Random Stinespring superchannel: converting channel queries into dilation isometry queries

TL;DR

This work introduces the random Stinespring superchannel, a channel-level analogue of random purification, which maps $ $ parallel uses of any channel with Choi rank $r$ to $n$ uses of a uniformly random Stinespring isometry via universal encoding/decoding. It provides a simple existence proof in the full-rank case and an explicit, efficient circuit for the construction based on Schur-Weyl duality, QFT, and Weingarten calculus. The authors prove that learning a quantum channel reduces to learning a Stinespring isometry, establishing a tight query complexity of $\Theta(d_A d_B r)$ for channel tomography without extra logarithmic factors, and provide matching lower bounds valid for broad query models. These results unify upper and lower bounds and imply that channel learning can be performed via isometry learning, with potential extensions to Gaussian channels and adaptive settings.

Abstract

The recently introduced random purification channel, which converts $n$ copies of an arbitrary mixed quantum state into $n$ copies of the same uniformly random purification, has emerged as a powerful tool in quantum information theory. Motivated by this development, we introduce a channel-level analogue, which we call the random Stinespring superchannel. This consists in a procedure to transform $n$ parallel queries of an arbitrary quantum channel into $n$ parallel queries of the same uniformly random Stinespring isometry, via universal encoding and decoding operations that are efficiently implementable. When the channel is promised to have Choi rank at most $r$, the procedure can be tailored to yield a Stinespring environment of dimension $r$. As a consequence, quantum channel learning reduces to isometry learning, yielding a simple channel learning algorithm, based on existing isometry learning protocols, that matches the performance of the two recently proposed channel tomography algorithms. Complementarily, whereas the optimality of these algorithms had previously been established only up to a logarithmic factor in the dimension, we close this gap by removing this logarithmic factor from the lower bound. Taken together, our results fully establish the optimality of these recently introduced channel learning algorithms, showing that the optimal query complexity of learning a quantum channel with input dimension $d_A$, output dimension $d_B$, and Choi rank $r$ is $Θ(d_A d_B r)$.

Figures (2)

• Figure 1: Schematic representation of the random Stinespring superchannel introduced in Theorem \ref{['thm1']}.
• Figure 2: Circuit implementation of the random Stinespring superchannel from Theorem \ref{['th:circuit']}.

Theorems & Definitions (24)

• Theorem 1: (Random Stinespring superchannel)
• Lemma 2: Chiribella2008
• Proposition 3
• Corollary 4
• proof
• proof : Proof of Proposition \ref{['prop:Choi']}.
• Lemma 5: (Explicit formula for random Stinespring isometry)
• proof
• Theorem 6: (Circuit implementing the random Stinespring superchannel)
• proof