Every state on the algebra $M_n$ of complex nxn matrices restricts to a any matrix system. Whereas restriction system is generally not open, we prove that every *-subalgebra open. This simplifies topology problems in theory and quantum information theory.

The set of quantum states consists density matrices order N, which are hermitian, positive and normalized by the trace condition. We analyze structure this in framework Euclidean geometry naturally arisingin space hermitian matrices. For $$ N\,\,=\,\,2 is Bloch ball, embedded \mathbb{R}^3 . N\,\,\geq \,\,3$$ dimensionality N^2 \,\,-\,1 has a much richer structure. study its properties at first advocate an apophatic approach, concentrates on characteristics not possessed set. also apply more...

We study a curve of Gibbsian families complex 3x3-matrices and point out new features, absent in commutative finite-dimensional algebras: discontinuous maximum-entropy inference, entropy distance non-exposed faces the mean value set. analyze these problems from various aspects including convex geometry, topology information geometry. This research is motivated by theory info-max principles, where we contribute computing first order optimality conditions distance.

We study many-party correlations quantified in terms of the Umegaki relative entropy (divergence) from a Gibbs family known as hierarchical model. derive these quantities maximum-entropy principle which was used earlier to define closely related irreducible correlation. point out differences between quantum states and probability vectors exist models, divergence model local maximizers this divergence. The are, respectively, missing factorization, discontinuity reduction uncertainty. discuss...

We study the continuity of an abstract generalization maximum-entropy inference - a maximizer. It is defined as right-inverse linear map restricted to convex body which uniquely maximizes on each fiber continuous function body. Using geometry we prove, amongst others, existence discontinuities maximizer at limits extremal points not being themselves and apply result quantum correlations. Further, use numerical range methods in case refers two observables. One complete characterization...

Voronoi cells of a discrete set in Euclidean space are known as generalized polyhedra. We identify polyhedral through direction cone. For an arbitrary we distinguish from non-polyhedral using inversion at sphere and theorem semi-infinite linear programming.

We define an information topology (I-topology) and a reverse (rI-topology) on the state space of C*-subalgebra Mat(n,C). These topologies arise from sequential convergence with respect to relative entropy. prove that open disks, entropy, base for them, while Csiszar has shown in 1967 analogue is wrong probability measures countably infinite set. The I-topology finer than norm topology, it disconnects convex into its faces. rI-topology intermediate between these topologies. complete two...

We show that for any energy observable every extreme point of the set quantum states with bounded is a pure state.This allows us to write state in terms continuous convex combination energy.Furthermore, we prove finite can be represented as same energy.We discuss examples from information theory.

Kippenhahn's Theorem asserts that the numerical range of a matrix is convex hull certain algebraic curve. Here, we show joint finitely many Hermitian matrices similarly semi-algebraic set. We discuss an analogous statement regarding dual cone to hyperbolicity and prove class bases these cones closed under linear operations. The result offers new geometric method analyze quantum states.

We investigate weak coin flipping, a fundamental cryptographic primitive where two distrustful parties need to remotely establish shared random bit. A cheating player can try bias the output bit towards preferred value. For flipping players have known opposite values. coin-flipping protocol has $\epsilon$ if neither force outcome their value with probability more than $\frac{1}{2}+\epsilon$. While it is that all classical protocols $\epsilon=\frac{1}{2}$, Mochon showed in 2007...

We study touching cones of a (not necessarily closed) convex set in finitedimensional real Euclidean vector space and we draw relationships to other concepts Convex Geometry. Exposed faces correspond normal by an antitone lattice isomorphism. Poonems generalize the former latter cones, these extensions are non-isomorphic, though. behavior lattices under projections affine subspaces intersections with subspaces. prove theorem that characterizes exposed assumptions about cones. For body K...

We discuss methods to analyze a quantum Gibbs family in the ultra-cold regime where norm closure of fails due discontinuities maximum-entropy inference. The current discussion inference and irreducible correlation area phase transitions is major motivation for this research. extend representation from finite temperatures absolute zero.

We analyze the smoothness of ground state energy a one-parameter Hamiltonian by studying differential geometry numerical range and continuity maximum-entropy inference. The domain inference map is range, convex compact set in plane. show that its boundary, viewed as manifold, has same order differentiability energy. prove discontinuities correspond to C1-smooth crossings with higher level. Discontinuities may appear only at points boundary range. exist all C2-smooth non-analytic are...

The state space of an operator system $n$-by-$n$ matrices has, in a sense, many normal cones. Merely this convex geometrical property implies smoothness qualities and clustering exposed faces. latter holds since each face is intersection maximal An isomorphism translates these results to the lattice ground projections system. We work on minimizing assumptions under which set has mentioned properties.

Views Icon Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Twitter Facebook Reddit LinkedIn Tools Reprints and Permissions Cite Search Site Citation Stephan Weis; Discontinuities in the maximum-entropy inference. AIP Conference Proceedings 21 August 2013; 1553 (1): 192–199. https://doi.org/10.1063/1.4820000 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Dropdown Menu input auto...

We extend the pre-image representation of exposed points numerical range a matrix to all extreme points. With that we characterize which are multiply generated, having at least two linearly independent pre-images, as Hausdorff limits flat boundary portions on ranges sequence converging given matrix. These studies address inverse map and maximum-entropy inference continuous functions except possibly certain generated This work also allows us describe closures subsets 3-by-3 matrices same shape range.