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arXiv:2103.06353 · Journal of Physics A Mathematical and Theoretical

Modular operators and entanglement in supersymmetric quantum mechanics

Rupak Chatterjee, Ting Yu

Published Apr 7, 2021• 3 citations
arXivHTMLPDF

Abstract

Abstract The modular operator approach of Tomita–Takesaki to von Neumann algebras is elucidated in the algebraic structure of certain supersymmetric (SUSY) quantum mechanical systems. A von Neumann algebra is constructed from the operators of the system. An explicit operator characterizing the dual infinite degeneracy structure of a SUSY two dimensional system is given by the modular conjugation operator. Furthermore, the entanglement of formation for these SUSY systems using concurrence is shown to be related to the expectation value of the modular conjugation operator in an entangled bi-partite supermultiplet state thus providing a direct physical meaning to this anti-unitary, anti-linear operator as a quantitative measure of entanglement. Finally, the theory is applied to the case of two-dimensional Dirac fermions, as is found in graphene, and a SUSY Jaynes Cummings model.

DOI
10.1088/1751-8121/abf585

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Abstract The modular operator approach of Tomita–Takesaki to von Neumann algebras is elucidated in the algebraic structure of certain supersymmetric (SUSY) quantum mechanical systems. A von Neumann algebra is constructed from the operators of the system. An explicit operator characterizing the dual infinite degeneracy structure of a SUSY two dimensional system is given by the modular conjugation operator. Furthermore, the entanglement of formation for these SUSY systems using concurrence is shown to be related to the expectation value of the modular conjugation operator in an entangled bi-partite supermultiplet state thus providing a direct physical meaning to this anti-unitary, anti-linear operator as a quantitative measure of entanglement. Finally, the theory is applied to the case of two-dimensional Dirac fermions, as is found in graphene, and a SUSY Jaynes Cummings model.
The quantum entanglement measure of concurrence is shown to be directly calculable from a Tomita- Takesaki modular operator framework constructed from the local von Neumann algebras of observables for two quantum systems. Specifically, the Tomita-Takesaki modular conjugation operator JJJ that links two separate systems with respect to their von Neumann algebras is related to the quantum concurrence CCC of a pure bivariate entangled state composed from these systems. This concurrence relation provides a direct physical meaning to JJJ as both a symmetry operator and a quantitative measure of entanglement. This procedure is then demonstrated for a supersymmetric quantum mechanical system and a real scalar field interacting with two entangled spin-12\frac{1}{2}21​ Unruh-DeWitt qubit detectors. For the latter system, the concurrence result is shown to be consistent with some known results on the Bell-CHSH inequality for such a system.
An operator-algebraic framework based on Tomita-Takesaki modular theory is used to study aspects of quantum entanglement via the application of the modular conjugation operator J . The entanglement structure of quantum fields is studied through the protocol of entanglement harvesting whereby quantum correlations evolve through the time evolution of qubit detectors coupled to a Bosonic field. Modular conjugation operators are constructed for Unruh-Dewitt type qubits interacting with a scalar field such that initially unentangled qubits become entangled. The entanglement harvested in this process is directly quantified by an expectation value involving J offering a physical application of this operator. The modular operator formalism is then extended to the Markovian open system dynamics of coupled qubits by expressing entanglement monotones as functionals of a state ρ and its modular reflection J ρ J . The second derivative of such functionals with respect to an external coupling parameter, termed the curvature of entanglement, provides a natural measure of entanglement sensitivity. At points of modular self-duality, the curvature of entanglement coincides with the quantum Fisher information measure. These results demonstrate that the modular conjugation operator J captures both the harvesting of entanglement from quantum fields and the curvature of entanglement in coupled qubit dynamics providing parallel modular structures that connect these systems.
In this thesis, we study aspects of entanglement theory of quantum field theories from an algebraic point of view. The main motivation is to gain insights about the general structure of the entanglement in QFT, towards a definition of an entropic version of QFT. In the opposite direction, we are also interested in exploring any consequence of the entanglement in algebraic QFT. This may help us to reveal unknown features of QFT, with the final aim of finding a dynamical principle which allows us to construct non-trivial and rigorous models of QFT. The algebraic approach is the natural framework to define and study entanglement in QFT, and hence, to pose the above inquiries. After a self-contained review of algebraic QFT and quantum information theory in operator algebras, we focus on our results. We compute, in a mathematically rigorous way, exact solutions of entanglement measures and modular Hamiltonians for specific QFT models, using algebraic tools from modular theory of von Neumann algebras. These calculations show explicitly non-local features of modular Hamiltonians and help us to solve ambiguities that arise in other non-rigorous computations. We also study aspects of entanglement entropy in theories having superselection sectors coming from global symmetries. We follow the algebraic perspective of Doplicher, Haag, and Roberts. In this way, we find an entropic order parameter that "measures" the size of the symmetry group, which is made out of a difference of two mutual informations. Moreover, we identify the main operators that take account of such a difference, and we obtain a new quantum information quantity, the entropic certainty relation, involving algebras containing such operators. This certainty relation keeps an intrinsic connection with subfactor theory of von Neumann algebras.
This preprint develops a rigorous operator-algebraic extension of Tomita-Takesaki modular theory to pseudo-Hermitian quantum field systems. We construct a twisted Tomita operator S_η for PT-symmetric fields equipped with a positive-definite metric operator η, establishing its closability and polar decomposition S_η = J_η Δ_η^(1/2) within the η-inner product GNS representation. The central result demonstrates that the twisted modular conjugation J_η squares not to the identity (as in standard theory) but to a non-trivial central unitary W = J_η², where W ∈ Z(A). We prove W is unitary and central using purely algebraic methods, generalizing the standard relation J² = I. A finite-dimensional construction explicitly exhibits W = -I for specific pseudo-Hermitian operator algebras, providing concrete evidence of this stable algebraic phase. This central modular phase represents an intrinsic degree of freedom absent in Hermitian quantum field theory, persisting under pseudo-Hermitian similarity transformations. We discuss potential physical implications, including connections to CP violation and baryon asymmetry, where the phase W may serve as an algebraic source of matter-antimatter asymmetry emerging from field algebra structure rather than explicit Lagrangian symmetry breaking. The framework maintains mathematical consistency with Tomita-Takesaki axioms while extending modular theory beyond Hermitian constraints. This work bridges pseudo-Hermitian similarity transformations and modular conjugation into unified algebraic language, opening directions for PT-symmetric quantum field theory and non-Hermitian approaches to fundamental physics. STATUS: Verison 1.0 and 2.0 completed rest will be available soon .
We derive a general relation between the bosonic and fermionic entanglement in the ground states of supersymmetric quadratic Hamiltonians.For this, we construct canonical identifications between bosonic and fermionic subsystems.Our derivation relies on a unified framework to describe both bosonic and fermionic Gaussian states in terms of so-called linear complex structures J.The resulting dualities apply to the full entanglement spectrum between the bosonic and the fermionic systems, such that the von Neumann entropy and arbitrary Renyi entropies can be related.We illustrate our findings in one-and two-dimensional systems, including the paradigmatic Kitaev honeycomb model.While typically supersymmetry preserves features like area law scaling of the entanglement entropies on either side, we find a peculiar phenomenon, namely, an amplified scaling of the entanglement entropy ("super area law") in bosonic subsystems when the dual fermionic subsystems develop almost maximally entangled modes.