Quantifying the quantumness of pure-state ensembles via coherence of Gram matrix

The Gram matrix of a set of pure states plays a crucial role in exploring the information content of quantum states. Since the Gram matrix of a pure-state ensemble can be regarded as a quantum state, we can quantify the quantumness of the corresponding quantum ensemble by means of the coherence of the Gram matrix. In particular, we introduce two natural measures of quantumness of an ensemble by means of the coherence of the Gram matrix based on Hilbert-Schmidt distance and skew information, respectively, and investigate their fundamental properties. In addition, notice that the average coherence and the maximum coherence are independent of the choice of a reference basis, we further propose to use the average coherence and the maximum coherence of the Gram matrix to quantify the generalized quantumness of the corresponding ensemble, and find that the generalized quantumness is not only related to the quantum states in the ensemble, but also to the probability distribution. We discuss the validity of these quantumness measures by calculating some important pure-state ensembles in quantum measurement and quantum cryptography. Finally, we compare these quantifiers with other existing quantifiers in the literatures and illustrate that different quantifiers can yield different features and quantumness orderings for quantum ensembles.