Geometric quantum discord through the Schatten 1-norm

Paula, F. M.; Oliveira, Thiago R. de; Sarandy, M. S.

doi:10.1103/physreva.87.064101

Cited by 242 publications

(266 citation statements)

References 40 publications

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“…(45) [122,123,80,78,124]. Because D p is only contractive for p = 1, the trace distance based geometric measure, alias "trace distance discord", is the only one of its class that directly obeys Requirements (ii), (iv), and (v), hence emerging as the most suitable choice out of all the Schatten D p norm induced geometric measures.…”

Section: Trace Distancementioning

confidence: 99%

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Measures and applications of quantum correlations

Adesso¹,

Bromley²,

Cianciaruso³

2016

J. Phys. A: Math. Theor.

395

416

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Abstract. Quantum information theory is built upon the realisation that quantum resources like coherence and entanglement can be exploited for novel or enhanced ways of transmitting and manipulating information, such as quantum cryptography, teleportation, and quantum computing. We now know that there is potentially much more than entanglement behind the power of quantum information processing. There exist more general forms of non-classical correlations, stemming from fundamental principles such as the necessary disturbance induced by a local measurement, or the persistence of quantum coherence in all possible local bases. These signatures can be identified and are resilient in almost all quantum states, and have been linked to the enhanced performance of certain quantum protocols over classical ones in noisy conditions. Their presence represents, among other things, one of the most essential manifestations of quantumness in cooperative systems, from the subatomic to the macroscopic domain. In this work we give an overview of the current quest for a proper understanding and characterisation of the frontier between classical and quantum correlations in composite states. We focus on various approaches to define and quantify general quantum correlations, based on different yet interlinked physical perspectives, and comment on the operational significance of the ensuing measures for quantum technology tasks such as information encoding, distribution, discrimination and metrology. We then provide a broader outlook of a few applications in which quantumness beyond entanglement looks fit to play a key role.

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Section: Trace Distancementioning

confidence: 99%

“…Some distances D δ between two quantum states ρ and σ. Details on the definitions and discussions of contractivity can be found in [77,78,79].…”

Section: (D)mentioning

confidence: 99%

Measures and applications of quantum correlations

Adesso¹,

Bromley²,

Cianciaruso³

2016

J. Phys. A: Math. Theor.

395

416

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“…These measures are based on different metrics quantifying the minimum distance of the quantum state from the set of all possible classicalquantum states [8,[35][36][37][38][39][40]. Although a collection of distance metrics have been used to characterize geometric measures of quantum correlations, it has been shown that out of all the Schatten p-norm distances, only the onenorm distance has properties that are rather similar to the QD as well as QWD [38][39][40]. But due to the difficulty in optimizing the measure, analytically closed forms of one-norm geometric discord has been obtained only for some special types of states, e.g., Bell diagonal states, and the X states [40].…”

Section: Constrained Quantum Correlations: Error In Estimationmentioning

confidence: 99%

Reducing computational complexity of quantum correlations

et al. 2015

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We address the issue of reducing the resource required to compute information-theoretic quantum correlation measures like quantum discord and quantum work deficit in two qubits and higher dimensional systems. We show that determination of the quantum correlation measure is possible even if we utilize a restricted set of local measurements. We find that the determination allows us to obtain a closed form of quantum discord and quantum work deficit for several classes of states, with a low error. We show that the computational error caused by the constraint over the complete set of local measurements reduces fast with an increase in the size of the restricted set, implying usefulness of constrained optimization, especially with the increase of dimensions. We perform quantitative analysis to investigate how the error scales with the system size, taking into account a set of plausible constructions of the constrained set. Carrying out a comparative study, we show that the resource required to optimize quantum work deficit is usually higher than that required for quantum discord. We also demonstrate that minimization of quantum discord and quantum work deficit is easier in the case of two-qubit mixed states of fixed ranks and with positive partial transpose in comparison to the corresponding states having non-positive partial transpose. Applying the methodology to quantum spin models, we show that the constrained optimization can be used with advantage in analyzing such systems in quantum information-theoretic language. For bound entangled states, we show that the error is significantly low when the measurements correspond to the spin observables along the three Cartesian coordinates, and thereby we obtain expressions of quantum discord and quantum work deficit for these bound entangled states.

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“…Clearly, Hilbert-Schmidt norm is not the unique distance which can be defined in the space of quantum states. Several distances are possible (trace distance, Bures distance, ...) with their own advantages and drawbacks and each one might be useful for some appropriate purpose [36,37,38,39].…”

Section: Introductionmentioning

confidence: 99%

Unified scheme for correlations using linear relative entropy

Daoud

Laamara²,

Kaydi³

2014

Physics Letters A

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A linearized variant of relative entropy is used to quantify in a unified scheme the different kinds of correlations in a bipartite quantum system. As illustration, we consider a two-qubit state with parity and exchange symmetries for which we determine the total, classical and quantum correlations. We also give the explicit expressions of its closest product state, closest classical state and the corresponding closest product state. A closed additive relation, involving the various correlations quantified by linear relative entropy, is derived.

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Geometric quantum discord through the Schatten 1-norm

Cited by 242 publications

References 40 publications

Measures and applications of quantum correlations

Measures and applications of quantum correlations

Reducing computational complexity of quantum correlations

Unified scheme for correlations using linear relative entropy

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