Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2501.17061

Two measurement bases are asymptotically informationally complete for any pure state tomography

Tianfeng Feng, Xiao, Tianqi, Yu Wang, Shengshi Pang, Farhan Hanif, Xiaoqi Zhou, Zhao, Qi, M. S. Kim, Jinzhao Sun

Published Jan 28, 2025• 0 citations
arXivHTMLPDF

Abstract

One of the fundamental questions in quantum information theory is to find how many measurement bases are required to obtain the full information of a quantum state. While a minimum of four measurement bases is typically required to determine an arbitrary pure state, we prove that for any states generated by finite-depth Clifford + T circuits, just two measurement bases are sufficient. More generally, we prove that two measurement bases are informationally complete for determining algebraic pure states whose state-vector elements represented in the computational basis are algebraic numbers. Since any pure state can be asymptotically approximated by a sequence of algebraic states with arbitrarily high precision, our scheme is referred to as asymptotically informationally complete for pure state tomography. Furthermore, existing works mostly construct the measurements using entangled bases. So far, the best result requires $O(n)$ local measurement bases for $n$-qubit pure-state tomography. Here, we show that two measurement bases that involve polynomial elementary gates are sufficient for uniquely determining sparse algebraic states. Moreover, we prove that two local measurement bases, involving single-qubit local operations only, are informationally complete for certain algebraic states, such as GHZ-like and W-like states. Besides, our two-measurement-bases scheme remains valid for mixed states with certain types of noises. We numerically test the uniqueness of the reconstructed states under two (local) measurement bases with and without measurement and depolarising types of noise. Our scheme provides a theoretical guarantee for pure state tomography in the fault-tolerant quantum computing regime.

DOI
10.48550/arxiv.2501.17061

Related papers

One local basis separates every countable state family

HyunHo Cha

In quantum theory, tomography addresses the problem of identifying an unknown state from the probabilities assigned to measurement outcomes. Understanding how much measurement data is needed for state identification is a fundamental question in tomography, and the answer depends on both the allowed state model and the allowed measurement structure. In multipartite systems, it was not known whether a constant number of local product bases suffices to distinguish all algebraic n-qubit pure states. We show the more general result that every fixed countable family of n-qubit states can be separated by one suitably chosen local product basis. Thus informational completeness for algebraic states admits a stronger local form than previously known. The result shows that for separation of algebraic or otherwise countable state models, locality by itself is not an obstruction.

•Jun 21, 2026•0 citations

Stable pure state quantum tomography from five orthonormal bases

Claudio Carmeli, Teiko Heinosaari, Michael Kech +2 more

For any finite-dimensional Hilbert space, we construct explicitly five orthonormal bases such that the corresponding measurements allow for efficient tomography of an arbitrary pure quantum state. This means that such measurements can be used to distinguish an arbitrary pure state from any other state, pure or mixed, and the pure state can be reconstructed from the outcome distribution in a feasible way. The set of measurements we construct is independent of the unknown state, and therefore our results provide a fixed scheme for pure state tomography, as opposed to the adaptive (state-dependent) scheme proposed by Goyeneche et al. ( Phys. Rev. Lett. , 115 (2015) 090401). We show that our scheme is robust with respect to noise, in the sense that any measurement scheme which approximates these measurements well enough is equally suitable for pure state tomography. Finally, we present two convex programs which can be used to reconstruct the unknown pure state from the measurement outcome distributions.

arXiv:1604.02970•Europhysics Letters (EPL)•Aug 1, 2016•34 citations

Quartic Certificates for Pure-State Tomography with Pauli Measurements

Hengjian Li, Baisong Sun, Bei Zeng

Full Pauli-basis tomography of an arbitrary $n$-qubit state uses the expectation values of all nonidentity Pauli observables. For multiqubit systems, prior knowledge that the state is pure allows some of these observables to be omitted while retaining uniqueness. We study two resulting notions of uniqueness: whether every pure state is uniquely determined among pure states (UDP), and whether every pure state is uniquely determined among all states, including mixed states (UDA). We derive quartic uniqueness certificates from the fourth elementary symmetric polynomial of operators in the real span of the omitted Pauli observables. An exact quartic identity shows that, for $n$-qubit systems with $n\geq 3$, every pure state is UDA whenever the omitted Pauli observables contain no four distinct operators that commute pairwise and multiply to $\pm I$. For three qubits, this condition is also necessary for UDP, yielding an exact characterization of both UDP and UDA. We further prove that UDP and UDA coincide for any number of qubits whenever the omitted Pauli observables commute pairwise. These results provide directly checkable structural criteria for certifying pure-state tomography from restricted Pauli measurements.

