Introduction Of Quantum Entanglement Measure Based On The Expectation Values Of Pauli Operators

TL;DR

This paper introduces a new measure of quantum entanglement based on the expectation values of Pauli operators, which can be experimentally tested and extended to higher-dimensional states.

Contribution

A novel entanglement measure called the 'separability index' derived from expectation values, applicable to multi-dimensional quantum states and experimentally measurable.

Findings

The measure distinguishes between separable and entangled states effectively.

Examples with qubits, qutrits, and qudits demonstrate the measure's applicability.

The measure correlates with the degree of entanglement in various states.

Abstract

In this paper, firstly considering that in separable states, the measurement of one particle has no effect on the measurement of the second particle, we show that Alice and Bob can find directions in which the results of their measurements on the spin of the particle are always maximized. In other words, the state of the particle is an eigenstate for the operator that is applied in that direction, so the sum of the spins of two particles can have a maximum value. We will argue that in entangled states, due to the effect of particle measurement results on each other, Alice and Bob cannot find the desired operators. Therefore, in such measurements, the total spin of the particles will always be less than the mentioned maximum. But we ask them to try and measure in directions that will get the most value. Because this value is maximum for separable states and minimum for fully entangled states, and for the rest of the states, it will be proportional to the degree of entanglement between the two maximum and minimum values, we set this parameter as We are calling it the "separability index". Then, based on this index, the measure of entanglement was introduced and extended to states with higher dimensions. In the end, examples of di-qubit states di-qutrit states, and di-qudit states were investigated and the efficiency of the measure was confirmed by the results of the examples. Considering that in this measure, the values of entanglement are calculated based on the expectation values and we can measure the expectation values in the experiment, we hope to be one step closer to the testability of the entanglement value.