irreversible

Operational Meaning of Quantum Discord

After we posted our paper, Operational interpretations of quantum discord by D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and A. Winter (arXiv:1008.3205), on the arXiv a similar note, Interpreting quantum discord through quantum state merging by V. Madhok and A. Datta (arXiv:1008.4135), appeared. Recently someone asked me to discuss the differences in the two papers.

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 $p_{ab}$ with marginal distribution $p_a=\sum_b p_{ab}$ and $p_b=\sum_a p_{ab}$ . The mutual information in this distribution is given by $I(a:b)=H_a+H_b-H_{ab}$ . $H_A$ is the Shannon’s entropy, which quantifies the ignorance of Alice and is defined as $H_a=\sum_a -p_a\log(p_a)$ . Now, there is an equivalent quantity called conditional information, $J(b|a)=H_b-H(b|a)$ , where the conditional entropy if defined as $H(b|a)=\sum_a p_a H(p_{ab}/p_a)$ . The conditional entropy quantifies the ignorance of Bob given some knowledge about Alice.

Classical $\rightarrow$ quantum: $H \rightarrow S$ , $p \rightarrow \rho$ , $a,b\rightarrow A,B$ .

Once again classically we have $I(a:b)=J(b|a)$ and there is no definition for the conditional entropy, $H(b|a)$ , in quantum mechanics, i.e. $S(B|A)$ . Well there two options.

1. Stick with $I(a:b)=J(b|a)$ even for the quantum case.

$S(B|A)=S_{AB}-S_A$ .

This quantity itself has a rich history. Classically this quantity is always positive but quantum mechanically it can be negative when $AB$ 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.

$S(B|\Pi)=\sum_i p_i S(\Pi_i \rho_{AB} \Pi_i/p_i),$

where $p_i=\mbox{Tr}[\Pi_i\rho_{AB}]$ . The only restrictions are $\Pi_i \geq 0$ and $\sum\Pi_i =I$ . In this case one can prove that $I(A:B)\geq J(B|\Pi)$ for any $\Pi$ and quantum discord is defined as

$D(B|A)=I(A:B)-\max_\Pi J(B|\Pi).$

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.

$D(B|A)=S(B|\Pi)-S(B|A)$ .

What our team was able to show was the following. We considered purification of the system $AB$ to system $ABC$ . In that case we found that $D(B|A)=\Gamma(B\rangle C)$ . Where $\Gamma(B\rangle C)$ 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), $S(B|A)$ , is depleted (inflated) by measurements on $A$ is equal to quantum discord. That is, $D(B|A)=S(B|\Pi)-S_A-S(B|A)+S_A$ .

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.

September 30, 2010 at 11:55 pm Leave a comment

‘Don’t become a scientist’

My brother sent me an article entitled ‘Don’t Become a Scientist!‘ last night. The article was written over ten years ago and reading it makes me think (as I am about to finish my first postdoc) that things have only gotten worse. Too late for me, I’m already stuck. Save yourself, go become a rock star or a car mechanics, anything useful!

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.

August 30, 2010 at 3:08 pm 4 comments

Operational interpretations of quantum discord

Quantum discord, a topic that I’ve worked on quite a bit, lacked any operational meaning thus far; this is ten years after its conception. I am happy to say that this is no longer the case, and even happier to say that I had something to with this. A coalition of six persons and five universities in five countries successfully found operational interpretation for quantum discord itself and the asymmetry of quantum discord, presented here arXiv:1008.3205. But still a lot of work remains in understanding this quantity with some depth. Nevertheless this is a boost for the quantum discord community, which growing rather quickly.

August 20, 2010 at 4:00 pm 1 comment

We Will Rock You

One of the greatest rock songs ever is Queen’s We Will Rock You, which was written by Brian May, the lead guitarist for Queen. I was just listening to an interview he recently did with Terry Gross of NPR’s Fresh Air [LINK]. When Gross asked May “… so how did you record ‘stomp-stomp-clap’ so it would sound grand and reverberating as oppose to four people stomping their feet and clapping?” To this May replied, “Well I am a physicist, you see…”

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

August 4, 2010 at 8:25 pm Leave a comment

Quantum in Qatar

This week I am visiting the group of Hyunchul Nha at Texas A&M University at Doha, Qatar. The campus is at the edge of the town and contains six US universities including Carnegie Mellon, Cornell, Georgetown University, Northwestern and Virginia Commonwealth. It’s an interesting campus, lot’s of space and lots of construction. The buildings are architecturally pleasing.

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

February 2, 2010 at 4:15 am 2 comments

Unification of Entanglement and Discord

We’ve been fairly busy lately on the equator. I for one have been buzzing around like a fly and seeing parts of Thailand, Indonesia, Malaysia, and Hong Kong. The last one being my favorite; what an incredible city!

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.

January 28, 2010 at 7:26 pm Leave a comment

From post to paper

Some of the first (three or four) blog posts on this blog have turned into a paper in collaboration with Cesar Rodriguez-Rosario and Alán Aspuru-Guzik of Chemistry Department at Harvard University. The paper titled Linear assignment maps for correlated system-environment states was just published in Physical Review A.

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.

January 26, 2010 at 3:58 pm 1 comment

Self dual elements

Following the Byrd and Khaneja [ref] any density matrix can be written as

$\rho=\frac{1}{d}\left(\mathbb{I}+\sqrt{\frac{d(d-1)}{2}}\vec{n}\cdot\vec{\lambda}\right),$

where $d$ is the dimension of the Hilber-Schimidt space, $\mathbb{I}$ is $d$ dimensional identity matrix, $\vec{n}$ is a real vector, and $\{\lambda\}$ are the generalized Pauli–Gell-Mann–Tilma matrices (generators of $SU(d)$ group).

The generators of $SU(d)$ group follow a nice trace condition, $\mbox{Tr}[\lambda_i \lambda_j]=2\delta_{ij}$ . What this means is that the linearly independent set of matrices that span the space of density matrices is $\{\mathbb{I},\lambda\}$ . And the corresponding dual set is also itself, almost. The dual set has to be normalized, giving $\left\{\frac{1}{d}\mathbb{I},\frac{1}{2}\lambda\right\}$ . 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.

September 24, 2009 at 2:57 pm Leave a comment

Dual set

In one of my earliest posts I talked about linearly spanning the Hilbert-Schmidt space using physical states. It turns out that to do this one needs a set projectors is larger than the set of orthogonal projections. But orthogonality is a nice property for sake of calculations. Therefore I (in the earlier post) relied on a set of matrices that I called the dual set satisfying $\mbox{Tr}[P_i D^\dag_j]=\delta_{ij}$ , where $\{P_i\}$ is the set of projectors that span the space and $\{D_i\}$ is the dual set.

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.

September 10, 2009 at 5:37 pm Leave a comment

death of a blog

Does anyone know what the growth rate of blogs is? How many are born each day? How many suffer death due to negligence? (I bet most die of neglect.) I will not let this happen to my blog. When time comes I shall kill it with my bear hands.

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’.

September 10, 2009 at 5:27 pm Leave a comment

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irreversible

Operational Meaning of Quantum Discord

‘Don’t become a scientist’

Operational interpretations of quantum discord

We Will Rock You

Quantum in Qatar

Unification of Entanglement and Discord

From post to paper

Self dual elements

Dual set

death of a blog

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