Distribution of entanglement and correlations in all finite dimensions

The physics of a many-particle system is determined by the correlations in its quantum state. Therefore, analyzing these correlations is the foremost task of many-body physics. Any 'a priori' constraint for the properties of the global vs. the local states-the so-called marginals-would help in order to narrow down the wealth of possible solutions for a given many-body problem, however, little is known about such constraints. We derive an equality for correlation-related quantities of any multipartite quantum system composed of finite-dimensional local parties. This relation defines a necessary condition for the compatibility of the marginal properties with those of the joint state. While the equality holds both for pure and mixed states, the pure-state version containing only entanglement measures represents a fully general monogamy relation for entanglement. These findings have interesting implications in terms of conservation laws for correlations, and also with respect to topology.

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Some Remarks on the Entanglement Number

Quanta ◽

10.12743/quanta.v9i1.140 ◽

2020 ◽

Vol 9 (1) ◽

pp. 22-36

Author(s):

George Androulakis ◽

Ryan McGaha

Keyword(s):

Point Of View ◽

Entanglement Measure ◽

Mixed States ◽

Interesting Point ◽

Finite Dimensional ◽

Entanglement Measures ◽

Pure States ◽

Tree Representations ◽

Natural Way ◽

Finite Dimensional Hilbert Space

Gudder, in a recent paper, defined a candidate entanglement measure which is called the entanglement number. The entanglement number is first defined on pure states and then it extends to mixed states by the convex roof construction. In Gudder's article it was left as an open problem to show that Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the entanglement number from pure to mixed states. We answer Gudder's question in the affirmative, and therefore we obtain that the entanglement number vanishes only on the separable states. More generally we show that OPSE exist for the convex roof extension of any function that is norm continuous on the pure states of a finite dimensional Hilbert space. Further we prove that the entanglement number is an LOCC monotone, (and thus an entanglement measure), by using a criterion that was developed by Vidal in 2000. We present a simplified proof of Vidal's result where moreover we use an interesting point of view of tree representations for LOCC communications. Lastly, we generalize Gudder's entanglement number by producing a monotonic family of entanglement measures which converge in a natural way to the entropy of entanglement.Quanta 2020; 9: 22–36.

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A PRIORI ESTIMATES FOR MANY-BODY HAMILTONIAN EVOLUTION OF INTERACTING BOSON SYSTEM

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891608001726 ◽

2008 ◽

Vol 05 (04) ◽

pp. 857-883 ◽

Cited By ~ 17

Author(s):

MANOUSSOS G. GRILLAKIS ◽

DIONISIOS MARGETIS

Keyword(s):

Particle Number ◽

A Priori Estimates ◽

Particle System ◽

A Priori ◽

Bbgky Hierarchy ◽

Particle Interactions ◽

Many Body ◽

Priori Estimates ◽

Reduced Density ◽

Hamiltonian Evolution

We study the evolution of a many-particle system whose wave function obeys the N-body Schrödinger equation under Bose symmetry. The system Hamiltonian describes pairwise particle interactions in the absence of an external potential. We derive a priori dispersive estimates that express the overall repulsive nature of the particle interactions. These estimates hold for a wide class of two-body interaction potentials which are independent of the particle number, N. We discuss applications of these estimates to the BBGKY hierarchy for reduced density matrices analyzed by Elgart, Erdős, Schlein and Yau.

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The Nuclear Many-Body Problem

10.1007/978-3-642-61852-9 ◽

1980 ◽

Cited By ~ 3144

Author(s):

Peter Ring ◽

Peter Schuck

Keyword(s):

Body Problem ◽

Many Body

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Many body dynamics

10.1093/oso/9780198708636.003.0018 ◽

2018 ◽

Author(s):

Joseph F. Boudreau ◽

Eric S. Swanson

Keyword(s):

Statistical Mechanics ◽

Body Problem ◽

Many Body ◽

Complex Objects ◽

Constrained Dynamics ◽

Gravitational Systems ◽

Body Dynamics ◽

System Temperature ◽

Speed Up ◽

Gravitating Systems

Specialized techniques for solving the classical many-body problem are explored in the context of simple gases, more complicated gases, and gravitating systems. The chapter starts with a brief review of some important concepts from statistical mechanics and then introduces the classic Verlet method for obtaining the dynamics of many simple particles. The practical problems of setting the system temperature and measuring observables are discussed. The issues associated with simulating systems of complex objects form the next topic. One approach is to implement constrained dynamics, which can be done elegantly with iterative methods. Gravitational systems are introduced next with stress on techniques that are applicable to systems of different scales and to problems with long range forces. A description of the recursive Barnes-Hut algorithm and particle-mesh methods that speed up force calculations close out the chapter.

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QLB for Quantum Many-Body and Quantum Field Theory

10.1093/oso/9780199592357.003.0033 ◽

2018 ◽

Author(s):

Sauro Succi

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Quantum Mechanics ◽

Quantum Computing ◽

Single Particle ◽

Body Problem ◽

Three Dimensions ◽

Quantum Field ◽

Many Body ◽

Quantum Particles

Chapter 32 expounded the basic theory of quantum LB for the case of relativistic and non-relativistic wavefunctions, namely single-particle quantum mechanics. This chapter goes on to cover extensions of the quantum LB formalism to the overly challenging arena of quantum many-body problems and quantum field theory, along with an appraisal of prospective quantum computing implementations. Solving the single particle Schrodinger, or Dirac, equation in three dimensions is a computationally demanding task. This task, however, pales in front of the ordeal of solving the Schrodinger equation for the quantum many-body problem, namely a collection of many quantum particles, typically nuclei and electrons in a given atom or molecule.

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Interaction between Different Kinds of Quantum Phase Transitions

Quantum Reports ◽

10.3390/quantum3020015 ◽

2021 ◽

Vol 3 (2) ◽

pp. 253-261

Author(s):

Angel Ricardo Plastino ◽

Gustavo Luis Ferri ◽

Angelo Plastino

Keyword(s):

Phase Transitions ◽

Nuclear Physics ◽

Body Problem ◽

Quantum Phase Transitions ◽

Quantum Phase ◽

Exactly Solvable Models ◽

Solvable Models ◽

Many Body ◽

Pairing Interaction ◽

Exactly Solvable

We employ two different Lipkin-like, exactly solvable models so as to display features of the competition between different fermion–fermion quantum interactions (at finite temperatures). One of our two interactions mimics the pairing interaction responsible for superconductivity. The other interaction is a monopole one that resembles the so-called quadrupole one, much used in nuclear physics as a residual interaction. The pairing versus monopole effects here observed afford for some interesting insights into the intricacies of the quantum many body problem, in particular with regards to so-called quantum phase transitions (strictly, level crossings).

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Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations

Symmetry ◽

10.3390/sym13061077 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1077

Author(s):

Yarema A. Prykarpatskyy

Keyword(s):

Conservation Laws ◽

Hamiltonian Structure ◽

Necessary Condition ◽

Type Representation ◽

Infinite Hierarchy ◽

Polynomial Coefficients ◽

Kdv Type Equations ◽

Special Case

Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems.

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