"You're Telling Me The Uncertainty Principle Is Wrong?" A play in 2 acts, of approximately 5-8 minutes. Players: Alice, Bob, and Eve - physicists Setting: at a chalkboard. Act I ----- Alice: Bob, our high precision measurement experiments have been a smashing success so far, certainly enough for another round of funding! Bob: yes, but lately I've become worried that we cannot continue to increase the precision of our position and momentum measurements forever, and we may already be at the limit. Have you heard of quantum mechanics? Alice: oh yes, quantum mechanics. But as far as I remember it only says that we can't simultaneously know both complementary observables to arbitrary precision. The uncertainty principle and all that. Bob: I've been doing some reading on this point, and it seems that we can't even simultaneously perform the two complementary measurements! Alice: let's leave that out of the grant proposal, shall we? but it's a good point. Maybe what it's saying is that we couldn't predict the outcome of either measurement to arbitrary precision. One or the other, sure, but never both. Eve: hey guys, are you talking about quantum mechanics? Let me tell you about this crazy thing called entanglement! [discussion ensues] Bob: Alice! doesn't entanglement imply the uncertainty principle is wrong somehow! An entangled state is nothing but a state in which a measurement on one system can be used to predict the outcome of a measurement on the other system, no matter what observables we're talking about. Alice: let me see if I understand you --- you're saying the uncertainty principle isn't valid if we've got some "side information", in this case the other quantum system, which is entangled with the system we're measuring. Is that it? For instance if we've got one half of an EPR state, we can certainty predict either the position or momentum of the other half perfectly. Bob: yes, I think that's it! Alice: I wonder if there's some kind of extension to the uncertainty principle which takes this into account? Wouldn't that be awesome? Act II ------ There is a generalized version of the uncertainty principle which takes quantum side information into account! It is formulated in terms of the conditional von Neumann entropy as follows: H(X|B) + H(Z|B) >= -2 log c + H(A|B), where X and Z are the two observables we're interested in measuring, A is the system we're measuring, and B is a quantum system representing the side information. Furthermore, H(X|B) is the conditional entropy of the X observable measurement given the side information B, and the constant c is the maximum absolute value of the overlap of pairs of the eigenvectors of X and Z. So if X and Z share an eigenvector, the quantum state of the system we're measuring might be this eigenvector, at which point either outcome would be certain, no matter what's going on with the side information. On the other hand, if the observables are conjugate (in the sense that the maximum overlap is 1/sqrt(d) for d the dimension of the system to be measured), then the entropy of X given B plus Z given B must be at least log d plus H(A|B). When the AB system is entangled, this last term is negative, and can therefore offset the effect of X and Z not commuting. The proof proceeds by using smoothed min and max entropies, and discussing it is wholely and entirely unsuitable for an informal talk of only several minutes. Ok, the meat of the proof is strong subadditivity. Thus, we have answered Alice's question! But what is it _really_ good for? For one thing, it gives us a different composable security criterion for secret keys, and in a way which is useful in quantum Shannon theory. In particular, it is easy to prove that if Alice and Bob share a state such that Bob's conditional entropies H(X|B) and H(Z|B) are small, say less than epsilon, then (along with the condition that the entropy of Alice's marginal state is less than -2 log c) Alice's system is decoupled from any other system E in the sense that the trace distance between the AE state and a product state having the same marginals is less than 2 sqrt(epsilon).