I Does decoherence mean the loss of entanglement?

[Haorong Wu]

TL;DR Summary

If a pure densitry matrix becomes mixed, does it lose some entanglement?

Hello, there. I am studying a model for decoherence of two entangled photons. The space for the first photon is 2 dimensional, while that for the other one is 6 dimensional. In total, the system will be in a 12 dimensional space.

Initially, they are set to one of the Bell states, such as $\left | 1,-1 \right > + \left | -1,1 \right>$, normalization omitted. The first photon can only take states 1 or -1, while the other photon can be in states $\pm 3, \pm 2, \pm 1$.

Now, I have the final density matrix whose $tr(\rho)=1$ and $tr(\rho^2)\approx 0.5$ that means the decoherence is quite strong.

I want to measure the entanglement of the system. I go through couples of papers and I find that almost no available quantities can be used in a mixed system with high dimensions except relative entropy of entanglement (REE), defined by $$E_R(\rho)=\min _{\sigma \in \rm{Sep}}\rm{tr}\rho (\log \rho- \log \sigma ) $$where Sep denotes all separable states. The program shows that the initial density matrix has a REE of 1, and 0.9485 for the final density matrix.

It seems that the entanglement is almost intact. But my advisor said it is not reasonable since the system is almost decoherent.

But I am still wondering is there any direct link between coherence and entanglement? Or, do you know any other suitable quantities that can measure entanglement for a mixed high-dimensional system?

Thanks!

 

[Demystifier]

Or, do you know any other suitable quantities that can measure entanglement for a mixed high-dimensional system?

There are many, see e.g. the review
https://arxiv.org/abs/quant-ph/0504163