There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.

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Entanglement in many-body systems

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

One of the best signatures of nonclassicality in a quantum system is the existence of correlations that have no classical counterpart. Different methods for quantifying the quantum and classical parts of correlations are among the more actively studied topics of quantum-information theory over the past decade. Entanglement is the most prominent of these correlations, but in many cases unentangled states exhibit nonclassical behavior too. Thus distinguishing quantum correlations other than entanglement provides a better division between the quantum and classical worlds, especially when considering mixed states. Here different notions of classical and quantum correlations quantified by quantum discord and other related measures are reviewed. In the first half, the mathematical properties of the measures of quantum correlations are reviewed, related to each other, and the classical-quantum division that is common among them is discussed. In the second half, it is shown that the measures identify and quantify the deviation from classicality in various quantum-information-processing tasks, quantum thermodynamics, open-system dynamics, and many-body physics. It is shown that in many cases quantum correlations indicate an advantage of quantum methods over classical ones.

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The classical-quantum boundary for correlations: Discord and related measures

a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

Geometry of Quantum States: Frontmatter

We examine the quantum discord between two spins in the exact ground state of finite spin-1/2 arrays with anisotropic XY couplings in a transverse field B. It is shown that in the vicinity of the factorizing field B{sub s}, the discord approaches a common finite non-negligible limit which is independent of the pair separation and the coupling range. An analytic expression of this limit is provided. The discord of a mixture of aligned pairs in two different directions, crucial for the previous results, is analyzed in detail, including the evaluation of coherence effects, relevant in small samples and responsible for a parity splitting at B{sub s}. Exact results for finite chains with first-neighbor and full-range couplings and their interpretation in terms of such mixtures are provided.

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Quantum discord in finite XY chains

We examine the inference of quantum density operators from incomplete information by means of the maximization of general nonadditive entropic forms. Extended thermodynamic relations are given. When applied to a bipartite spin 1 / 2 system, the formalism allows one to avoid fake entanglement for data based on the Bell-Clauser-Horne-Shimony-Holt observable, and, in general, on any set of Bell constraints. Particular results obtained with the Tsallis entropy and with an introduced exponential entropic form are also discussed.

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Generalized nonadditive entropies and quantum entanglement

We discuss the entropic criterion for separability of compound quantum systems for general nonadditive entropic forms based on arbitrary concave functions f. For any separable state, the generalized entropy of the whole system is shown to be not smaller than that of the subsystems, for any choice of f, providing thus a necessary criterion for separability. Nevertheless, the criterion is not sufficient and examples of entangled states with the same property are provided. This entails, in particular, that the conjecture about the positivity of the conditional Tsallis entropy for all q, a more stringent requirement than the positivity of the conditional von Neumann entropy, is actually a necessary but not sufficient condition for separability in general. The direct relation between the entropic criterion and the largest eigenvalues of the full and reduced density operators of the system is also discussed.

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Generalized entropic criterion for separability

We propose a general measure of nonclassical correlations for bipartite systems based on generalized entropic functions and majorization properties. Defined as the minimum information loss due to a local measurement, in the case of pure states it reduces to the generalized entanglement entropy, i.e., the generalized entropy of the reduced state. However, in the case of mixed states it can be nonzero in separable states, vanishing just for states diagonal in a general product basis, like the quantum discord. Simple quadratic measures of quantum correlations arise as a particular case of the present formalism. The minimum information loss due to a joint local measurement is also discussed. The evaluation of these measures in simple relevant cases is as well provided, together with comparison with the corresponding entanglement monotones.

http://www.cbpf.br/~compsyst/Tsallis70_Atualizacao/Apresentacoes/Dia_3/Raul_Rossignoli_Tsallis70_2013.pdf

