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Journal of Shaanxi Normal University

Quantum correlation of a pair of quantum ensembles

Xueqi Cen

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

Quantum correlation in a pair{e,η}of quantum ensembles is studied and computed.In quantum information theory,aquantum state of a quantum system lies in some stateρiat a certain probability pithrough a quantum measurement.Thus,quantum statesρ1,ρ2,…,ρnand corresponding probabilities p1,p2,…,pnare obtained and so an ensemblee={pi,ρi}ni=1is presented.To do this,a 4-partite quantum stateρe,ηis constructed,and based on quantum relative entropy,a measure function is defined in order to reveal the quantum correlation of the pair.Some properties of the measure function are proved.

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•Aug 12, 2026•0 citations
Quantum correlation in a pair{e,η}of quantum ensembles is studied and computed.In quantum information theory,aquantum state of a quantum system lies in some stateρiat a certain probability pithrough a quantum measurement.Thus,quantum statesρ1,ρ2,…,ρnand corresponding probabilities p1,p2,…,pnare obtained and so an ensemblee={pi,ρi}ni=1is presented.To do this,a 4-partite quantum stateρe,ηis constructed,and based on quantum relative entropy,a measure function is defined in order to reveal the quantum correlation of the pair.Some properties of the measure function are proved.
We present a measure of quantum correlation of an ensemble of multiparty states. It is based on the idea of minimal entropy production in a measurement whose elements are locally distinguishable. It is shown, for pure ensembles, to be a relative entropy distance from a set of ensembles. For pure bipartite ensembles, which span the whole bipartite Hilbert space, the measure is bounded below by the average entanglement. We naturally obtain a monotonicity axiom for any measure of quantum correlation of ensembles for single systems. We evaluate the measure for certain cases. Subsequently we use this measure to propose a complementarity relation between our measure and the accessible information obtainable about the ensemble under local operations. The measure is well defined even for the case of a single system, where the complementarity relation is seen to be yet another face of the ``Heisenberg uncertainty relation.''
As a peculiar resource of quantum mechanics, quantum correlation has been applied to many aspects. In quantum information processing and quantum computing, the quantum correlation plays an extremely important role, and it has been a subject of further studies, principally due to the general belief that it is a fundamental resource for different quantum information processing tasks. In addition, correlation measure is a very important physical quantity in studying the quantum correlation. A well-defined correlation measure needs to have some necessary properties. By proving these necessary properties, we can deepen our understanding of correlation measure. As one of the key concepts of quantum information theory, relative entropy is always used to measure the uncertainty contained in the state of physical system. In order to better understand the properties and applications of correlation measure based on relative entropy, in this paper, according to the properties of the min relative entropy, we give the quantum correlation measure based on min relative entropy for two-partition and k-partition. Furthermore, we prove that it satisfies some necessary properties of quantum correlation measures, including the nonnegativity, the invariance under local unitary operators, and the monotonicity under completely positive trace-preserving. By proving these properties, we show that the given correlation measure is well-defined. Security of communication has received much attention since ancient times. In today's society, the internet, instant messaging and e-commerce applications are all related to the information security, and the information security is related to the vital interests of everyone. The information encryption is one of the important methods to ensure information security. As an important way to ensure information security, quantum channel has received more and more attention. At the end of the paper, we introduce the concept of quantum channel, and discuss the influence of quantum channel on the correlation measure based on min relative entropy under k-partition. By proposing a new correlation measure and proving the effect quantum channel on the measure, we can better describe the uncertainty contained in the state of physical system.
This paper introduces a generalized definition of quantum information entropy, aiming to provide a more comprehensive framework for quantifying information in complex quantum systems. The core of this work lies in extending the conventional concept of entropy, traditionally applied to classical information, to the realm of quantum mechanics. We explore various types of quantum entropy, including von Neumann entropy, relative entropy, and a newly proposed generalized entropy, denoted as *Eg*. The definition of *Eg* is based on a novel approach that considers the entanglement structure and correlations within quantum states. We establish relationships between these different entropy measures, demonstrating their interconnectedness within the broader context of quantum information theory. The paper further outlines a mathematical formalism for calculating *Eg*, providing a tool for analyzing and characterizing quantum information. Our generalized entropy provides a more flexible and potentially more accurate measure of quantum information, particularly in scenarios involving complex quantum correlations and multi-partite quantum systems. ---
