Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeAug 16th 2010
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexThis is mathematically much simpler than the classical Gleason's Theorem, but I added it to [[Gleason's theorem]] anyway.

   This is mathematically much simpler than the classical Gleason’s Theorem, but I added it to Gleason’s theorem anyway.

2. • CommentRowNumber2.
   • CommentAuthorIan_Durham
   • CommentTimeAug 16th 2010
   • PermaLink
   Author: Ian_Durham
   Format: MarkdownItexI'm half-thinking outloud here with this question but, if the POVM version of Gleason's theorem is weaker than the classical theorem, does that imply that POVMs are inherently weaker than Hermitian operators in some regard?

   I’m half-thinking outloud here with this question but, if the POVM version of Gleason’s theorem is weaker than the classical theorem, does that imply that POVMs are inherently weaker than Hermitian operators in some regard?

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeAug 17th 2010
   • (edited Aug 17th 2010)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexPOVMs are inherently more general than Hermitian operators. This is nothing deep; every Hermitian operator defines a POVM, but not every POVM comes from a Hermitian operator. So if the hypothesis of a theorem has to be true for all POVMs, then this is a stronger hypothesis than merely assuming it for all Hermitian operators, and therefore the theorem is weaker. Physically, the point of interest is that, at least according to some quantum information theorists (and possibly other physicsts, I don't know), thinking of observables as Hermitian operators is too restrictive, and we ought to count every POVM as an observable. Really, that's the only reason that one even cares about the theorem for POVMs.

   POVMs are inherently more general than Hermitian operators. This is nothing deep; every Hermitian operator defines a POVM, but not every POVM comes from a Hermitian operator. So if the hypothesis of a theorem has to be true for all POVMs, then this is a stronger hypothesis than merely assuming it for all Hermitian operators, and therefore the theorem is weaker.

   Physically, the point of interest is that, at least according to some quantum information theorists (and possibly other physicsts, I don’t know), thinking of observables as Hermitian operators is too restrictive, and we ought to count every POVM as an observable. Really, that’s the only reason that one even cares about the theorem for POVMs.

4. • CommentRowNumber4.
   • CommentAuthorIan_Durham
   • CommentTimeAug 18th 2010
   • PermaLink
   Author: Ian_Durham
   Format: MarkdownItex> at least according to some quantum information theorists Well, _some_ quantum information theorists. There's a whole sub-class of them that don't think that way - in fact, in some sub-circles of QI POVMs have become passé. But that's beside the point. The point is: "duh." I'm an idiot. I knew that. Thank you for reminding me, though. ;)

   at least according to some quantum information theorists

   Well, some quantum information theorists. There’s a whole sub-class of them that don’t think that way - in fact, in some sub-circles of QI POVMs have become passé.

   But that’s beside the point. The point is: “duh.” I’m an idiot. I knew that. Thank you for reminding me, though. ;)