gentleunwashed
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Actually, I have a little more to say:
Another way to think about higher-rank density matrices is as probability distributions over pure states; I think this is what Charlie Steiner's comment is alluding to.
So, the rank-2 matrix from my previous comment, can be thought of as
, i.e., an equal probability of observing each of . And, because for any orthonormal vectors , again there's nothing special about using the standard basis here (this is mathematically equivalent to the argument I made in the above comment about why you can use any basis for your measurement).
I always hated this point of view; it felt really hacky, and I always found it ugly and unmotivated to go from... (read more)
The usual story about where rank > 1 density matrices come from is when your subsystem is entangled with an environment that you can't observe.
The simplest example is to take a Bell state, say
|00> + |11> (obviously I'm ignoring normalization) and imagine you only have access to the first qubit; how should you represent this state? Precisely because it's entangled, we know that there is no |Psi> in 1-qubit space that will work. The trace method alluded to in the post is to form the (rank-1) density matrix of the Bell state, and then "trace out" the second system; if you think of the density matrix as living in M_2 tensor M_2,... (read 417 more words →)
I liked this post a lot!
I can't be the only person who read this and thought of the three major varieties of moral theory: Decision Theory roughly corresponding to naive Act Utilitarianism; Policy Theory roughly matching Rule Utilitarianism or Deontology; and Agent Theory being sort of Virtue Ethics-y.
One thing that would be interesting, is to think about what features of the world would lead us to prefer different levels of analysis: my impression is, the move from CDT to EDT/FDT/UDT etc. is driven by Newcomb-like problems and game theory-ish situations, where your decision process can affect the outcomes you will face; usually this seems to arise in situations where there are agents... (read more)
Every year, a handful of small children, and visiting tourists, and people chasing stray pets, and people who get lost in the dark, accidentally wander up into the monster's cave and into its mouth.
How big does this number have to be before it's worth whipping up the village to kill the monster by all walking in together?
What if the monster has only recently settled in the mountain, so no wayward children have been eaten yet. The town holds a vote: either we can commit to drilling it into our children, and our stumbling drunks, and our visiting tourists to never, ever, ever, ever go up the mountain, or we can go up as one and get rid of the monster now. How do you think most people will vote?
"It seems like everyone will pick red pill"
-- but in the actual poll, they didn't! So, something has gone wrong in the reasoning here, even if there is some normative sense in which it should work.
My understanding is that 70% of Twitter respondents chose "blue", and I'd expect the Twitter poll was both seen by, and responded to, at higher rates by people with an interest in game theory and related topics, i.e. the people more likely to understand the principles necessary to arrive at "red" as an answer.
Obviously a Twitter poll isn't the real life situation, but a) it is far from clear that "blue"s are committing suicide and b) if you find yourself arguing that a supermajority of humanity is below your intellectual threshold of concern, I think that's a good sign in general to reflect on how much you really mean that.
You can switch back and forth between the two views, obviously, and sometimes you do, but I think the most natural reason is because the operators you get are trace 1 positive semidefinite matrices, and there's a lot of theory on PSD matrices waiting for you. Also, the natural maps on density matrices, the quantum channels or trace preserving completely positive maps have a pretty nice representation in terms of conjugation when you think of density matrices as matrices: \rho \mapsto \sum_i K_i \rho K_i^* for some operators K_i that satisfy \sum_i K_i^*K_i = I
Obviously all of these translates to the (0,2) tensor view, but a lot of theory was already built for thinking of these as linear maps on matrix spaces (or c* algebras or whatever fancier generalizations mathematicians had already been looking at)