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Partial quantum information - can be negative
..if this sounds paradoxical, it must be, since negative information can occur only in the strange world of quantum mechanics. In the classical world of our experience, information, i.e. the amount of communication required to 'inform', must always be positive or zero; and this is even true of partial information, which is the communication effort to inform a receiver who already has some (statistical) partial knowledge.
How different the matter becomes with quantum information, was recently discovered by Michael Horodecki (Gdansk), Jonathan Oppenheim (Cambridge) and Andreas Winter (Bristol) in their paper Partial quantum information, Nature 436, pp. 673-676.
The idea is very simple: let two players, Alice and Bob, obtain states from a source of quantum information. I.e., the states are states of the composite systems AB. How much quantum communication from Alice to Bob is required such that he has the full state? To isolate the quantum aspect, we allow the parties free classical communication, and to obtain a nice information theoretical model, let us assume that many independent samples from the source are taken - the minimal rate of communication is the 'partial quantum information' Bob lacks about the joint source.
The main result is that the partial information is given by the conditional entropy S(A|B) = S(AB) - S(B), i.e. the von Neumann entropy of the global source minus the von Neumann entropy of Bob's reduced state. This answer is beautifully intuitive when one recalls that the entropy of AB is the total available information, and that that of B would then be how much Bob already knows. On closer look, however, it is very mysterious, as the above quantity can be positive or negative, depending on the source, as was noted earlier, e.g. by Cerf and Adami In the case of no prior knowledge of Bob (in other words, the source doesn't give him any state), the above result reduces to Schumacher's data compression and the communication cost is the von Neumann entropy S(A).
However, in cases of actual correlation between A and B the required communication S(A|B) is less than S(A). If S(A|B) becomes negative, the protocol (which we call 'state merging') does not require any quantum communication at all - of course, in general it will still involve classical communication; instead it produces -S(A|B) many EPR states |0>|0>+|1>|1>, which, by virtue of teleportation provide the potential to communicate quantum information.
The upshot is that the conditional entropy quantifies partial quantum information in the above sense, regardless of its sign: if it is positive, we have to spend entanglement to merge the state, if it is negative, merging produces the corresponding amount of entanglement.
Of course, the above explanation is only a short teaser: please try reading the full article which also contains several applications of the result to quantum information theory (or the preprint version quant-ph/0505062 on the arXiv, which has some juicy bits that were left out of the published paper), or Patrick Hayden's Views article in the same issue, Nature 436, pp. 633-634 or in Andreas Trabesinger's account of it in Nature Physics.
We've also attempted a few popularisations, the best of which may be this one here.
(Andreas Winter, September 2005). The cartoon illustrates state merging and is (c) Roberta Rodriquez 2005.
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