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arXiv:2002.03819 · Journal of Physics A Mathematical and Theoretical

Macroscopic approach to N-qudit systems

C Muñoz, Isabel Sainz, A B Klimov

Published Apr 21, 2020• 2 citations
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Abstract

Abstract We develop a general scheme for an analysis of macroscopic N -qudit systems which includes: (a) a scheme to organize the information obtained from collective measurements in the form of distribution functions in a discrete low-dimensional space; (b) a set of collective operators appropriate for the characterization of N -qudit states; (c) two collective tomographic protocols both for general and fully symmetric N -qudit states. The example of N -qutrits is analyzed in detail and compared to the N -qubit case.

DOI
10.1088/1751-8121/ab8b77

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Abstract We develop a general scheme for an analysis of macroscopic N -qudit systems which includes: (a) a scheme to organize the information obtained from collective measurements in the form of distribution functions in a discrete low-dimensional space; (b) a set of collective operators appropriate for the characterization of N -qudit states; (c) two collective tomographic protocols both for general and fully symmetric N -qudit states. The example of N -qutrits is analyzed in detail and compared to the N -qubit case.
We introduce a discrete QQQ function of an NNN-qubit system projected into the space of symmetric measurements as a tool for analyzing general properties of quantum systems in the macroscopic limit. For known states the projected QQQ function helps to visualize the results of collective measurements, and for unknown states it can be approximately reconstructed by measuring the lowest moments of the collective variables.
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.
The machinery of qubit-portraits of qudit states, recently presented, is consider here in more details in order to characterize the presence of quantum correlations in bipartite qudit states. In the tomographic representation of quantum mechanics, Bell-like inequalities are interpreted as peculiar properties of a family of classical joint probability distributions which describe the quantum state of two qudits. By means of the qubit-portraits machinery a semigroup of stochastic matrices can be associated to a given quantum state. The violation of the CHSH inequalities is discussed in this framework with some examples, we found that quantum correlations in qutrit isotropic states can be detected by the suggested method while it cannot in the case of qutrit Werner states.
The machinery of qubit-portraits of qudit states, recently presented, is consider here in more details in order to characterize the presence of quantum correlations in bipartite qudit states. In the tomographic representation of quantum mechanics, Bell-like inequalities are interpreted as peculiar properties of a family of classical joint probability distributions which describe the quantum state of two qudits. By means of the qubit-portraits machinery a semigroup of stochastic matrices can be associated to a given quantum state. The violation of the CHSH inequalities is discussed in this framework with some examples, we found that quantum correlations in qutrit isotropic states can be detected by the suggested method while it cannot in the case of qutrit Werner states.
Quantum tomography is an important tool for obtaining information about the quantum state from experimental data. In this study, we conduct a comparative analysis of various quantum tomography protocols, including protocols based on highly symmetric sets of state vectors, on mutually unbiased bases (MUB), and two-level transformations. Using the universal distribution for the fidelity of quantum state tomography, we estimate the fidelity and efficiency of quantum measurements for qudits of various dimensions as applied to a set of random states uniformly distributed over the Haar measure.
Quantum tomography is an important tool for obtaining information about the quantum state from experimental data. In this study, we conduct a comparative analysis of various quantum tomography protocols, including protocols based on highly symmetric sets of state vectors, on mutually unbiased bases (MUB), and two-level transformations. Using the universal distribution for the fidelity of quantum state tomography, we estimate the fidelity and efficiency of quantum measurements for qudits of various dimensions as applied to a set of random states uniformly distributed over the Haar measure.