quant-pharXiv:2610.12320•Oct 9, 2026•0 citations

Two-qubit pure state tomography by five product orthonormal bases

Yu Wang, Yun Shang

In this paper, we focus on two-qubit pure state tomography. For an arbitrary unknown two-qubit pure state, separable or entangled, it has been found that the measurement probabilities of 16 projections onto the tensor products of Pauli eigenstates are enough to uniquely determine the state. Moreover, these corresponding product states are arranged into five orthonormal bases. We design five quantum circuits, which are decomposed into the common gates in universal quantum computation, to simulate the five projective measurements onto these bases. At the end of each circuit, we measure each qubit with the projective measurement {|0⟩⟨0|,|1⟩⟨1|}. Then, we consider the open problem whether three orthonormal bases are enough to distinguish all two-qubit pure states. A necessary condition is given. Suppose that there are three orthonormal bases { B 1 , B 2 , B 3 } . Denote the unitary transition matrices from B 1 to { B 2 , B 3 } as U 1 and U 2 . All 32 elements of matrices U 1 and U 2 should not be zero. If not, these three bases cannot distinguish all two-qubit pure states.

Chinese Physics B•Oct 1, 2018•1 citation

$d+1$ Measurement Bases are Sufficient for Determining $d$-Dimensional Quantum States: Theory and Experiment

Xiao, Tianqi, Wang, Yaxin, Xia, Ying +2 more

A long-standing problem in quantum physics is to determine the minimal number of measurement bases required for the complete characterization of unknown quantum states, a question of particular relevance to high-dimensional quantum information processing. Here, we propose a quantum state tomography scheme that requires only $d+1$ projective measurement bases to fully reconstruct an arbitrary $d$-dimensional quantum state. As a proof-of-principle, we experimentally verified this scheme on a silicon photonic chip by reconstructing quantum states for $d=6$, in which a complete set of mutually unbiased bases does not exist. This approach offers new perspectives for quantum state characterization and measurement design, and holds promise for future applications in quantum information processing.

arXiv:2507.11204•Jul 15, 2025•0 citations

Efficient Circuit-Based Quantum State Tomography via Sparse Entry Optimization

Chi-Kwong Li, Wu, Kevin Yipu, Zherui Zhang

Many quantum states arising in algorithms and physical systems occupy only a small, structured subset of the exponentially large Hilbert space, yet standard quantum state tomography fails to exploit this structure. We present an efficient circuit-based tomography framework for pure quantum states that are sparse in a computational basis. For an $n$-qubit state supported on $k$ basis elements, the protocol reconstructs all amplitudes using $1 + 2(k-1)$ measurement settings. The method admits both entanglement-assisted and entanglement-free implementations, enabling explicit tradeoffs between two-qubit gate usage and statistical noise. We derive bounds on the required number of CNOT gates from the combinatorial structure of the state support and analyze their effect on reconstruction infidelity. The framework extends naturally to closed-system process tomography and is validated via numerical simulations using Qiskit.

