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Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant

Malte Gerhold, Marcel Scherer, Orr Shalit

TL;DR

The paper tackles the open problem of the universal commuting dilation constant $C_2$ for pairs of contractions by combining matrix-range theory, levelwise convergence results, and a semidefinite-programming–based algorithm to compute dilation constants. It shows that for $d=2$ the current finite-size bound $C_2(n) \le \sqrt{2+2\sin(\tfrac{\pi}{2}(1-\tfrac{1}{2n}))}$ remains strictly below $2$, and provides strong numerical evidence suggesting $c(u_f,u_0) = \lim_{N\to\infty} c(U^{(N)},u_0) = \sqrt{2}$, which in turn yields $C_2 \le 2\sqrt{2/3} < 2$ via a dilation-triangle inequality. The work additionally develops and validates an SDP-based method to approximate $c(\\mathsf{U},\\mathsf{N})$ by discretizing the target commuting normals, enabling scalable computation of dilation constants for matrix tuples. Overall, the results tighten the known bounds on universal commuting dilation constants and provide a practical computational tool for exploring dilation phenomena in random and structured unitary ensembles. The approach and findings have potential implications for operator system theory, quantum information, and numerical dilation methods.

Abstract

For a tuple $T$ of Hilbert space operators, the 'commuting dilation constant' is the smallest number $c$ such that the operators of $T$ are a simultaneous compression of commuting normal operators of norm at most $c$. We present numerical experiments giving a strong indication that the commuting dilation constant of a pair of independent random $N{\times}N$ unitary matrices converges to $\sqrt2$ as $N \to \infty$ almost surely. Under the assumption that this is the case, we prove that the commuting dilation constant of an arbitrary pair of contractions is strictly smaller than $2$. Our experiments are based on a simple algorithm that we introduce for the purpose of computing dilation constants between tuples of matrices.

Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant

TL;DR

The paper tackles the open problem of the universal commuting dilation constant for pairs of contractions by combining matrix-range theory, levelwise convergence results, and a semidefinite-programming–based algorithm to compute dilation constants. It shows that for the current finite-size bound remains strictly below , and provides strong numerical evidence suggesting , which in turn yields via a dilation-triangle inequality. The work additionally develops and validates an SDP-based method to approximate by discretizing the target commuting normals, enabling scalable computation of dilation constants for matrix tuples. Overall, the results tighten the known bounds on universal commuting dilation constants and provide a practical computational tool for exploring dilation phenomena in random and structured unitary ensembles. The approach and findings have potential implications for operator system theory, quantum information, and numerical dilation methods.

Abstract

For a tuple of Hilbert space operators, the 'commuting dilation constant' is the smallest number such that the operators of are a simultaneous compression of commuting normal operators of norm at most . We present numerical experiments giving a strong indication that the commuting dilation constant of a pair of independent random unitary matrices converges to as almost surely. Under the assumption that this is the case, we prove that the commuting dilation constant of an arbitrary pair of contractions is strictly smaller than . Our experiments are based on a simple algorithm that we introduce for the purpose of computing dilation constants between tuples of matrices.
Paper Structure (12 sections, 6 theorems, 57 equations, 5 figures)

This paper contains 12 sections, 6 theorems, 57 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and background
  3. Overview of this paper
  4. Levelwise convergence and matrix range distance
  5. Computing dilation constants --- algorithmic aspects
  6. Approximation of the dilation constant
  7. Algorithm for computing c(U,N)
  8. Experimental results
  9. Experiment description and collected data
  10. Code
  11. How reliable are the results
  12. A rigorous bound for the dilation constant C2(n)

Key Result

Theorem 2.1

Let $(\xi^{(N)})_{N\in\mathbb N}$ and $(\eta^{(N)})_{N\in\mathbb N}$ be sequences of operator $d$-tuples whose matrix ranges converge levelwise to the matrix ranges of operator $d$-tuples $\xi^{(\infty)}$ and $\eta^{(\infty)}$, respectively. Then and

Figures (5)

  • Figure 1: Histograms of calculated dilation constants plotted separately for different matrix size $N$. As we expected, as $N$ grows, the values of the dilation constants accumulate closely to a value slightly above $\sqrt{2}$ (left dotted line) and below $\sqrt{2}/\cos(\pi/k)$ (right dotted line).
  • Figure 2: The mean and standard deviations of the values of dilations constants as functions of $N$.
  • Figure 3: The values of the dilation constant calculated for a single sample of size $N \times N$ for $N = 10, 25, \ldots, 100$, with $k=20$ (left) and $k=30$ (right). Note that the values obtained for $N\geq 40$ are always below $\sqrt{2}/\cos(\pi/k)$ (upper dotted line).
  • Figure 4: Dilation constants for a single sample of size $N \times N$ for $N = 25, 50, \ldots, 300$, with $k=8$ (left) alongside the histogram of 100 additional random trials for $N = 125$ (right). All values are well below $\sqrt{2}/\cos(\pi/k) \approx 1.53$.
  • Figure 5: Ten random sequences $\{c(U^{(N)},\mathsf{N}) : N = 25, 50, \ldots, 300\}$.

Theorems & Definitions (15)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 5 more