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Entanglement and quantum information theory in the context of higher dimensional spin systems.

Christopher Hadley

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Quantum information theory is an exciting, inter-disciplinary field, combining elements of condensed matter theory, quantum mechanics and information theory. In this thesis, I shall make a modest contribution to this field by examining entanglement in many-body systems with more than two levels. In the first section, I consider the dynamics of a system of qutrits three-level quantum systems which are coupled through an SU(3)-invariant permutation Hamiltonian. Each term in this Hamil- tonian is a nearest-neighbour permutation operator, and thus this Hamiltonian may be considered a generalisation of the standard SU (2)-invariant Heisenberg Hamiltonian, in which every term (up to the addition of the identity operator) is a nearest-neighbour permutation operator for two-level system. The system considered has the topology of a cross, and thus may be considered (to a limited extent) analogous to a beam-splitter. The aim of the study is to establish a Bell singlet state between two distant parties. Building on this work, I shall go on to consider the ground state of a system made up of many-level systems coupled by the same Hamiltonian I shall show that this state is a generalisation of the two-level singlet to many levels and many systems. It thus has a high degree of symmetry. I will consider its application in entanglement distribution through measurements (localisable entanglement), and discuss how it may be physically implemented in systems of ultracold atoms, through the Hubbard model. I shall also show that in the famous valence bond solid (the ground state of the Affleck-Kennedy- Lieb-Tasaki spin chain), all the entanglement present in the state may be extracted from a single copy of the chain this is in contrast to gapless, critical chains, in which only half the total entanglement is extractable from a single copy.

Related papers

Entanglement in Many Body Quantum Systems

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THESIS SUMMARYTEXT:This thesis is made of two parts. In the first one, the issue of entanglement in many body systems is addressed. The concept of entanglement and some of the recent progress on the study of entropy of entanglement in many body quantum systems are reviewed. Emphasis is placed on the scaling properties of entropy for one-dimensional models at quantum phase transitions. Then, we focus on the area-law scaling of the entanglement entropy. An explicit computation in arbitrary dimensions of the entanglement entropy of the ground state of a discretized scalar free field theory that shows the expected area law result is also presented. For this system, it is shown that area law scaling is a manifestation of a deeper reordering of the vacuum produced by majorization relations.To finish this first part, the issue of how simple can a quantum system be such as to give a highly entangled ground state is addressed. In particular, we propose a Hamiltonian of a XX model with a ground state whose entropy scales linearly with the size of the block. It provides a simple example of a one dimensional system of spin-1/2 particles with nearest neighbour interactions that violates area-law for the entanglement entropy.The second part of this thesis deals with the problem of simulating quantum mechanics for highly entangled systems. Two different approaches to this issue are considered. One consists of using ultra-cold atoms systems as quantum simulators. With this aim, some experimental techniques related to cold atoms that allow to simulate strongly correlated many body quantum systems are reviewed an explicit example of simulation is presented. In particular, we analyze how to achieve a Mott state of Laughlin wave functions in an optical lattice and study the consequences of considering anharmonic corrections to each single site potential expansion that were not taken into account until now.Finally, a different approach to simulate strongly correlated systems is considered: to use small quantum computers to simulate them. An explicit quantum algorithm that creates the Laughlin state for an arbitrary number of particles n in the case of falling fraction equal to one is presented. We further prove the optimality of the circuit using permutation theory arguments and we compute exactly how entanglement develops along the action of each gate. We also discuss its experimental feasibility decomposing the qudits and the gates in terms of qubits and two qubit-gates as well as the generalization to arbitrary falling fraction.KEYWORDS: Entanglement, Many body quantum systems, Quantum Information Condensed Matter, Cold atoms, Spin chains, Quantum simulator, Quantum computation.

