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arXiv:1812.07925 · Physical review. D/Physical review. D.

N -level ghost spins and entanglement

Dileep P. Jatkar, Kedar S. Kolekar, Krishnan Narayan

Published May 7, 2019• 4 citations
arXivHTMLPDF

Abstract

Ghost spins, 2-level spinlike variables with indefinite norm have been studied in previous work. Here we explore various $N$-level generalizations of ghost spins. First we discuss a flavored generalization comprising $N$ copies of the ghost-spin system, as well as certain ghost-spin chains which in the continuum limit lead to 2-dim $bc$-ghost conformal field theorys with $O(N)$ flavor symmetry. Then we explore a symplectic generalization that involves antisymmetric inner products, and finally a ghost-spin system exhibiting $N$ irreducible levels. We also study entanglement properties. In all these cases, we show the existence of positive norm ``correlated ghost spin'' states in two copies of ghost-spin ensembles obtained by entangling identical ghost spins from each copy: these exhibit positive entanglement entropy.

DOI
10.1103/physrevd.99.106003

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Ghost spins, 2-level spinlike variables with indefinite norm have been studied in previous work. Here we explore various NNN-level generalizations of ghost spins. First we discuss a flavored generalization comprising NNN copies of the ghost-spin system, as well as certain ghost-spin chains which in the continuum limit lead to 2-dim bcbcbc-ghost conformal field theorys with O(N)O(N)O(N) flavor symmetry. Then we explore a symplectic generalization that involves antisymmetric inner products, and finally a ghost-spin system exhibiting NNN irreducible levels. We also study entanglement properties. In all these cases, we show the existence of positive norm ``correlated ghost spin'' states in two copies of ghost-spin ensembles obtained by entangling identical ghost spins from each copy: these exhibit positive entanglement entropy.
We study patterns of quantum entanglement in systems of spins and ghost-spins regarding them as simple quantum mechanical toy models for theories containing negative norm states. We define a single ghost-spin as in [20] as a 2-state spin variable with an indefinite inner product in the state space. We find that whenever the spin sector is disentangled from the ghost-spin sector (both of which could be entangled within themselves), the reduced density matrix obtained by tracing over all the ghost-spins gives rise to positive entanglement entropy for positive norm states, while negative norm states have an entanglement entropy with a negative real part and a constant imaginary part. However when the spins are entangled with the ghost-spins, there are new entanglement patterns in general. For systems where the number of ghost-spins is even, it is possible to find subsectors of the Hilbert space where positive norm states always lead to positive entanglement entropy after tracing over the ghost-spins. With an odd number of ghost-spins however, we find that there always exist positive norm states with negative real part for entanglement entropy after tracing over the ghost-spins.
We study one-dimensional chains of ghost-spins with nearest neighbor interactions amongst them, developing further the study of ghost-spins in previous work, defined as 2-state spin variables with indefinite norm. First we study finite ghost-spin chains with Ising-like nearest neighbor interactions: this helps organize and clarify the study of entanglement earlier, and we develop this further. Then we study a family of infinite ghost-spin chains with a different Hamiltonian containing nearest neighbor hopping-type interactions. By defining fermionic ghost-spin variables through a Jordan-Wigner transformation, we argue that these ghost-spin chains lead in the continuum limit to the bcbcbc-ghost conformal field theories.
In arXiv [K. Narayan, arXiv:1501.03019.], the areas of certain complex extremal surfaces in de Sitter space were found to have resemblance with entanglement entropy in appropriate dual Euclidean nonunitary CFTs, with the area being real and negative in dS4{\mathrm{dS}}_{4}dS4​. In this paper, we study some toy models of 2-dim ghost conformal field theories with negative central charge with a view to exploring this further from the CFT point of view. In particular we consider bcbcbc-ghost systems with central charge $c=\ensuremath{-}2$ and study the replica formulation for entanglement entropy for a single interval, and associated issues arising in this case, notably pertaining to (i) the SL(2)SL(2)SL(2) vacuum coinciding with the ghost ground state, and (ii) the background charge inherent in these systems which leads to particular forms for the norms of states (involving zero modes). This eventually gives rise to negative entanglement entropy. We also discuss a (logarithmic) CFT of anticommuting scalars, with similarities in some features. Finally we discuss a simple toy model of two ``ghost-spins'' which mimics some of these features.
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.
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.
We consider the single-copy entanglement as a quantity to assess quantum correlations in the ground state in quantum many-body systems. We show for a large class of models that already on the level of single specimens of spin chains, criticality is accompanied with the possibility of distilling a maximally entangled state of arbitrary dimension from a sufficiently large block deterministically, with local operations and classical communication. These analytical results---which refine previous results on the divergence of block entropy as the rate at which maximally entangled pairs can be distilled from many identically prepared chains---are made quantitative for general isotropic translationally invariant spin chains that can be mapped onto a quasifree fermionic system, and for the anisotropic XYXYXY model. For the XXXXXX model, we provide the asymptotic scaling of $\ensuremath{\sim}(1∕6){\mathrm{log}}_{2}(L)$, and contrast it with the block entropy.