Psy-Kosh: TrA just means the operation that “traces out” (i.e., discards) the A subsystem, leaving only the B subsystem. So for example, if you applied TrA to the state |0〉|1〉, you would get |1〉. If you applied it to |0〉|0〉+|1〉|1〉, you would get a classical probability distribution that’s half |0〉 and half |1〉. Mathematically, it means starting with a density matrix for the joint quantum state ρAB, and then producing a new density matrix ρB for B only by summing over the A-indices (sort of like tensor contraction in GR, if that helps).
Eliezer: The best way I can think of to explain a density matrix is, it’s what you’d inevitably come up with if you tried to encode all information locally available to you about a quantum state (i.e., all information needed to calculate the probabilities of local measurement outcomes) in a succinct way. (In fact it’s the most succinct possible way.)
You can see it as the quantum generalization of a probability distribution, where the diagonal entries represent the probabilities of various measurement outcomes if you measure in the “standard basis” (i.e., whatever basis the matrix happens to be presented in). If you measure in a different orthogonal basis, identified with some unitary matrix U, then you have to “rotate” the density matrix ρ to UρU before measuring it (where U is U’s conjugate transpose). In that case, the “off-diagonal entries” of ρ (which intuitively encode different pairs of basis states’ “potential for interfering with each other”) become relevant.
If you understand (1) why density matrices give you back the usual Born rule when ρ=|ψ〉〈ψ| is a pure state, and (2) why an equal mixture of |0〉 and |1〉 leads to exactly the same density matrix as an equal mixture of |0〉+|1〉 and |0〉-|1〉, then you’re a large part of the way to understanding density matrices.
One could argue that density matrices must reflect part of the “fundamental nature of QM,” since they’re too indispensable not to. Alas, as long as you insist on sharply distinguishing between the “really real” from the “merely mathematical,” density matrices might always cause trouble, since (as we were discussing a while ago) a density matrix is a strange sort of hybrid of amplitude vector with probability distribution, and the way you pick apart the amplitude vector part from the probability distribution part is badly non-unique. Think of someone who says: “I understand what a complex number does—how to add and multiply one, etc. -- but what does it mean?” It means what it does, and so too with density matrices.
Think of someone who says: “I understand what a complex number does—how to add and multiply one, etc. -- but what does it mean?” It means what it does, and so too with density matrices.
But a complex number does mean something intuitive: it represents a rotation, or something like a rotation in whatever system is at hand. Indeed once you understand this it becomes so much easier to work with complex numbers… I too would like an intuition for density matrices that matches.
Psy-Kosh: TrA just means the operation that “traces out” (i.e., discards) the A subsystem, leaving only the B subsystem. So for example, if you applied TrA to the state |0〉|1〉, you would get |1〉. If you applied it to |0〉|0〉+|1〉|1〉, you would get a classical probability distribution that’s half |0〉 and half |1〉. Mathematically, it means starting with a density matrix for the joint quantum state ρAB, and then producing a new density matrix ρB for B only by summing over the A-indices (sort of like tensor contraction in GR, if that helps).
Eliezer: The best way I can think of to explain a density matrix is, it’s what you’d inevitably come up with if you tried to encode all information locally available to you about a quantum state (i.e., all information needed to calculate the probabilities of local measurement outcomes) in a succinct way. (In fact it’s the most succinct possible way.)
You can see it as the quantum generalization of a probability distribution, where the diagonal entries represent the probabilities of various measurement outcomes if you measure in the “standard basis” (i.e., whatever basis the matrix happens to be presented in). If you measure in a different orthogonal basis, identified with some unitary matrix U, then you have to “rotate” the density matrix ρ to UρU before measuring it (where U is U’s conjugate transpose). In that case, the “off-diagonal entries” of ρ (which intuitively encode different pairs of basis states’ “potential for interfering with each other”) become relevant.
If you understand (1) why density matrices give you back the usual Born rule when ρ=|ψ〉〈ψ| is a pure state, and (2) why an equal mixture of |0〉 and |1〉 leads to exactly the same density matrix as an equal mixture of |0〉+|1〉 and |0〉-|1〉, then you’re a large part of the way to understanding density matrices.
One could argue that density matrices must reflect part of the “fundamental nature of QM,” since they’re too indispensable not to. Alas, as long as you insist on sharply distinguishing between the “really real” from the “merely mathematical,” density matrices might always cause trouble, since (as we were discussing a while ago) a density matrix is a strange sort of hybrid of amplitude vector with probability distribution, and the way you pick apart the amplitude vector part from the probability distribution part is badly non-unique. Think of someone who says: “I understand what a complex number does—how to add and multiply one, etc. -- but what does it mean?” It means what it does, and so too with density matrices.
But a complex number does mean something intuitive: it represents a rotation, or something like a rotation in whatever system is at hand. Indeed once you understand this it becomes so much easier to work with complex numbers… I too would like an intuition for density matrices that matches.