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:2108.05752

Determination of All Unknown Pure Quantum States with Two Observables

Wang, Yu

Published Aug 12, 2021• 0 citations
arXivHTMLPDF

Abstract

Efficiently extracting information from pure quantum states using minimal observables on the main system is a longstanding and fundamental issue in quantum information theory. Despite the inability of probability distributions of position and momentum to uniquely specify a wavefunction, Peres conjectured a discrete version wherein two complementary observables, analogous to position and momentum and realized as projective measurements onto orthogonal bases, can determine all pure qudits up to a finite set of ambiguities. Subsequent findings revealed the impossibility of uniquely determining $d$-dimenisonal pure states even when neglecting a measure-zero set with any two orthogonal bases, and Peres's conjecture is also correct for $d=3$ but not for $d=4$. In this study, we show that two orthogonal bases are capable of effectively filtering up to $2^{d-1}$ finite candidates by disregarding a measure-zero set, without involving complex numbers in the bases' coefficients. Additionally, drawing inspiration from sequential measurements to directly calculate the target coefficients of the wavefunction using two complementary observables, we show that almost all pure qudits can be uniquely determined by adaptively incorporating a POVM in the middle, followed by measuring the complementary observable.

DOI
10.48550/arxiv.2108.05752

Related papers

Minimal determination of a pure qutrit state and four-measurement protocol for pure qudit state

Liang-Liang Sun, Sixia Yu, Zeng‐Bing Chen

Abstract Finding the least measurement settings to determine an arbitrary pure state has been long known as the Pauli problem. Peres conjectured that two unbiased bases may be sufficient to determine a pure state up to some finite ambiguities. Here we find that the Peres conjecture is correct in the case d = 3 where a pure state is determined up to at most six candidates states and is incorrect in the case of d = 4 for which a counterexample is constructed. We observe that the target state can be picked out from candidates states by an adaptive two-outcome measurement. We thus provided a minimal qutrit tomography protocol constituted of three measurements in contrast to the four measurements required in previous non-adaptive method. We also reduce a five-measurement protocol for a pure qudit state into three full-dimensional measurements and one two-outcome measurement protocol, which exhibits robustness to the usual white noise thus also applies to nearly pure state.

Journal of Physics A Mathematical and Theoretical•Dec 20, 2019•2 citations

Minimal Determination of a Pure State through Adaptive Tomography

Liang-Liang Sun, Yingqiu, Mao, Fei-Lei Xiong +2 more

Finding the least measurement settings to determine an arbitrary pure state has been long known as the Pauli problem. In the fixed measurement scheme four orthonormal bases are required even though there are far less parameters in a pure state. Peres conjectured that two unbiased bases suffice to determine a pure state up to some finite ambiguities. Here we shall at first prove Peres conjecture in the case of $d=3,4$, namely, two unbiased measurements determine a pure state up to to 6 and 16 candidates for a qutrit and ququad, respectively. And then, taking Peres' conjecture for established, we propose an adaptive 3-measurement scheme involving the minimal number of measurements, based on the observation that the ambiguities can be removed by an adaptive two-outcome projective measurement. With the help of this observation, we simplify a recent five-basis protocol $[Phys. Rev. Lett. 115, 090401 (2015)]$ to a three-basis one at the cost of the extra dichotomic measurement.

arXiv:1802.00140•Feb 1, 2018•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

Estimation of Pure States Using Three Measurement Bases

Leonardo Zambrano, Luciano Pereira, Daniel Martínez +3 more

We introduce a method to estimate unknown pure $d$-dimensional quantum states using the probability distributions associated with only three measurement bases. The measurement results of $2d$ projectors are used to generate a set of ${2}^{d\ensuremath{-}1}$ states, the value of the likelihood function of which is evaluated using the measurement results of the remaining $d$ projectors. The state with the highest value of the likelihood function is the estimate of the unknown state. The method estimates all pure states except for a null-measure set, which can be overcome by adapting two bases. The viability of the protocol is experimentally demonstrated using two different and complementary high-dimensional quantum information platforms. First, by exploring the photonic path-encoding strategy, we validate the method on a single eight-dimensional quantum system. Then, we resort to the five-superconducting-qubit IBM quantum processor to demonstrate the high performance of the method in the multipartite scenario.

arXiv:2006.03219•Physical Review Applied•Dec 1, 2020•14 citations

Five Measurement Bases Determine Pure Quantum States on Any Dimension

Dardo Goyeneche, Gustavo Cañas, S. Etcheverry +4 more

A long-standing problem in quantum mechanics is the minimum number of observables required for the characterization of unknown pure quantum states. The solution to this problem is especially important for the developing field of high-dimensional quantum information processing. In this work we demonstrate that any pure d-dimensional state is unambiguously reconstructed by measuring five observables, that is, via projective measurements onto the states of five orthonormal bases. Thus, in our method the total number of different measurement outcomes (5d) scales linearly with d. The state reconstruction is robust against experimental errors and requires simple postprocessing, regardless of d. We experimentally demonstrate the feasibility of our scheme through the reconstruction of eight-dimensional quantum states, encoded in the momentum of single photons.

