discord

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Three papers published

When it rains it pours. I had three papers published in the last week. One experimental paper and two papers about entanglement.

  1. Experimental violation of the Leggett–Garg inequality in a three-level system. A cool experimental project with IQC’s liquid state NMR group.    Check out the outreach article  about this experiment.
  2. Extrapolated quantum states, void states and a huge novel class of distillable entangled states. My first collaboration with Tal Mor and Michel Boyer and my first paper to appear in a bona fide CS journal (although the content is really mathematical physics). It took about 18 months to get the first referee reports.
  3. Entanglement and deterministic quantum computing with one qubit. This is a follow up to the paper above, although it appeared on arXiv  a few months earlier.
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Some updates

These last four and a half months have been exciting in may ways. Three papers submitted to arXiv: The first on Entanglement in DQC1, the second, a Leggett Garg experiment in liquid state NMR; and the third, a book chapter titled Why should we care about quantum discord? I also had two papers published one on quantum money and the second on  sequential measurements.

In August I organized a workshop on Semi-quantum computing and recently wrote about it on the IQC blog.  I also attended a workshop on Entanglement and quantumnes  in Montréal.

Earlier this month I got sucked into a discussion about publishing.

 

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Discord and completely positive maps

After over two years of work  we finally published our results showing that the connection between discord and complete positivity is quite weak, and probably has no operational significance. But let me start at the beginning:

In the beginning… and then there was a big discussion/argument about possible maps describing the evolution of a quantum system interacting with the environment. In the case where they are initially correlated this discussion is still not settled. In 2007 came a paper by Cesar Rodriguez-Rosario Kavan Modi, Aik-meng Kuah, Anil Shaji and ECG Sudarshan titled “Completely positive maps and classical correlations“, I call it Cesar and Kavan’s paper. They examined a situation where some initial family of system-environement, $$mathcal{SE}$$ states is classically correlated (has zero discord). It turns out that such a family of states is in the consistency domain of  a completely positive assignment map. In slightly less technical language: given a family of classically correlated $$mathcal{SE}$$ states it is possible to describe the evolution of the system using a completely positive map. Without going into details this comes with some caveats. Cesar, Kavan and Alán Aspuru-Guzik explained these caveats in another paper.

About a year after Cesar and Kavan’s paper Alireza Shabani and Daniel Lidar published a paper titled “Vanishing Quantum Discord is Necessary and Sufficient for Completely Positive Maps” This result was published in PRL, I will call it the SL paper. It made a lot of waves and has since been cited around 150 times. Unfortunately no one really understands it. I don’t know who should be blamed here, the authors for writing an unreadable paper (I assume they can read it), the editor for accepting an unreadable paper, or the referees who thought the paper was readable. But as it stands this paper was accepted, and since it was published in a prestigious journal and has such a bombastic title, people love to cite it. Especially to justify their research on discord. I guess I could start a rant but it’s nothing new so let us return to the story.

In September 2010 I had the extreme pleasure of attending the “Quantum Coherenece and Decoherence” workshop in Benasque where I met Cesar and Animesh Datta. After a short conversation about discord and interesting results in the field we discovered that although we have all cited SL we don’t know what they actually claim. We all assumed it was the “necessary” part of Cesar and Kavan’s “sufficient” result for completely positive maps but none of us could really explain the bottom line. After spending a few days in trying to understand the paper together we finally gave up, and instead came up with a counterexample. That is, we found a family of discordant states which is consistent with a completely positive assignment map.

A few weeks later I met Kavan in Singapore and we discussed this result further…

[missing reel]

.. and finally  Ángel Rivas joined our jolly group. The work was very slow, mostly due to us being on 4 different continents. By the time time I was at IQC we had a draft. When Kavan came to visit we finalized the paper.

The final version is much more then a comment on SL’s result. We showed that the problem of finding the map that correctly describes the evolution is a matter of how the problem is stated. More to the point, we showed that in at least thee sensible frameworks for approaching this problem there is at best a one way connection between positivity of the map and discord. Presumably there might be a framework where zero discord is both necressary and sufficient for completely positive maps. Unfortunately we were unable to identify this framework.

 

Vanishing quantum discord is not necessary for completely-positive maps
Aharon Brodutch, Animesh Datta, Kavan Modi, Ángel Rivas, César A. Rodríguez-Rosario arXiv:1212.4387, Phys. Rev. A 87, 042301

 

