In the mathematical formulation of quantum mechanics, physical quantities that classical mechanics had treated as real-valued variables become self-adjoint operators on a Hilbert space. These operators, called " observables ", can fail to commute, in which case they are called "incompatible": [ A ^ ^ , B ^ ^ ] := A ^ ^ B ^ ^ − − B ^ ^ A ^ ^ ≠ ≠ 0 ^ ^ . Incompatible observables cannot have a complete set of common eigenstates; there can be some simultaneous eigenstates of A ^ ^ and B ^ ^ , but not enough in number to constitute a complete basis. The canonical commutation relation [ x ^ ^ , p ^ ^ ] = i ℏ ℏ implies that this applies to position and momentum. In a Bohrian view, this is a mathematical statement that position and momentum are complementary aspects. Likewise, an analogous relationship holds for any two of the spin observables defined by the Pauli matrices ; measurements of spin along perpendicular axes are complementary. The Pauli spin observables are defined for a quantum system described by a two-dimensional Hilbert space; mutually unbiased bases generalize these observables to Hilbert spaces of arbitrary finite dimension. Two bases { | a j ⟩ ⟩ } and { | b k ⟩ ⟩ } for an N -dimensional Hilbert space are mutually unbiased when | ⟨ ⟨ a j | b k ⟩ ⟩ | 2 = 1 N for all j , k = 1 , . . . N − − 1.