Gleason's theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 31/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Gleason%27s_theorem


00:02:33 1 Overview
00:13:21 2 Implications
00:17:08 3 Outline of Gleason's proof
00:18:16 4 Generalizations



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SUMMARY
=======
Gleason's theorem (named after Andrew M. Gleason) is a mathematical result which shows that the rule one uses to calculate probabilities in quantum physics follows logically from particular assumptions about how measurements are represented mathematically.  More specifically, 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.  This result is of particular importance for the field of quantum logic.  Furthermore, it was historically significant for the role it played in showing that local hidden variable theories are inconsistent with quantum physics.  The theorem states:

Theorem. Suppose H is a separable Hilbert space. A measure on H is a function f that assigns a nonnegative real number to each closed subspace of H in such a way that, if 
  
    
      
        {
        
          A
          
            i
          
        
        }
      
    
    {\textstyle \{A_{i}\}}
   is a countable collection of mutually orthogonal subspaces of H, and the closed linear span of this collection is B, then 
  
    
      
        f
        (
        B
        )
        =
        
          ∑
          
            i
          
        
        f
        (
        
          A
          
            i
          
        
        )
      
    
    {\textstyle f(B)=\sum _{i}f(A_{i})}
  . If the Hilbert space H has dimension at least three, then every measure f can be written in the form 
  
    
      
        f
        (
        A
        )
        =
        
          T
          r
        
        (
        W
        
          P
          
            A
          
        
        )
      
    
    {\textstyle f(A)=\mathrm {Tr} (WP_{A})}
  , where W is a positive semidefinite trace class operator and 
  
    
      
        
          P
          
            A
          
        
      
    
    {\textstyle P_{A}}
   is the orthogonal projection onto A.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 measurement outcomes is generated by some quantum state.