• Corpus ID: 90261209

Explicit construction of the density matrix in Gleason's theorem

@article{Rajan2019ExplicitCO,
  title={Explicit construction of the density matrix in Gleason's theorem},
  author={Del Rajan and Matt Visser},
  journal={arXiv: Quantum Physics},
  year={2019}
}
Gleason's theorem is a fundamental 60 year old result in the foundations of quantum mechanix, setting up and laying out the surprisingly minimal assumptions required to deduce the existence of quantum density matrices and the Born rule. Now Gleason's theorem and its proof have been continuously analyzed, simplified, and revised over the last 60 years, and we will have very little to say about the theorem and proof themselves. Instead, we find it useful, (and hopefully interesting), to make some… 
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References

SHOWING 1-10 OF 18 REFERENCES
On Gleason’s Theorem without Gleason
The original proof of Gleason’s Theorem is very complicated and therefore, any result that can be derived also without the use of Gleason’s Theorem is welcome both in mathematics and mathematical
Quantum states and generalized observables: a simple proof of Gleason's theorem.
  • P. Busch
  • Physics
    Physical review letters
  • 2003
TLDR
A simple proof of the result, analogous to Gleason's theorem, that any quantum state is given by a density operator, and a von Neumann-type argument against noncontextual hidden variables is obtained.
Gleason's Theorem Has a Constructive Proof
TLDR
The issues raised by discussions in this journal regarding the possibility of a constructive proof of Gleason"s theorem in light of the recent publication of such a proof are examined.
Infinite and finite Gleason’s theorems and the logic of indeterminacy
In the first half of the paper I prove Gleason’s lemma: Every non-negative frame function on the set of rays in R3 is continuous. This is the central and most difficult part of Gleason’s theorem. The
Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements
We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal
A Gleason-type theorem for qubits based on mixtures of projective measurements
We derive Born's rule and the density-operator formalism for quantum systems with Hilbert spaces of dimension two or larger. Our extension of Gleason's theorem only relies upon the consistent
Gleason's theorem is not constructively provable
TLDR
There exists a positive self-adjoint operator W of trace class such that, for every closed subspace A, 12(A) = Tr(WPA), where PA is the projection operator of ~ onto A.
An elementary proof of Gleason's theorem
Gleason's theorem characterizes the totally additive measures on the closed sub-spaces of a separable real or complex Hilbert space of dimension greater than two. This paper presents an elementary
A Constructive Formulation of Gleason's Theorem
In this paper I wish to show that we can give a statement of a restricted form of Gleason's Theorem that is classically equivalent to the standard formulation, but that avoids the counterexample that
A Constructive Proof of Gleason's Theorem
Abstract Gleason's theorem states that any totally additive measure on the closed subspaces, or projections, of a Hilbert space of dimension greater than two is given by a positive operator of trace
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