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arXiv:2202.01626

Target Space Entanglement in Quantum Mechanics of Fermions and Matrices

Sotaro Sugishita

Published Jun 1, 2022• 2 citations
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Abstract

Quantum entanglement is closely related to the structure of spacetime in quantum gravity. For quantum field theories or statistical models, we usually consider base space entanglement. However, target space instead of base space sometimes directly connects to our spacetime. In these cases, it is natural to consider a concept of target space entanglement. To define the target space entanglement, we consider a generalized definition of entanglement entropy based on an algebraic approach. This approach is reviewed and is applied to the first quantized particles, in particular, fermions. This article is based on the paper JHEP 08 (2021) 046 [arXiv:2105.13726].

DOI
10.1142/9789811261633_0016

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Quantum entanglement is closely related to the structure of spacetime in quantum gravity. For quantum field theories or statistical models, we usually consider base space entanglement. However, target space instead of base space sometimes directly connects to our spacetime. In these cases, it is natural to consider a concept of target space entanglement. To define the target space entanglement, we consider a generalized definition of entanglement entropy based on an algebraic approach. This approach is reviewed and is applied to the first quantized particles, in particular, fermions. This article is based on the paper JHEP 08 (2021) 046 [arXiv:2105.13726].
Abstract We define a notion of target space entanglement entropy. Rather than partitioning the base space on which the theory is defined, we consider partitions of the target space. This is the physical case of interest for first-quantized theories, such as worldsheet string theory. We associate to each subregion of the target space a suitably chosen subalgebra of observables A \mathcal{A} A A . The entanglement entropy is calculated as the entropy of the density matrix restricted to A \mathcal{A} A A . As an example, we illustrate our framework by computing spatial entanglement in first-quantized many-body quantum mechanics. The algebra A \mathcal{A} A A is chosen to reproduce the entanglement entropy obtained by embedding the state in the fixed particle sub-sector of the second-quantized Hilbert space. We then generalize our construction to the quantum field-theoretical setting.
Abstract We consider entanglement of first-quantized identical particles by adopting an algebraic approach. In particular, we investigate fermions whose wave functions are given by the Slater determinants, as for singlet sectors of one-matrix models. We show that the upper bounds of the general Rényi entropies are N log 2 for N particles or an N × N matrix. We compute the target space entanglement entropy and the mutual information in a free one-matrix model. We confirm the area law: the single-interval entropy for the ground state scales as 13 \frac{1}{3} 31​ 1 3 log N in the large N model. We obtain an analytical O(N0) \mathcal{O}\left({N}^0\right) O(N0) O N 0 expression of the mutual information for two intervals in the large N expansion.
We review aspects of entanglement entropy in the quantum mechanics of N×N matrices, i.e., matrix quantum mechanics (MQM), at large N. In doing so, we review standard models of MQM and their relation to string theory, D-brane physics, and emergent non-commutative geometries. We overview, in generality, definitions of subsystems and entanglement entropies in theories with gauge redundancy and discuss the additional structure required for definining subsystems in MQMs possessing a U(N) gauge redundancy. In connecting these subsystems to non-commutative geometry, we review several works on ‘target space entanglement,’ and entanglement in non-commutative field theories, highlighting the conditions in which target space entanglement entropy displays an ‘area law’ at large N. We summarize several example calculations of entanglement entropy in non-commutative geometries and MQMs. We review recent work in connecting the area law entanglement of MQM to the Ryu–Takayanagi formula, highlighting the conditions in which U(N) invariance implies a minimal area formula for the entanglement entropy at large N. Finally, we make comments on open questions and research directions.
We consider defining a fuzzy space by a specific state in a fermionic field theory in terms of which all the observables for the space can be evaluated. This allows for a definition of entanglement for a fuzzy space by direct integration of the fields over a certain region. Even though the resulting formula for the entanglement entropy (EE) is similar to what has been used in the quantum Hall effect, our derivation provides a novel perspective. We also review and strengthen the arguments for the EE to be described by a generalized Chern-Simons form.
Abstract We consider the target space entanglement in quantum mechanics of non-interacting fermions at finite temperature. Unlike pure states investigated in [1], the (Rényi) entanglement entropy for thermal states does not follow a simple bound because all states in the infinite-dimensional Hilbert space are involved. We investigate a general formula of the target space Rényi entropy for N fermions at finite temperature, and present numerical results of the entropy in a one-dimensional model. We also argue the large N behaviors with a comparison to the grand canonical ensemble.
Quantum Field Theory (QFT) represents a vast generalization of Quantum Mechanics (QM), as it deals with systems that have an infinite number of degrees of freedom. The Stone-von Neumann theorem, which establishes the equivalence of irreducible representations of the canonical commutation relations (CCR) in QM, does not extend to QFT. Consequently, QFT admits multiple inequivalent irreducible representations, leading to a much richer algebraic structure. This essay aims to explore the physics of QFT from the operator algebra perspective, particularly focusing on entanglement entropy. We discuss the role of von Neumann algebras of different types in QFT, describe the local operator algebra approach to QFT, and explain how entanglement entropy can be defined in terms of the algebra of observables. Additionally, we explore the benefits of this approach in concrete applications, specifically in quantum field theory on curved spacetime.