So here it goes.

Let’s start by defining quantum discord. To do so, we have to first discuss bipartite mutual information. Some folks noticed, about ten years, the two classical equivalent definitions of mutual information are not the same in quantum mechanics. Suppose we have a shared (between Alice and Bob) probability distribution with marginal distribution and . The mutual information in this distribution is given by . is the Shannon’s entropy, which quantifies the ignorance of Alice and is defined as . Now, there is an equivalent quantity called conditional information, , where the conditional entropy if defined as . The conditional entropy quantifies the ignorance of Bob given some knowledge about Alice.

Classical quantum: , , .

Once again classically we have and there is no definition for the conditional entropy, , in quantum mechanics, i.e. . Well there two options.

1. Stick with even for the quantum case.

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This quantity itself has a rich history. Classically this quantity is always positive but quantum mechanically it can be negative when is entangled. It also has a nice operational interpretation for the task known as quantum state merging.

2. Define conditional entropy by measurements on Alice’s side.

where . The only restrictions are and . In this case one can prove that for any and quantum discord is defined as

It is this quantity that we hope to give some operational meaning to.

What both papers noticed was that quantum discord is not only the difference in two definitions of (classically equivalent) mutual information, it is also the difference in the two conditional information.

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What our team was able to show was the following. We considered purification of the system to system . In that case we found that . Where total cost of entanglement in (extended) state merging between Bob and Charlie with the final state with Charlie.

Madhok and Datta showed that the state merging resource (cost), , is depleted (inflated) by measurements on is equal to quantum discord. That is, .

At the superficial level that is the difference in two papers. We are both on to very similar ideas but the results look different, at least for now. I will try to write up something on state merging and extended state merging before I expand the discussion above.

Context: My brother is working on his PhD at the University of Texas at Austin (the same place as I got my PhD). He’s having a tough time in lab right now as experiments are not working, etc.

How cool is that? I never expected that answer. May recently got his Ph.D. in Physics as was explained in the interview.

Here are some pictures to illustrate my point.

View outside the campus

Construction along the way

Entrance to A&M Qatar

A view from the inside

Fantastic lunch for 3 USD

Yet, in the mean time, we manage to get a paper accepted in Physical Review Letters that deals with putting entanglement and other correlations, like quantum discord, on an equal-footing.   The whole thing worked out to be rather nice as it is perfectly applicable for arbitrary dimensional multipartite systems as well.

We used the concept of relative entropy as the basis of work.  Relative entropy of entanglement is a well known measure.  Using the same approach we defined quantum discord, quantum dissonance, and classical correlations.  Quantum dissonance is new quantity that we introduced, which is similar quantum discord but excludes entanglement.

Our work simplifies comparing different correlations.  Till now, there was no clear method to compare quantum discord with entanglement for instance.  One can compare concurrence and discord, but what does that mean?  With our methods relative entropy of entanglement is computed in the same way as quantum discord, and so comparing the two is perfectly legitimate.

Our definition of discord differs from the original definition.  We show that the definitions are closely related and therefore we are not diverting too much from the original spirt of quantum discord.  I think the results in this paper will enable many to do new computations that are pertinent.

Seems like some of those blog entries did pay off in the end.  Re-reading at some of them I noted not how different the paper is from the original ideas, but how true it is to them.  The difference between the paper and the blog entries is that the paper is much more refined version of the blog.  Cesar has a real nice discussion on his blog, MinusTwoFish, about this project and open science that is very much worth looking at.

where is the dimension of the Hilber-Schimidt space, is dimensional identity matrix, is a real vector, and are the generalized Pauli–Gell-Mann–Tilma matrices (generators of group).

The generators of group follow a nice trace condition, . What this means is that the linearly independent set of matrices that span the space of density matrices is .  And the corresponding dual set is also itself, almost.  The dual set has to be normalized, giving .  All matrices from the two sets follow orthonormality condition.

Now it is true that this is not what we are after.  I would like to have a set of linearly independent projections that span the space of density matrices.  And construct the corresponding dual set.

We are half way there.  We can use the above formalism to construct the set of linearly independent projections first and the hopefully be able to construct the dual set after wards.

When talking to people about such a construction they always ask ‘can such dual set always be constructed?’  I will try to prove that the answer is yes by constructing the dual set in general.  This will probably take a few posts over next couple of weeks.

This blog did go into coma for the last three months.  There are reasons for this.  For one, I stopped doing serious research for some time.  And therefore had nothing much say or nothing I wanted to say.  I won’t claim ‘I’m back baby!’ but let’s just call it ‘attempt two’.