Measures of coherence-generating power for quantum unital operations

Abstract

Given an orthonormal basis $B$ in a $d$-dimensional Hilbert space and a unital quantum operation $\mathcal{E}$ acting on it, one can define a nonlinear mapping that associates with $\mathcal{E}$ a $d\ifmmode\times\else\texttimes\fi{}d$ real-valued matrix that we call the coherence matrix of $\mathcal{E}$ with respect to $B$. This is the Gram matrix of the coherent part of the basis projections of $B$ under $\mathcal{E}$. 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 $\mathcal{E}$ from the set of incoherent unital operations. The probabilistic approach to CGP discussed in P. Zanardi et al. [Phys. Rev. A 95, 052306 (2017).] can be reformulated and generalized, introducing, alongside the coherence matrix, another $d\ifmmode\times\else\texttimes\fi{}d$ 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.