Generalized entropic measures of quantum correlations

A systematic study of the pairing correlations derived from various particle-number projection methods is performed in an exactly soluble cranked-deformed shell model Hamiltonian. It is shown that most of the approximate particle-number projection methods including the method of Lipkin and Nogami, which is used quite extensively in nuclear structure studies, break down in the weak pairing limit. The results obtained from the recently formulated number-projected Hartree-Fock-Bogoliubov ~PHFB! equations, on the other hand, are in complete agreement with the exact solutions of the model Hamiltonian. The pairing energy calculated from the PHFB method is shown to be finite in all the studied limits. Pairing correlations play a central role in describing the properties of atomic nuclei @1,2#. The bulk of the nuclear phenomena, for instance, the odd-even mass differences and moments of inertia of deformed nuclei can be described by considering nucleons to be in a superfluid paired phase. The recent discovery of halo nuclei have also elucidated the im- portance of pairing correlations in the formation of such structures @3-5#. In the traditional nuclear structure models, the pairing correlations are treated in Bardeen-Cooper- Schrieffer ~BCS! or Hartree-Fock-Bogoliubov ~HFB! ap- proximations @2#. These approaches use a simple ansatz of a product state of quasiparticles for the ground-state wave function. The product ansatz leads to a set of nonlinear equations in the BCS theory or to the diagonalization of a matrix in the case of HFB approach. There exist now very fast numerical methods to find accurate solutions of these problems. The major advantage in the mean-field models is that the number of nonlinear equations or the dimensionality of the matrix to be diagonalized is quite small as compared to the exact analysis, which is impossible to apply in a realistic configuration space of medium and heavy mass nuclei. Nevertheless, the product ansatz of the mean-field ap- proach breaks the symmetries that the orginal many-body Hamiltonian obeys. For instance, in the case of BCS theory, the product state violates the gauge symmetery associated with the particle number. The BCS product state of quasipar- ticles does not have a well defined particle number. The most undesirable feature associated with broken symmetry is that it leads to sharp phase transitions. These phase transitions may be reasonable for a system with a very large number of particles, for example, a metallic superconductor. But for a finite mesoscopic system these phase transitions are washed out by fluctuations and are not observed in the experimental data or in the exact study of toy models. The sharp transition is an artifact of the simple ansatz of the product state, which does not contain fluctuations. The symmetry restoration in the mean-field studies is a major challenge to the nuclear structure models. Although the projection methods that restore the broken symmetries have been known for more than 30 years, but have been studied only in simple model cases due to the numerical difficulties. In most of the realistic studies, approximate pro- jection methods have been employed @6#. Only recently, the gradient method in the projected energy surface has been employed in realistic calculations with Gogny forces @7# .I t has been an unsolved problem whether simple HFB-like equations can be obtained with the projected energy func- tional. This would allow the usage of fast numerical algo- rithms for diagonalization of matrices as is done in the ordi- nary HFB theory. This problem has been recently addressed and it has been shown that the variation of an arbitrary en- ergy functional, which is completely expressed in terms of the HFB density matrix r and the pairing tensor k, results in the HFB like equation @8#. It is noted that the projected en- ergy functional can also be expressed in terms of the HFB density matrices r and k, and therefore one obtains HFB- like equations with modified expressions for the pairing po- tential and the Hartree-Fock field. In the present work, we shall limit ourselves to the resto- ration of the gauge symmetry associated with the particle- number projection, which is important in the discussion of the pairing correlations. Several approximate particle- number projection methods have been developed and the most popular of these is the method of Lipkin and Nogami @9-12#. The purpose of the present work is to perform a systematic study of these various approximate projection methods and to compare them with the recently developed number-projected HFB ~PHFB! approach. It is shown that most of the approximate methods including the Lipkin- Nogami method break down in the weak pairing limit as has been shown in earlier studies @13#. The PHFB method, on the other hand, gives finite pair correlations in all the cases stud- ied. The paper is organized as follows: a brief survey of the the particle-number projection methods is given in the next section and in the Appendix. The model Hamitonian used in the present work is expressed in Sec. III, the numerical re-

Raúl Dante Rossignoli

Papers

Quantum discord in finite XY chains

Generalized nonadditive entropies and quantum entanglement

Generalized entropic criterion for separability

Generalized entropic measures of quantum correlations

Pairing correlations and particle-number projection methods