Quantum dissipative systems. III. Definition and algebraic structure

Ilya Prigogine

Peng Qi

This paper presents a rigorous derivation pipeline for the quantum simulation of sparse polynomial nonlinear ordinary differential equations. The framework proceeds in three stages: (i) mapping the nonlinear dynamics to an exact infinite-dimensional linear system via Carleman–Koopman lifting; (ii) applying a graph-theoretic truncation and compression scheme that exploits the sparsity and locality of the dependency graph; and (iii) implementing Schr\"odingerisation for the resulting class of dissipative, Hermitian positive definite generators to yield a unitary-overlap formula compatible with standard quantum primitives. The study provides a comprehensive mathematical construction of these steps, establishing the algebraic and analytical foundations required for structure-preserving quantum simulation.

M. Courbage

Netta Engelhardt, Hong Liu

A bstract We propose an algebraic definition of ER=EPR in the G N → 0 limit, which associates bulk spacetime connectivity/disconnectivity to the operator algebraic structure of a quantum gravity system. The new formulation not only includes information on the amount of entanglement, but also more importantly the structure of entanglement . We give an independent definition of a quantum wormhole as part of the proposal. This algebraic version of ER=EPR sheds light on a recent puzzle regarding spacetime disconnectivity in holographic systems with $$ \mathcal{O} $$ O (1 /G N ) entanglement. We discuss the emergence of quantum connectivity in the context of black hole evaporation and further argue that at the Page time, the black hole-radiation system undergoes a transition involving the transfer of an emergent type III 1 subalgebra of high complexity operators from the black hole to radiation. We argue this is a general phenomenon that occurs whenever there is an exchange of dominance between two competing quantum extremal surfaces.

Rolando P. Hong Enriquez

Quantum Darwinism identifies a system's objective observables with the center of its observable algebra, but says nothing about which algebra a given open, dissipative dynamics selects, or how fast objectivity forms. This paper answers both. For Markovian open dynamics with a faithful steady state, we show that the conserved observables form a definite algebra — the commutant of the dissipation — and that the objective observables are exactly its center, making objectivity a computable invariant of the dissipator. Three consequences follow. Objectivity has a rate, set by the dynamics' "dissipative gap," so a pointer becomes objective at a definite speed — information a static pointer-basis account cannot supply. The protected "dark" states that resist decoherence are exactly the non-objective complement — precisely what the environment cannot witness. And objectivity can be engineered: one can design a dissipator whose fixed-point center is a chosen observable broadcast to a chosen redundancy, with an objectification time that scales inversely with the dissipation strength. The structural results are proved in finite dimension from standard, clearly cited facts of open-system and algebra theory, and the dephasing core is additionally verified by proof-checking software; the objectification time is a falsifiable prediction for dissipative quantum simulators. In short: a dissipative environment fixes which fact becomes classical and how fast — and you can design both, making a chosen observable objective at a chosen rate.

Mudassir Moosa

The number of non-Clifford gates needed to perform a task, or simply \textit{magic}, is a resource for fault-tolerant quantum computation. Von Neumann algebras provide a formal mathematical structure to describe infinite-dimensional quantum systems, such as those in quantum field theory or quantum statistical mechanics. A particularly important class of von Neumann algebras is called Type III algebras, for which the standard notions of finite-dimensional systems such as density matrices and traces break down. In this work, we argue that Type III von Neumann algebras fundamentally require an infinite amount of magic. Specifically, we consider the thermodynamic limit of finite-dimensional quantum systems, such as lattice systems, and show that if the states in the thermodynamic limit possess only a bounded amount of magic, the resulting local von Neumann algebra cannot be of Type III. Our result has direct implications for the quantum simulations of quantum field theories, for which the algebra of a local subregion is known to be of Type III.