•Mar 18, 2010•7 citations

Entanglement in many-body systems

Federico Rottoli

One of the most important predictions of quantum mechanics is the existence of quantum entanglement. Since the inception of quantum theory, the study of entanglement has become ubiquitous in several branches of physics, from high energy to condensed matter physics and quantum information. To this avail, different measures of entanglement in quantum systems have been introduced. The most successful ones are the von Neumann and the Rényi entanglement entropies, which have been heavily studied both in lattice models and in quantum field theories (QFTs). A different, more complete characterization of entanglement is provided by the entanglement (or modular) Hamiltonian, defined as the logarithm of the reduced density matrix. One of the key property of this operator is that, under generic conditions, in 1+1-dimensional conformal field theories (CFTs) presents a local and few-body structure. Beyond just being of theoretical interest, this property has been recently used to efficiently construct the ground state of spin chains in quantum simulators. A recently introduced generalisation of the entanglement Hamiltonian is the negativity Hamiltonian, a non-Hermitian operator which is more useful for mixed states and tripartite entanglement. In this thesis we study these quantifications in many-body quantum systems and in QFTs. In the first part of the thesis, we focus on fermionic systems at equilibrium, and we obtain analytical expressions for the entanglement and the negativity Hamiltonians in the massless Dirac fermion QFT, both in presence of boundaries and at finite temperature. We compare these predictions in the continuum QFT with numerical calculations on the lattice, showing that a proper comparison requires a non-trivial continuum limit of the lattice expression, which we detail. We also investigate a lattice realisation of a non-unitary CFT, the bc-ghost theory, showing that the entanglement Hamiltonian presents an additional term with respect to the unitary case. In the second part, we study the entanglement entropies in an integrable QFT, the massless renormalisation group flow from the tricritical to the critical Ising CFT, demonstrating that the entropy is monotonically decreasing under the renormalisation group flow, as expected from the entropic c-theorem. In the third part we consider out-of-equilibrium problems. We study the melting of a domain wall configuration, computing analytically the entanglement Hamiltonian and showing that the insertion of a defect dramatically modifies the entanglement properties, changing the growth of the entropies from logarithmic to linear in time. Finally, we study generic quantum quenches in free fermionic theories, using the celebrated quasiparticle picture to obtain a prediction for the evolution of the entanglement Hamiltonian during the quench.

•Sep 16, 2024•0 citations

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We consider the valence bond solid (VBS) state built of singlet pairs of fundamental representations and projected onto adjoint representations of the exceptional Lie group G _2 2 . The two-point correlation function in the VBS state is non-vanishing only for nearest neighbours, but possesses finite string order. We construct a parent Hamiltonian for the VBS state, which constitutes the G _2 2 analog of the famous AKLT chain.

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Olivier Gauthé, Didier Poilblanc

Two-dimensional (spin-2) Affleck-Kennedy-Lieb-Tasaki (AKLT) type valence bond solids on a square lattice are known to be symmetry-protected topological (SPT) gapped spin liquids [S. Takayoshi, P. Pujol, and A. Tanaka Phys. Rev. B 94, 235159 (2016)]. Using the projected entangled pair state framework, we extend the construction of the AKLT state to the case of $\text{SU}(3)$, relevant for cold atom systems. The entanglement spectrum is shown to be described by an alternating $\text{SU}(3)$ chain of ``quarks'' and ``antiquarks'', subject to exponentially decaying (with distance) Heisenberg interactions, in close similarity with its $\text{SU}(2)$ analog. We discuss the SPT feature of the state.

arXiv:1708.00848•Physical review. B./Physical review. B•Sep 25, 2017•7 citations

A study of entanglement in quantum information theory

Frank Verstraete

Although the concept of quantum entanglement has been known for about seventy years, it only recently quit the realms of meta-theoretical discussions when it was discovered how entanglement can be exploited to compute and communicate with an unprecedented power. The primary motivation of the work presented in this thesis has been to contribute to the big e#ort that has been done during the last decade to understand and quantify quantum entanglement. We have developed advanced techniques of linear and multilinear algebra to investigate and classify entangled pure and mixed quantum states, and discussed some novel applications in the field of quantum information theory. The results presented in this thesis are mainly of interest from a fundamental point a view: entanglement is the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought [186]. It is however a real privilege that fundamental research in quantum information theory bears the tools of tomorrow's electrical engineers: the ongoing miniaturization of electronic components will soon reach a scale where quantum mechanical e#ects play a major role.