arXiv:1411.2789•Physical Review Letters•Aug 28, 2015•71 citations

Quantum state tomography from a sequential measurement of two variables in a single setup

Antonio Di Lorenzo

We demonstrate that the task of determining an unknown quantum state can be accomplished efficiently by making a sequential measurement of two observables, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}$, the eigenstates of which form bases connected by a discrete Fourier transform. The state can be pure or mixed, the dimension of the Hilbert space and the coupling strength are arbitrary, and the experimental setup is fixed. The concept of Moyal quasicharacteristic function is introduced for finite-dimensional Hilbert spaces.

arXiv:1303.7154•Physical Review A•Oct 17, 2013•13 citations
Efficiently extracting information from pure quantum states using minimal observables on the main system is a longstanding and fundamental issue in quantum information theory. Despite the inability of probability distributions of position and momentum to uniquely specify a wavefunction, Peres conjectured a discrete version wherein two complementary observables, analogous to position and momentum and realized as projective measurements onto orthogonal bases, can determine all pure qudits up to a finite set of ambiguities. Subsequent findings revealed the impossibility of uniquely determining ddd-dimenisonal pure states even when neglecting a measure-zero set with any two orthogonal bases, and Peres's conjecture is also correct for d=3d=3d=3 but not for d=4d=4d=4. In this study, we show that two orthogonal bases are capable of effectively filtering up to 2d−12^{d-1}2d−1 finite candidates by disregarding a measure-zero set, without involving complex numbers in the bases' coefficients. Additionally, drawing inspiration from sequential measurements to directly calculate the target coefficients of the wavefunction using two complementary observables, we show that almost all pure qudits can be uniquely determined by adaptively incorporating a POVM in the middle, followed by measuring the complementary observable.
Abstract Finding the least measurement settings to determine an arbitrary pure state has been long known as the Pauli problem. Peres conjectured that two unbiased bases may be sufficient to determine a pure state up to some finite ambiguities. Here we find that the Peres conjecture is correct in the case d = 3 where a pure state is determined up to at most six candidates states and is incorrect in the case of d = 4 for which a counterexample is constructed. We observe that the target state can be picked out from candidates states by an adaptive two-outcome measurement. We thus provided a minimal qutrit tomography protocol constituted of three measurements in contrast to the four measurements required in previous non-adaptive method. We also reduce a five-measurement protocol for a pure qudit state into three full-dimensional measurements and one two-outcome measurement protocol, which exhibits robustness to the usual white noise thus also applies to nearly pure state.
Finding the least measurement settings to determine an arbitrary pure state has been long known as the Pauli problem. In the fixed measurement scheme four orthonormal bases are required even though there are far less parameters in a pure state. Peres conjectured that two unbiased bases suffice to determine a pure state up to some finite ambiguities. Here we shall at first prove Peres conjecture in the case of d=3,4d=3,4d=3,4, namely, two unbiased measurements determine a pure state up to to 6 and 16 candidates for a qutrit and ququad, respectively. And then, taking Peres' conjecture for established, we propose an adaptive 3-measurement scheme involving the minimal number of measurements, based on the observation that the ambiguities can be removed by an adaptive two-outcome projective measurement. With the help of this observation, we simplify a recent five-basis protocol [Phys.Rev.Lett.115,090401(2015)][Phys. Rev. Lett. 115, 090401 (2015)][Phys.Rev.Lett.115,090401(2015)] to a three-basis one at the cost of the extra dichotomic measurement.
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.
We introduce a method to estimate unknown pure ddd-dimensional quantum states using the probability distributions associated with only three measurement bases. The measurement results of 2d2d2d projectors are used to generate a set of ${2}^{d\ensuremath{-}1}$ states, the value of the likelihood function of which is evaluated using the measurement results of the remaining ddd projectors. The state with the highest value of the likelihood function is the estimate of the unknown state. The method estimates all pure states except for a null-measure set, which can be overcome by adapting two bases. The viability of the protocol is experimentally demonstrated using two different and complementary high-dimensional quantum information platforms. First, by exploring the photonic path-encoding strategy, we validate the method on a single eight-dimensional quantum system. Then, we resort to the five-superconducting-qubit IBM quantum processor to demonstrate the high performance of the method in the multipartite scenario.
A long-standing problem in quantum mechanics is the minimum number of observables required for the characterization of unknown pure quantum states. The solution to this problem is especially important for the developing field of high-dimensional quantum information processing. In this work we demonstrate that any pure d-dimensional state is unambiguously reconstructed by measuring five observables, that is, via projective measurements onto the states of five orthonormal bases. Thus, in our method the total number of different measurement outcomes (5d) scales linearly with d. The state reconstruction is robust against experimental errors and requires simple postprocessing, regardless of d. We experimentally demonstrate the feasibility of our scheme through the reconstruction of eight-dimensional quantum states, encoded in the momentum of single photons.
We demonstrate that the task of determining an unknown quantum state can be accomplished efficiently by making a sequential measurement of two observables, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}$, the eigenstates of which form bases connected by a discrete Fourier transform. The state can be pure or mixed, the dimension of the Hilbert space and the coupling strength are arbitrary, and the experimental setup is fixed. The concept of Moyal quasicharacteristic function is introduced for finite-dimensional Hilbert spaces.