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

• URL: http://arxiv.org/abs/2601.10683v1
• Date: Thu, 15 Jan 2026 18:45:59 GMT
• Title: Optimal lower bound for quantum channel tomography in away-from-boundary regime
• Abstract summary: Heisenberg scaling $(d2/varepsilon)$ is achievable.<n>In particular, this lower bound fully settles the query complexity for the commonly studied case of equal input and output dimensions.
• Score: 32.904052887092284
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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