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arXiv:1905.07522

Properties of quantum reactivity for a multipartite state

Shahabeddin Mostafanazhad aslmarand, Warner A. Miller, Tahereh Razaei, Paul M. Alsing, Verinder S. Rana

Published May 13, 2019• 1 citation
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Abstract

We discuss the properties of quantum state reactivity as a measure for quantum correlation. This information geometry-based definition is a generalization of the two qubit construction of Schumacher to multipartite quantum states. It requires a generalization of information distance to information areas as well as to higher-dimensional volumes. The reactivity is defined in the usual chemistry way as a ratio of surface area to volume. The reactivity is an average over all detector settings. We show that this measure posses the key features required for a measure of quantum correlation. We show that it is invariant under local unitary transformations, non{increasing under local operations and classical communication, and monotonic. Its maximum bound can't be obtained using only classical correlation. Furthermore, reactivity is an analytic function of measurement probabilities and easily extendable to higher multipartite states.

DOI
10.1117/12.2524224

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We discuss the properties of quantum state reactivity as a measure for quantum correlation. This information geometry-based definition is a generalization of the two qubit construction of Schumacher to multipartite quantum states. It requires a generalization of information distance to information areas as well as to higher-dimensional volumes. The reactivity is defined in the usual chemistry way as a ratio of surface area to volume. The reactivity is an average over all detector settings. We show that this measure posses the key features required for a measure of quantum correlation. We show that it is invariant under local unitary transformations, non{increasing under local operations and classical communication, and monotonic. Its maximum bound can't be obtained using only classical correlation. Furthermore, reactivity is an analytic function of measurement probabilities and easily extendable to higher multipartite states.
Geometry is often a valuable guide to complex problems in physics. In this paper, we introduce a novel geometric quantity called quantum reactivity (QR) to probe quantum correlations in higher-dimensional quantum systems. Much like quantum discord, QR is not a measure of quantum entanglement but can be useful in quantum information processes where a notion of quantum correlation in higher dimensions is needed. Both quantum discord and QR are extendable to an arbitrarily large number of qubits; however, unlike discord, QR satisfies the invariance under unitary operations. Our approach parallels Schumacher's singlet state triangle inequality, which used an information geometry-based entropic distance. We use a generalization of information distance to area, volume, and higher-dimensional volumes and then use these to define a quantity that we call QR, which is the familiar ratio of surface area to volume. We examine a spectrum of multipartite states (Werner, W, GHZ, randomly generated density matrices, etc.) and demonstrate that QR can provide an ordering of these quantum states as to their degree of quantum correlation.
Everett's concept of relative state can be viewed as a map that contains information about correlations between measurement outcomes on two quantum systems. We demonstrate how geometric properties of the relative state map can be used to develop operationally well-defined measures of the total correlation in bipartite quantum systems of arbitrary state space dimension. These measures are invariant under local unitary transformations and non-increasing under local operations. We show that some known correlation measures have a natural interpretation in terms of relative states.
Everett's concept of relative state can be viewed as a map that contains information about correlations between measurement outcomes on two quantum systems. We demonstrate how geometric properties of the relative state map can be used to develop operationally well-defined measures of the total correlation in bipartite quantum systems of arbitrary state space dimension. These measures are invariant under local unitary transformations and non-increasing under local operations. We show that some known correlation measures have a natural interpretation in terms of relative states.
We compute the quantum maximal correlation for bipartite Gaussian states of continuous-variable systems. Quantum maximal correlation is a measure of correlation with the monotonicity and tensorization properties that can be used to study whether an arbitrary number of copies of a resource state can be locally transformed into a target state without classical communication, known as the local state transformation problem. We show that the required optimization for computing the quantum maximal correlation of Gaussian states can be restricted to local operators that are linear in terms of phase-space quadrature operators. This allows us to derive a closed-form expression for the quantum maximal correlation in terms of the covariance matrix of Gaussian states. Moreover, we define Gaussian maximal correlation based on considering the class of local Hermitian operators that are linear in terms of phase-space quadrature operators associated with local homodyne measurements. This measure satisfies the tensorization property and can be used for the Gaussian version of the local state transformation problem when both resource and target states are Gaussian. We also generalize these measures to the multipartite case. Specifically, we define the quantum maximal correlation ribbon and then characterize it for multipartite Gaussian states.
Chirality is a fundamental property of many topological phases, yet it lacks a general information-theoretic formulation. In this work, we introduce a notion of chirality for generic quantum states, defined by the impossibility of transforming a state into its complex conjugate under local unitary operations. We propose several quantitative measures of chirality, including a faithful metric called the chiral log-distance, and a family of nested commutators of modular Hamiltonians. We show that chirality, although not a resource in the traditional sense, is intrinsically linked to two major classes of quantum resources: magic and quantum correlations. In particular, we demonstrate that (i) qubit stabilizer states are always non-chiral, (ii) the chiral log-distance provides a lower bound for several magic monotones, and (iii) a nested commutator-based chirality measure is lower bounded by a variant of interferometric power.
We introduce a new approach to evaluating entangled quantum networks using information geometry. Quantum computing is powerful because of the enhanced correlations from quantum entanglement. For example, larger entangled networks can enhance quantum key distribution (QKD). Each network we examine is an n-photon quantum state with a degree of entanglement. We analyze such a state within the space of measured data from repeated experiments made by n observers over a set of identically-prepared quantum states – a quantum state interrogation in the space of measurements. Each observer records a 1 if their detector triggers, otherwise they record a 0. This generates a string of 1’s and 0’s at each detector, and each observer can define a binary random variable from this sequence. We use a well-known information geometry-based measure of distance that applies to these binary strings of measurement outcomes,1–3 and we introduce a generalization of this length to area, volume and higher-dimensional volumes.4 These geometric equations are defined using the familiar Shannon expression for joint and mutual entropy.5 We apply our approach to three distinct tripartite quantum states: the |GHZi state, the |Wi state, and a separable state |Pi. We generalize a well-known information geometry analysis of a bipartite state to a tripartite state. This approach provides a novel way to characterize quantum states, and it may have favorable scaling with increased number of photons.