Gesamtzustände

Gesamtzustände (total states) is a term used in quantum mechanics to describe the complete state of a system, including all subsystems. For a composite system composed of subsystems A and B, the total state resides in the tensor product of their Hilbert spaces, H = Ha ⊗ Hb. If the total state can be written as a product |Ψ⟩ = |ψA⟩ ⊗ |ψB⟩, the subsystems are uncorrelated; in general, the total state may be entangled and cannot be factorized into subsystem states.

A total state can be described by a state vector (a pure state) |Ψ⟩ or, more generally, by a density operator ρ acting on the global Hilbert space, which also covers statistical mixtures of pure states. The evolution of the total state follows the Schrödinger equation for pure states or the quantum Liouville equation for ρ. Observables related to the whole system are evaluated by expectation values ⟨O⟩ = Tr(ρO). Subsystems are described by reduced states obtained via the partial trace: ρA = TrB(ρ). Entanglement of the total state leads to correlations that cannot be explained by classical means and has observable consequences in measurements on the subsystems.

Examples include two spin-1/2 particles, where the singlet state |Ψ−⟩ = (|01⟩ − |10⟩)/√2 is a maximally entangled total state, in contrast to a simple product state |↑⟩ ⊗ |↓⟩. The concept is central in quantum information, quantum computing, and many-body physics, where the properties of the Gesamtzustand determine dynamics, correlations, and information processing.