Quantum state

A mathematical description of a quantum system that encodes probabilities for measurement outcomes and the relations among them. It can be represented by a state vector or, more generally, a density operator.

Hilbert space: A complete inner-product vector space used to represent quantum states and observables. State vectors live in this mathematical space, whose inner product determines measurement probabilities.
State vector: A vector in Hilbert space that represents a pure quantum state, up to an overall phase. It is the standard compact representation when a system is in a pure state.
Pure state: A quantum state represented by a single state vector, or equivalently a rank-one density operator. Pure states are the special case with maximal information about preparation.
Pusey–Barrett–Rudolph theorem: A theorem showing that certain assumptions make distinct pure quantum states correspond to distinct underlying physical states. It constrains interpretations that treat quantum states as purely statistical descriptions.
Quantum superposition: A linear combination of possible quantum states that can produce interference. A state can combine alternatives whose amplitudes interfere before measurement.
Density matrix: An operator representing a quantum state, including both pure states and statistical mixtures. It handles uncertainty about preparation and subsystems of entangled systems.
Mixed state: A quantum state represented by a statistical mixture of pure states. Mixed states distinguish ensemble uncertainty from a single vector description.
Quantum state discrimination: The task of identifying an unknown quantum state using measurements on one or more systems. Nonorthogonal states cannot be perfectly distinguished from a single copy.
Born rule: The rule that converts quantum amplitudes into probabilities for measurement outcomes. It extracts outcome probabilities from a state and the measurement basis.
Schrödinger equation: A differential equation governing the time evolution of a quantum state. For an isolated system, it specifies how the state changes continuously over time.

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