The paper quant-ph/050562 has been out for a while now, and I kept meaning to comment on it, but somehow never got around to it. Now it’s hit big time. This month in Nature has the article article, “Partial Quantum Information” by Michal Horodecki, Jonathan Oppenheim and Andreas Winter, along with the a nice intro written by Patrick Hayden. Johnathan Oppenheim also has a really really good popular explanation here . Normally I would write a little explaining this result, but Patrick’s intro and Johnathan Oppenheim’s web page are so good that, well, you should just get your info from the horse’s mouth (so to speak.) But I will tell a story related to this work.
When I was an undergraduate I spent the summer between my junior and senior years working for Nicholas Cerf and Chris Adami. Specifically I was working on trying to find quantum algorithms to efficiently solve NP-complete problems. I call this my summer spent banging my head against the wall. Not a very effective summer, research wise, but I learned a lot about quantum computing over that summer. Anyways, Cerf and Adami worked for Professor Steve Koonin (I think they were both postdocs, but I’m not sure if I remember this correctly.) When I was a freshman at Caltech, I was wandering around campus a few weeks after I showed up on campus, and I saw these two guys having a really animated conversation outside on a bench. One of the guys was really really tall, and the other guy was fairly short and looked EXACTLY like
Steve Rick Moranis. I mean exactly. This being Los Angeles, and me being a hick from the sticks, I was only a few feet away from asking the shorter guy for an autograph, when I chickened out. Which is a good thing, because it turns out that this guy was none other than Steve Koonin! (Koonin, by the way, was an undergrad at Caltech. He went to MIT and got his Ph.D. in three years. Three years! Now that’s the power of a Caltech education 😉 ) Some of you will find it funny that the second guy on that bench, I later learned, was Jeff Kimble. Many of you will know Professor Kimble from his demonstration of a quantum logic gate in a cavity QED system in 1995 (as well as a plethora of other cool cavity QED results.)
Back to the story at hand! So I worked for Cerf and Adami. At the time, they had this really strange paper (see here and here) where they talked about negative quantum information. In particular they noticed that if you took the conditional Shannon entropy and converted it over to the quantum mechanical conditional von Neumann entropy, then while the classical conditional entropy was always positive, the quantum mechanical conditional entropy could be negative. For example, a Bell pair had a conditional von Neumann entropy of negative one (bit). This was strange, and one question that their work raised is what exactly this negative number might mean. Another question along these lines arose a little later when the conditional entropy showed up in the quantum channel capacity. Here, however, when the conditional entropy showed up as negative, the capacity was set to zero (creating the so-called mutual information.) Having a negative capacity for a noisy quantum channel didn’t seem like a reasonable thing! What the current work describe above does is to actually given an operational meaning to this mysterious quantity the conditional quantum entropy. By operational, this means that those negative numbers now have an intpretation in terms of a protocl which operates at a rate related to those negative numbers. Read the article to see how this all works out. It is very nice. The author promise a more detail paper (with a date stamp of 2005, so soon) which I’m looking forward to.
(Thanks to Saheli and Grant for pointing me to Johnathan’s web page. Grant, who is a student in the class I am teaching, emailed the class mailing list with links to the negative quantum information web pages, and the comment “It’s possible we may leave this class knowing less than when we started!!!” Hopefully, the same effect hasn’t happened to you after reading this post 😉 )