Triorthogonal uniqueness theorem and its relevance to the interpretation of quantum mechanics

Many-world, decoherence, and modal interpretations of quantum mechanics suffer from a ‘‘basis degeneracy problem’’ arising from the nonuniqueness of some biorthogonal decompositions. We prove that when a quantum state can be written in the triorthogonal form Ψ=${\mathit{tsum}}_{\mathit{i}}$ ${\mathit{c}}_{\mathit{i}}$‖${\mathit{A}}_{\mathit{i}}$〉⊗‖${\mathit{B}}_{\mathit{i}}$〉⊗‖${\mathit{C}}_{\mathit{i}}$〉, then, even if some of the ${\mathit{c}}_{\mathit{i}}$’s are equal, no alternative bases exist such that Ψ can be rewritten ${\mathit{tsum}}_{\mathit{i}}$ ${\mathit{d}}_{\mathit{i}}$‖${\mathit{A}}_{\mathit{i}}$ ’〉⊗‖ ${\mathit{B}}_{\mathit{i}}$ ’〉⊗‖${\mathit{C}}_{\mathit{i}}$ ’〉. Therefore the triorthogonal decomposition picks out a ‘‘special’’ basis. We can use this preferred basis to address the basis degeneracy problem.