------------------------------------------------------------ S1c. What is the meaning of the entries of a density matrix? ------------------------------------------------------------ Density matrices are a convenient way of describing states of quantum systems in contact with an environment. (State vectors = wave functions are appropriate only for isolated systems at zero absolute temperature, though they can be used in an approximate way in thermally isolated contexts. But contact with an environment means positive temperature.) If the quantum system has only a finite number n of levels, the density matrix is an n x n matrix; otherwise it is a linear operator on Hilbert space (but nevertheless called a matrix). The real use for density matrices is to compute expectations = trace (rho f) for quantities f of interest. Indeed, rho is just a collection of numbers enabling one to calculate these expectations. The fact that the constant 1 must have expectation 1 leads to the restriction that sum_k rho_kk = trace rho = 1. Apart from that, rho must be a Hermitian, positive semidefinite matrix, to satisfy the requirements of statistics. (See quant-ph/0303047 for details.) For small systems, all such density matrices can indeed be approximately realized in practice. Since diagonal entries of a semidefiniteness are always nonnegative, the p_k:=rho_kk are nonnegative numbers summing to 1 and thus look like probabilities. What the components mean depends on the basis used. In particluar, if the basis consists of eigenstates of a Hamiltonian, and the eigenvalues E_k are all nondegenerate, a diagonal element rho_kk can be interpreted as the probability that upon measuring the energy of the system one will find the value E_k. If f is a function of the Hamiltonian H, and the basis used consists of eigenstates |k> of H, with H|k>=E_k|k> then the density matrix rho has entries rho_jk = . If one now calculates the expectation of a function f(H), the equation f(H)|k>=f(E_k)|k> implies that = trace (rho f(H)) = sum_k = sum_k = sum_k f(E_k) = sum_k rho_kk f(E_k). If we average the results f(E) of a number of measurements of the energy, where the energy E_k is measured with probability p_k, we get = sum_k p_k f(E_k). Thus, to match the expectations no matter which function we are averaging, we need to take p_k=rho_kk. This gives the claimed probability interpretation of the diagonal entries. Off-diagonal elements have no simple interpretation. Usually one does not look at off-diagonal elements at all, but they are important in intermediate steps of calculations. Close to absolute zero temperature, and assuming the absence of degeneracy, (but also in certain other, well prepared nearly isolated systems), quantum state have the property that all columns of the density matrix are nearly parallel to a wave function psi that is conventionally normalized to have norm 1, psi^*psi=1. (In Dirac language, this says =1; see the FAQ entry for bras and kets.). This vector psi, which is clearly determined only up to a complex number of absolute value 1, is called the wave vector (or, in infinite dimensions, the wave function) of the state. Idealizing this situation, one describes such quantum systems by states in which all columns of the density matrix are exactly parallel to some nonzero wave vector psi. (Such matrices are called rank 1 matrices; the wave vector, also referred to as a wave function, is defined only up to a phase factor.) Then the k-th column is a multiple c_k psi of psi. The fact that rho is Hermitian forces each row to be a multiple of psi^*. But this implies that c_k is a multiple of phi^*_k, so that rho is a multiple of psi psi^*. Since psi is normalized, the multiplication factor is just the trace, and since the trace is 1 we find rho = psi psi^* for any rank 1 density matrix. If we now calculate the probability of measuring the energy E_k, we find p_k = rho_kk = = = , and since is just the complex conjugate of , we end up with p_k = ||^2. This is Born's squared amplitude formula for calculating probabilities. Thus one sees that the traditional wave vector calculus is just a special case of the density matrix calculus, appropriate (only) for the study of tiny, well-prepared nearly isolated systems and for systems close to zero absolute temperature. For the study of ordinary matter under ordinary conditions, one needs to represent states by density matrices. Everything that is done with wave vectors can also be done with density matrices, or equivalently with the associated expectation mapping. Indeed, everything becomes simpler that way, much closer to classical mechanics, and much less weird-looking. See quant-ph/0303047 for an exposition of the foundations of quantum mechanics (including the probability interpretation, uncertainty relations, nonlocality, and Bell's theorem) in terms of expectations.