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.

• Received 15 July 1993

DOI:https://doi.org/10.1103/PhysRevA.49.4213