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Entanglement Generation via Hamiltonian Dynamics Having Limited Resources

Moein Naseri

TL;DR

The paper addresses how fast bipartite entanglement can be generated via Hamiltonian dynamics under finite energetic resources, measured by the relative entropy of entanglement. It derives a closed-form instantaneous rate $\Gamma(\psi,H)$ in the Schmidt basis, showing that only the imaginary part of the Hamiltonian $H_I$ dictates growth, and that a bound on energy variance is required to obtain a finite maximum. The authors obtain a complete characterization of optimal states and Hamiltonians in the no-ancilla setting, revealing a connection between surprisal variance of the Schmidt coefficients and the maximal rate, and they present explicit forms for the optimal state and Hamiltonian. When local ancillas are allowed, a matrix-analytic framework yields an explicit optimization problem with an enhanced achievable rate, captured by $\Lambda^2 = \sup \mathrm{Tr}(|C^T K - K^T C|^2 (C^T C)^{-1})$, offering a concrete method to quantify ancilla-assisted gains. Overall, the work provides operational limits on entanglement generation under energetic constraints and lays out future directions for mixed states, higher-order energetics, and many-body scenarios.

Abstract

We investigate the fundamental limits of entanglement generation under bipartite Hamiltonian dynamics when only finite physical resources-specifically, bounded energy variance-are available. Using the relative entropy of entanglement, we derive a closed analytical expression for the instantaneous entanglement generation rate for arbitrary pure states and Hamiltonians expressed in the Schmidt basis. We find that constraints based solely on the mean energy of the Hamiltonian are insufficient to bound the entanglement generation rate, whereas imposing a variance constraint ensures a finite and well-defined maximum. We fully characterize the Hamiltonians that achieve this optimal rate, establishing a direct relation between their imaginary components in the Schmidt basis and the structure of the optimal initial states. For systems without ancillas, we obtain a closed-form expression for the maximal rate in terms of the surprisal variance of the Schmidt coefficients and identify the family of optimal states and Hamiltonians. We further extend our analysis to scenarios where Alice and Bob may employ local ancillary systems: using a matrix-analytic framework and a refined description of the Hamiltonians allowed by the physical constraints, we derive an explicit optimization formula and characterize the attainable enhancement in entanglement generation.

Entanglement Generation via Hamiltonian Dynamics Having Limited Resources

TL;DR

The paper addresses how fast bipartite entanglement can be generated via Hamiltonian dynamics under finite energetic resources, measured by the relative entropy of entanglement. It derives a closed-form instantaneous rate in the Schmidt basis, showing that only the imaginary part of the Hamiltonian dictates growth, and that a bound on energy variance is required to obtain a finite maximum. The authors obtain a complete characterization of optimal states and Hamiltonians in the no-ancilla setting, revealing a connection between surprisal variance of the Schmidt coefficients and the maximal rate, and they present explicit forms for the optimal state and Hamiltonian. When local ancillas are allowed, a matrix-analytic framework yields an explicit optimization problem with an enhanced achievable rate, captured by , offering a concrete method to quantify ancilla-assisted gains. Overall, the work provides operational limits on entanglement generation under energetic constraints and lays out future directions for mixed states, higher-order energetics, and many-body scenarios.

Abstract

We investigate the fundamental limits of entanglement generation under bipartite Hamiltonian dynamics when only finite physical resources-specifically, bounded energy variance-are available. Using the relative entropy of entanglement, we derive a closed analytical expression for the instantaneous entanglement generation rate for arbitrary pure states and Hamiltonians expressed in the Schmidt basis. We find that constraints based solely on the mean energy of the Hamiltonian are insufficient to bound the entanglement generation rate, whereas imposing a variance constraint ensures a finite and well-defined maximum. We fully characterize the Hamiltonians that achieve this optimal rate, establishing a direct relation between their imaginary components in the Schmidt basis and the structure of the optimal initial states. For systems without ancillas, we obtain a closed-form expression for the maximal rate in terms of the surprisal variance of the Schmidt coefficients and identify the family of optimal states and Hamiltonians. We further extend our analysis to scenarios where Alice and Bob may employ local ancillary systems: using a matrix-analytic framework and a refined description of the Hamiltonians allowed by the physical constraints, we derive an explicit optimization formula and characterize the attainable enhancement in entanglement generation.

Paper Structure

This paper contains 6 sections, 11 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hamiltonians with Bounded Mean and Variance, and Generation of Entanglement
  3. Maximal Entanglement Generation Rate without Ancilla
  4. The Case with Ancilla
  5. Conclusion
  6. S5. Proof of Theorem \ref{['Charac-With-Ancilla']}

Key Result

Proposition 1

The bipartite entanglement generation rate $\Gamma(\psi, H)$ between subsystems $A$ and $B$, with Hilbert space dimensions $d_A$ and $d_B$, can be expressed as where $\ket{\psi}_{AB} \;=\; \sum_{i=1}^{\min(d_A,d_B)} C_i \, \ket{i}_A \ket{i}_B$ and $\hat{H} \;=\; \sum_{k,l=1}^{d_A} \sum_{m,n=1}^{d_B} H_{kl,mn} \, \ket{k}\bra{l}_A \otimes \ket{m}\bra{n}_B$ which are expanded in a Schmidt basis. Mor

Figures (1)

  • Figure 1: Conceptual overview of entanglement generation mechanisms: Entanglement can be generated from separable inputs via non-separable quantum channels $\Lambda$

Theorems & Definitions (11)

  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Proposition 2
  • Theorem 3
  • Proposition 4
  • Lemma 3
  • Proposition 5
  • Theorem 6
  • Theorem 7
  • ...and 1 more