# Optimal lower bounds for quantum state tomography

arXiv: https://arxiv.org/abs/2510.07699 · HTML: https://mlsanity.com/paper/2510.07699/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2025.

## 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)
- [Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography](https://mlsanity.com/paper/2609.04165/) (2026)

## Abstract (arXiv)

We show that n = Omega(rd/varepsilon2) copies are necessary to learn a rank r mixed state ρ ∈ mathbbCd × d up to error varepsilon in trace distance. This matches the upper bound of n = O(rd/varepsilon2) from prior work, and therefore settles the sample complexity of mixed state tomography. We prove this lower bound by studying a special case of full state tomography that we refer to as projector tomography, in which ρ is promised to be of the form ρ = P/r, where P ∈ mathbbCd × d is a rank r projector. A key technical ingredient in our proof, which may be of independent interest, is a reduction which converts any algorithm for projector tomography which learns to error varepsilon in trace distance to an algorithm which learns to error O(varepsilon) in the more stringent Bures distance.