# Continuous Variable Entanglement Measure for the Statistically Mixed State

Can anybody tell me, which is the best entanglement measure for the Continuous Variable Entanglement of a Statistically Mixed State ? I have read that Schmidt decomposition is not valid in this situation. Hence I cannot use von-Neumann entropy of the reduced density matrix. The system that I study has dissipation terms and the states are no longer statistically pure states.

• Indeed, the von Neumann entropy of the reduced matrix is not an entanglement measure for statistically mixed states, since it is very high for the tensor product of two noisy states. In general, entanglement measures for statistically mixed state are plenty and have different physical meaning (e.g. entanglement of distillation is not the entanglement of formation, which is not the negativity, an so on.). So, your question does not have a single valid answer, because we do not know enough on your system and why you want to look at it's entanglement. – Frédéric Grosshans May 28 '14 at 15:07
• So a few questions : are you interested in the entanglement of a 2-mode system or a bigger one ? Do you want an entanglement witness or an entanglement measure ? Are your state supposed to be Gaussian or not ? Depending on your answer to these question, the answer could be different. – Frédéric Grosshans May 28 '14 at 15:10
• But if you only want a simple to compute entanglement measure, (logarithmic) negativity is your friend – Frédéric Grosshans May 28 '14 at 15:13
• Dear Frédéric Grosshans thanks for the fast response. My system is a two coupled harmonic oscillator with dissipation. Evolution of the density matrix can be modeled by a Linbald Equation. But the states are no longer statistally pure. My states are also non-gaussian, I study continuous variable entanglement. I have found that the logarithmic negativity is valid in dimensions less than six. Since the states are non-gaussian I cannot use covariance matrix method to find the negativity. – Sijo Joseph May 28 '14 at 16:09
• My research is exploring the connection between entanglement and classical chaos. Chaos is at its best when the system is dissipative then the quantum entanglement is complicated !!! – Sijo Joseph May 28 '14 at 16:16