Quantum Bayesian Networks

December 7, 2012

Synergetic Ears

Filed under: Uncategorized — rrtucci @ 9:48 pm

The topic of monogamy has been recently in the news, and I’m not referring to the mormons. John Preskill, our own quantum information theory Jedi master, has extolled the virtues of monogamy in a recent blog post of his about the black hole information paradox. And Eric Verlinde, a recipient of 30 million dollars in research grants, courtesy of the Dutch government, at a time when Europe is on the brink of bankruptcy, for his so-far-only-a-pipe-dream theory of entropic quantum gravity, has recently advocated monogamy in a paper of his. But wait…What’s this? Lubos Motl thinks Eric Verlinde’s paper is junk. All this drama is a great excuse for me to preach about monogamy, a virtue which you might agree with even if you are clueless about black holes and quantum gravity.

I’ll use the ampersand symbol \& to denote entanglement, because this symbol looks pretty tangled to me, and, besides, so many other things in physics (like energy and electric field) are already represented by the letter E. Besides, the ampersand looks like an E because it was originally meant to abbreviate the Latin “Et”, which means “and”.

The CMI or squashed entanglement for the “random variables” \underline{a} and \underline{b} in the mixed state \rho_{\underline{a},\underline{b}} is defined by

\&_{\rho_{\underline{a},\underline{b}}} (\underline{a},\underline{b}) = \frac{1}{2} {\rm min}_{\underline{\Omega}} \;\; S_{\rho_{\underline{a},\underline{b},\underline{\Omega}}} (\underline{a}:\underline{b}|\underline{\Omega}).

Here S_{\rho_{\underline{a},\underline{b}, \underline{\Omega}}} (\underline{a}:\underline{b}|\underline{\Omega}) is the quantum CMI. The minimum is over all density matrices \rho_{\underline{a},\underline{b},\underline{\Omega}} which yield \rho_{\underline{a},\underline{b}} when you trace over \underline{\Omega}. If you think about it, this is basically minimizing over “all possible” random variables \underline{\Omega}.

The 1/2 factor in the definition of squashed entanglement is a normalization convention, used so that \&(\underline{a},\underline{b})=S(\underline{a})=S(\underline{b}) when \rho_{\underline{a},\underline{b}} is a pure state. Indeed, when \rho_{\underline{a},\underline{b}} is a pure state, S(\underline{a})=S(\underline{b}) (by the Schmidt decomposition). By definition,

S(\underline{a}:\underline{b}|\underline{\Omega}) =S(\underline{a}|\underline{\Omega}) + S(\underline{b}|\underline{\Omega})-S(\underline{a},\underline{b}| \underline{\Omega}).

Minimizing both sides of this equation over \underline{\Omega} when \rho_{\underline{a},\underline{b}} is a pure state yields

2\&(\underline{a},\underline{b})= S(\underline{a}) + S(\underline{b}) + 0= 2S(\underline{a}).

Next we shall consider the following inequality, which was first proposed by Coffman, Kundu and Wootters (CKW) in a 1999 paper:

\&(\underline{a}:\underline{b_1}) + \&(\underline{a}:\underline{b_2}) \leq \&(\underline{a}: \underline{b_1},\underline{b_2})

CKW proved this inequality for pure states. They didn’t prove it for mixed states because squashed entanglement had not been invented yet. Nowadays we know that squashed entanglement satisfies the CKW inequality for mixed states.

The proof that squashed entanglement satisfies the CKW inequality for mixed states is a very simple one line proof. Here it is: By the chain rule,

S(\underline{a}:\underline{b_1},\underline{b_2}|\underline{\Omega})= S(\underline{a}:\underline{b_1}|\underline{b_2},\underline{\Omega}) + S(\underline{a}:\underline{b_2}|\underline{\Omega}).

Minimizing both sides of this equation over \underline{\Omega} immediately yields the CKW inequality.

The CKW inequality implies the monogamy principle, although it implies much more than that. The monogamy principle says that if \underline{a} is maximally entangled with \underline{b_1}, then \underline{a} must have zero entanglement with any other random variable \underline{b_2}. To see how this principle must follow from the CKW inequality, note that if \underline{a} is maximally entangled with \underline{b_1}, then \&(\underline{a}:\underline{b_1})= \&(\underline{a}:\underline{b_1},\underline{b_2})=S(\underline{a}) so CKW implies that \&(\underline{a}:\underline{b_2})=0.

I like to think of squashed entanglement \&(\underline{a}:\underline{b}) as the correlation (or transmitted information) between two people (or events) \underline{a} and \underline{b} conditioned over and minimized over all possible observers \underline{\Omega}. In terms of this narrative, what the CKW inequality is saying is that a message is “heard” better if person \underline{a} sends a single message to a joint delegation of two people \underline{b_1} and \underline{b_2}, instead of sending separate messages to \underline{b_1} and \underline{b_2}. It’s what people call a synergetic or superadditive effect.

Here is a CMI cartoon representation of the CKW inequality.

The message sender \underline{a} is given a megaphone and the receivers (listeners) \underline{b_1} and \underline{b_2} are given ears. A single receiver listening alone has just one ear. Two receivers listening as a joint delegation have 3 ears! I like to call this picture of the CKW inequality “Synergetic Ears”.

1 Comment »

  1. Great. In such a case I used to suggest to apply here http://fqxi.org/grants/large/initial

    Comment by Alexander Vlasov — December 10, 2012 @ 3:34 pm

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