Simple criteria for noise resistance of two qudit entanglement

TL;DR

This paper introduces a simple geometric criterion based on correlation tensors to determine the noise resistance of entanglement in bipartite qudit systems, applicable under various noise models.

Contribution

It presents a novel, simplified criterion for detecting entanglement in high-dimensional systems using correlation tensor eigenvalues, extending analysis to multiple noise types.

Findings

The criterion effectively detects entanglement under white, colored, depolarizing, and damping noise.

Analytical results are provided for high-dimensional limits as d approaches infinity.

Critical noise thresholds for Bell inequality violations are identified for maximally entangled states.

Abstract

Too much noise kills entanglement. This is the main problem in its production and transmission. We use a handy approach to indicate noise resistance of entanglement of a bi-partite system described by $d\times d$ Hilbert space. Our analysis uses a geometric approach based on the fact that if a scalar product of a vector $\vec{s}$ with a vector $\vec {e}$ is less than the square of the norm of $\vec{e}$, then $\vec{s}\neq\vec{e}$. We use such concepts for correlation tensors of separable and entangled states. As a general form correlation tensors for pairs of qudits, for $d>2$, is very difficult to obtain, because one does not have a Bloch sphere for pure one qudit states, we use a simplified approach. The criterion reads: if the largest Schmidt eigenvalue of a correlation tensor is smaller than the square of its norm, then the state is entangled. this criterion is applied in the case of various types of noise admixtures to the initial (pure) state. These include white noise, colored noise, local depolarizing noise and amplitude damping noise. A broad set of numerical and analytical results is presented. As the other simple criterion for entanglement is violation of Bell's inequalities, we also find critical noise parameters to violate specific family of Bell inequalities (CGLMP), for maximally entangled states. We give analytical forms of our results for $d$ approaching infinity.