[Paper Review] Purification of Lindblad dynamics, geometry of mixed states and geometric phases

David Viennot|arXiv (Cornell University)|Aug 10, 2015

Nonlinear Photonic Systems48 references4 citations

TL;DR

This paper proposes a nonlinear Schrödinger equation in an enlarged Hilbert space with an ancilla that purifies Lindblad dynamics, ensuring the partial trace of its solution reproduces the open quantum system's density matrix evolution. The theory unifies geometric phases for open systems and reveals a complex geometric structure based on higher gauge theory, specifically categorical principal bundles with connective structures.

ABSTRACT

We propose a nonlinear Schr\\"odinger equation in a Hilbert space enlarged with an ancilla such that the partial trace of its solution obeys to the Lindblad equation of an open quantum system. The dynamics involved by this nonlinear Schr\\"odinger equation constitutes then a purification of the Lindbladian dynamics. This nonlinear equation is compared with other Schr\\"odinger like equations appearing in the theory of open systems. We study the (non adiabatic) geometric phases involved by this purification and show that our theory unifies several definitions of geometric phases for open systems which have been previously proposed. We study the geometry involved by this purification and show that it is a complicated geometric structure related to an higher gauge theory, i.e. a categorical bibundle with a connective structure.

Motivation & Objective

• To unify disparate definitions of geometric phases in open quantum systems by constructing a unified dynamical framework.
• To provide a purification of Lindblad dynamics using a nonlinear Schrödinger equation in an extended Hilbert space with an ancilla.
• To reveal the underlying geometric structure of mixed states as a stratified categorical principal bundle with connective structure.
• To establish a link between decoherence in open systems and entanglement in the purified pure-state dynamics.
• To generalize geometric phase theories from pure states to mixed states using C*-module and higher gauge theory formalisms.

Proposed method

• Introduce an enlarged Hilbert space H_S ⊗ H_A, where H_A is an ancilla Hilbert space, to embed the density matrix ρ as a partial trace of a pure state Ψ.
• Derive a nonlinear Schrödinger equation for Ψ such that tr_H_A(|Ψ⟩⟨Ψ|) satisfies the Lindblad equation.
• Construct a purification bundle over the space of density matrices, stratified by rank, with a C*-module structure and C*-adjoint compatibility.
• Define left and right principal categorical bundles to model the geometric structure of mixed states, with connective structures encoding dynamical connections.
• Use the nonlinear dynamics in the purified space to define operator-valued geometric phases, generalizing Berry and Uhlmann phases.
• Relate the theory to existing stochastic formulations (e.g., quantum state diffusion and piecewise deterministic processes) by showing their dynamics emerge as approximations within the purified framework.

Experimental results

Research questions

• RQ1 How can Lindblad dynamics for open quantum systems be purified into a unitary-like nonlinear Schrödinger equation in an enlarged Hilbert space?
• RQ2 What geometric structure underlies the space of mixed quantum states when viewed through the lens of purification and non-Abelian gauge theory?
• RQ3 How do geometric phases in open systems unify previously proposed definitions (e.g., Uhlmann, Sjöqvist, and adiabatic C*-module phases)?
• RQ4 What is the role of the ancilla Hilbert space in encoding decoherence and relaxation as entanglement in the purified dynamics?
• RQ5 How does the connective structure on the categorical principal bundle relate to the dynamics of the purified system and its geometric phases?

Key findings

• The nonlinear Schrödinger equation in the enlarged Hilbert space successfully purifies Lindblad dynamics, with the partial trace of its solution reproducing the original density matrix evolution.
• The geometric structure of mixed states is identified as a stratified principal composite bibundle, with regular and singular strata corresponding to different ranks of density matrices.
• The theory realizes a higher gauge theory structure: a categorical principal bundle with left and right connective structures, generalizing standard principal bundles to category-theoretic frameworks.
• The operator-valued geometric phases derived from the purified dynamics unify Uhlmann, Sjöqvist, and adiabatic C*-module geometric phases into a single coherent framework.
• The formalism connects to stochastic open system dynamics (e.g., quantum state diffusion and piecewise deterministic processes), showing they describe short-time approximations of the full nonlinear purified dynamics.
• The effective Hamiltonian in the purified dynamics includes a nonlinear term proportional to γ^k ‖Γ_k ψ‖², which captures relaxation and decoherence effects as entanglement with the ancilla.