In Section 5.6, we compare pure and mixed states and discuss the ensemble interpretation of mixed states. The density operator generalizes the orthogonal projection operator onto the subspace spanned by a “pure state” ψ. We describe this new situation by a density operator ρ (Section 5.5). If the bipartite system is in an entangled state, then any subsystem is in a statistical mixture of states. In general, the state of a subsystem cannot be described by a state vector in the Hilbert space of the subsystem. This information describes correlations between the subsystems. An entangled state of a composite system encodes information about the system as a whole that cannot be measured locally (that is, by measurements on the subsystems alone). The theory presented here has applications not only to atomic physics but also to quantum information theory. Entanglement cannot be created by local measurements or manipulations of the subsystems, but usually, an interaction between the subsystems immediately leads to entanglement (Sections 5.3 and 5.4). In general, these states are entangled, that is, they cannot be written as simple products. The Hilbert space of a composite system contains not only product states, but also their linear combinations (Section 5.2). An abstract formulation of this method is given by the tensor product of Hilbert spaces. ![]() ![]() This shows us how to construct two-particle states as products of one-particle states (Section 5.1). As a first example of a two-particle system we consider the Schrödinger equation for two free particles. Real physics starts where at least two particles are involved. Up to now, we have only considered single-particle systems.
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