Matches in DBpedia 2016-04 for { ?s ?p "Gleason's theorem (named after Andrew M. Gleason) is a mathematical result which is of particular importance for the field of quantum logic. It proves that the Born rule for the probability of obtaining specific results for a given measurement follows naturally from the structure formed by the lattice of events in a real or complex Hilbert space. The theorem states:Theorem. Suppose H is a separable Hilbert space of complex dimension at least 3. Then for any quantum probability measure on the lattice Q of self-adjoint projection operators on H there exists a unique trace class operator W such that P(E) = Tr(W E) for any self-adjoint projection E in Q.The lattice of projections Q can be interpreted as the set of quantum propositions, each proposition having the form \"a ≤ A ≤ b\", where A is the measured value of some observable on H (given by a self-adjoint linear operator). The trace-class operator W can be interpreted as the density matrix of a quantum state. Effectively, the theorem says that any legitimate probability measure on the space of allowable propositions is generated by some quantum state. This implies that the Standard Quantum Logic can be viewed as a manifold of interlocking perspectives that cannot be embedded into a single perspective [Edwards]. Hence, the perspectives cannot be viewed as perspectives on one real world. So, even considering one world as a methodological principle breaks down in the quantum micro-domain."@en }
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- Gleasons_theorem abstract "Gleason's theorem (named after Andrew M. Gleason) is a mathematical result which is of particular importance for the field of quantum logic. It proves that the Born rule for the probability of obtaining specific results for a given measurement follows naturally from the structure formed by the lattice of events in a real or complex Hilbert space. The theorem states:Theorem. Suppose H is a separable Hilbert space of complex dimension at least 3. Then for any quantum probability measure on the lattice Q of self-adjoint projection operators on H there exists a unique trace class operator W such that P(E) = Tr(W E) for any self-adjoint projection E in Q.The lattice of projections Q can be interpreted as the set of quantum propositions, each proposition having the form \"a ≤ A ≤ b\", where A is the measured value of some observable on H (given by a self-adjoint linear operator). The trace-class operator W can be interpreted as the density matrix of a quantum state. Effectively, the theorem says that any legitimate probability measure on the space of allowable propositions is generated by some quantum state. This implies that the Standard Quantum Logic can be viewed as a manifold of interlocking perspectives that cannot be embedded into a single perspective [Edwards]. Hence, the perspectives cannot be viewed as perspectives on one real world. So, even considering one world as a methodological principle breaks down in the quantum micro-domain.".