arXiv:2407.20298•Jul 29, 2024•0 citations
One of the fundamental questions in quantum information theory is to find how many measurement bases are required to obtain the full information of a quantum state. While a minimum of four measurement bases is typically required to determine an arbitrary pure state, we prove that for any states generated by finite-depth Clifford + T circuits, just two measurement bases are sufficient. More generally, we prove that two measurement bases are informationally complete for determining algebraic pure states whose state-vector elements represented in the computational basis are algebraic numbers. Since any pure state can be asymptotically approximated by a sequence of algebraic states with arbitrarily high precision, our scheme is referred to as asymptotically informationally complete for pure state tomography. Furthermore, existing works mostly construct the measurements using entangled bases. So far, the best result requires O(n)O(n)O(n) local measurement bases for nnn-qubit pure-state tomography. Here, we show that two measurement bases that involve polynomial elementary gates are sufficient for uniquely determining sparse algebraic states. Moreover, we prove that two local measurement bases, involving single-qubit local operations only, are informationally complete for certain algebraic states, such as GHZ-like and W-like states. Besides, our two-measurement-bases scheme remains valid for mixed states with certain types of noises. We numerically test the uniqueness of the reconstructed states under two (local) measurement bases with and without measurement and depolarising types of noise. Our scheme provides a theoretical guarantee for pure state tomography in the fault-tolerant quantum computing regime.
In quantum theory, tomography addresses the problem of identifying an unknown state from the probabilities assigned to measurement outcomes. Understanding how much measurement data is needed for state identification is a fundamental question in tomography, and the answer depends on both the allowed state model and the allowed measurement structure. In multipartite systems, it was not known whether a constant number of local product bases suffices to distinguish all algebraic n-qubit pure states. We show the more general result that every fixed countable family of n-qubit states can be separated by one suitably chosen local product basis. Thus informational completeness for algebraic states admits a stronger local form than previously known. The result shows that for separation of algebraic or otherwise countable state models, locality by itself is not an obstruction.
For any finite-dimensional Hilbert space, we construct explicitly five orthonormal bases such that the corresponding measurements allow for efficient tomography of an arbitrary pure quantum state. This means that such measurements can be used to distinguish an arbitrary pure state from any other state, pure or mixed, and the pure state can be reconstructed from the outcome distribution in a feasible way. The set of measurements we construct is independent of the unknown state, and therefore our results provide a fixed scheme for pure state tomography, as opposed to the adaptive (state-dependent) scheme proposed by Goyeneche et al. ( Phys. Rev. Lett. , 115 (2015) 090401). We show that our scheme is robust with respect to noise, in the sense that any measurement scheme which approximates these measurements well enough is equally suitable for pure state tomography. Finally, we present two convex programs which can be used to reconstruct the unknown pure state from the measurement outcome distributions.
Full Pauli-basis tomography of an arbitrary nnn-qubit state uses the expectation values of all nonidentity Pauli observables. For multiqubit systems, prior knowledge that the state is pure allows some of these observables to be omitted while retaining uniqueness. We study two resulting notions of uniqueness: whether every pure state is uniquely determined among pure states (UDP), and whether every pure state is uniquely determined among all states, including mixed states (UDA). We derive quartic uniqueness certificates from the fourth elementary symmetric polynomial of operators in the real span of the omitted Pauli observables. An exact quartic identity shows that, for nnn-qubit systems with n≥3n\geq 3n≥3, every pure state is UDA whenever the omitted Pauli observables contain no four distinct operators that commute pairwise and multiply to ±I\pm I±I. For three qubits, this condition is also necessary for UDP, yielding an exact characterization of both UDP and UDA. We further prove that UDP and UDA coincide for any number of qubits whenever the omitted Pauli observables commute pairwise. These results provide directly checkable structural criteria for certifying pure-state tomography from restricted Pauli measurements.
In this paper, we focus on two-qubit pure state tomography. For an arbitrary unknown two-qubit pure state, separable or entangled, it has been found that the measurement probabilities of 16 projections onto the tensor products of Pauli eigenstates are enough to uniquely determine the state. Moreover, these corresponding product states are arranged into five orthonormal bases. We design five quantum circuits, which are decomposed into the common gates in universal quantum computation, to simulate the five projective measurements onto these bases. At the end of each circuit, we measure each qubit with the projective measurement {|0⟩⟨0|,|1⟩⟨1|}. Then, we consider the open problem whether three orthonormal bases are enough to distinguish all two-qubit pure states. A necessary condition is given. Suppose that there are three orthonormal bases { B 1 , B 2 , B 3 } . Denote the unitary transition matrices from B 1 to { B 2 , B 3 } as U 1 and U 2 . All 32 elements of matrices U 1 and U 2 should not be zero. If not, these three bases cannot distinguish all two-qubit pure states.
d+1d+1d+1 Measurement Bases are Sufficient for Determining ddd-Dimensional Quantum States: Theory and Experiment
A long-standing problem in quantum physics is to determine the minimal number of measurement bases required for the complete characterization of unknown quantum states, a question of particular relevance to high-dimensional quantum information processing. Here, we propose a quantum state tomography scheme that requires only d+1d+1d+1 projective measurement bases to fully reconstruct an arbitrary ddd-dimensional quantum state. As a proof-of-principle, we experimentally verified this scheme on a silicon photonic chip by reconstructing quantum states for d=6d=6d=6, in which a complete set of mutually unbiased bases does not exist. This approach offers new perspectives for quantum state characterization and measurement design, and holds promise for future applications in quantum information processing.
Many quantum states arising in algorithms and physical systems occupy only a small, structured subset of the exponentially large Hilbert space, yet standard quantum state tomography fails to exploit this structure. We present an efficient circuit-based tomography framework for pure quantum states that are sparse in a computational basis. For an nnn-qubit state supported on kkk basis elements, the protocol reconstructs all amplitudes using 1+2(k−1)1 + 2(k-1)1+2(k−1) measurement settings. The method admits both entanglement-assisted and entanglement-free implementations, enabling explicit tradeoffs between two-qubit gate usage and statistical noise. We derive bounds on the required number of CNOT gates from the combinatorial structure of the state support and analyze their effect on reconstruction infidelity. The framework extends naturally to closed-system process tomography and is validated via numerical simulations using Qiskit.