An optimal tradeoff between entanglement and copy complexity for state tomography
Sitan Chen, Jerry Li, Allen Liu
TL;DR
This work resolves the tomography copy-complexity tradeoff under practical, partially entangled measurements by establishing a smooth interpolation: for batch size $t$ up to $d^2$, the required total copies scale as $\widetilde{\Theta}\left( \frac{d^3}{\sqrt{t}\,\varepsilon^2} \right)$. The authors introduce a two-stage learning strategy that linearizes near the maximally mixed state and uses quantum splitting to handle general spectra, enabling a cohesive approach that bridges single-copy and fully entangled protocols. A key technical advance is the linearization of $\rho^{\otimes t}$ around $I_d/d$, paired with Keyl’s POVM and Schur-Weyl analysis to control information gain via the unfolded matrix $G_1(z)$, plus a lower-bound framework based on posterior anticoncentration. The results yield a first-ever smooth, provable entanglement-copy tradeoff for a natural quantum learning task, with implications for near-term quantum devices that exploit modest memory while performing tomography.
Abstract
There has been significant interest in understanding how practical constraints on contemporary quantum devices impact the complexity of quantum learning. For the classic question of tomography, recent work tightly characterized the copy complexity for any protocol that can only measure one copy of the unknown state at a time, showing it is polynomially worse than if one can make fully-entangled measurements. While we now have a fairly complete picture of the rates for such tasks in the near-term and fault-tolerant regimes, it remains poorly understood what the landscape in between looks like. In this work, we study tomography in the natural setting where one can make measurements of $t$ copies at a time. For sufficiently small $ε$, we show that for any $t \le d^2$, $\widetildeΘ(\frac{d^3}{\sqrt{t}ε^2})$ copies are necessary and sufficient to learn an unknown $d$-dimensional state $ρ$ to trace distance $ε$. This gives a smooth and optimal interpolation between the known rates for single-copy and fully-entangled measurements. To our knowledge, this is the first smooth entanglement-copy tradeoff known for any quantum learning task, and for tomography, no intermediate point on this curve was known, even at $t = 2$. An important obstacle is that unlike the optimal single-copy protocol, the optimal fully-entangled protocol is inherently biased and thus precludes naive batching approaches. Instead, we devise a novel two-stage procedure that uses Keyl's algorithm to refine a crude estimate for $ρ$ based on single-copy measurements. A key insight is to use Schur-Weyl sampling not to estimate the spectrum of $ρ$, but to estimate the deviation of $ρ$ from the maximally mixed state. When $ρ$ is far from the maximally mixed state, we devise a novel quantum splitting procedure that reduces to the case where $ρ$ is close to maximally mixed.