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arXiv:1903.05709 · Optics Letters

Set of 4d–3 observables to determine any pure qudit state

Quimey Pears Stefano, Lorena Rebón, Silvia Adriana Ledesma, Claudio Iemmi

Published May 10, 2019• 12 citations
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Abstract

We present a tomographic method which requires only 4d-3 measurement outcomes to reconstruct any pure quantum state of arbitrary dimension d. Using the proposed scheme, we have experimentally reconstructed a large number of pure states of dimension d=7, obtaining a mean fidelity of 0.94. Moreover, we performed numerical simulations of the reconstruction process, verifying the feasibility of the method for higher dimensions. In addition, the a priori assumption of purity can be certified within the same set of measurements, which represents an improvement with respect to other similar methods and contributes to answering the question of how many observables are needed to uniquely determine any pure state.

DOI
10.1364/ol.44.002558

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We present a tomographic method which requires only 4d-3 measurement outcomes to reconstruct any pure quantum state of arbitrary dimension d. Using the proposed scheme, we have experimentally reconstructed a large number of pure states of dimension d=7, obtaining a mean fidelity of 0.94. Moreover, we performed numerical simulations of the reconstruction process, verifying the feasibility of the method for higher dimensions. In addition, the a priori assumption of purity can be certified within the same set of measurements, which represents an improvement with respect to other similar methods and contributes to answering the question of how many observables are needed to uniquely determine any pure state.
We present a proof-of-principle demonstration of a method to characterize any pure spatial qudit of arbitrary dimension ddd, which is based on the classic phase-shift interferometry technique. In the proposed scheme a total of only 4d4d4d measurement outcomes are needed, implying a significant reduction with respect to the standard schemes for quantum-state tomography which require on the order of d2{d}^{2}d2. By using this technique, we have experimentally reconstructed a large number of states ranging from d=2d=2d=2 up to 14 with mean fidelity values higher than 0.97. For that purpose the qudits were codified in the discretized transverse-momentum position of single photons, once they are sent through an aperture with ddd slits. We provide an experimental implementation of the method based in a Mach-Zehnder interferometer, which allows one to reduce the number of measurement settings to four since the ddd slits can be measured simultaneously. Furthermore, it can be adapted to consider the reconstruction of the unknown state from the outcome frequencies of $4d\ensuremath{-}3$ fixed projectors independently of the encoding or the nature of the quantum system, allowing one to implement the reconstruction method in a general experiment.
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.
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.
We examine the problem of finding the minimum number of Pauli measurements needed to uniquely determine an arbitrary nnn-qubit pure state among all quantum states. We show that only 11 Pauli measurements are needed to determine an arbitrary two-qubit pure state compared to the full quantum state tomography with 16 measurements, and only 31 Pauli measurements are needed to determine an arbitrary three-qubit pure state compared to the full quantum state tomography with 64 measurements. We demonstrate that our protocol is robust under depolarizing error with simulated random pure states. We experimentally test the protocol on two- and three-qubit systems with nuclear magnetic resonance techniques. We show that the pure-state tomography protocol saves us a number of measurements without considerable loss of fidelity. We compare our protocol with same-size sets of randomly selected Pauli operators and find that our selected set of Pauli measurements significantly outperforms those random sampling sets. As a direct application, our scheme can also be used to reduce the number of settings needed for pure-state tomography in quantum optical systems.
We establish methods for quantum state tomography based on compressed sensing. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rdlog²d) measurement settings, compared to standard methods that require d² settings. Our methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements, use fast convex optimization, are stable against noise, and can be applied to states that are only approximately low rank. The acquired data can be used to certify that the state is indeed close to pure, so no a priori assumptions are needed.