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Multipartite entanglement sharing and characterization of many-body systems using entanglement

Das, Anirban (author)

Published Oct 7, 2013• 0 citations
Publisher

Abstract

In this thesis I study multipartite entanglement sharing, and use entanglement to examine quantum phase transitions (QPT) and equilibrium in closed quantum systems. ? In the first part, I discuss on replacing concurrence by convex-roof extended negativity (CREN) for studying monogamy of entanglement (MOE). I show that all proven MOE relations using concurrence can be rephrased in terms of CREN. Furthermore, I show that higher-dimensional (qudit) extensions of MOE in terms of CREN are not disproven by any of the counterexamples used to disprove qudit extensions of MOE in terms of concurrence. I further test the CREN version of MOE for qudits by considering fully or partially coherent mixtures of a qudit W-class state with the vacuum and show that the CREN version of MOE for qudits is satisfied in this case as well. Hence I prove that the CREN version of MOE for qudits is a strong conjecture with no obvious counterexamples. ? In the second part, I show how entanglement entropy can be used to characterize phases and study equilibrium dynamics in closed quantum systems. As an example I study the Ising spin chain with a Dzyaloshinskii-Moriya interaction, focusing on the static and dynamic properties of the entanglement entropy. I show that the effects of the additional anisotropic interaction on the phase diagram and on the dynamics of the system are captured by the properties of the entanglement entropy. In particular, the model provides a way to study the quench dynamics in a system with an energy current. I consider quenches starting from an initial excited state of the Ising spin chain, and I analyze the effects of different initial conditions.

DOI
10.25549/usctheses-c3-335012

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In this thesis I study multipartite entanglement sharing, and use entanglement to examine quantum phase transitions (QPT) and equilibrium in closed quantum systems. ? In the first part, I discuss on replacing concurrence by convex-roof extended negativity (CREN) for studying monogamy of entanglement (MOE). I show that all proven MOE relations using concurrence can be rephrased in terms of CREN. Furthermore, I show that higher-dimensional (qudit) extensions of MOE in terms of CREN are not disproven by any of the counterexamples used to disprove qudit extensions of MOE in terms of concurrence. I further test the CREN version of MOE for qudits by considering fully or partially coherent mixtures of a qudit W-class state with the vacuum and show that the CREN version of MOE for qudits is satisfied in this case as well. Hence I prove that the CREN version of MOE for qudits is a strong conjecture with no obvious counterexamples. ? In the second part, I show how entanglement entropy can be used to characterize phases and study equilibrium dynamics in closed quantum systems. As an example I study the Ising spin chain with a Dzyaloshinskii-Moriya interaction, focusing on the static and dynamic properties of the entanglement entropy. I show that the effects of the additional anisotropic interaction on the phase diagram and on the dynamics of the system are captured by the properties of the entanglement entropy. In particular, the model provides a way to study the quench dynamics in a system with an energy current. I consider quenches starting from an initial excited state of the Ising spin chain, and I analyze the effects of different initial conditions.
We propose replacing concurrence by convex-roof extended negativity (CREN) for studying monogamy of entanglement (MOE). We show that all proven MOE relations using concurrence can be rephrased in terms of CREN. Furthermore, we show that higher-dimensional (qudit) extensions of MOE in terms of CREN are not disproven by any of the counterexamples used to disprove qudit extensions of MOE in terms of concurrence. We further test the CREN version of MOE for qudits by considering fully or partially coherent mixtures of a qudit WWW-class state with the vacuum and show that the CREN version of MOE for qudits is satisfied in this case as well. The CREN version of MOE for qudits is thus a strong conjecture with no obvious counterexamples.
Abstract It is well known that entanglement is one of the distinctive aspects of the quantum world. The measurement and quantification of entanglement is one of the major problems in understanding and applying this resource. This research aims to investigate entanglement measures and their properties in high-dimensional systems. One such property is entanglement monogamy, viz. quantum correlations are not shareable, different from the classical ones. On the other hand, the study of entanglement depends on the definitions of its measures. Here we examine the applicability of concurrence, an entanglement measure known for its effectiveness in the analysis of qubit entanglement. While the entanglement of two-level systems or qubits is widely understood, its extension to higher dimensions remains challenging. Our research establishes equivalence between different concurrence definitions for bipartite qubit pure states and extends them to qudits. Subsequently, we investigate the monogamous relations of these measurements to explore their characteristics. This approach might enhance our comprehension of quantum traits in qudit systems and advance our understanding of entanglement in broader quantum contexts.
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.
Monogamy relations characterize the distributions of entanglement in multipartite systems. We investigate monogamy relations related to the concurrence CCC, the entanglement of formation EEE, negativity Nc{N}_{c}Nc​, and Tsallis-qqq entanglement Tq{T}_{q}Tq​. Monogamy relations for the $\ensuremath{\alpha}\mathrm{th}$ power of entanglement have been derived, which are tighter than the existing entanglement monogamy relations for some classes of quantum states. Detailed examples are presented.
This thesis investigates the properties of entanglement in one-dimensional many-body systems. In the first part, the non-equilibrium dynamics following a sudden global quench are exploited for the purpose of generating long-range entanglement. A number of initial states are considered. It is shown that the dynamics following the considered quench can be mapped to the problem of a state transfer. The quench can then be optimised by exploiting the literature about quantum state transfer to generate maximal long-range entanglement and maximal block entropy. \nIn the second part of the thesis, a spin chain emulation of the two-channel, Kondo (2CK) model is proposed. Studying the local magnetisation and susceptibility we show that the spin-only emulation truly represent the two-channel Kondo model and extract the Kondo temperature. A detailed entanglement analysis is presented. Using density matrix renormalisation group (DMRG), which allow for real space analysis, Kondo temperature and Kondo length are evaluated. An entanglement measure, namely the negativity, as well as the Schmidt gap are used as possible order parameters predicting the critical point. An extensive analysis of the block entropy of the system is presented for different limiting values of Kondo coupling. A universal scaling of the impurity contribution to the entropy is found and the 2CK residual entropy is extracted. \nThe last part explores quench dynamics in Kondo systems using time-dependent DMRG. For a quench in the Kondo coupling a travelling and breathing clouds are ob-served. A measurement-induced dynamics lead to an oscillation between an effective singlet and triplet states of the impurity and the Kondo cloud. Kondo temperature can be extracted from the frequency of the oscillation.