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arXiv:2006.03219 · Physical Review Applied

Estimation of Pure States Using Three Measurement Bases

Leonardo Zambrano, Luciano Pereira, Daniel Martínez, Gustavo Cañas, G. Lima, A. Delgado

Published Dec 1, 2020• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevapplied.14.064004

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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 2d\ensuremath−1{2}^{d\ensuremath{-}1}2d\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.
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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.
We present an analytical method to estimate pure quantum states using a minimum of three measurement bases in any finite-dimensional Hilbert space. This is optimal as two bases are insufficient to construct an informationally complete positive operator-valued measurement (IC-POVM) for pure states. We demonstrate our method using a binary tree structure, providing an algorithmic path for implementation. The performance of the method is evaluated through numerical simulations, showcasing its effectiveness for quantum state estimation.
Abstract We introduce an inductiven-qubit pure-state estimation method based on projective measurements onmn + 1 separable bases ormentangled bases plus the computational basis, withm ≥ 2. The method exhibits a favorable scaling in the number of qubits compared to other estimation schemes. The use of separable bases makes our estimation method particularly well suited for applications in noisy intermediate-scale quantum computers, where entangling gates are much less accurate than local gates. Our method is also capable of estimating the purity of mixed states generated by the action of white noise on pure states. Monte Carlo simulations show that the method achieves a high estimation fidelity. Besides, the fidelity can be improved by increasingmabove 2. We experimentally demonstrate the method on the IBM’s quantum processors by estimating up to 10-qubit separable and entangled states. In particular, a 4-qubit GHZ is estimated with experimental fidelity of 0.875.
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.