Birkhoff-von Neumann theorem

**physics**, mathematical physics, philosophy of physics
## Surveys, textbooks and lecture notes
* _(higher) category theory and physics_
* _geometry of physics_
* books and reviews, physics resources
***
theory (physics), model (physics)
experiment, measurement, computable physics
* **mechanics**
* mass, charge, momentum, angular momentum, moment of inertia
* dynamics on Lie groups
* rigid body dynamics
* field (physics)
* Lagrangian mechanics
* configuration space, state
* action functional, Lagrangian
* covariant phase space, Euler-Lagrange equations
* Hamiltonian mechanics
* phase space
* symplectic geometry
* Poisson manifold
* symplectic manifold
* symplectic groupoid
* multisymplectic geometry
* n-symplectic manifold
* spacetime
* smooth Lorentzian manifold
* special relativity
* general relativity
* gravity
* supergravity, dilaton gravity
* black hole
* **Classical field theory**
* classical physics
* classical mechanics
* waves and optics
* thermodynamics
* **Quantum Mechanics**
* in terms of †-compact categories
* quantum information
* Hamiltonian operator
* density matrix
* Kochen-Specker theorem
* Bell's theorem
* Gleason's theorem
* **Quantization**
* geometric quantization
* deformation quantization
* path integral quantization
* semiclassical approximation
* **Quantum Field Theory**
* Axiomatizations
* algebraic QFT
* Wightman axioms
* Haag-Kastler axioms
* operator algebra
* local net
* conformal net
* Reeh-Schlieder theorem
* Osterwalder-Schrader theorem
* PCT theorem
* Bisognano-Wichmann theorem
* modular theory
* spin-statistics theorem
* boson, fermion
* functorial QFT
* cobordism
* (∞,n)-category of cobordisms
* cobordism hypothesis-theorem
* extended topological quantum field theory
* Tools
* perturbative quantum field theory, vacuum
* effective quantum field theory
* renormalization
* BV-BRST formalism
* geometric ∞-function theory
* particle physics
* phenomenology
* models
* standard model of particle physics
* fields and quanta
* Grand Unified Theories, MSSM
* scattering amplitude
* on-shell recursion, KLT relations
* Structural phenomena
* universality class
* quantum anomaly
* Green-Schwarz mechanism
* instanton
* spontaneously broken symmetry
* Kaluza-Klein mechanism
* integrable systems
* holonomic quantum fields
* Types of quantum field thories
* TQFT
* 2d TQFT
* Dijkgraaf-Witten theory
* Chern-Simons theory
* TCFT
* A-model, B-model
* homological mirror symmetry
* QFT with defects
* conformal field theory
* (1,1)-dimensional Euclidean field theories and K-theory
* (2,1)-dimensional Euclidean field theory and elliptic cohomology
* CFT
* WZW model
* 6d (2,0)-supersymmetric QFT
* gauge theory
* field strength
* gauge group, gauge transformation, gauge fixing
* examples
* electromagnetic field, QED
* electric charge
* magnetic charge
* Yang-Mills field, QCD
* Yang-Mills theory
* spinors in Yang-Mills theory
* topological Yang-Mills theory
* Kalb-Ramond field
* supergravity C-field
* RR field
* first-order formulation of gravity
* general covariance
* supergravity
* D'Auria-Fre formulation of supergravity
* gravity as a BF-theory
* sigma-model
* particle, relativistic particle, fundamental particle, spinning particle, superparticle
* string, spinning string, superstring
* membrane
* AKSZ theory
* String Theory
* string theory results applied elsewhere
* number theory and physics
* Riemann hypothesis and physics

Zoran: there are several things called “Birkhoff’s theorem” in various field of mathematics and mathematical physics, and belong even to at least 2 different classical Birkhoff’s. Even wikipedia has pages for more than one such theorem. To me the first which comes to mind is Birkhoff’s factorization theorem, now also popular in Kreimer-Connes-Marcolli work and in connection to loop groups (cf. book by Segal nad Pressley). I would like that the $n$lab does not mislead by distinguishing one of the several famous Bikhoff labels without mentioning and directing to 2-3 others.

Ian Durham: Good point. I think this probably ought to be renamed the “Birkhoff-von Neumann theorem.” Is that a good enough label or should we get more specific with it?

*Toby*: I have moved it. See also the new page Birkhoff's theorem, which is basically just Zoran's comment above.

Given a permutation $\sigma \in S_n$, the permutation matrix that is associated with $\sigma$ is the $n \times n$ matrix $P_{\sigma}(j,k)$, where $1 \le j,k \le n$, whose entries are given by

$P_{\sigma}(j,k)= \left \{ \begin{aligned} 1 & if \sigma (j)=k \\ 0 & otherwise. \end{aligned} \right.$

An $n \times n$ doubly stochastic matrix $D$ is a square matrix whose elements are real and whose rows and columns sum to unity. Given such a matrix, there exist nonnegative weights $\{w_{\sigma}:\sigma \in S_n\}$ such that

$\sum_{\sigma \in S_n} w_{\sigma}=1$ and $\sum_{\sigma \in S_n} w_{\sigma} P_{\sigma}=D$.

The set of such matrices of order $n$ is said to form the convex hull of permutation matrices of the same order where the latter are the vertices (extreme points) of the former.

In the quantum context, doubly stochastic matrices become doubly stochastic channels, i.e. completely positive maps preserving both the trace and the identity. Quantum mechanically we understand the permutations to be the unitarily implemented channels. That is, we expect doubly stochastic quantum channels to be convex combinations of unitary channels (that is channels that can be represented by some combination of unitary transformations). Unfortunately it is well-known that some quantum channels $cannot$ be written that way.

However, large tensor powers of a channel may be easier to represent in this way, because one need not use only product unitaries in the decomposition. Thus one proposed solution to the problem is to denote, for any doubly stochastic channel $T$, by $d_{B}(T)$ the **Birkhoff defect**, defined as the cb-norm distance from $T$ to the convex hull of the unitarily implemented channels. Then the key is to determine whether $d_{B}(T^{\otimes n})$ goes to zero as $n\to\infty$.

David Roberts: I’m guessing you wanted $T^{\otimes n}$ instead of $T\otimes n$.

Ian Durham: Yep. My bad.

Categorically, one possible way to approach this problem is to determine the relationship between the category of quantum channels, $QChan$, and the category of unitary transformations.

Ian Durham: Any suggestions concerning the last point?

David Roberts: There may be a functor between the two which is 'nice' in some sense (has an adjoint, say) which encapsulates the next best possible result.

Ian Durham: Hmmm. What I think I’m going to do is to try to tally up all the category theoretic properties of each. That should make it easier to see if there’s a functor between them, I would think.

Revised on April 11, 2010 18:34:59
by Toby Bartels
(98.19.62.150)