5,089 research outputs found
Decoherence free algebra
We consider the decoherence free subalgebra which satisfies the minimal
condition introduced by Alicki. We show the manifest form of it and relate the
subalgebra with the Kraus representation. The arguments also provides a new
proof for generalized L\"{u}ders theorem.Comment: To appear in Physics Letters A v2.minor chang
On Free Completely Iterative Algebras
For every finitary set functor F we demonstrate that free algebras carry a canonical partial order. In case F is bicontinuous, we prove that the cpo obtained as the conservative completion of the free algebra is the free completely iterative algebra. Moreover, the algebra structure of the latter is the unique continuous extension of the algebra structure of the free algebra.
For general finitary functors the free algebra and the free completely iterative algebra are proved to be posets sharing the same conservative completion. And for every recursive equation in the free completely iterative algebra the solution is obtained as the join of an ?-chain of approximate solutions in the free algebra
The convex Positivstellensatz in a free algebra
Given a monic linear pencil L in g variables let D_L be its positivity
domain, i.e., the set of all g-tuples X of symmetric matrices of all sizes
making L(X) positive semidefinite. Because L is a monic linear pencil, D_L is
convex with interior, and conversely it is known that convex bounded
noncommutative semialgebraic sets with interior are all of the form D_L. The
main result of this paper establishes a perfect noncommutative
Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative
polynomial p is positive semidefinite on D_L if and only if it has a weighted
sum of squares representation with optimal degree bounds: p = s^* s + \sum_j
f_j^* L f_j, where s, f_j are vectors of noncommutative polynomials of degree
no greater than 1/2 deg(p). This noncommutative result contrasts sharply with
the commutative setting, where there is no control on the degrees of s, f_j and
assuming only p nonnegative, as opposed to p strictly positive, yields a clean
Positivstellensatz so seldom that such cases are noteworthy.Comment: 22 page
- …