•Jan 1, 2002•22 citations

Entanglement spectra of the two-dimensional Affleck-Kennedy-Lieb-Tasaki model: Correspondence between the valence-bond-solid state and conformal field theory

Jie Lou, Shu Tanaka, Hosho Katsura +1 more

We investigate the entanglement properties of the valence-bond-solid (VBS) state defined on two-dimensional lattices, which is the exact ground state of the Affleck-Kennedy-Lieb-Tasaki model. It is shown that the entanglement entropy obeys an area law and the nonuniversal prefactor of the leading term is strictly less than $\mathrm{ln}2$. The analysis of entanglement spectra for various lattices reveals that the reduced density matrix associated with the VBS state is closely related to a thermal density matrix of a holographic spin chain, the spectrum of which is reminiscent of that of the spin-1/2 Heisenberg chain. This correspondence is further supported by comparing the entanglement entropy in the holographic spin chain with conformal field theory predictions.

arXiv:1107.3888•Physical Review B•Dec 27, 2011•47 citations
Quantum information theory is an exciting, inter-disciplinary field, combining elements of condensed matter theory, quantum mechanics and information theory. In this thesis, I shall make a modest contribution to this field by examining entanglement in many-body systems with more than two levels. In the first section, I consider the dynamics of a system of qutrits three-level quantum systems which are coupled through an SU(3)-invariant permutation Hamiltonian. Each term in this Hamil- tonian is a nearest-neighbour permutation operator, and thus this Hamiltonian may be considered a generalisation of the standard SU (2)-invariant Heisenberg Hamiltonian, in which every term (up to the addition of the identity operator) is a nearest-neighbour permutation operator for two-level system. The system considered has the topology of a cross, and thus may be considered (to a limited extent) analogous to a beam-splitter. The aim of the study is to establish a Bell singlet state between two distant parties. Building on this work, I shall go on to consider the ground state of a system made up of many-level systems coupled by the same Hamiltonian I shall show that this state is a generalisation of the two-level singlet to many levels and many systems. It thus has a high degree of symmetry. I will consider its application in entanglement distribution through measurements (localisable entanglement), and discuss how it may be physically implemented in systems of ultracold atoms, through the Hubbard model. I shall also show that in the famous valence bond solid (the ground state of the Affleck-Kennedy- Lieb-Tasaki spin chain), all the entanglement present in the state may be extracted from a single copy of the chain this is in contrast to gapless, critical chains, in which only half the total entanglement is extractable from a single copy.
THESIS SUMMARYTEXT:This thesis is made of two parts. In the first one, the issue of entanglement in many body systems is addressed. The concept of entanglement and some of the recent progress on the study of entropy of entanglement in many body quantum systems are reviewed. Emphasis is placed on the scaling properties of entropy for one-dimensional models at quantum phase transitions. Then, we focus on the area-law scaling of the entanglement entropy. An explicit computation in arbitrary dimensions of the entanglement entropy of the ground state of a discretized scalar free field theory that shows the expected area law result is also presented. For this system, it is shown that area law scaling is a manifestation of a deeper reordering of the vacuum produced by majorization relations.To finish this first part, the issue of how simple can a quantum system be such as to give a highly entangled ground state is addressed. In particular, we propose a Hamiltonian of a XX model with a ground state whose entropy scales linearly with the size of the block. It provides a simple example of a one dimensional system of spin-1/2 particles with nearest neighbour interactions that violates area-law for the entanglement entropy.The second part of this thesis deals with the problem of simulating quantum mechanics for highly entangled systems. Two different approaches to this issue are considered. One consists of using ultra-cold atoms systems as quantum simulators. With this aim, some experimental techniques related to cold atoms that allow to simulate strongly correlated many body quantum systems are reviewed an explicit example of simulation is presented. In particular, we analyze how to achieve a Mott state of Laughlin wave functions in an optical lattice and study the consequences of considering anharmonic corrections to each single site potential expansion that were not taken into account until now.Finally, a different approach to simulate strongly correlated systems is considered: to use small quantum computers to simulate them. An explicit quantum algorithm that creates the Laughlin state for an arbitrary number of particles n in the case of falling fraction equal to one is presented. We further prove the optimality of the circuit using permutation theory arguments and we compute exactly how entanglement develops along the action of each gate. We also discuss its experimental feasibility decomposing the qudits and the gates in terms of qubits and two qubit-gates as well as the generalization to arbitrary falling fraction.KEYWORDS: Entanglement, Many body quantum systems, Quantum Information Condensed Matter, Cold atoms, Spin chains, Quantum simulator, Quantum computation.
