Friday, June 26, 2015

What is so great about the von Neumann entropy?

I got a referee report for a paper submitted to PhysChemChemPhys that looks at the quantum entanglement of electronic and nuclear degrees of freedom in molecules. The paper goes beyond the calculations considered here, and explores subtle issues about how entanglement may or may not be related to the breakdown of the Born Oppenheimer approximation.

One referee asked a good but basic question, "Why is the von Neumann entropy the appropriate measure of entanglement to consider here?"

Here is my answer. I think experts could do better and so I welcome suggestions.

The von Neumann entropy is widely accepted as the best measure of quantum entanglement for pure quantum states defined on bipartite systems, such as that considered here. This is because the von Neumann entropy satisfies certain desired criteria, including vanishing for separable states, monotonicity (it does not increase under local operations or classical communication between the subsystems), additivity, convexity, and continuity.

It was a bit of work to come up with this answer, because this is all second nature to people who work in quantum information. It is hard to find a place where this is clearly stated and discussed in detail. The Quantiki wiki entry on entanglement measures  and the axiomatic approach, along with the review article by the Horodecki family helps.

Anyone suggest a place where this basic issue is discussed and worked out in detail?

The big challenge is defining entanglement measures for multi-partite systems and for mixed quantum states.


  1. Not an expert on this, but I note that your first two criteria, "vanishing for separable states" and "monotonicity", are part of the definition of an entanglement measure (according to Quantiki), so they are distinct from the following three properties, "additivity, convexity, and continuity", which are optional, so to speak. Can we interpret the referee as asking whether these last three are appropriate requirements when considering an asymmetric system of nuclear and electronic DoFs? In other words, is he asking a /subtle/ leading question, not something elementary?
    Congratulations on your mention on Physics World,, BTW.

    1. Thanks for the comment.
      Based on the context of other comments from the refer she/he is a straight chemist who is asking an elementary question.
      However, as you point out and I discovered there is some subtle dimensions to the question in general.

    2. Thanks for also for pointing out the Physics World write up. I was completely unaware of it.

  2. I liked the discussion in "Geometry of Quantum States" by Bengtsson & Zyczkowski. I have a copy if you want to borrow it.