Properties of Nonfreeness: An Entropy Measure of Electron Correlation

Gottlieb, Alex D.; Mauser, Norbert J.

doi:10.1142/s0219749907003201

Cited by 26 publications

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“…After publication of the paper we learned that the relative entropy was already proposed by A. D. Gottlieb and N. J.Mauser [36] as a means to quantify correlations of a many-fermion state by comparing its density operator with that of the corresponding free state.…”

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Quantification of Correlations in Quantum Many-Particle Systems

et al. 2012

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We introduce a well-defined and unbiased measure of the strength of correlations in quantum many-particle systems which is based on the relative von Neumann entropy computed from the density operator of correlated and uncorrelated states. The usefulness of this general concept is demonstrated by quantifying correlations of interacting electrons in the Hubbard model and in a series of transition-metal oxides using dynamical mean-field theory.PACS numbers: 74.70. Tx, 71.10.Fd, 71.30.+h Correlations in solids are the origin of many surprising phenomena such as Mott-insulating behavior and hightemperature superconductivity [1][2][3]. During the last few years the investigation of correlations between ultracold atoms in optical lattices has opened another fascinating field of research [4].In many-body physics correlations are conventionally defined as the effects which go beyond factorization approximations such as Hartree-Fock theory [5]. The actual strength of correlations in a given system is usually quantified by an interaction strength U relative to an energy unit such as the bandwidth W , or by comparing expectation values of particular operators, e.g., the interaction or the total energy [6][7][8][9][10]. However, there are many other quantities which can in principle be used to measure the correlation strength. Indeed "correlations" are by definition a relative concept since they always require the comparison with some reference system. Any approach which employs the expectation value of a particular set of operators for comparison will be biased. This raises a fundamental question: Is there an objective way to quantify the correlations of a system, which even allows one to compare the correlation strength of different systems?Maximal information about a general quantum state is provided by the corresponding density operator ("statistical operator")ρ = i p i |ψ i ψ i |, where p i is the probability for the quantum state |ψ i to be present in the mixture [11]. If there exists a basis in whichρ can be completely factorized (ρ →ρ PS , where PS refers to "product state"), the corresponding state is by definition uncorrelated [12]. To quantify the correlation strength of a quantum state one has to compareρ withρ PS in a suitable way.In this Letter we propose to quantify correlations within a statistical approach which is based on the concept of the von Neumann entropy [11,13]. We will show that the relative entropy [14] of a quantum state with respect to an uncorrelated product state provides a well-defined, unbiased and useful measure of the correlation strength in that system. This concept not only allows one to quantify correlations, but even to compare the correlation strength of different quantum states. In quantum theory the von Neumann entropy is defined as S(ρ) = − lnρ ρ = −Tr(ρ lnρ). It quantifies the degree to which a quantum system is in a mixed state. In this way one can also define a relative entropy [14]provided the support [15] ofρ is contained in the support ofσ; otherwise ∆S(ρ||σ) is infinite. ...

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Quantification of Correlations in Quantum Many-Particle Systems

et al. 2012

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“…Apparently, this problem is about the geometry of a multifermion Hilbert space. It naturally provides a geometric measure [25][26][27] of the entanglement between the fermions. Compared to those measures based on Schmidt decomposition or dividing a multipartite system into subsystems, it has the advantage of identical particles being treated identically, namely, on an equal footing [28].…”

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Optimal Slater-determinant approximation of fermionic wave functions

Zhang

Mauser

2016

Phys. Rev. A

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We study the optimal Slater-determinant approximation of an N -fermion wave function analytically. That is, we seek the Slater-determinant (constructed out of N orthonormal single-particle orbitals) wave function having largest overlap with a given N -fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the Greenberger-Horne-Zeilinger state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the optimal Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the optimal Slater approximation of some special target wave functions.

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“…[8,9,10,11,12,13,14,15] In Refs. [16,17], these concepts were extended to mixed states, which requires the full density matrix instead of the 1PDM. The resulting measure of correlation entropy, called "nonfreeness," [17] is for a pure state equivalent to the particle-hole symmetric correlation entropy introduced in Ref.…”

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“…With this γ ρ , Ref. [17] defines as "nonfreeness" the relative entropy [18] C(ρ) ≡ −Tr ρ log Γ γρ + Tr ρ log ρ .…”

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Physics behind the minimum of relative entropy measures for correlations

Held

Mauser²

2013

Eur. Phys. J. B

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Abstract. The relative entropy of a correlated state and an uncorrelated reference state is a reasonable measure for the degree of correlations. A key question is however which uncorrelated state to compare to. The relative entropy becomes minimal for the uncorrelated reference state that has the same oneparticle density matrix as the correlated state. Hence, this particular measure, coined nonfreeness, is unique and reasonable. We demonstrate that for relevant physical situations, such as finite temperatures or a correlation enhanced orbital splitting, other choices of the uncorrelated state, even educated guesses, overestimate correlations. e.g. in high-temperature superconductors. However, correlations are particularly difficult to deal with in theory and even a universally agreed definition, or measure, of "correlation" is hitherto lacking. There is a general agreement that a Hartree-Fock Slater determinant represents an uncorrelated state, even though such a wave function includes formally something one might call "correlations", which originate from the antisymmetrization of the wave function. Hence a correlation measure typically considers the difference of the correlated state vs. an uncorrelated Hartree-Fock calculation. In this situation, the questions are: For what quantity should one consider the difference? To which uncorrelated (possibly mixed) state should one compare to?One possibility to quantify correlation is to look at the energy differencebetween the (correlated) state investigated (E corr ) and an uncorrelated (or free) state E free , which is also coined correlation energy. This is e.g. the typical quantity considered in quantum chemistry or density functional theory (DFT) [4,5,6]. The exchange-correlation energy E xc is, as the difference to the Hartree energy, also readily accessible in DFT, at least within e.g. the local density approximation (LDA). This requires, however, still a separation into exchange and correlation part. In many-body theory on the other hand, one often considers two particle correlation functions of the type [7]

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Properties of Nonfreeness: An Entropy Measure of Electron Correlation

Cited by 26 publications

References 27 publications

Quantification of Correlations in Quantum Many-Particle Systems

Quantification of Correlations in Quantum Many-Particle Systems

Optimal Slater-determinant approximation of fermionic wave functions

Physics behind the minimum of relative entropy measures for correlations

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