# Tight Lower Bounds for State Tomography with Limited Entanglement

arXiv: https://arxiv.org/abs/2609.05718 · HTML: https://mlsanity.com/paper/2609.05718/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 23 references that are in the map.
- First posted on arXiv 4 Sep 2026.

## Builds on (most cited first, 6 of 6)

- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [When Does Adaptivity Help for Quantum State Learning?](https://mlsanity.com/paper/2206.05265/) (2022)
- [A proof of the Caffarelli contraction theorem via entropic regularization](https://mlsanity.com/paper/1904.06053/) (2019)
- [Lower Bounds for Learning Quantum States with Single-Copy Measurements](https://mlsanity.com/paper/2207.14438/) (2022)
- [An optimal tradeoff between entanglement and copy complexity for state tomography](https://mlsanity.com/paper/2402.16353/) (2024)
- [Instance-optimal high-precision shadow tomography with few-copy measurements: A metrological approach](https://mlsanity.com/paper/2602.04952/) (2026)

## Built on by (most cited first, 2 of 2)

- [Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements](https://mlsanity.com/paper/2609.10514/) (2026)
- [When are bosonic Gaussian states classical to learn?](https://mlsanity.com/paper/2609.26705/) (2026)

## Abstract (arXiv)

We study state tomography when each measurement acts on at most k fresh copies and no quantum memory is retained between blocks. We prove a lower bound matching the upper bound in [arXiv:2510.07788]. Thus the copy complexity of estimating an arbitrary d-dimensional state to trace distance ε is, up to absolute constant factors, maxd3/(√kε2),d2/ε2 for every k and all sufficiently small ε. This removes the earlier restriction that k be small as a function of the accuracy. The lower bound applies to arbitrary measurements within each block and adaptive choices between blocks. The lower bound already applies in a small neighborhood of any state whose smallest eigenvalue is of order 1/d, even when the center is known. The main ingredient is a uniform Fisher information bound for one measurement block that depends only on the smallest eigenvalue of the state. The proof avoids the perturbative expansion responsible for the restriction in [arXiv:2402.16353]. Fano's inequality for metric balls and a log-Sobolev comparison between mutual and Fisher information then reduce the adaptive protocol to this block bound [arXiv:1607.00550, arXiv:1902.08582].