Entanglement Measures and Asymptotic Continuity

[Ulquiorra15]

I'm finishing my master studies in Theoretical Physics and my thesis is being about entanglement measures. I've read several papers in which all kind of properties that one would desire to have in a good entanglement measure are exhibit. One of these properties that one would consider fundamental is the asymptotic continuity of a entanglement measure. In particular, it's defined as follows:

"One entanglement measure E in d dimensions is asymptotic continuous if for two arbitrary states ρ and σ we've in the asymptotic regime n---->∞ that

||ρ - σ|| ---> 0

with

[E(ρ) - E( σ)]/log(d) ---> 0 "

I can understand that for two states which are close to each other, the value for the entanglement measure is expected to be similar, but what I don't understand is why we have to use the asymptotic regime. I've searched in various papers about entanglement measures but I haven't found a clear answer for the question.

Thank you everyone for reading me and I will be very grateful if you can help me with any contribution.

 

[eljose79]

Dear fellow scientist,

First of all, congratulations on completing your master studies in Theoretical Physics and focusing on entanglement measures for your thesis. It is a fascinating and important area of research.

To answer your question, let me first clarify the concept of asymptotic continuity. Asymptotic continuity is a property of a function or measure that describes how it behaves as its input values become very large, or approach infinity. In the context of entanglement measures, it refers to how the measure behaves as the number of particles or subsystems involved in the entangled state increases.

Now, why is asymptotic continuity important for entanglement measures? The reason lies in the nature of entanglement itself. Entanglement is a phenomenon that arises when two or more particles or subsystems are connected in such a way that their properties are strongly correlated. This correlation can be measured by entanglement measures, which quantify the amount of entanglement present in a given state.

In many practical applications, we are interested in studying entanglement in systems with a large number of particles or subsystems. For example, in quantum computing, we may need to manipulate entangled states with thousands or even millions of qubits. In such cases, it is not feasible to calculate the exact value of an entanglement measure for every possible state. Instead, we need a measure that can be approximated efficiently and accurately, even for large systems.

This is where asymptotic continuity comes in. By requiring that the measure behaves similarly for states that are close to each other in the asymptotic regime, we can use it as an approximation for a wide range of entangled states. This is because in many practical applications, we are interested in the overall trend of the entanglement measure, rather than its exact value for a specific state.

In conclusion, the use of asymptotic continuity in entanglement measures is motivated by the need for efficient and accurate approximations in the study of large entangled systems. I hope this helps clarify the concept for you. Best of luck with your thesis and future research endeavors.