 Biorthogonal Decomposition Theorem:  Given a vector \(\ket{\psi}\) in a tensor-product Hilbert space \(\mathcal{H}^1 \otimes \mathcal{H}^2\), there exist bases \(\{\ket{a_i}\}\) and \(\{\ket{p_i}\}\) for \(\mathcal{H}^1\) and \(\mathcal{H}^2\) respectively, such that \(\ket{\psi}\) can be written as a linear combination of terms of the form \(\ket{a_i} \otimes \ket{p_i}\). If the absolute values (modulus) of the coefficients in this linear combination are all unequal, then the bases are unique (see, for example, Schrödinger 1935 for a proof). 