11,900 research outputs found
Quantum channels as a categorical completion
We propose a categorical foundation for the connection between pure and mixed
states in quantum information and quantum computation. The foundation is based
on distributive monoidal categories.
First, we prove that the category of all quantum channels is a canonical
completion of the category of pure quantum operations (with ancilla
preparations). More precisely, we prove that the category of completely
positive trace-preserving maps between finite-dimensional C*-algebras is a
canonical completion of the category of finite-dimensional vector spaces and
isometries.
Second, we extend our result to give a foundation to the topological
relationships between quantum channels. We do this by generalizing our
categorical foundation to the topologically-enriched setting. In particular, we
show that the operator norm topology on quantum channels is the canonical
topology induced by the norm topology on isometries.Comment: 12 pages + ref, accepted at LICS 201
The Physics and Mathematics of the Second Law of Thermodynamics
The essential postulates of classical thermodynamics are formulated, from
which the second law is deduced as the principle of increase of entropy in
irreversible adiabatic processes that take one equilibrium state to another.
The entropy constructed here is defined only for equilibrium states and no
attempt is made to define it otherwise. Statistical mechanics does not enter
these considerations. One of the main concepts that makes everything work is
the comparison principle (which, in essence, states that given any two states
of the same chemical composition at least one is adiabatically accessible from
the other) and we show that it can be derived from some assumptions about the
pressure and thermal equilibrium. Temperature is derived from entropy, but at
the start not even the concept of `hotness' is assumed. Our formulation offers
a certain clarity and rigor that goes beyond most textbook discussions of the
second law.Comment: 93 pages, TeX, 8 eps figures. Updated, published version. A summary
appears in Notices of the Amer. Math. Soc. 45 (1998) 571-581, math-ph/980500
Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations
We consider the inverse problem of the simultaneous reconstruction of the
dielectric permittivity and magnetic permeability functions of the Maxwell's
system in 3D with limited boundary observations of the electric field. The
theoretical stability for the problem is provided by the Carleman estimates.
For the numerical computations the problem is formulated as an optimization
problem and hybrid finite element/difference method is used to solve the
parameter identification problem.Comment: in Inverse Problems and Imaging Volume: 9, Number: 1 February 2015.
arXiv admin note: text overlap with arXiv:1510.0752
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