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Entanglement in Bipartite Quantum Systems with Fast Local Unitary Control

Emanuel Malvetti

TL;DR

The work addresses entanglement control in bipartite quantum systems under fast local unitaries by reducing the full bilinear dynamics to a tractable evolution of the state's singular values on the Schmidt sphere. It develops reduced control systems tied to symmetric Lie algebras (AIII for distinguishable, CI/DIII for bosonic/fermionic) and proves an Equivalence Theorem that preserves controllability, stabilizability, and reachability between the full and reduced descriptions. This yields fundamental results: the full system is controllable and stabilizable whenever the reduced system is, and explicit quantum speed limits arise from bounds on the induced vector fields; these conclusions extend coherently to bosonic and fermionic subsystems. The framework provides a principled approach to designing entanglement-generation and state-transfer protocols in both distinguishable and indistinguishable settings, with clear links to Autonne–Takagi and Hua factorizations and to symmetric Lie-algebra structure.

Abstract

The well-known Schmidt decomposition, or equivalently, the complex singular value decomposition, states that a pure quantum state of a bipartite system can always be brought into a "diagonal" form using local unitary transformations. In this work we consider a finite-dimensional closed bipartite system with fast local unitary control. In this setting one can define a reduced control system on the singular values of the state which is equivalent to the original control system. We explicitly describe this reduced control system and prove equivalence to the original system. Moreover, using the reduced control system, we prove that the original system is controllable and stabilizable and we deduce quantum speed limits. We also treat the fermionic and bosonic cases in parallel, which are related to the Autonne-Takagi and Hua factorization respectively.

Entanglement in Bipartite Quantum Systems with Fast Local Unitary Control

TL;DR

The work addresses entanglement control in bipartite quantum systems under fast local unitaries by reducing the full bilinear dynamics to a tractable evolution of the state's singular values on the Schmidt sphere. It develops reduced control systems tied to symmetric Lie algebras (AIII for distinguishable, CI/DIII for bosonic/fermionic) and proves an Equivalence Theorem that preserves controllability, stabilizability, and reachability between the full and reduced descriptions. This yields fundamental results: the full system is controllable and stabilizable whenever the reduced system is, and explicit quantum speed limits arise from bounds on the induced vector fields; these conclusions extend coherently to bosonic and fermionic subsystems. The framework provides a principled approach to designing entanglement-generation and state-transfer protocols in both distinguishable and indistinguishable settings, with clear links to Autonne–Takagi and Hua factorizations and to symmetric Lie-algebra structure.

Abstract

The well-known Schmidt decomposition, or equivalently, the complex singular value decomposition, states that a pure quantum state of a bipartite system can always be brought into a "diagonal" form using local unitary transformations. In this work we consider a finite-dimensional closed bipartite system with fast local unitary control. In this setting one can define a reduced control system on the singular values of the state which is equivalent to the original control system. We explicitly describe this reduced control system and prove equivalence to the original system. Moreover, using the reduced control system, we prove that the original system is controllable and stabilizable and we deduce quantum speed limits. We also treat the fermionic and bosonic cases in parallel, which are related to the Autonne-Takagi and Hua factorization respectively.
Paper Structure (15 sections, 25 theorems, 63 equations)

This paper contains 15 sections, 25 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Outline
  4. Control Systems
  5. Bilinear Control System
  6. Related Matrix Decompositions
  7. Reduced Control System
  8. Global Phases
  9. Some Applications
  10. Controllability and Stabilizability
  11. Speed Limit and Control Time
  12. Relation to Symmetric Lie Algebras
  13. Complex Singular Value Decomposition (Type AIII)
  14. Autonne--Takagi Factorization (Type CI)
  15. Hua Factorization (Type DIII)

Key Result

Proposition 2.5

Let $H_0\in\mathrm{i}\mkern1mu\mathfrak{u}(d_1d_2)$ denote an arbitrary coupling Hamiltonian and let $E_k\in\mathrm{i}\mkern1mu\mathfrak{u}(d_1)$ and $F_k\in\mathrm{i}\mkern1mu\mathfrak{u}(d_2)$ for $k=1,\ldots,r$ be given such that $H_0=\sum_{k=1}^r E_k\otimes F_k$. Then the induced vector field $H Now assume additionally that $d:=d_1=d_2$ and that $H_0\in\mathrm{i}\mkern1mu\mathfrak{u}^s(d^2)$ i

Theorems & Definitions (54)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Corollary 2.7
  • Definition 2.8: Reduced control systems
  • Remark 2.9
  • ...and 44 more