A good week for me is where I understand something knew, even if this is just clarifying some mis-understanding about something I should have "known". Here is what I learnt this week. It will be obvious to people who have mastered their linear algebra and quantum information, but for some reason I was confused.
Consider a state |psi> in a composite Hilbert space A+B of dimension 2 x n where n >2. Then the reduced density matrix rho_A (which traces over B) is a 2 x 2 matrix and rho_B is an n x n matrix. The former has 2 eigenvalues [say x and 1-x] and rho_B has n eigenvalues. For some mistaken reason I thought that it might be possible that rho_B has many non-zero eigenvalues (after all if n is large it will be a large matrix!). But this is not the case, it only has two [x and 1-x]. In other words, rho_A and rho_B has the same rank (i.e, number of non-zero eigenvalues).
This is actually "obvious" if |psi> is written according to the Schmidt decomposition. It can only contain at most 2 terms.
This was all motivated by trying to quantify entanglement between electronic and nuclear degrees of freedom in molecules. I will post more about that later...
I thank Jeff Reimers for correcting my misunderstanding.
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Yes. This is a very important piece of information. It seems to have repercussions for e.g. Chemistry. For example, this seems to imply that e.g. QMMM simulations cannot represent pure states of a super system in a multi-state problem. If one thinks of the boundary as a bipartition, then there should be different environment states for each quantum state of the subsystem (I.e. Each state should have it's own forcefield. This is potentially damning for many current attempts to model photochemistry in interacting environments, because the assumption that one environment potential can be used for all subsystem states is clearly wrong. How wrong? I don't know but at least I'm asking the question... Most don't.
ReplyDeleteThe statement that the branching space of a two-state conical intersection is a 2D plane can be viewed as a consequence of the fact that the the Schmidt decomposition divides the entanglement entropy equally between electrons and vibrations. Like many strande things in quantum mechanics, this is all a simple consequence of the mathematics of the singlar value decomposition!