To echo Scott, a density matrix is a probability distribution over quantum states—over a set of basis states, specifically. But if you pick a new basis, the same density matrix resolves into a different probability distribution over a different set of quantum states. If you believe that reality does reduce to amplitude flows in configuration space, then that means that one basis, the position basis, is the real one (since it corresponds to different possible states of reality, i.e. distributions of amplitude in configuration space); you can think of a density matrix as a probability distribution, period, and you’re done. Density matrices will mean trouble if and only if you want to think of various incompatible choices of basis as equally real.
I would suggest that understanding superdense coding is another test of whether one’s explanation of the significance of entanglement works. There is no purely quantum communication at a distance, but there can be a quantum rider on an otherwise classical communication channel.
Not exactly. By writing down a density matrix you specify a special basis (via the eigenvectors) and cannot change this basis and still have some meaning. Eliezer, back in the beginning of the series you wrote about classical phase space and that it is an optional addition to classical mechanics, while QM inherently takes place in configuration space. Well, the density matrix is sort of the same addition to QM. Whether we have (in some basis) a simple state or a linear combination thereof is a physical fact that does not say anything about our beliefs. But if we want to express that a system could be in one of several states with different probabilities (or, equivalently, a lot of equally prepared systems are in one state each and the percentage of systems in a particular state is given), you use a density matrix. Because a hermitian matrix encodes exactly two informations: it picks an orthogonal basis (the eigenvectors) and stores a number for each of those basis vectors (the eigenvalue). In the case of the density matrix these are the possible states with their probabilities.
By the Way, that’s also why we use hermitian operators/matrices to represent observables: they specify the states the measurement is designed to distinguish together with the results of the measurement in each case.
To echo Scott, a density matrix is a probability distribution over quantum states—over a set of basis states, specifically. But if you pick a new basis, the same density matrix resolves into a different probability distribution over a different set of quantum states. If you believe that reality does reduce to amplitude flows in configuration space, then that means that one basis, the position basis, is the real one (since it corresponds to different possible states of reality, i.e. distributions of amplitude in configuration space); you can think of a density matrix as a probability distribution, period, and you’re done. Density matrices will mean trouble if and only if you want to think of various incompatible choices of basis as equally real.
I would suggest that understanding superdense coding is another test of whether one’s explanation of the significance of entanglement works. There is no purely quantum communication at a distance, but there can be a quantum rider on an otherwise classical communication channel.
Not exactly. By writing down a density matrix you specify a special basis (via the eigenvectors) and cannot change this basis and still have some meaning. Eliezer, back in the beginning of the series you wrote about classical phase space and that it is an optional addition to classical mechanics, while QM inherently takes place in configuration space. Well, the density matrix is sort of the same addition to QM. Whether we have (in some basis) a simple state or a linear combination thereof is a physical fact that does not say anything about our beliefs. But if we want to express that a system could be in one of several states with different probabilities (or, equivalently, a lot of equally prepared systems are in one state each and the percentage of systems in a particular state is given), you use a density matrix. Because a hermitian matrix encodes exactly two informations: it picks an orthogonal basis (the eigenvectors) and stores a number for each of those basis vectors (the eigenvalue). In the case of the density matrix these are the possible states with their probabilities. By the Way, that’s also why we use hermitian operators/matrices to represent observables: they specify the states the measurement is designed to distinguish together with the results of the measurement in each case.