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Disentangling tensor product structures

Antoine Soulas

TL;DR

Addresses whether a fixed TPS can disentangle the time evolution of a finite-dimensional quantum system. Uses a constructive two-qubit C-NOT gate example to demonstrate a disentangling TPS exists in that case, and proves nonexistence for most other evolutions. From a Hamiltonian perspective, derives a spectral criterion for when a disentangling TPS is possible, relating separability of the Hamiltonian to the structure of its spectrum. The results highlight fundamental limits of TPS-based disentanglement and point toward approximate strategies and Hamiltonian-driven TPS selection in quantum foundations.

Abstract

As a contribution to the field of quantum mereology, we study how a change of tensor product structure in a finite-dimensional Hilbert space affects its entanglement properties. In particular, we ask whether, given a time-evolving state, there exists a tensor product structure in which no entanglement is generated. We give a concrete, constructive example of disentangling tensor product structure in the case of a C-NOT gate evolution between two qbits, before showing that this cannot be achieved for most time-evolving quantum states.

Disentangling tensor product structures

TL;DR

Addresses whether a fixed TPS can disentangle the time evolution of a finite-dimensional quantum system. Uses a constructive two-qubit C-NOT gate example to demonstrate a disentangling TPS exists in that case, and proves nonexistence for most other evolutions. From a Hamiltonian perspective, derives a spectral criterion for when a disentangling TPS is possible, relating separability of the Hamiltonian to the structure of its spectrum. The results highlight fundamental limits of TPS-based disentanglement and point toward approximate strategies and Hamiltonian-driven TPS selection in quantum foundations.

Abstract

As a contribution to the field of quantum mereology, we study how a change of tensor product structure in a finite-dimensional Hilbert space affects its entanglement properties. In particular, we ask whether, given a time-evolving state, there exists a tensor product structure in which no entanglement is generated. We give a concrete, constructive example of disentangling tensor product structure in the case of a C-NOT gate evolution between two qbits, before showing that this cannot be achieved for most time-evolving quantum states.

Paper Structure

This paper contains 7 sections, 3 theorems, 18 equations.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Definitions
  4. Existence for the C-NOT gate
  5. Non-existence in most cases
  6. The Hamiltonian perspective
  7. Conclusion

Key Result

Proposition 3.1

Let $\mathcal{H} = \mathbb{C}^2 \otimes \mathbb{C}^2$ be the standard TPS of local observables associated with two qbits, and $\ket{\Psi(t)} = \frac{1}{\sqrt{2}} $ for $t \in [0, \frac{\pi}{2}]$ be the trajectory, as written in the basis $\mathcal{B} = (\ket{00}, \ket{01}, \ket{10}, \ket{11})$, desc

Theorems & Definitions (8)

  • Definition 2.1: Tensor product structure
  • Definition 2.2: Disentangling tensor product structure
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Proposition 5.1
  • proof