Measures of coherence generating power for quantum unital operations

TL;DR

This paper introduces a geometric framework using coherence matrices to quantify the coherence generating power of unital quantum operations, generalizing existing probabilistic measures and exploring their properties.

Contribution

It defines coherence matrices and new measures of CGP, providing a geometric interpretation and extending probabilistic approaches with a focus on unital quantum operations.

Findings

Coherence matrices effectively quantify CGP of unital operations.

New measures relate to the separation from incoherent operations.

Additive measures are developed for unitary operations with tensor product structures.

Abstract

Given an orthonormal basis in a $d$-dimensional Hilbert space and a unital quantum operation $\cal E$ acting on it one can define a non-linear mapping that associates to $\cal E$ a $d\times d$ real-valued matrix that we call the Coherence Matrix of $\cal E$ with respect to $B$. We show that one can use this coherence matrix to define vast families of measures of the coherence generating power (CGP) of the operation. These measures have a natural geometrical interpretation as separation of $\cal E$ from the set of incoherent unital operations. The probabilistic approach to CGP discussed in P. Zanardi et al., arXiv:1610.00217 can be reformulated and generalized introducing, alongside the coherence matrix, another $d\times d$ real-valued matrix, the Simplex Correlation Matrix. This matrix describes the relevant statistical correlations in the input ensemble of incoherent states. Contracting these two matrices one finds CGP measures associated with the process of preparing the given incoherent ensemble and processing it with the chosen unital operation. Finally, in the unitary case, we discuss how these concepts can be made compatible with an underlying tensor product structure by defining families of CGP measures that are additive.