Optimal lower bound for quantum channel tomography in away-from-boundary regime
Kean Chen, Zhicheng Zhang, Nengkun Yu
TL;DR
This work proves an optimal quantum-channel-tomography lower bound of $\\Omega\\left(\\frac{r d_1 d_2}{\\varepsilon^2}\\right)$ queries in the away-from-boundary regime $r d_2 \ge 2 d_1$, matching the known upper bounds and fully resolving the typical case $d_1=d_2=d$, $r\\ge 2$. The authors reduce tomography to discriminating a carefully constructed hard set of isometries, leveraging quantum combs and testers to bound distinguishability, and instantiate the construction using a recent $\\varepsilon$-net of isometries. Their analysis shows a phase transition from Heisenberg scaling ($r=1$) to classical scaling ($r\\ge 2$) in this regime, and closes the gap without logarithmic factors that appeared in prior lower-bound results. The results provide precise resource estimates for quantum process tomography and highlight the power of the combs/testers framework in establishing tight information-theoretic limits.
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 $Ω(rd_1d_2/\varepsilon^2)$ 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 $Θ(d^2/\varepsilon)$ is achievable.