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:1212.4387Phys. Rev. A 87, 042301



One thought on “Discord and completely positive maps

  1. Pingback: Writing a PRL comment. | Wikian Theological Foundation

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