Optimal lower bound for quantum channel tomography in away-from-boundary regime

TL;DR

This paper establishes an optimal lower bound on the number of queries needed for quantum channel tomography in the away-from-boundary regime, matching existing upper bounds and clarifying the complexity for channels with certain rank and dimension constraints.

Contribution

It proves a tight lower bound on quantum channel tomography query complexity in the away-from-boundary regime, resolving the complexity for channels with equal input and output dimensions.

Findings

Lower bound matches the upper bound, confirming optimality.

Query complexity scales as ^2/\u03b5^2 in the equal-dimension case.

Contrast with the unitary case where Heisenberg scaling is possible.

Abstract

Consider quantum channels with input dimension $d_1$, output dimension $d_2$ and Kraus rank at most $r$. Any such channel must satisfy the constraint $rd_2\geq d_1$, and the parameter regime $rd_2=d_1$ is called the boundary regime. In this paper, we show an optimal query lower bound $\Omega(rd_1d_2/\varepsilon^2)$ for quantum channel tomography to within diamond norm error $\varepsilon$ in the away-from-boundary regime $rd_2\geq 2d_1$, matching the existing upper bound $O(rd_1d_2/\varepsilon^2)$. In particular, this lower bound fully settles the query complexity for the commonly studied case of equal input and output dimensions $d_1=d_2=d$ with $r\geq 2$, in sharp contrast to the unitary case $r=1$ where Heisenberg scaling $\Theta(d^2/\varepsilon)$ is achievable.