A metastable state m subjected to a couple of pulses transits to an excited level α of an atom with probabilities p1 and p2, respectively. One photon is emitted in the transition from the level α to a lower atomic level η. A second photon is emitted in the transition from the level η to a lowest atomic level β. The first photon must be emitted before than the second one so they are both time ordered (time-bin) and entangled. It is calculated that the concurrence of the two produced time-bin photons as a function of p1, p2 and the branching ratios of the two transitions α→η and (η→β). It is found that the two produced photons are maximally entangled if the branching ratios are greater than 80%. It is also found that certain values of p1, p2 and the branching ratios preclude entanglement between the two produced photons. Intervals of values for p1 and p2 where the concurrence is large enough and then two-qubit quantum information processing protocols can be optimally implemented are found.
Temporal Rate Ontology (TRO) proposes that the locally realized rate of temporal progression is more fundamental than spacetime geometry. In its simplest representation, dτ = ρ(x,t) dt, where ρ is the local temporal-rate field and spacetime geometry is interpreted as an effective representation of relations among locally accumulated temporal rates. This paper develops a phenomenological, falsifiable test of TRO at the interface between temporal-rate structure and quantum measurement, and formulates the proposal around three explicit mathematical consistency conditions rather than a single postulated weighting law. The primitive object is a physically instantiated history γ, admissible only as a member of a decoherent partition C = {γ_1,…,γ_n} in the sense of consistent- or decoherent-histories quantum mechanics. Within such a partition, ordinary quantum-measure theory already supplies well-defined probabilities p_Q(γ_i|C), satisfying Σ_i p_Q(γ_i|C) = 1; this paper does not modify Sorkin's quantum measure itself, which is not in general additive outside a decoherent partition, but reweights the ordinary probabilities that quantum measure theory already assigns within one. The proposed TRO probability is P_TRO(γ_i|C) = p_Q(γ_i|C) exp(ηΘ[γ_i]) / Σ_{γ_j∈C} p_Q(γ_j|C) exp(ηΘ[γ_j]), where Θ[γ] = ∫_γ dτ = ∫_γ ρ dt is the accumulated temporal measure along γ, well-defined only for a timelike physical history. We require three consistency conditions. First, shift invariance: normalized probabilities must be unchanged under a common additive shift of every history's Θ value. Under positivity and mild regularity, this fixes the weighting function uniquely up to normalization as F(Θ)=C exp(ηΘ). Second, history-partition invariance: two admissible decoherent partitions describing the same physically coarse-grained outcome must assign it the same TRO probability. At a single coarse-graining step this forces the effective temporal composition law Θ_eff[Γ]=(1/η) ln[(Σ_i p_i e^{ηΘ_i})/(Σ_i p_i)], which reduces to the ordinary probability-weighted mean proper time as η→0 and is associative under arbitrary finite regrouping. Third, no-signalling: temporal weighting must not allow a spacelike-separated measurement choice to alter a remote marginal. A local-reweighting plus constrained information-projection construction guarantees no-signalling marginals by definition but encounters a support obstruction for maximally correlated states. For binary full-support quantum tables, the constrained reconstruction can nevertheless be reduced exactly to an odds-ratio equation for the reconstructed correlation E_xy; the resulting correlation differs from the quantum value only at second order in the local temporal biases, so CHSH deviations begin at O(η²) for fixed finite temporal contrasts. This does not establish quantum realizability or the Tsirelson bound, which remain additional consistency tests. The antecedent physical premise, that an accumulated temporal history may enter outcome probabilities at all, is not derived here from the deeper dynamics of ρ. It is the phenomenological TRO hypothesis tested by this paper; the propositions below determine what mathematical form and consistency constraints such a dependence must satisfy if it exists. We also state explicitly, that in the minimal model developed here temporal weighting does not modify coherent unitary evolution; it enters only in the probability assigned to members of an already-decohered history set, after ordinary quantum dynamics and decoherence have run their course. We separate this proposal from the decoherence-based proper-time witness of Zych, Costa, Pikovski, and Brukner (2011), note that a photon cannot carry Θ since dτ = 0 identically on a null worldline, and record the dimensional content [η] = T⁻¹, with η = κ/τ* for dimensionless κ and characteristic temporal scale τ*.