What is the physical interpretation of the off-diagonal elements in a density matrix?

The density matrix, often denoted by ρ, is a mathematical representation of the quantum state of a system. Unlike a wave function, which describes a pure quantum state, the density matrix can also describe mixed states, which are statistical ensembles of pure states. In the density matrix, the diagonal elements, ρᵢᵢ, represent the probabilities of finding the system in the corresponding basis state |i⟩. The off-diagonal elements, ρᵢⱼ (where i ≠ j), represent the coherence between the basis states |i⟩ and |j⟩. Coherence refers to the existence of quantum superposition and interference effects between different states. Specifically, the off-diagonal elements quantify the degree to which the system exists in a superposition of the states |i⟩ and |j⟩. A non-zero off-diagonal element indicates that there is a well-defined phase relationship between the states |i⟩ and |j⟩, which allows for quantum interference. The magnitude of the off-diagonal element is related to the strength of the coherence, and the phase of the off-diagonal element determines the relative phase between the states. If all the off-diagonal elements are zero, the system is in a mixed state with no coherence, meaning that it is simply a statistical mixture of the basis states without any quantum superposition or interference. The decay of the off-diagonal elements over time corresponds to decoherence, the loss of quantum coherence due to interactions with the environment. Therefore, the off-diagonal elements of the density matrix provide crucial information about the quantum coherence and superposition properties of the system, indicating the presence and strength of quantum interference effects.