QIPIRC

But this is not the end of the story. In fact, the rate at which the error probability goes down to 0 can be used as a measure of distinguishability between distributions or quantum states. Indeed: the faster the rate, the easier these distributions or states can be distinguished. The error rate therefore provides a genuine distance measure, with a clear operational meaning and with many other desirable properties.

However nice this distance measure might be, it would be useless if one could not calculate it. Indeed, it is defined as the limit of the rate when the number of copies goes to infinity, and cannot be calculated directly (unless your PC has an infinite amount of memory, and infinite speed). In the statistical setting, this problem was overcome when Herman Chernoff, in a classical paper published in 1952, calculated this limit analytically and found a simple closed-form expression for the error rate, which now bears the name of Chernoff divergence, in his honour. The illustration on the left depicts the simple case of comparing two coins and shows how the error rate (blue) tends to the Chernoff divergence log Q (red), in the limit of large n.

So what about the corresponding problem in quantum state discrimination? Koenraad Audenaert from Imperial College London and Frank Verstraete, now at the University of Vienna , in collaboration with the group of Emili Bagan This question turned out to be surprisingly hard and remained open for a long time, with many failed attempts at answering it. The breakthrough finally came in 2006, when in a very short time span two papers appeared that together completely solve the quantum Chernoff problem.

In July 2006, Michael Nussbaum from Cornell University and Arleta Szkoła from the Max Planck Institute for Mathematics in the Sciences, Leipzig , showed that the so-called minimal relative Renyi entropy – one of the proposed generalisations of the Chernoff divergence to the quantum case – is an upper bound to the error rate. 

Then, just two months later, Koenraad Audenaert from Imperial College London and Frank Verstraete, now at the University of Vienna, in collaboration with the group of Emili Bagan  in Barcelona, finally settled the issue by proving that not only does the minimal relative Renyi entropy provide an upper bound to the error rate, it is actually equal to it. Hence, it is the correct quantum generalisation of the Chernoff divergence. Their proof heavily relies on state-of-the-art matrix-analytical methods and beautifully illustrates the power of interdisciplinarity in research.