived for symmetric and mixed states.

## Abstract

Coherence, the superposition of orthogonal quantum states, is indispensable in various quantum processes. Inspired by the polynomial invariant for classifying and quantifying entanglement, we first define polynomial coherence measure and systematically investigate its properties. Except for the qubit case, we show that there is no polynomial coherence measure satisfying the criterion that its value takes zero if and only if for incoherent states. Then, we release this strict criterion and obtain a necessary condition for polynomial coherence measure. Furthermore, we give a typical example of polynomial coherence measure for pure states and extend it to mixed states via a convex-roof construction. Analytical formula of our convex-roof polynomial coherence measure is obtained for symmetric states which are invariant under arbitrary basis permutation. Consequently, for general mixed states, we give a lower bound of our coherence measure.