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arXiv:2406.15900 · Physical review. D/Physical review. D.

Tomita-Takesaki theory and quantum concurrence

Rupak Chatterjee

Published Sep 16, 2024• 5 citations
arXivHTMLPDF

Abstract

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 $J$ that links two separate systems with respect to their von Neumann algebras is related to the quantum concurrence $C$ of a pure bivariate entangled state composed from these systems. This concurrence relation provides a direct physical meaning to $J$ 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-$\frac{1}{2}$ 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.

DOI
10.1103/physrevd.110.065012

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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.
A bstract The interaction between Unruh-De Witt spin 1/2 detectors and a real scalar field is scrutinized by making use of the Tomita-Takesaki modular theory as applied to the Von Neumann algebra of the Weyl operators. The use of the modular theory enables to evaluate in an exact way the trace over the quantum field degrees of freedom. The resulting density matrix is employed to the study of the Bell-CHSH correlator. It turns out that, as a consequence of the interaction with the quantum field, the violation of the Bell-CHSH inequality exhibits a decreasing as compared to the case in which the scalar field is absent.
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.
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.
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 .
The general scope of this PhD thesis is to establish relations between Tomita-Takesaki modular theory, in particular modular Hamiltonians and relative modular Hamiltonians, and physically better understood quantities, such as the stress-energy tensor, energy densities or other modular data in cases where it is known explicitly. In Chapter 1, we will present the basics of operators algebras, Tomita-Takesaki theory, symplectic spaces, and link these concepts to discuss quasi-equivalence of Gaussian states. In Chapter 2, we will discuss algebraic QFTs both in the framework of the Haag-Kastler axioms and of locally covariant QFT, to then specialize to scalar fields and present the stress-energy tensor and the Hadamard condition. In Chapter 3, focussing on Gaussian states of the Weyl-CCR algebra, we study relations between quasi-equivalence of their GNS-representations, differences of the total energy (on some Cauchy surface), and differences between functions of the modular Hamiltonians. Following qualitative relations by Araki and Yamagami, Verch and Longo, we investigate quantitative relations, including in particular estimates of differences between functions of modular Hamiltonians in terms of energy differences. These results will make use of inequalities between p-norms of operators, which we will discuss in Appendix C. In Chapter 4, generalizing the picture beyond Gaussian states, we exploit locality properties of general algebraic QFTs to construct a scheme that allows to estimate relative modular Hamiltonians for suitable pairs of states on some local algebra in terms of modular Hamiltonians of a reference state on bigger or smaller algebras, which might be better understood. For pairs of states susceptible to this scheme, there exist unitaries linking them to the reference state which do not enable superluminal signalling in the sense of Sorkin's paradox. Such unitaries may not however compose the whole class of non-signalling operators, raising the question whether the existence of a non-signalling unitary is a sufficient condition to show the applicability of our estimate. To investigate the strength of our estimates we then consider coherent states for CCR systems, focussing particularly on free scalar fields, where the difference between the upper and lower bounds can be made arbitrarily small, recovering an exact result by squeezing. This yields an independent proof for the relative entropy formula in cases where the relative modular Hamiltonian has not been computed exactly and corroborates the strength of our general estimates. For these results, we will make use of several commutation properties of operators on Fock spaces, which we will present in Appendix A. Results about operator monotone functions will be used throughout this thesis and are explained in Appendix B.
We provide rigorous numerical verification of the modular time definition through Tomita–Takesaki theory and establish a scaling relation between entanglement time τ_ent = dS/dt and modular time τ_mod = √Var(K_ρ) in Heisenberg XXX spin chains. Extensive simulations for system sizes N = 6–20 and systematic robustness checks reveal that τ_ent ∝ τ_mod^γ with exponent γ∞ = 2.07 ± 0.04 for N ≥ 14, consistent with diffusion-like growth of entanglement in modular time. We verify that the modular flow satisfies all four axioms of a Tomita–Takesaki automorphism group to numerical precision ~10⁻¹⁵. Furthermore, we demonstrate that emergent time is intensive (non-additive): τ(2N)/(2τ(N)) = 0.76 ± 0.17 ≠ 1. These results provide strong numerical evidence that different notions of time in quantum many-body systems are related by a scale-invariant law, with potential implications for emergent gravity.