States, bases, and Hilbert space

Write a two-level state as $|\psi\rangle=c_0|0\rangle+c_1|1\rangle$, where the reference states form an orthonormal basis. A ket is a state vector; its bra is the conjugate transpose. The inner product conjugates the first vector, which makes $\langle\psi|\psi\rangle$ a nonnegative real number. Normalization sets that norm squared to one. A nonzero vector can be normalized by dividing by its norm.

A Hilbert space is a complete complex inner-product space: every Cauchy sequence, whose sufficiently late terms are all arbitrarily close in norm, has a limit inside the space. Finite-dimensional examples automatically have this property. For a particle on a line, square-integrable wavefunctions provide an infinite-dimensional example. Ideal exact position states and plane waves require generalized, delta-normalized notation; they are not ordinary normalized vectors in that space. Physical wavepackets avoid this difficulty by having finite norm.

An amplitude for finding the normalized state $|a\rangle$ is the overlap $\langle a|\psi\rangle$. A new orthonormal basis reorganizes the same state instead of physically changing it. Coordinate columns can look very different after this change, while every consistently computed probability stays the same. Unitary basis transformations preserve inner products, which is why they preserve normalization and distinguishability.

For example, $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ and $|-\rangle=(|0\rangle-|1\rangle)/\sqrt{2}$ define another orthonormal basis. A state that is definite in the plus/minus basis is generally a superposition in the zero/one basis. Consequently, saying only “the system is in a superposition” is incomplete: ask which alternatives are being compared. A superposition need not mean spatial separation.

A coherent pure state $|+\rangle$ and a preparation that randomly supplies $|0\rangle$ or $|1\rangle$ equally often have the same zero/one probabilities. They differ in a plus/minus measurement: the pure state always yields plus, whereas the mixture yields each result half the time. A density operator represents this distinction: the mixture averages projectors, not state vectors. Averaging vectors would invent coherence that the preparation did not supply.

Two systems use a tensor-product space. If each has two independent basis states, the joint space has four basis states, not two. Some joint vectors cannot be written as a product of one vector for each subsystem; these are entangled pure states. This mathematical fact does not let a local observer send messages instantaneously. It says that a complete description may belong to the pair rather than to two independently assigned pure states.