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Quantum information can be negative

Based on work with M. Horodecki, and A. Winter

Even the most ignorant cannot know less than nothing. After all, negative information makes no sense. But, although this may be true in the everyday world we are accustomed to, negative information does exist in the quantum world. Small objects such as atoms, molecules and electrons behave radically different than larger objects -- they obey the laws of quantum mechanics.

What could negative information possibly mean? In short, after I send you negative information, you will know less. Such strange situations can occur because what it means to know something is very different in the quantum world. In the quantum world, we can know too much, and it is in these situations where one finds negative information. Negative information turns out to be precisely the right amount to cancel the fact that we know too much.

While all this might appear to be very mysterious (not to mention, an abuse of the word know!), negative information, can be put on a rigorous footing. I will try to explain how to do so here, in a manner which I hope is accessible to all. This description is intended for those who have an interest in this subject, but may not have a background in quantum information theory.  Most of this text should be understandable by anyone willing to put in a bit of effort, and the rest should be understandable by anyone with some knowledge of quantum mechanics (or by anyone willing to put in a lot of effort).  So if there are parts which continue to be unclear after some time, please let me know [email J.Oppenheim (at) damtp.cam.ac.uk], and I can modify this text to make things clearer. An executive summary of the result can be found at the end of this text here if you get impatient.

In order to concentrate on the main points, I have sacrificed some precision, and even some accuracy, so those with a modest background in physics should first try our recent article on negative information, which is available in Nature here.  It was written with Michal Horodecki, and Andreas Winter.   Patrick Hayden has written a commentary on it here, and a short description of the contents can be found in this piece by Andreas Trabesinger. The original version of the paper can be obtained at the pre-print arxiv here. It has a cartoon and George Orwell quotes, which were deemed inappropriate for Nature. We will shortly finish a more technical account which has the full proofs, calculations and details.
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