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Dual Block Gradient Ascent for Entropically Regularised Quantum Optimal Transport

Marvin Randig, Max von Renesse

TL;DR

This work addresses entropically regularised quantum optimal transport by formulating a dual block gradient ascent on the variables $(U,V)$ and proving a linear convergence rate. The primal problem minimizes the energy plus entropy over joint quantum states with fixed marginals, while the dual maximises a tractable objective involving the matrix exponential. The authors establish strong concavity of the dual on a suitable subspace, derive explicit gradient expressions and step-size rules, and demonstrate linear convergence both theoretically and numerically. The results provide a Sinkhorn-like, scalable method for quantum OT with potential applications in quantum information processing and entangled-state transport. All mathematical constructs are rigorously tied to density operators, partial traces, and matrix exponentials, offering a principled framework for entropic quantum transport computations.

Abstract

We present a block gradient ascent method for solving the quantum optimal transport problem with entropic regularisation similar to the algorithm proposed in [D. Feliciangeli, A. Gerolin, L. Portinale: J. Funct. Anal. 285 (2023), no. 4, 109963] and [E. Caputo, A. Gerolin, N. Monina, L. Portinale: arXiv:2409.03698]. We prove a linear convergence rate based on strong concavity of the dual functional and present some results of numerical experiments of an implementation.

Dual Block Gradient Ascent for Entropically Regularised Quantum Optimal Transport

TL;DR

This work addresses entropically regularised quantum optimal transport by formulating a dual block gradient ascent on the variables and proving a linear convergence rate. The primal problem minimizes the energy plus entropy over joint quantum states with fixed marginals, while the dual maximises a tractable objective involving the matrix exponential. The authors establish strong concavity of the dual on a suitable subspace, derive explicit gradient expressions and step-size rules, and demonstrate linear convergence both theoretically and numerically. The results provide a Sinkhorn-like, scalable method for quantum OT with potential applications in quantum information processing and entangled-state transport. All mathematical constructs are rigorously tied to density operators, partial traces, and matrix exponentials, offering a principled framework for entropic quantum transport computations.

Abstract

We present a block gradient ascent method for solving the quantum optimal transport problem with entropic regularisation similar to the algorithm proposed in [D. Feliciangeli, A. Gerolin, L. Portinale: J. Funct. Anal. 285 (2023), no. 4, 109963] and [E. Caputo, A. Gerolin, N. Monina, L. Portinale: arXiv:2409.03698]. We prove a linear convergence rate based on strong concavity of the dual functional and present some results of numerical experiments of an implementation.

Paper Structure

This paper contains 16 sections, 13 theorems, 63 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setting and presentation of the results
  3. Entropy-regularised Quantum Optimal Transport
  4. Sets of operators on Hilbert spaces
  5. Tensor sum and partial traces
  6. Liftings of functions
  7. Primal and dual problem
  8. Gradients and marginal errors
  9. Description of the algorithms
  10. Convergence theorem
  11. Proof of Lemma \ref{['lem:DstronglyConcave']} and Theorem \ref{['thm:Convergence']}
  12. Numerical Experiments
  13. Acknowledgements
  14. Derivatives of the matrix exponential
  15. Eigenvalue estimates
  16. ...and 1 more sections

Key Result

Theorem 2.1

There is a unique minimiser $\Gamma \in \mathfrak{P}(\mathcal{H}_1 \otimes \mathcal{H}_2)$ of the primal problem. It holds where $U, V$ are maximisers of the dual problem with the restrictions of the density matrices to the complement of their kernel ($\rho |_{(\ker \rho)^\perp}$ and $\sigma |_{(\ker \sigma)^\perp}$) as problem parameters.

Figures (1)

  • Figure 1: These plots show the evolution of the errors of the first and second variable, dual functional, first and second marginal and result for a run of the first version of the algorithm for two $2 \times 2$ matrices as marginals and a $4\times 4$ matrix as Hamiltonian. In each of the plots, the corresponding value is plotted whenever $U$ or $V$ changes. The first 500 out of 6169 values are shown.

Theorems & Definitions (22)

  • Theorem 2.1: Duality
  • Theorem 2.2: Convergence of Algorithm \ref{['alg:bga']}
  • Lemma 2.2: Concavity of the dual functional
  • Lemma 3.1: Spectral bounds
  • proof
  • Lemma 3.1: Concavity of the dual functional
  • proof
  • Corollary 3.2: Dual functional improvement
  • Proposition 3.3: Relation between marginal error and dual functional
  • proof
  • ...and 12 more