Abstract Bub and Clifton (1996) proved a uniqueness theorem for ‘no col lapse’ interpretations of quantum mechanics. The proof was repeated, with minor modifications, in Bub (1997). The original proof involved a ‘weak separability’ assumption (introduced to avoid a dimensionality constraint) that required several preliminary definitions and considerably complicated the formulation of the theorem. Sheldon Goldstein has pointed out that the proof goes through without this assumption. The question at issue is this: Consider an arbitrary ‘pure’ quantum state represented by a ray e on a Hilbert space 1t and the Boolean algebra or lattice, B(R), generated by the eigenspaces of a single observable R on Ji. The probabilities defined bye for the ranges of values of R can be represented by a probability measure over the 2valued homomorphisms on B(R). What is the maximal
Presenting a realistic interpretation of quantum mechanics and, in particular, a realistic view of quantum waves, this book defends, with one exception, Schrodinger's views on quantum mechanics. Johansson goes on to defend the view that the collapse of a wave function during a measurement is a real physical collapse of a wave and argues that the collapse is a consequence of quantisation of interaction. Lastly Johansson argues for a revised principle of individuation in the quantum domain and that this principle enables a sort of explanation of non-local phenomena.
We investigate structural properties of quantum proof systems by establishing collapse results that uncover simplifications in their complexity landscape. We extend classical results such as the Karp-Lipton theorem to quantum polynomial hierarchy with quantum proofs and establish uniqueness preservation for quantum-classical probabilistically checkable proof systems. Our main contributions are threefold. First, we prove that restricting quantum-classical PCP systems to uniqueness does not reduce computational power:
UniqueQCPCP=QCPCP under
BQ-operator and randomized reductions, demonstrating robustness similar to the
UniqueQCMA=QCMA result. Second, we establish a non-uniform quantum analogue of the Karp-Lipton theorem, showing that if
QMA⊆BQP/qpoly, then
QPH⊆QΣ2/qpoly, extending the classical collapse theorem to quantum complexity with quantum advice. Third, we introduce a consistent variant of the quantum polynomial hierarchy (
CQPH) with consistency constraints across interaction rounds while maintaining product-state proofs, proving its unconditional collapse
CQPH=CQΣ2. This contrasts with prior work on quantum-entangled polynomial hierarchy, showing that consistency rather than entanglement drives the collapse. These results contribute to understanding structural boundaries in quantum complexity theory and the interplay between constraint types in quantum proof systems.