Abstract The Gram matrix of an ensemble of pure states can be regarded as a quantum state, and the quantumness of the ensemble can be quantified by the coherence of the Gram matrix. By using the affinity between mixed states, the concept of Gram matrix of pure-state ensembles can be extended to the one of mixed-state ensembles. By utilizing the generalized α - z -relative Rényi entropy of coherence of Gram matrices, we present a new quantifier of quantumness of mixed-sate ensembles and further reveal its peculiar properties. To illustrate our quantumness of mixed-sate ensembles, we also calculate the quantumness for some detailed mixed-sate ensembles by deriving their analytical formulae.
The Gram matrix of a set of vectors, which encapsulates the relations between the constituent vectors, plays an important role in the exploration of both geometric and information-theoretic aspects of quantum state space. In view of its usefulness and importance, we study the Gram matrix of an ensemble of pure states (a set of pure states with a prior probability distribution) and reveal its fundamental properties. We highlight and exploit the fact that the Gram matrix of an ensemble can be formally regarded as a bona fide quantum state. The key idea is to employ coherence (with respect to the computational basis) of the Gram matrix (regarded as a quantum state) to quantify quantumness of the corresponding ensemble. In particular, we propose to use the l 1 -norm of coherence and the relative entropy of coherence, of the Gram matrix, as two significant quantifiers of quantumness. We illustrate the effectiveness and power of these quantifiers of quantumness by evaluating them for several important ensembles arising from quantum measurement and quantum cryptography. We further compare the quantumness based on the Gram matrix with various other quantifiers of quantumness in the literature, and show that although they are closely related in general, they have subtle differences and capture different aspects of ensembles, which shed light on the complexity of quantum ensembles.
This work presents a generalization of the Koashi-Winter relation for
α-Renyi entropies. This result is based on the Renyi\apos s entropy version of quantum Jensen Shannon divergence. By means of this definition, a classical correlations quantifier
Cα(ρAB)=supξABMBQα(ξABMB) is proposed, where the optimization is taken over the ensembles
ξABMB created by the outputs of the local measurement process. The main result is applied to the capacity of a quantum classical channel over a tripartite pure state
ψABE, that is rated above in function of the probability of success to discriminate the states in the ensemble
ξAEME, created by the local dephasing over partition
E, and the asymptotic log generalized robustness of partition
AB. Some analytical results are calculated for classical correlations and entanglement of formation.