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Results in Physics

Qudit quantum state tomography

Naser Karimi, Hadi Z. Olyaei, Marziyeh Yahyavi, M. A. Jafarizadeh

Published Jun 20, 2023• 4 citations
Publisher

Abstract

In this study, we present a new procedure for quantum state reconstruction of qudit quantum state based on unambiguous state discrimination (USD) measurement. In our recent paper (Karimi et al., 2022), we considered the reconstruction of a single-qubit state via USD measurement. In this paper, we extend this method to the reconstruction of the qudit quantum state.

DOI
10.1016/j.rinp.2023.106641

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In this study, we present a new procedure for quantum state reconstruction of qudit quantum state based on unambiguous state discrimination (USD) measurement. In our recent paper (Karimi et al., 2022), we considered the reconstruction of a single-qubit state via USD measurement. In this paper, we extend this method to the reconstruction of the qudit quantum state.
We study the determination of an unknown mixed state of a ddd-dimensional quantum system by means of unambiguous state discrimination. We show that optimal and nonoptimal unambiguous state discrimination can be used to reconstruct unknown states of a qubit. This result is extended to the case of a qudit by a sequence of reconstructions in two-dimensional subspaces. The total number of projections scales approximately as 2d22{d}^{2}2d2 for ddd large; this is twice as much as in the case of tomography based on mutually unbiased bases or symmetric informationally complete positive-operator-valued measures, and less than the d3{d}^{3}d3 projections required by standard tomography.
Recently quantum tomography has been proposed as a fundamental tool for prototyping a few qubit quantum device. It allows the complete reconstruction of the state produced from a given input into the device. From this reconstructed density matrix, relevant quantum information quantities such as the degree of entanglement and entropy can be calculated. Generally, orthogonal measurements have been discussed for this tomographic reconstruction. In this paper, we extend the tomographic reconstruction technique to two new regimes. First, we show how nonorthogonal measurements allow the reconstruction of the state of the system provided the measurements span the Hilbert space. We then detail how quantum-state tomography can be performed for multiqudits with a specific example illustrating how to achieve this in one- and two-qutrit systems.
Abstract In this paper, we introduce an analytical framework for the reconstruction of quantum states. The reconstruction of an unknown quantum state requires the information of a complete set of observables, obtained through experimental measurements of Hermitian operators usually defined as positive-operator-valued measures (POVMs). The scheme involves a single-qubit unambiguous state discrimination POVM, which can be generalized to perform n-qubit measurements. We also use maximum likelihood estimation as a method in the reconstruction of the density matrix from experimental data and show that the expected value of the cleaner is independent of the parameter of the density operator.
We propose an efficient quantum state tomography method inspired by compressed sensing and threshold quantum state tomography that can drastically reduce the number of measurement settings to reconstruct the density matrix of an NNN-qudit system. We validate our algorithm with demonstrations on IBMQ and show the efficient and accurate reconstruction of $N\ensuremath{\le}7$ qubit systems, reproducing GHZ, W\mathrm{W}W, and random states with O(1)O(1)O(1), O(N2)O({N}^{2})O(N2), and O(N)O(N)O(N) settings.