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Physical Review A

Quantifying the quantumness of ensembles

Nan Li, Shunlong Luo, Yuanyuan Mao

Published Aug 25, 2017• 17 citations
Publisher

Abstract

We quantify the quantumness of quantum ensembles in terms of noncommutativity between pairs of the square root of the constituent states. We prove that this measure satisfies desirable and intuitive properties required naturally for a measure of quantumness, such as positivity, unitary invariance, subadditivity, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Some applications and implications are indicated.

DOI
10.1103/physreva.96.022132

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We quantify the quantumness of quantum ensembles in terms of noncommutativity between pairs of the square root of the constituent states. We prove that this measure satisfies desirable and intuitive properties required naturally for a measure of quantumness, such as positivity, unitary invariance, subadditivity, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Some applications and implications are indicated.
The quantification of the quantumness of a quantum ensemble has theoretical and practical significance in quantum information theory. We propose herein a class of measures of the quantumness of quantum ensembles using the unitary similarity invariant norms of the commutators of the constituent density operators of an ensemble. Rigorous proof shows that they share desirable properties for a measure of quantumness, such as positivity, unitary invariance, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Several specific examples illustrate the applications of these measures of quantumness in studying quantum information.
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We study quantum coarse-grained entropy and demonstrate that the gap in entropy between local and global coarse-grainings is a natural generalization of entanglement entropy to mixed states and multipartite systems. This ``quantum correlation entropy'' Sqc{S}^{q}cSqc is additive over independent systems, is invariant under local unitary operations, measures total nonclassical correlations (vanishing on states with strictly classical correlation), and reduces to the entanglement entropy for bipartite pure states. It quantifies how well a quantum system can be understood via local measurements and ties directly to nonequilibrium thermodynamics, including representing a lower bound on the quantum part of thermodynamic entropy production. We discuss two other measures of nonclassical correlation to which this entropy is equivalent and argue that together they provide a unique thermodynamically distinguished measure.
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We study the local indistinguishability problem of quantum states. By introducing an easily calculated quantity, non-commutativity, we present an criterion which is both necessary and sufficient for the local indistinguishability of a complete set of pure orthogonal product states. A constructive distinguishing procedure to obtain the concrete local measurements and classical communications is given. The non-commutativity of ensembles can be also used to characterize the quantumness for classical-quantum or quantum-classical correlated states.
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.