One of the most important predictions of quantum mechanics is the existence of quantum entanglement. Since the inception of quantum theory, the study of entanglement has become ubiquitous in several branches of physics, from high energy to condensed matter physics and quantum information. To this avail, different measures of entanglement in quantum systems have been introduced. The most successful ones are the von Neumann and the Rényi entanglement entropies, which have been heavily studied both in lattice models and in quantum field theories (QFTs). A different, more complete characterization of entanglement is provided by the entanglement (or modular) Hamiltonian, defined as the logarithm of the reduced density matrix. One of the key property of this operator is that, under generic conditions, in 1+1-dimensional conformal field theories (CFTs) presents a local and few-body structure. Beyond just being of theoretical interest, this property has been recently used to efficiently construct the ground state of spin chains in quantum simulators. A recently introduced generalisation of the entanglement Hamiltonian is the negativity Hamiltonian, a non-Hermitian operator which is more useful for mixed states and tripartite entanglement. In this thesis we study these quantifications in many-body quantum systems and in QFTs. In the first part of the thesis, we focus on fermionic systems at equilibrium, and we obtain analytical expressions for the entanglement and the negativity Hamiltonians in the massless Dirac fermion QFT, both in presence of boundaries and at finite temperature. We compare these predictions in the continuum QFT with numerical calculations on the lattice, showing that a proper comparison requires a non-trivial continuum limit of the lattice expression, which we detail. We also investigate a lattice realisation of a non-unitary CFT, the bc-ghost theory, showing that the entanglement Hamiltonian presents an additional term with respect to the unitary case. In the second part, we study the entanglement entropies in an integrable QFT, the massless renormalisation group flow from the tricritical to the critical Ising CFT, demonstrating that the entropy is monotonically decreasing under the renormalisation group flow, as expected from the entropic c-theorem. In the third part we consider out-of-equilibrium problems. We study the melting of a domain wall configuration, computing analytically the entanglement Hamiltonian and showing that the insertion of a defect dramatically modifies the entanglement properties, changing the growth of the entropies from logarithmic to linear in time. Finally, we study generic quantum quenches in free fermionic theories, using the celebrated quasiparticle picture to obtain a prediction for the evolution of the entanglement Hamiltonian during the quench.
A brief note on the G2_22​ Affleck-Kennedy-Lieb-Tasaki chain
Two-dimensional (spin-2) Affleck-Kennedy-Lieb-Tasaki (AKLT) type valence bond solids on a square lattice are known to be symmetry-protected topological (SPT) gapped spin liquids [S. Takayoshi, P. Pujol, and A. Tanaka Phys. Rev. B 94, 235159 (2016)]. Using the projected entangled pair state framework, we extend the construction of the AKLT state to the case of SU(3)\text{SU}(3)SU(3), relevant for cold atom systems. The entanglement spectrum is shown to be described by an alternating SU(3)\text{SU}(3)SU(3) chain of ``quarks'' and ``antiquarks'', subject to exponentially decaying (with distance) Heisenberg interactions, in close similarity with its SU(2)\text{SU}(2)SU(2) analog. We discuss the SPT feature of the state.
Although the concept of quantum entanglement has been known for about seventy years, it only recently quit the realms of meta-theoretical discussions when it was discovered how entanglement can be exploited to compute and communicate with an unprecedented power. The primary motivation of the work presented in this thesis has been to contribute to the big e#ort that has been done during the last decade to understand and quantify quantum entanglement. We have developed advanced techniques of linear and multilinear algebra to investigate and classify entangled pure and mixed quantum states, and discussed some novel applications in the field of quantum information theory. The results presented in this thesis are mainly of interest from a fundamental point a view: entanglement is the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought [186]. It is however a real privilege that fundamental research in quantum information theory bears the tools of tomorrow's electrical engineers: the ongoing miniaturization of electronic components will soon reach a scale where quantum mechanical e#ects play a major role.
We investigate the entanglement properties of the valence-bond-solid (VBS) state defined on two-dimensional lattices, which is the exact ground state of the Affleck-Kennedy-Lieb-Tasaki model. It is shown that the entanglement entropy obeys an area law and the nonuniversal prefactor of the leading term is strictly less than ln2\mathrm{ln}2ln2. The analysis of entanglement spectra for various lattices reveals that the reduced density matrix associated with the VBS state is closely related to a thermal density matrix of a holographic spin chain, the spectrum of which is reminiscent of that of the spin-1/2 Heisenberg chain. This correspondence is further supported by comparing the entanglement entropy in the holographic spin chain with conformal field theory predictions.