Optimized tomography for pure quantum states

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.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.

quant-pharXiv:2511.15806Nov 21, 20250 citations

Quantum state tomography for generic pure states

Shilin Huang, Jianxin Chen, Youning Li +1 more

arXiv:1711.10878Science China Physics Mechanics and AstronomyAug 27, 201811 citations

Nearly Time-Optimal Pure State Tomography with Pauli Measurements

Sabee Grewal, Meghal Gupta, William He +2 more

We give an algorithm for pure state tomography with near-optimal copy and time complexity using only single-qubit measurements. Specifically, given $\widetilde{O}(2^n/ε)$ copies of an unknown $n$ -qubit pure state $|ψ\rangle$ , the algorithm performs only nonadaptive Pauli measurements, runs in time $\widetilde{O}(2^n/ε)$ , and outputs $|\widehatψ \rangle$ with fidelity at least $1-ε$ with $|ψ\rangle$ with high probability. This is the first algorithm for pure state tomography that achieves near-optimal running time.We give an algorithm for pure state tomography with near-optimal copy and time complexity using only single-qubit measurements. Specifically, given $\widetilde{O}(2^n/ε)$ copies of an unknown $n$-qubit pure state $|ψ\rangle$, the algorithm performs only nonadaptive Pauli measurements, runs in time $\widetilde{O}(2^n/ε)$, and outputs $|\widehatψ \rangle$ with fidelity at least $1-ε$ with $|ψ\rangle$ with high probability. This is the first algorithm for pure state tomography that achieves near-optimal running time.

quant-pharXiv:2601.04444Jan 9, 20260 citations

Efficient quantum state tomography

Gregory S. Kanter, Prem Kumar

Nature PhotonicsOct 31, 20237 citations

Efficient quantum state tomography

M. Cramer, Martin Bodo Plenio, Steven T. Flammia +6 more

arXiv:1101.4366Nature CommunicationsDec 21, 2010671 citations

Optimal quantum-state tomography via weak value

Xuanmin Zhu, Dezheng Zhang, Runping Gao +3 more

arXiv:2402.11484Chinese Journal of PhysicsJul 21, 20250 citations

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.

We give an algorithm for pure state tomography with near-optimal copy and time complexity using only single-qubit measurements. Specifically, given

\widetilde{O}(2^n/ε)

copies of an unknown

n

-qubit pure state

|ψ\rangle

, the algorithm performs only nonadaptive Pauli measurements, runs in time

\widetilde{O}(2^n/ε)

, and outputs

|\widehatψ \rangle

with fidelity at least

1-ε

with

|ψ\rangle

with high probability. This is the first algorithm for pure state tomography that achieves near-optimal running time.