PHYS771 Lecture 11: Decoherence and Hidden VariablesMixed in with the above reasonable sounding words is some moderately serious math, none of which I could follow. For me the discussion affirmed that the 'hidden variable' interpretations of QM do collide with the spirit of special relativity.
... if you teach an introductory course on quantum mechanics, and the students don't have nightmares for weeks, tear their hair out, wander around with bloodshot eyes, etc., then you probably didn't get the point across. So rather than deny this aspect of quantum mechanics -- rather than cede the field to the hucksters and charlatans, the Deepak Chopras and Brian Josephsons -- shouldn't we map it out ourselves...
...But why should we care about multiple-time probabilities? For me, it has to do with the reliability of memory. The issue is this: does the "past" have any objective meaning? ... Or does the past only "exist" insofar as it's reflected in memories and records in the present?
The latter view is certainly the more natural one in quantum mechanics. But as John Bell pointed out, if we take it seriously, then it would seem difficult to do science! For what could it mean to make a prediction if there's no logical connection between past and future states -- if by the time you finish reading this sentence, you might as well find yourself deep in the Amazon rainforest, with all the memories of your trip there conveniently inserted, and all the memories of sitting at a computer reading quantum computing lecture notes conveniently erased?
... if we accept the usual picture of quantum mechanics, then in a certain sense the situation is far worse: the world (as you experience it) might as well not have existed 10-43 seconds ago...
... When I was talking before about the fragility of quantum states -- how they're so easy to destroy, so hard to put back together -- you might have been struck by a parallel with the Second Law of Thermodynamics. Obviously that's just a coincidence, right? Duhhh, no. The way people think about it today, decoherence is just one more manifestation of the Second Law....
At this point the sharp-eyed reader might notice a problem: won't the branches have to collide eventually, when the tree "runs out of room to expand"? The answer is yes. Firstly, if the Hilbert space is finite-dimensional, then obviously the parallel universes can only branch off a finite number times before they start bumping into each other. But even in an infinite-dimensional Hilbert space, we need to think of each universe as having some finite "width" (think of Gaussian wavepackets for example), so again we can only have a finite number of splittings.
The answer of decoherence theory is that yes, eventually the branches of the multiverse will start interfering with each other -- just like eventually the universe will reach thermal equilibrium. But by that time we'll presumably all be dead....
The idea of hidden-variable theories is simple. If we think of quantum mechanics as describing this vast roiling ocean of parallel universes, constantly branching off, merging, and cancelling each other out, then we're now going to stick a little boat in that ocean. We'll think of the boat's position as representing the "real," "actual" state of the universe at a given point in time, and the ocean as just a "field of potentialities" whose role is to buffet the boat around. For historical reasons, the boat's position is called a hidden variable...
.... Thinking about hidden variables seems scientifically fruitful: it led Einstein, Podolsky, and Rosen to the EPR experiment, Bell to Bell's Inequality, Kochen and Specker to the Kochen-Specker Theorem, and me to the collision lower bound...... Hidden-variable theories will give me a perfect vehicle for discussing other issues in quantum foundations -- like nonlocality, contextuality, and the role of time...
Now, you might've heard of a little thing called Bell's Inequality. As it turns out, Bell's Inequality doesn't quite rule out hidden-variable theories satisfying the two axioms above, but a slight strengthening of what Bell proved does the trick.
So what is Bell's Inequality?
...since I'm not a member of the Physics Popularizers' Guild, I'm now going to break that profession's time-honored bylaws, and just tell you the conceptual point directly.
We've got two players, Alice and Bob, and they're playing the following game. Alice flips a fair coin; then, based on the result, she can either raise her hand or not. Bob flips another fair coin; then, based on the result, he can either raise his hand or not. What both players want is that exactly one of them should raise their hand, if and only if both coins landed heads. If that condition is satisfied then they win the game; if it isn't then they lose. (This is a cooperative rather than competitive game.)
Now here's the catch: Alice and Bob are both in sealed rooms (possibly even on different planets), and can't communicate with each other at all while the game is in progress.
The question that interests us is: what is the maximum probability with which Alice and Bob can win the game?
Well, certainly they can win 75% of the time. Why?
Right: they can both just decide never to raise their hands, regardless of how the coins land! In that case, the only way they'll lose is if both of the coins land heads.
Exercise: Prove that this is optimal. In other words, any strategy of Alice and Bob will win at most 75% of the time.
Now for the punchline: suppose that Alice and Bob share the entangled state with Alice holding one half and Bob holding the other half. In that case, there exists a strategy by which they can win the game with probability
To be clear, having the state .... does not let Alice and Bob send messages to each other faster than the speed of light -- nothing does! What it lets them do is to win this particular game more than 75% of the time. Naïvely, we might have thought that would require Alice and Bob to "cheat" by sending each other messages, but that simply isn't true -- they can also cheat by using entanglement!
So that was Bell's Inequality.... It follows that, if we want it to agree with quantum mechanics, then any hidden-variable theory has to allow "instantaneous communication" between any two points in the universe. Once again, this doesn't mean that quantum mechanics itself allows instantaneous communication (it doesn't), or that we can exploit hidden variables to send messages faster than light (we can't). It only means that, if we choose to describe quantum mechanics using hidden variables, then our description will have to involve instantaneous communication.
... So what we've learned, from Alice and Bob's coin-flipping game, is that any attempt to describe quantum mechanics with hidden variables will necessarily lead to tension with relativity. Again, none of this has any experimental consequences, since it's perfectly possible for hidden-variable theories to violate the "spirit" of relativity while still obeying the "letter." Indeed, hidden-variable fans like to argue that all we're doing is unearthing the repressed marital tensions between relativity and quantum mechanics themselves!
Saturday, April 07, 2007
Quantum reality, with the math: Decoherence and Hidden Variables
I've been reading some popular physics books in my latest attempt to figure out how bad reality is looking these days. The answer is, pretty bad -- even worse than your average middle-aged American. So a post on a new blog I'm following caught my eye...
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