Geometric measures of quantum correlations: characterization, quantification, and comparison by distances and operations

Abstract

We investigate and compare three distinguished geometric measures of bipartite quantum correlations that have been recently introduced in the literature: the geometric discord, the measurement-induced geometric discord, and the discord of response, each one defined according to three contractive distances on the set of quantum states, namely the trace, Bures, and Hellinger distances. We establish a set of exact algebraic relations and inequalities between the different measures. In particular, we show that the geometric discord and the discord of response based on the Hellinger distance are easy to compute analytically for all quantum states whenever the reference subsystem is a qubit. These two measures thus provide the first instance of discords that are simultaneously fully computable, reliable (since they satisfy all the basic Axioms that must be obeyed by a proper measure of quantum correlations), and operationally viable (in terms of state distinguishability). We apply the general mathematical structure to determine the closest classical-quantum state of a given state and the maximally quantum-correlated states at fixed global state purity according to the different distances, as well as a necessary condition for a channel to be quantumness breaking.

36 pages, 5 figures; the third version differs from the first one by some added references and marginal modifications, and from the second one by some minor corrected typos

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Cited by in corpus (33)

• Quantum Coherence as a Resource
• Measures and applications of quantum correlations
• Quantum discord and its allies: a review
• Strong Quantum Darwinism and Strong Independence is equivalent to Spectrum Broadcast Structure
• Family of coherence measures and duality between quantum coherence and path distinguishability
• Versatile Gaussian probes for squeezing estimation
• Coherence number as a discrete quantum resource
• Convergence rates for quantum evolution & entropic continuity bounds in infinite dimensions
• Device-independent quantum reading and noise-assisted quantum transmitters
• Gaussian State-Based Quantum Illumination with Simple Photodetection
• Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance
• Dissipative dynamics of quantum correlation quantifiers under decoherence channels
• Assigning Temperatures to Eigenstates
• Entanglement-Coherence and Discord-Coherence analytical relations for X states
• Random density matrices: Analytical results for mean root fidelity and mean square Bures distance
• Hierarchy of correlation quantifiers comparable to negativity
• Measurement - induced qudit geometric discord
• Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance
• Two-qubit trace-norm geometric discord: the complete solution
• Correlation Minor Norms, Entanglement Detection and Discord
• Characterizing entanglement using quantum discord over state extensions
• Geometric quantum discord of an arbitrary two-qudit state: the exact value and general upper bounds
• Geometric monotones of violations of quantum realism
• Algebraic Probability-Theoretic Characterization of Quantum Correlations
• Finite-temperature quantum discordant criticality
• Affinity-based geometric discord and quantum speed limits of its creation and decay
• Simulating the discrete-time quantum walk dynamics with simultaneous coin and shift operators
• Mimicking the Hadamard discrete-time quantum walk with a time-independent Hamiltonian
• Geometric measures of quantum nonlocality: characterization, quantification, and comparison by distances and operations
• Characterising two-sided quantum correlations beyond entanglement via metric-adjusted f-correlations
• Entanglement as the cross-symmetric part of quantum discord
• Geometric measures of discordlike quantum correlations based on Tsallis relative entropy
• Distance-based approach to quantum coherence and nonclassicality