“The idea that density matrices summarize locally invariant entanglement information is certainly helpful, but I still don’t know how to start with a density matrix and visualize some physical situation, nor can I take your proof and extract back out an argument that would complete the demonstration in this blog post. I confess this is strictly a defect of my own education, but...”
From what I understand (which is admittedly not much; I could well be wrong), a density matrix is the thingy that describes the probability distribution of the quantum system over all possible states. Suppose that you have a set of quantum states A1...An. The density matrix is a way of describing a system that, say, has a 75% chance of being in A1 and a 25% chance of being in A2, or a 33% chance of being in A1 or A2 or A4, or whatever. You can then plug the density matrix into the standard quantum equations, but everything you get back will have one extra dimension, to account for the fact that the system you are discussing is described by a distribution rather than a pure quantum state.
The gist of Scott Aaronson’s proof is (again, if I understand correctly): Suppose that you have two quantum systems, A and B. List the Cartesian product over all possible states of A and B (A1B1, A2B1, A3B1, etc., etc.). Use a density matrix to describe a probability distribution over these states (10% chance of A1B1, 5% chance of A1B2, whatever). Suppose that you are physically located at system A, and you fiddle with the density matrix using some operator Q. Using some mathematical property of Q which I don’t really understand, you can show that, after Q has been applied, another person’s observations at B will be the same as their earlier observations at B (ie, the density matrix after Q acts the same as it did before Q, so long as you only consider B).
“The idea that density matrices summarize locally invariant entanglement information is certainly helpful, but I still don’t know how to start with a density matrix and visualize some physical situation, nor can I take your proof and extract back out an argument that would complete the demonstration in this blog post. I confess this is strictly a defect of my own education, but...”
From what I understand (which is admittedly not much; I could well be wrong), a density matrix is the thingy that describes the probability distribution of the quantum system over all possible states. Suppose that you have a set of quantum states A1...An. The density matrix is a way of describing a system that, say, has a 75% chance of being in A1 and a 25% chance of being in A2, or a 33% chance of being in A1 or A2 or A4, or whatever. You can then plug the density matrix into the standard quantum equations, but everything you get back will have one extra dimension, to account for the fact that the system you are discussing is described by a distribution rather than a pure quantum state.
The gist of Scott Aaronson’s proof is (again, if I understand correctly): Suppose that you have two quantum systems, A and B. List the Cartesian product over all possible states of A and B (A1B1, A2B1, A3B1, etc., etc.). Use a density matrix to describe a probability distribution over these states (10% chance of A1B1, 5% chance of A1B2, whatever). Suppose that you are physically located at system A, and you fiddle with the density matrix using some operator Q. Using some mathematical property of Q which I don’t really understand, you can show that, after Q has been applied, another person’s observations at B will be the same as their earlier observations at B (ie, the density matrix after Q acts the same as it did before Q, so long as you only consider B).