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arXiv:1911.10208 · Physics Letters B

Gauge fields and quantum entanglement

Jakub Mielczarek A B, Tomasz Trześniewski

Published Sep 25, 2020• 4 citations
arXivHTMLPDF

Abstract

The purpose of this letter is to explore the relation between gauge fields, which are at the base of our understanding of fundamental interactions, and the quantum entanglement. To this end, we investigate the case of SU(2) gauge fields. It is first argued that holonomies of the SU(2) gauge fields are naturally associated with maximally entangled two-particle states. Then, we provide some evidence that the notion of such gauge fields can be deduced from the transformation properties of maximally entangled two-particle states. This new insight unveils a possible relation between gauge fields and spin systems, as well as contributes to understanding of the relation between tensor networks (such as MERA) and spin network states considered in loop quantum gravity. In consequence, our results turn out to be relevant in the context of the emerging Entanglement/Gravity duality.

DOI
10.1016/j.physletb.2020.135808

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The purpose of this letter is to explore the relation between gauge fields, which are at the base of our understanding of fundamental interactions, and the quantum entanglement. To this end, we investigate the case of SU(2) gauge fields. It is first argued that holonomies of the SU(2) gauge fields are naturally associated with maximally entangled two-particle states. Then, we provide some evidence that the notion of such gauge fields can be deduced from the transformation properties of maximally entangled two-particle states. This new insight unveils a possible relation between gauge fields and spin systems, as well as contributes to understanding of the relation between tensor networks (such as MERA) and spin network states considered in loop quantum gravity. In consequence, our results turn out to be relevant in the context of the emerging Entanglement/Gravity duality.
In the context of quantum gravity, we clarify entanglement calculations on spin networks: we distinguish the gauge-invariant entanglement between intertwiners located at the nodes and the entanglement between spin states located on the network's links. We compute explicitly these two notions of entanglement between neighboring nodes and show that they are always related to the typical ln(2j+1)\mathrm{ln}(2j+1)ln(2j+1) term depending on the spin jjj living on the link between them. This ln(2j+1)\mathrm{ln}(2j+1)ln(2j+1) contribution comes from looking at nongauge-invariant states; thus we interpret it as gauge breaking and unphysical. In particular, this confirms that pure spin network basis states do not carry any physical entanglement, so that true entanglement and correlations in loop quantum gravity comes from spin or intertwiner superpositions.
Research at the intersection of quantum gravity and quantum information theory has seen significant success in describing the emergence of spacetime and gravity from quantum states whose entanglement entropy approximately obeys an area law. In a different direction, the Kaluza-Klein proposal aims to recover gauge symmetries by means of dimensional reduction of higher-dimensional gravitational theories. Integrating both, gravitational and gauge degrees of freedom in 3+13+13+1 dimensions may be obtained upon dimensional reduction of higher-dimensional emergent gravity. To this end, we show that entangled systems of two and three qubits can be associated with 5+15+15+1 and 9+19+19+1 dimensional spacetimes respectively, which are reduced to 3+13+13+1 dimensions upon singling out a preferred complex direction. In the latter case, this reduction is invariant under a residual SU(3)×SU(2)×U(1)/Z6SU(3) \times SU(2) \times U(1) /\mathbb{Z}_6SU(3)×SU(2)×U(1)/Z6​ symmetry, the Standard Model gauge group. This motivates a picture in which spacetime emerges from the area law-contribution to the entanglement entropy, while gauge and matter degrees of freedom are due to area law-violating terms. We remark on a possible natural origin of the chirality of the weak force in the given construction. Furthermore, we highlight the possibility of using this construction in quantum simulations of Standard Model fields.
This thesis consists of four research papers and an introduction covering the most important concepts appearing in the papers. The papers deal with applications of gauge/gravity dualities in the study of various physical quantities and systems. Gauge/gravity dualities are equivalences between certain quantum field theories and classical theories of gravity. These dualities can be used as computational tools in a wide range of applications across different fields of physics, and as such they have garnered much attention in the last two decades. The great promise of these new tools is the ability to tackle difficult problems in strongly interacting quantum field theories by translating them to problems in classical gravity, where progress is much easier to make. Quantum information theory studies the information contained in quantum systems. Entanglement is the fundamental property of quantum mechanics that sets it apart from classical theories of physics. Entanglement is commonly quantified by entanglement entropy, a quantity which is difficult to compute in interacting quantum field theories. Gauge/gravity dualities provide a practical way for computing the entanglement entropy via the Ryu-Takayanagi formula. The primary focus of this thesis is to use this formula for computing various entanglement measures in strongly interacting quantum field theories via their gravity duals. The purpose of this thesis is to introduce quantum information theory concepts that have been important in our research. When applicable, quantities of interest are first introduced in the classical setting in order to build intuition about their behaviour. Quantum properties of entanglement measures are discussed in detail, along with their holographic counterparts, and remarks are made concerning their applications.
A bstract We study the existence and limitations of hyperinvariant tensor networks incorporating a local SU(2) symmetry. As discrete implementations of the anti de-Sitter/conformal field theory (AdS/CFT) correspondence, such networks have created bridges between the fields of quantum information theory and quantum gravity. Adding SU(2) symmetry to the tensor network allows a direct connection to spin network states, a basis of the kinematic Hilbert space of loop quantum gravity (LQG). We consider a particular situation where the states can be interpreted as kinematic quantum states for three-dimensional quantum gravity. We demonstrate an entanglement-geometry correspondence realized in certain quantum states of the gravitational field in LQG, showing that models similar to the ones introduced by [F. Pastawski et al., JHEP 06, 149 (2015)] can be obtained as quantum gravitational models from first principles. We provide examples of hyperinvariant tensor networks, but also prove constraints on their existence in the form of no-go theorems that exclude absolutely maximally entangled states as well as general holographic codes from local SU(2)-invariance, as long as the logical index also transforms covariantly under the gauge group. We calculate surface areas as expectation values of the LQG area operator and discuss further possible constraints as a consequence of a decay of correlations on the boundary.
The work presented in this thesis is a contribution towards the understanding of holo- graphic entanglement from an operational perspective. Following the interpretation which is most natural for quantum information theory, entanglement is viewed as a resource that can be produced, stored, transferred and used for practical purposes. The first chapter introduces the main concepts which are necessary throughout the discussion. Quantum entanglement, as opposed to merely classical correlation, is presented in detail within the framework of quantum mechanics. This is followed by a brief overview on the current state of knowledge about entanglement in quantum field theory and more specifically in gauge/gravity duality. The second chapter investigates a particular measure of entanglement, known as nega- tivity, in the context of holographic field theories. This is further explored in the following (third) chapter, where an interesting dependence of the entanglement between a region and its complement on the topology of their interface is presented. The forth chapter investigates qubits systems and compares equivalence classes of entanglement structures to known properties of holographic states. The fifth chapter focuses on the behaviour of the tripartite information for highly en- tangled states, both in a bipartite and multipartite sense, in relation to the sign definiteness imposed by holography. A final chapter comments on future directions of investigation within this program.
This is an idiosyncratic colloquium-style review of the idea that spacetime and gravity can emerge from entanglement. Drawing inspiration from the conjectured duality between quantum gravity in anti de Sitter space and certain conformal field theories, we argue that tensor networks can be used to define a discrete geometry that encodes entanglement geometrically. With the additional assumption that a continuum limit can be taken, the resulting geometry necessarily obeys Einstein's equations. The discussion takes the point of view that the emergence of spacetime and gravity is a mysterious phenomenon of quantum many-body physics that we would like to understand. We also briefly discuss possible experiments to detect emergent gravity in highly entangled quantum systems.