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Quantum discord

After a long an eventful month that included a visit by Kavan Modi to IQC and my visit to Israel (I’m posting from Israel), it’s time I got back to writing something. This time I’ll say something about my work for the past four years (as promised). One of the main subjects of my research is quantum correlations, and their role in defining the difference between quantum and classical (not quantum) systems.
Imagine a piece of information shared between two people Alice and Bob. Now think of a way to quantify the correlations between them. One way to quantify correlations is to ask what can Alice know about Bob’s part of the information by looking at her own part.
For example lets say Alice and Bob are each given a queen from a chess board. Alice then looks at her queen and sees it is white. She now knows Bob’s queen must be black. Alice and Bob are strongly correlated, since Alice always knows Bob’s piece by looking at her own.
For the second example Alice and Bob are each given a queen, but this time from a Deck of cards. If Alice sees a red queen she can say that it is more likely that Bob has a black queen, but she has no certainty. Correlations are lower in this case then in the chess example.
There is another way to account for correlations. We can ask about difference between the information in Alice and Bob’s hands individually and the information in their hands together. In the chess example Alice and Bob can each get one of two types of queens: black or white Together they also have two options Black White or White Black.
It turns out that both options for quantifying correlations are the same. To see this in the example we need to quantify the information in bits. Since Bob has two options in his hand “black queen” or “white queen” he has one bit of information. The amount of information Alice can discover about Bob is precisely this one bit. So they have one bit of correlations. Alternatively we can say that Alice has one bit of information: “black queen” or “white queen”; Bob has one bit of information: “black queen” or “white queen” and together they also have one bit “black white” or “white black”. The difference (1+1)-1 is again 1 bit so there is one bit of correlations.
Since i’m avoiding maths you will have to take my word that both methods give the same result in all cases… in the classical world. In the quantum world things are a bit different.
There are two essential (and related) aspects of quantum theory that make these two ideas about how to to quantify correlations give different results. 1) Measurements affect the system. If Alice wants to know the color of her queen, she needs to make a measurement, this measurement can change the state of the system; and 2) Quantum systems can be correlated in a much stronger way then classical systems, a phenomenon known as entanglement.
Before discussing the first aspect in detail, I will say a bit about entanglement. Entanglement was a term coined by Schrodinger in his famous “cat” paper, this paper was inspired by the earlier “paradox” of Einstein Podolski and Rosen (EPR). They showed that quantum mechanics predicts a situation where a system shared by two parties is in a well defined state although locally it is not defined. A system is in a well defined state if making a measurement on this system will give some result with certainty. So if I give Alice and Bob an entangled system I can predict the result of a measurement made on the whole system, but I cannot predict the result of a measurement made by Alice and Bob separately.
Entanglement is the most remarkable prediction of quantum mechanics, and in one way of another it is the driving force behind most of the really cool quantum phenomena. From quantum computers to Schrodinger’s cat. Nevertheless entanglement does not account for all the non-classical features of the theory. At least not directly. When discussing correlations, measurements and their effects on the system play a crucial role in describing non-classicality. To explain quantum measurements we can imagine a quantum system as an arrow pointing to some direction, X, in the simplest case we can think of this problem in two dimensions.
A quantum measurement is a question regarding the direction of the arrow. Is the arrow pointing in direction A? This has one of two results either yes or no. The probability is given by the angle between the “actual” direction and the direction in question. The effect of the measurement is that the arrow will now point in the same direction as the result. If the answer is yes it will point in direction A if the answer is no, it will point in the opposite direction.

 

A quantum measurement will "collapse" the state X into A or Not A.

A quantum measurement will “collapse” the state X into A or Not A.

Ok so what does this have to do with correlations? Well lets go back to the two definitions for correlations. The first was “What information can Alice find about Bob by looking at her own system”. In the quantum case this is no a clear question, we need to also say what measurement Alice is making. Different measurements will reveal different information about Bob. The second definition for correlations is what is the difference between the information in Alice’s hands plus the information in Bob’s hands and the information in their joint system” This is not directly related to measurement, so clearly it is not the same as the first definition. The difference between these definitions in the quantum case is the quantum discord. It is a measure of the “quantumness” of correlations.
As it turns out discord can be found in many interesting quantum systems and paradigms, but it is not yet clear what this means…

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I’m an expert

Two expected, but long awaited events happened on my birthday. One: I found out that my Phd was approved, so no more bureaucratic shit regarding that. Two: The review paper on discord and similar quantities was finally published. This sums up about one and a half years’ work spent on reading, writing and rewriting this review with my collaborators.  Two versions have been posted on  arXiv  since the end of last year. The latest one, posted in August, is pretty much the published version.

I will soon post something longer about discord and non-classical correlations, for now it is enough to say that quantum theory allows more general correlations then a classical theory.  Entanglement is the best example of these types of correlations, but as it turns out there are unentangled systems with non-classical correlations. Quantum discord captures entanglement and more (but not everything).

Since the beginning of the century (i.e 12 years ago) people started studying these kinds of correlations “beyond” entanglement in various forms and physical scenarios. The area exploded about 5 years ago and discord became a “hot” topic. The review includes almost all the work done on the subject until the end of 2011. discord was studied in so many different scenarios like quantum information, thermodynamics, many body systems, relativistic quantum information and others which made work on this review so much fun, on the one hand, but a lot of work on the other.