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Macroscopic Entanglement and Phase Transitions

Janet Anders, Vlatko Vedral

Published Oct 31, 2006• 9 citations
OpenAlex

Abstract

This paper summarises the results of our research on macroscopic entanglement in spin systems and free Bosonic gases. We explain how entanglement can be observed using entanglement witnesses which are themselves constructed within the framework of thermodynamics and thus macroscopic observables. These thermodynamical entanglement witnesses result in bounds on macroscopic parameters of the system, such as the temperature, the energy or the susceptibility, below which entanglement must be present. The derived bounds indicate a relationship between the occurrence of entanglement and the establishment of order, possibly resulting in phase transition phenomena. We give a short overview over the concepts developed in condensed matter physics to capture the characteristics of phase transitions in particular in terms of order and correlation functions. Finally we want to ask and speculate whether entanglement could be a generalised order concept by itself, relevant in (quantum induced) phase transitions such as BEC, and that taking this view may help us to understand the underlying process of high-T superconductivity.

Related papers

Macroscopic Entanglement and Phase Transitions

Janet Anders, Vlatko Vedral

This paper summarises the results of our research on macroscopic entanglement in spin systems and free Bosonic gases. We explain how entanglement can be observed using entanglement witnesses which are themselves constructed within the framework of thermodynamics and thus macroscopic observables. These thermodynamical entanglement witnesses result in bounds on macroscopic parameters of the system, such as the temperature, the energy or the susceptibility, below which entanglement must be present. The derived bounds indicate a relationship between the occurrence of entanglement and the establishment of order, possibly resulting in phase transition phenomena. We give a short overview over the concepts developed in condensed matter physics to capture the characteristics of phase transitions in particular in terms of order and correlation functions. Finally we want to ask and speculate whether entanglement could be a generalised order concept by itself, relevant in (quantum induced) phase transitions such as BEC, and that taking this view may help us to understand the underlying process of high-T superconductivity.

arXiv:quant-ph/0610268•Open Systems & Information Dynamics•Mar 1, 2007•18 citations

Detecting entanglement with a thermometer

Janet Anders, Dagomir Kaszlikowski, Christian Lunkes +2 more

We present a general argument showing that the temperature as well as other thermodynamical state variables can qualify as entanglement witnesses (EWs) for spatial entanglement. This holds for a variety of systems and we exemplify our ideas using a simple free non-interacting bosonic gas. We find that entanglement can exist at arbitrarily high temperatures, provided that we can probe smaller and smaller regions of space. We then discuss the relationship between the occurrence of Bose–Einstein condensation and our conditions for the presence of entanglement and compare the respective critical temperatures. We close with a short discussion of the idea of seeing entanglement as a macroscopic property in thermodynamical systems and its possible relation to phase transitions in general.

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Detecting entanglement with a thermometer

Janet Anders, Dagomir Kaszlikowski, Christian Lunkes +2 more

Abstract. We present a general argument showing that the temperature as well as other thermodynamical state variables qualify as entanglement witnesses for spatial entanglement. This holds for a variety of systems and we exemplify our ideas using a simple non-interacting Bosonic gas. We find that entanglement can exist at arbitrarily high temperatures, provided that we can probe smaller and smaller regions of space. We close with a short discussion of the relationship between the occurrence of Bose-Einstein condensation and our conditions for the presence of entanglement. PACS numbers: Submitted to: New Journal of Physics Keywords: macroscopic entanglement, entanglement witnesses, bosonic gas, BEC, continuous variable systemsDetecting entanglement with a thermometer 2 1.

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Entanglement as an Observer-Dependent Concept: An Application to Quantum Phase Transitions

Gerardo Ortíz, Rolando D. Somma, Howard Barnum +2 more

This paper addresses the following main question: Do we have a theoretical understanding of entanglement applicable to a full variety of physical settings? It is clear that not only the assumption of distinguishability, but also the few-subsystem scenario, are too narrow to embrace all possible physical settings. In particular, the need to go beyond the traditional subsystem-based framework becomes manifest when one tries to apply the conventional concept of entanglement to the physics of matter, since the constituents of a quantum many-body system are indistinguishable particles. We shall discuss here a notion of generalized entanglement, which can be applied to any operator language (fermions, bosons, spins, etc.) used to describe a physical system and which includes the conventional entanglement settings introduced to date in a unified fashion. This is realized by noticing that entanglement is an observer-dependent concept, whose properties are determined by the expectations of a distinguished set of observables without reference to a preferred subsystem decomposition, i.e., it depends on the physically relevant point of view. This viewpoint depends in turn upon the relationship between different sets of observables that determine our ability to control the system of interest. Indeed, the extent to which entanglement is present depends on the observables used to measure a system and describe its states. This represents a most conspicuous advantage as will be highlighted by the condensed-matter application we will discuss.

arXiv:quant-ph/0403043•Mar 4, 2004•4 citations

Witnessing entanglement by proxy

Stefan Bäuml, Dagmar Bruß, Marcus Huber +2 more

Entanglement is a ubiquitous feature of low temperature systems and believed to be highly relevant for the dynamics of condensed matter properties and quantum computation even at higher temperatures. The experimental certification of this paradigmatic quantum effect in macroscopic high temperature systems is constrained by the limited access to the quantum state of the system. In this paper we show how macroscopic observables beyond the mean energy of the system can be exploited as proxy witnesses for entanglement detection. Using linear and semi-definite relaxations we show that all previous approaches to this problem can be outperformed by our proxies, i.e. entanglement can be certified at higher temperatures without access to any local observable. For an efficient computation of proxy witnesses one can resort to a generalised grand canonical ensemble, enabling entanglement certification even in complex systems with macroscopic particle numbers.

arXiv:1504.04575•New Journal of Physics•Dec 29, 2015•4 citations

An entanglement perspective on phase transitions, conventional and topological order

Johannes Helmes

The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.

