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

TL;DR

The Pauli problem asks how many measurement bases are needed to uniquely determine a quantum state. This work shows that two measurement bases suffice asymptotically for pure states when one restricts to algebraic states, a dense subset that includes states generated by finite-depth Clifford + T circuits; it also provides explicit two-basis constructions using a computational basis and a transformed basis with algebraic phase shifts, and proves uniqueness via a Lindemann–Weierstrass-based linear-independence framework. The approach extends to two local bases that uniquely determine sparse algebraic states such as GHZ-like and W-like states, and remains robust under certain noise models, with numerical demonstrations up to 20 qubits. While not strictly informationally complete for all pure states, the method offers a practically achievable tomography scheme with strong theoretical guarantees in fault-tolerant regimes and potential extensions to mixed-state tomography. Overall, the paper delivers a principled, asymptotically complete two-basis tomography protocol for a physically relevant class of quantum states, with explicit constructions and numerical validation demonstrating its feasibility and resilience.

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.

Figures (8)

• Figure 1: Two-measurement-base scheme.a. Scheme for a qudit state: $F$ is the standard quantum Fourier transform, $F=\sum_{i,j=0}^{d-1} \frac{1}{\sqrt{d}} \omega^{ij}\ket{i}\bra{j}$, where $\omega =e^{\frac{2\pi i}{d}}$, and the diagonal operation is $D = \operatorname{Diag}\{ e^{i \theta_0}, ..., e^{i \theta_{d-1}}\}$ with angle $\theta_i$. b. Two measurement bases for multi-qubit systems. If $\rho$ is a sparse state, i.e. the number of non-zero components of $r_k$ is $\mathcal{O}(\text{poly}(n))$, $D$ can be reduced to be a diagonal operation with $\mathcal{O}(\text{poly}(n))$ phase shifts. c. Illustration of the concept of uniqueness: The pure quantum state $\in \mathbb{P}_A$ has a one-to-one correspondence to the measurement outcomes $\mathbf{P}$ and $\mathbf{Q}$.
• Figure 2: Investigation on our local-measurement-bases scheme. We study the error in probability distribution $\varepsilon_{\textrm{Prob}}$ and the average fidelity and their relationship in the state reconstruction. The algebraic numbers and transcendental numbers are simulated on a computer using low-precision numbers and high-precision numbers, respectively, as computers can only store data in finite decimal places. The numbers of decimal places for the modulus $r_k$, the phases $\cos \varphi_k$ and $\sin \varphi_k$ are denoted as $C_1$, $C_2$, and $C_3$, respectively, and are set to be $3$. The numbers of decimal places for the second measurement basis and the measured probability distributions are set to be $15$. The target state $\ket{\psi}$ and the simulated state $\ket{\psi'}$ are set at the same level of precision. a, The error in probability distribution $\varepsilon_{\textrm{Prob}}$ in the reconstruction process for random states and 20-qubit W-like state. We randomly generate 40 quantum states with dimension 8, and reconstruct each quantum state 2500 times using two local measurement bases scheme. To find the state, we use the Monte Carlo method to sample an initial guess and use the simulated annealing method to optimise $\varepsilon_{\textrm{Prob}}$. The dark blue dots in the figure indicate the average values of $\varepsilon_{\textrm{Prob}}$, and the shaded area represent the standard deviation of the random states during the reconstruction. The figure inset shows the fidelity during the reconstruction process of the 20-qubit W-like state. b, The horizontal coordinate is the 40 states we randomly generated and the vertical coordinate is the $\varepsilon_{\textrm{Prob}}$. The blue dots represent all reconstruction results with fidelity less than or equal to 0.99, while the orange dots represent all reconstruction results with fidelity greater than 0.99. The red line represents $\varepsilon _{Prob}=8\times 10^{-5}$.c, The horizontal coordinate is the 40 states we randomly generated, and the vertical coordinate is the fidelity between a target pure state and a reconstructed pure state. The red and blue dots represent the average fidelity of all reconstruction results for a fixed target state with $\varepsilon_{\textrm{Prob}} < 10^{-5}$, and $10^{-2}<\varepsilon_{\textrm{Prob}} < 10^{-1}$, respectively. The upper and lower bounds of the shaded area are the locations that deviate from the average values by one standard deviation.
• Figure 3: Verification of our local measurement bases scheme with measurement and device statistical error. 40 quantum states (of dimension 8) are randomly generated, and measurement results under two measurement bases are produced. Each state is reconstructed 2500 times. The setting for the decimal numbers is set the same as in \ref{['fig:M2_Numerous_results']}. a, The bit-flip noise is added to the measurement results.b, The depolarising noise is added to the state. The red, blue, and green dots represent the average fidelity of all reconstruction results for a fixed target state with $\varepsilon_{\textrm{Prob}} < 0.8 \times 10^{-5}$, $\varepsilon_{\textrm{Prob}} < 4 \times 10^{-5}$, and $\varepsilon_{\textrm{Prob}} < 10^{-4}$, respectively.
• Figure 4: The number of decimal places of the second measurement base $X^{\prime}$ and the simulated phases $C_2^{\prime}$ during the reconstruction process are varied respectively, in order to test their effects on the $\varepsilon_{\textrm{Prob}}$ and average fidelity of the reconstructed 3-qubit quantum state by using two local measurement bases scheme. The numbers of decimal places for the amplitudes are $C_1=C_{1}^{\prime}=3$, for the phases are $C_2=3$ and $C_{2}^{\prime}$ changing from $3$ to $8$. The numbers of decimal places for the second measurement base are $X=15$ and $X^{\prime}$ changing from $6$ to $15$, and for the measured probability distributions are $Y=Y^{\prime}=15$. Forty 3-qubit quantum states are randomly generated and each quantum state is reconstructed 2500 times. (a)(b) We choose the reconstruction result with the smallest $\varepsilon_{\textrm{Prob}}$ for each state and average the $\varepsilon_{\textrm{Prob}}$ and fidelity over the 40 states.a, The horizontal coordinate is the number of decimal places of the second measurement base $X^{\prime}$, the blue dots reflect the values of $\varepsilon_{\textrm{Prob}}$, while the red dots reflect the values of the absolute values of 1 minus the average fidelity.b, The horizontal coordinate is the number of decimal places of the simulated phases $C_2^{\prime}$, the blue dots reflect the values of $\varepsilon_{\textrm{Prob}}$, while the red dots reflect the values of the absolute values of 1 minus the average fidelity.
• Figure 5: Fidelity improvement when adding the third basis. A 5-qubit pure state is randomly generated and depolarising noise is added to produce its probability distributions under three measurement bases. Then we use the measurement results to reconstruct the pure state. The horizontal coordinate is the coefficient of the depolarising noise added to the randomly generated pure state, the blue dots reflect the values of $\varepsilon_{\textrm{Prob}}$, while the red dots reflect the values of the absolute values of 1 minus the fidelity.

Theorems & Definitions (11)

• Theorem 1: Two measurements are asymptotically informationally complete for any qudit and qubits states
• Theorem 2: For a certain class of pure states, two local measurements are strictly informationally complete
• Remark 1: Asymptotic equivalence between $\mathbb{P}_A$ and $\mathbb{P}$
• proof
• Remark 2
• Definition 1: Information completeness with respect to a given set heinosaari2013quantum
• Lemma 1: Lindemann–Weierstrass theorem lindemann1882zahlweierstrass1885lindemannnesterenko2021lindemann
• Lemma 2: Linear Independence Theorem
• proof
• Theorem 3: The existence of multi-solutions with the same measurement outcomes, i.e., not strictly informationally complete with respect to $\mathbb{P}_A$