Quantumness of ensemble via coherence of Gram matrix

¹ School of Mathematical Sciences, Nanjing Normal University - Nanjing 210023, China
² Academy of Mathematics and Systems Science, Chinese Academy of Sciences - Beijing 100190, China
³ School of Mathematical Sciences, University of Chinese Academy of Sciences - Beijing 100049, China

^(a) luosl@amt.ac.cn (corresponding author)

Received: 27 February 2021
Accepted: 3 April 2021

Abstract

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 l₁-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.