 Eigenstate

An eigenstate of an observable is a quantum state with a sharp value of that observable. In the discrete pure-state case, it is represented by a normalized eigenvector satisfying $A|\psi ⟩=a|\psi ⟩$; the Born rule assigns probability one to outcome $a$.

Having a definite value of one observable does not make all quantities definite. A Pauli Z gate eigenstate gives equal probabilities for the two Pauli X gate outcomes. If the eigenvalue is degenerate, different vectors in the same eigenspace have the same sharp outcome, and mixtures supported entirely in that eigenspace also have it. Idealized sharp continuous-spectrum states, such as a whole-line momentum eigenstate, require generalized rather than normalizable eigenvectors.