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A fixed-point algorithm for matrix projections with applications in quantum information

Shrigyan Brahmachari, Roberto Rubboli, Marco Tomamichel

TL;DR

This work introduces a fixed-point algorithm to compute the Bures-distance projection of a fixed matrix onto the set of group-invariant positive semidefinite matrices, extending the known fixed-point method for Bures–Wasserstein barycenters. The update map $K(S) = S^{-1/2}(\mathcal{E}((S^{1/2}RS^{1/2})^{1/2}))^2S^{-1/2}$ yields a monotonically decreasing Bures distance to $R$ and, under strong convexity, converges exponentially fast to the unique optimal $T$. The analysis combines a Hölder-based contraction with a Polyak–Łojasiewicz-type inequality, yielding problem-dependent convergence rates that become dimension-independent in commuting-invariant cases. The framework unifies several quantum information tasks under a common projection viewpoint, with numerical evidence showing substantial practical advantages over SDP-based methods, and points to broad future directions including stochastic variants and extensions to other quantum divergences.

Abstract

We develop a fixed-point iterative algorithm that computes the matrix projection with respect to the Bures distance on the set of positive definite matrices that are invariant under some symmetry. We prove that the fixed-point iteration algorithm converges exponentially fast to the optimal solution in the number of iterations. Moreover, it numerically shows fast convergence compared to the off-the-shelf semidefinite program solvers. Our algorithm, for the specific case of Bures-Wasserstein barycenter, recovers the fixed-point iterative algorithm originally introduced in (Álvarez-Esteban et al., 2016). Our proof is concise and relies solely on matrix inequalities. Finally, we discuss several applications of our algorithm in quantum resource theories and quantum Shannon theory.

A fixed-point algorithm for matrix projections with applications in quantum information

TL;DR

This work introduces a fixed-point algorithm to compute the Bures-distance projection of a fixed matrix onto the set of group-invariant positive semidefinite matrices, extending the known fixed-point method for Bures–Wasserstein barycenters. The update map yields a monotonically decreasing Bures distance to and, under strong convexity, converges exponentially fast to the unique optimal . The analysis combines a Hölder-based contraction with a Polyak–Łojasiewicz-type inequality, yielding problem-dependent convergence rates that become dimension-independent in commuting-invariant cases. The framework unifies several quantum information tasks under a common projection viewpoint, with numerical evidence showing substantial practical advantages over SDP-based methods, and points to broad future directions including stochastic variants and extensions to other quantum divergences.

Abstract

We develop a fixed-point iterative algorithm that computes the matrix projection with respect to the Bures distance on the set of positive definite matrices that are invariant under some symmetry. We prove that the fixed-point iteration algorithm converges exponentially fast to the optimal solution in the number of iterations. Moreover, it numerically shows fast convergence compared to the off-the-shelf semidefinite program solvers. Our algorithm, for the specific case of Bures-Wasserstein barycenter, recovers the fixed-point iterative algorithm originally introduced in (Álvarez-Esteban et al., 2016). Our proof is concise and relies solely on matrix inequalities. Finally, we discuss several applications of our algorithm in quantum resource theories and quantum Shannon theory.
Paper Structure (20 sections, 9 theorems, 73 equations, 2 figures, 2 tables, 2 algorithms)

This paper contains 20 sections, 9 theorems, 73 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Fixed-point algorithm for matrix projections with respect to the Bures distance
  4. Relationship with the Bures-Wasserstein barycenter problem
  5. The fixed-point equation
  6. Proof of Theorem 1
  7. First inequality
  8. Uniform lower bounds for strong convexity
  9. Stronger bounds for commuting invariant matrices
  10. Stronger bounds for commuting invariant matrices
  11. Second inequality
  12. Combining the inequalities
  13. Proof of Corollary \ref{['corollary Bures-Wasserstein barycenter']}
  14. Application to quantum information theory
  15. Numerical comparison between the SDP solver and the fixed-point algorithm
  16. ...and 5 more sections

Key Result

Theorem 1

Let $R$ be a positive definite matrix. Consider Algorithm Fixed-point algo to solve the convex optimization problem first problem and $T$ represent the optimal solution. Then the sequence $\{B(R,S_n)\}$ decreases monotonically, and satisfies where Moreover, when the invariant matrices commute, we have $\xi = (\lambda_{\max}(\mathcal{E}(R))/ \lambda_{\min}(R))^\frac{3}{2}$.

Figures (2)

  • Figure 1: We compare the runtime of the fixed-point iterative algorithm (red) and the SDP solver (blue). We consider two cases, namely the max-conditional entropy (a) and fidelity of coherence (b). We defined these two functions in equation \ref{['optimization problems']}. We check the performance of 100 randomly generated quantum states for several values of the dimension. We use a logarithmic scale on the y-axis, and we represent the data using a Whisker chart. The colored areas represent the interquartile regions, while the vertical lines extend to the minimum and maximum values. We considered bipartite quantum systems with subsystem dimensions ranging from 2 to 7 for the max-conditional entropy case, and quantum states of dimensions ranging from 4 to 36 for the fidelity of coherence. We run the iterative algorithm for enough iterations so it finds a state whose value is at least $10^{-5}$ close to the one returned by the SDP.
  • Figure 2: The plot shows the value of $\sum_j \omega_j B(X_j, S_n)^2$ as a function of the iteration number $n$, displayed on a logarithmic scale for the three algorithms discussed in the main text: the fixed-point algorithm (FP), projected gradient descent (PGD), and Riemannian gradient descent with non-unit step size (RGD (non-unit)). The example considers an average of $m = 3$ matrices in dimension $d = 4$. The plot illustrates that both RGD and PGD converge more slowly, primarily due to the use of very small step sizes in higher-dimensional settings.

Theorems & Definitions (17)

  • Theorem 1
  • Corollary 2
  • Lemma 3
  • proof
  • Theorem 4: bhatia2018strong
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Remark 1
  • ...and 7 more