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Approximate pushforward designs and image bounds on approximations

Jakub Czartowski, Adam Sawicki, Karol Życzkowski

TL;DR

The paper analyzes how approximation errors in $t$-designs propagate under pushforward mappings to image spaces in quantum settings. It develops Lipschitz-based bounds for the resulting $\delta'_p$ in simplex, mixed-state, and channel designs, leveraging moment-operator formalisms and symmetry considerations to obtain tighter estimates. By connecting to frame potentials and Welch bounds, the authors provide computable, dimension-dependent bounds that improve over naive partial-trace arguments. Numerical simulations corroborate the theory, showing near-optimal performance in low-dimensional scenarios, and pave the way for practical construction of approximate pushforward designs in quantum tomography and related tasks.

Abstract

We extend the framework of quantum pushforward designs to the approximate setting, where averaging is achieved only up to finite precision. Using Schatten $p$-norms and Lipschitz continuity arguments, we derive bounds on the approximation parameters of pushforward designs obtained from complex projective spaces, including simplices, mixed states, and quantum channels. In the mixed-state case, we refine the bounds by exploiting the symmetric subspace structure, leading to asymptotically tighter estimates. Numerical simulations support our theoretical results, showing near-optimality in low-dimensional scenarios.

Approximate pushforward designs and image bounds on approximations

TL;DR

The paper analyzes how approximation errors in -designs propagate under pushforward mappings to image spaces in quantum settings. It develops Lipschitz-based bounds for the resulting in simplex, mixed-state, and channel designs, leveraging moment-operator formalisms and symmetry considerations to obtain tighter estimates. By connecting to frame potentials and Welch bounds, the authors provide computable, dimension-dependent bounds that improve over naive partial-trace arguments. Numerical simulations corroborate the theory, showing near-optimal performance in low-dimensional scenarios, and pave the way for practical construction of approximate pushforward designs in quantum tomography and related tasks.

Abstract

We extend the framework of quantum pushforward designs to the approximate setting, where averaging is achieved only up to finite precision. Using Schatten -norms and Lipschitz continuity arguments, we derive bounds on the approximation parameters of pushforward designs obtained from complex projective spaces, including simplices, mixed states, and quantum channels. In the mixed-state case, we refine the bounds by exploiting the symmetric subspace structure, leading to asymptotically tighter estimates. Numerical simulations support our theoretical results, showing near-optimality in low-dimensional scenarios.

Paper Structure

This paper contains 8 sections, 6 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounds on $\delta$-pushforward designs
  4. Simplex designs from Complex projective designs
  5. Mixed state designs
  6. Channel designs
  7. Numerical experiment
  8. Concluding remarks

Key Result

Theorem 1

Consider partial trace map $\Tr_B:\mathcal{L}\qty(\mathcal{H}^A\otimes\mathcal{H}^B)\ni Q \mapsto Q_A \in \mathcal{H}^{d_A}$. It is Lipschitz-continuous with respect to arbitrary Schatten $p$-norm $\norm{\cdot}_p$ with a (non-optimal) Lipschitz constant $L = d_B^{(p-1)/p}$.

Figures (1)

  • Figure 1: Monte Carlo trajectories of Schatten-$p$ norm convergence for $t=2$ approximate designs with subsystem dimensions $d_A, d_B \leq 3$. Each trajectory corresponds to $1 \leq M \leq 10^4$ samples, repeated over 100 runs. The empirical results (black curves) are compared against the bound from Observation 1 (dashed blue), the bound from Theorem 4 (dashed cyan), and the averaged Monte Carlo trajectory (solid red). The plots illustrate that the new bounds leveraging multicopy structure improve upon those derived from partial trace alone, with tightness for $d_A = d_B = 2$ and potential improvements for larger subsystem dimensions.

Theorems & Definitions (16)

  • Definition 1: Exact $t$-design
  • Definition 2: Exact $t$-design in terms of moment operator
  • Definition 3
  • Definition 4: $\delta_p$-approximate $t$-design
  • Definition 5: Lipschitz continuity
  • Theorem 1: Prop. 1 in rastegin2012relations
  • Theorem 2: Cor. 1 in rastegin2012relations
  • Theorem 3
  • proof
  • proof
  • ...and 6 more