# 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…

## 2 Citations

Kochen-Specker theorem revisited

- Physics
- 2017

The Kochen-Specker theorem is a basic and fundamental 50 year old non-existence result affecting the foundations of quantum mechanix, strongly implying the lack of any meaningful notion of "quantum…

Quantum Entanglement in Time

- Physics, Computer Science
- 2020

This thesis describes how the entanglement in time provides the quantum advantage over a classical blockchain, namely a quantum blockchain, and explores this temporal effect within the study of quantum information and its foundations as well as through relativistic quantum information.

## References

SHOWING 1-10 OF 18 REFERENCES

On Gleason’s Theorem without Gleason

- Mathematics
- 2009

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.

- PhysicsPhysical review letters
- 2003

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

- MathematicsJ. Philos. Log.
- 2000

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

- Mathematics
- 1998

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

- Mathematics
- 2004

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

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2019

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

- MathematicsJ. Philos. Log.
- 1993

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

- Mathematics
- 1985

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

- MathematicsJ. Philos. Log.
- 1997

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

- Mathematics
- 1999

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…