•Aug 16, 2017•3 citations
This paper summarises the results of our research on macroscopic entanglement in spin systems and free Bosonic gases. We explain how entanglement can be observed using entanglement witnesses which are themselves constructed within the framework of thermodynamics and thus macroscopic observables. These thermodynamical entanglement witnesses result in bounds on macroscopic parameters of the system, such as the temperature, the energy or the susceptibility, below which entanglement must be present. The derived bounds indicate a relationship between the occurrence of entanglement and the establishment of order, possibly resulting in phase transition phenomena. We give a short overview over the concepts developed in condensed matter physics to capture the characteristics of phase transitions in particular in terms of order and correlation functions. Finally we want to ask and speculate whether entanglement could be a generalised order concept by itself, relevant in (quantum induced) phase transitions such as BEC, and that taking this view may help us to understand the underlying process of high-T superconductivity.
This paper summarises the results of our research on macroscopic entanglement in spin systems and free Bosonic gases. We explain how entanglement can be observed using entanglement witnesses which are themselves constructed within the framework of thermodynamics and thus macroscopic observables. These thermodynamical entanglement witnesses result in bounds on macroscopic parameters of the system, such as the temperature, the energy or the susceptibility, below which entanglement must be present. The derived bounds indicate a relationship between the occurrence of entanglement and the establishment of order, possibly resulting in phase transition phenomena. We give a short overview over the concepts developed in condensed matter physics to capture the characteristics of phase transitions in particular in terms of order and correlation functions. Finally we want to ask and speculate whether entanglement could be a generalised order concept by itself, relevant in (quantum induced) phase transitions such as BEC, and that taking this view may help us to understand the underlying process of high-T superconductivity.
We present a general argument showing that the temperature as well as other thermodynamical state variables can qualify as entanglement witnesses (EWs) for spatial entanglement. This holds for a variety of systems and we exemplify our ideas using a simple free non-interacting bosonic gas. We find that entanglement can exist at arbitrarily high temperatures, provided that we can probe smaller and smaller regions of space. We then discuss the relationship between the occurrence of Bose–Einstein condensation and our conditions for the presence of entanglement and compare the respective critical temperatures. We close with a short discussion of the idea of seeing entanglement as a macroscopic property in thermodynamical systems and its possible relation to phase transitions in general.
Abstract. We present a general argument showing that the temperature as well as other thermodynamical state variables qualify as entanglement witnesses for spatial entanglement. This holds for a variety of systems and we exemplify our ideas using a simple non-interacting Bosonic gas. We find that entanglement can exist at arbitrarily high temperatures, provided that we can probe smaller and smaller regions of space. We close with a short discussion of the relationship between the occurrence of Bose-Einstein condensation and our conditions for the presence of entanglement. PACS numbers: Submitted to: New Journal of Physics Keywords: macroscopic entanglement, entanglement witnesses, bosonic gas, BEC, continuous variable systemsDetecting entanglement with a thermometer 2 1.
This paper addresses the following main question: Do we have a theoretical understanding of entanglement applicable to a full variety of physical settings? It is clear that not only the assumption of distinguishability, but also the few-subsystem scenario, are too narrow to embrace all possible physical settings. In particular, the need to go beyond the traditional subsystem-based framework becomes manifest when one tries to apply the conventional concept of entanglement to the physics of matter, since the constituents of a quantum many-body system are indistinguishable particles. We shall discuss here a notion of generalized entanglement, which can be applied to any operator language (fermions, bosons, spins, etc.) used to describe a physical system and which includes the conventional entanglement settings introduced to date in a unified fashion. This is realized by noticing that entanglement is an observer-dependent concept, whose properties are determined by the expectations of a distinguished set of observables without reference to a preferred subsystem decomposition, i.e., it depends on the physically relevant point of view. This viewpoint depends in turn upon the relationship between different sets of observables that determine our ability to control the system of interest. Indeed, the extent to which entanglement is present depends on the observables used to measure a system and describe its states. This represents a most conspicuous advantage as will be highlighted by the condensed-matter application we will discuss.
Entanglement is a ubiquitous feature of low temperature systems and believed to be highly relevant for the dynamics of condensed matter properties and quantum computation even at higher temperatures. The experimental certification of this paradigmatic quantum effect in macroscopic high temperature systems is constrained by the limited access to the quantum state of the system. In this paper we show how macroscopic observables beyond the mean energy of the system can be exploited as proxy witnesses for entanglement detection. Using linear and semi-definite relaxations we show that all previous approaches to this problem can be outperformed by our proxies, i.e. entanglement can be certified at higher temperatures without access to any local observable. For an efficient computation of proxy witnesses one can resort to a generalised grand canonical ensemble, enabling entanglement certification even in complex systems with macroscopic particle numbers.
The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.