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arXiv:2511.15806

Mixed state tomography reduces to pure state tomography

Angelos Pelecanos, Jack Spilecki, Ewin Tang, John Wright

quant-phSubmitted Nov 19, 2025• Announced Nov 21, 2025• 0 citations
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Abstract

A longstanding belief in quantum tomography is that estimating a mixed state is far harder than estimating a pure state. This is borne out in the mathematics, where mixed state algorithms have always required more sophisticated techniques to design and analyze than pure state algorithms. We present a new approach to tomography demonstrating that, contrary to this belief, state-of-the-art mixed state tomography follows easily and naturally from pure state algorithms. We analyze the following strategy: given $n$ copies of an unknown state $ρ$, convert them into copies of a purification $|ρ\rangle$; run a pure state tomography algorithm to produce an estimate of $|ρ\rangle$; and output the resulting estimate of $ρ$. The purification subroutine was recently discovered via the "acorn trick" of Tang, Wright, and Zhandry. With this strategy, we obtain the first tomography algorithm which is sample-optimal in all parameters. For a rank-$r$ $d$-dimensional state, it uses $n = O((rd + \log(1/δ))/\varepsilon)$ samples to output an estimate which is $\varepsilon$-close in fidelity with probability at least $1-δ$. This algorithm also uses poly$(n)$ gates, making it the first gate-efficient tomography algorithm which is sample-optimal even in terms of the dimension $d$ alone. Moreover, with this method we recover essentially all results on mixed state tomography, including its applications to tomography with limited entanglement, classical shadows, and quantum metrology. Our proofs are simple, closing the gap in conceptual difficulty between mixed and pure tomography. Our results also clarify the role of entangled measurement in mixed state tomography: the only step of the algorithm which requires entanglement across copies is the purification step, suggesting that, for tomography, the reason entanglement is useful is for consistent purification.

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We study the complexity of two closely related learning problems, one quantum and one classical. In the quantum setting, we consider agnostic tomography for the natural class of product mixed states. Given NNN copies of an nnn-qubit state ρρρ, the goal is to output a nearly optimal product mixed state approximation in trace distance. While recent work has focused on pure-state ansatz (e.g., product or stabilizer states), no polynomial-time guarantees were previously known for mixed-state ansatz. In the classical setting, we study robust learning of binary product distributions: given samples from an unknown distribution on 0,1n{0,1}^n0,1n, the goal is to output a nearly optimal product approximation. Our main contributions are as follows. (1) We give a semi-agnostic tomography algorithm for product mixed states with polynomial sample and computational complexity achieving error O(optlog⁡(1/opt))O(\mathrm{opt}\log(1/\mathrm{opt}))O(optlog(1/opt)), where opt\mathrm{opt}opt is the trace distance to the best product approximation. This is the first efficient algorithm with any nontrivial agnostic guarantee for mixed-state ansatz, using only single-qubit, single-copy measurements. We also prove a Quantum Statistical Query lower bound showing near-optimality, and an unconditional lower bound demonstrating that adaptivity is necessary under single-qubit measurements. (2) We give a semi-agnostic algorithm for robustly learning binary product distributions with matching guarantees and establish a Statistical Query lower bound, essentially resolving the efficient robust learnability of this class and improving on prior work since Diakonikolas et al. (2016).