A more generalized two-qubit symmetric quantum joint measurement

TL;DR

This paper introduces a generalized family of elegant joint measurements (EJM) for two qubits, expanding the original EJM framework with additional parameters, and explores their symmetry properties and potential applications in quantum information processing.

Contribution

It proposes a three-parameter generalization of the EJM, providing quantum circuits and analyzing the symmetry and state properties, enhancing the understanding of multi-setting quantum measurements.

Findings

The generalized EJM reduces to the original with specific parameters.

Not all unit vectors in cylindrical coordinates form valid EJM basis.

The reduced states form mirror-image tetrahedrons, preserving symmetry.

Abstract

A standard two-qubit joint measurement is the well-known Bell state measurement (BSM), in which each reduced state (traced out one qubit) is the completely mixed state. Recently, a novel quantum joint measurement named elegant joint measurement (EJM) has been proposed, where the reduced states of the EJM basis have tetrahedral symmetry. In this work, we first suggest a five-parameter entangled state and reveal its inherent symmetry. Based on this, we define a more generalized EJM parameterized by $z$, $\varphi$ and $\theta$, and provide the quantum circuits for preparing and detecting these basis states. There are three main results: (i) the previous single-parameter EJM can be directly obtained by specifying the parameters $z$ and $\varphi$; (ii) the initial unit vectors related to the four vertices of the regular tetrahedron are not limited to the original choice and not all the unit vectors in cylindrical coordinates are suitable for forming the EJM basis; and (iii) the reduced states of the present EJM basis can always form two mirrorimage tetrahedrons, robustly preserving its elegant properties. We focus on figuring out what kind of states the EJM basis belongs to and providing a method for constructing the more generalized three-parameter EJM, which may contribute to the multi-setting measurement and the potential applications for quantum information processing.