extremal quantum channel

**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

The set of all quantum channels on $\mathcal{M}_{d}$ is convex and compact meaning it may be decomposed as

$T=\sum_{i}p_{i}T_{i}$

where the $p$’s are probabilities and the $T_{i}$’s are $extremal$ unital channels, that is channels that may not be further decomposed.

Channels with a single Kraus operator are pure channels and the extremal points in the convex set of channels are precisely the pure channels. Here $T$ represents the set of $all$ channels on the particular space, not necessarily copies of the same one, i.e. the $T_{i}$ may not represent the same channel.

$T$, with Kraus operators $\{A\}_i$, is extremal if and only if the set

$\left \{A_{k}^{\dagger}A_{l} \right \}_{k,l\ldots N}$

is linearly independent.

In the case where $T$ is unital, it is extremal if and only if the set

$\left \{A_{k}^{\dagger}A_{l} \oplus A_{l}A_{k}^{\dagger} \right \}_{k,l \ldots N}$

is linearly independent.

Ian Durham: One major conundrum is to determine whether extremality is preserved over tensor products, i.e. given an extremal quantum channel, if you take $n$ copies of it (which amounts to tensoring it $n$ times with itself), as a whole are these $n$ copies still extremal? It would be nice to see if category theory can shed some light on this problem since it is at the root of a particularly gnarly problem in quantum information theory..

Christian B. Mendl and Michael M. Wolf, *Unital Quantum Channels - Convex Structure and Revivals of Birkhoff’s Theorem* (pdf).

L. J. Landau and R. F. Streater, *On Birkhoff ’s theorem for doubly stochastic completely positive maps of matrix algebras*, Lin. Alg. Appl., 193:107–127, 1993.

Revised on March 28, 2010 02:23:10
by Eric Forgy
(61.92.132.172)