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Comment by tedsanders | Hacker News Reader

I'm having a difficult time following his explanation. Here are a few of my stumbling points. Maybe you can help me out.

1) What does it mean that entangled particles are supercorrelated? How can S(A|B) = 2 or -1?

2) The formula for Von Neumann entropy is S = -Tr(p log(p)), where p is the density matrix. How do you take a log of a matrix?

3) Why does the interpretation rely on the density matrix? I've always thought of the density matrix as something that gives information about our ignorance rather than the system (this is why the density matrix mixes classical probabilities in alongside quantum amplitudes/probabilities - observer uncertainty is classical). Therefore, to me, any good interpretation of quantum mechanics ought to work without density matrices.

Unfortunately his paper here (http://www.flownet.com/ron/QM.pdf) doens't seem to have significantly more information than his talk.

Thanks for any help you can offer.

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Without reading the paper or watching the video.

> 1) What does it mean that entangled particles are supercorrelated? How can S(A|B) = 2 or -1?

The density matrix does not have the same properties of a joint probability distribution, so quantum entropy doesn't really have the same properties of classical entropy. [1]

> 2) The formula for Von Neumann entropy is S = -Tr(p log(p)), where p is the density matrix. How do you take a log of a matrix?

The log of a matrix is defined as the inverse of the matrix exponential. The matrix exponential can be defined as the usual power series, only using matrices rather than scalars.

What one usually does is: diagonalize the matrix (if possible), take the log of the eigenvalues and rotate again. Obviously taking the log of a matrix is a little trickier than the exponential because if you have negative or complex-valued eigenvalues you have to be a bit more careful.

[1] http://arxiv.org/abs/quant-ph/9610005

Thanks for the link - it's exactly what I needed!

Good answers. Just a small comment: quantum entropy is a generalization of classical entropy. In particular, if A and B are classically correlated, then S(A|B) has all the properties of a classical entropy (Shannon's, in this case), for instance, it is non-negative.

What does classically correlated mean? Is it different than quantum correlated? (Is this analogous to the difference between pure and mixed states?)

Didn't watch but,

1. Super-correlation is entanglement.

2. Tr (ρ log_2 ρ ) = Σ_k of λ_k log2 λ_k where λ_k is the k-th eigenvalue of ρ. Sometimes the log2 is instead ln.

3. Density matrix indicates degree of entanglement. You can measure L or R polarized light but you have an entanglement of the two. Mind you that detecting a bell state without loopholes has, to my knowledge, never been done. A systems density matrix for a system can be measured statistically (Quantum tomography?).

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