Stephan Weis

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• Dr. rer. nat. Dipl.-Math. MSc
• Research Fellow at Czech Technical University in Prague

We study the continuity of an abstract generalization of the maximum-entropy inference - a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discont...

We quantify higher-order correlations in a composite quantum system in terms of the divergence from a family of Gibbs states with well-specified interaction patterns, known as hierarchical model. We begin with a review of factoring in a classical hierarchical model. This is just one aspect of our critical discussion of the divergence from a hierarc...

We study a curve of Gibbsian families of complex 3x3-matrices and point out new features, absent in commutative finite-dimensional algebras: a discontinuous maximum-entropy inference, a discontinuous entropy distance and non-exposed faces of the mean value set. We analyze these problems from various aspects including convex geometry, topology and i...

The faces of a convex set owe their relevance to an interplay between convexity and topology that is systematically studied in the work of Rockafellar. Infinite-dimensional convex sets are excluded from this theory as their relative interiors may be empty. Shirokov and the present author answered this issue by proving that every point in a convex s...

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

The faces of a convex set owe their relevance to an interplay between convexity and topology that is systematically studied in the work of Rockafellar. Infinite-dimensional convex sets are excluded from this theory as their relative interiors may be empty. Shirokov and the present author answered this issue by proving that every point in a convex s...

The lattice of faces of the convex set of reduced density matrices is essential for the construction of the information projection to a hierarchical model. The lattice of faces is also important in quantum state tomography. Yet, the description and computation of these faces is elusive in the simplest examples. Here, we study the face lattice of th...

Many problems of quantum information theory rely on the set of quantum marginals. A precise knowledge of the faces of this convex set is necessary, for example, in the reconstruction of states from their marginals or in the evaluation of complexity measures of many-body systems. Yet, even the two-body marginals of just three qubits were only descri...

We analyze faces generated by points in an arbitrary convex set and their relative algebraic interiors, which are nonempty as we shall prove. We show that by intersecting a convex set with a sublevel or level set of a generalized affine functional, the dimension of the face generated by a point may decrease by at most one. We apply the results to t...

Here we prove that every symmetric separable state admits a convex decomposition into symmetric pure product states. The same proof shows that every antisymmetric state is entangled. The decomposition sheds new light on numerical ranges useful to study ground state problems of infinite bosonic systems.

We analyze faces generated by points in an arbitrary convex set and their relative algebraic interiors, which are nonempty as we shall prove. We show that by intersecting a convex set with a sublevel or level set of a generalized affine functional, the dimension of the face generated by a point may decrease by at most one. We apply the results to t...

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

Kippenhahn's Theorem asserts that the numerical range of a matrix is the convex hull of a certain algebraic curve. Here, we show that the joint numerical range of finitely many hermitian matrices is similarly the convex hull of a semi-algebraic set. We discuss an analogous statement regarding the dual convex cone to a hyperbolicity cone and prove t...

We investigate weak coin flipping, a fundamental cryptographic primitive where two distrustful parties need to remotely establish a shared random bit. A cheating player can try to bias the output bit towards a preferred value. For weak coin flipping the players have known opposite preferred values. A weak coin-flipping protocol has a bias є if neit...

We analyze the smoothness of the ground state energy of a one-parameter Hamiltonian by studying the differential geometry of the numerical range and continuity of the maximum-entropy inference. The domain of the inference map is the numerical range, a convex compact set in the plane. We show that its boundary, viewed as a manifold, has the same ord...

We investigate weak coin flipping, a fundamental cryptographic primitive where two distrustful parties need to remotely establish a shared random bit. A cheating player can try to bias the output bit towards a preferred value. For weak coin flipping the players have known opposite preferred values. A weak coin-flipping protocol has a bias $\epsilon...

Kippenhahn discovered a real algebraic plane curve whose convex hull is the numerical range of a matrix. The correctness of this theorem was called into question when Chien and Nakazato found an example where the spatial analogue fails. They showed that the mentioned plane curve indeed lies inside the numerical range. We prove the easier converse d...

We prove that the ground space projections of a subspace of energy operators in a matrix *-algebra are the greatest projections of the algebra under certain operator cone constraints. The lattice of ground space projections being coatomistic, we also discuss its maximal elements as building blocks. We demonstrate the results with (commutative) two-...

The ground spaces of a vector space of hermitian matrices, partially ordered by inclusion, form a lattice constructible from top to bottom in terms of intersections of maximal ground spaces. In this paper we characterize the lattice elements and the maximal lattice elements within the set of all subspaces using constraints on operator cones. Our re...

Predicting quantum phase transitions by signatures in finite models has a long tradition. Here we consider the numerical range $W$ of a finite dimensional one-parameter Hamiltonian, which is a planar projection of the convex set of density matrices. We propose the new geometrical signature of non-analytic points of class $C^2$ on the boundary of $W...

We study the inverse problem of inferring the state of a finite-level quantum system from expected values of a fixed set of observables, by maximizing a continuous ranking function. We have proved earlier that the maximum-entropy inference can be a discontinuous map from the convex set of expected values to the convex set of states because the imag...

We extend the pre-image representation of exposed points of the numerical range of a matrix to all extreme points. With that we characterize extreme points which are multiply generated, having at least two linearly independent pre-images, as the extreme points which are Hausdorff limits of flat boundary portions on numerical ranges of a sequence co...

The joint numerical range of three hermitian matrices of order three is a convex and compact subset W ⊂ R^3 which is an image of the unit sphere S^5 ⊂ C^3 under the hermitian form defined by the three matrices. We label classes of the analyzed set W by pairs of numbers counting the exposed faces of dimension one and two. Generically, W belongs to t...

The state space of an operator system of $n$-by-$n$ matrices has, in a sense, many normal cones. Merely this convex geometrical property implies smoothness qualities and a clustering property of exposed faces. The latter holds since each exposed face is an intersection of maximal exposed faces. An isomorphism translates these results to the lat...

The joint numerical range $W(F)$ of three hermitian $3$-by-$3$ matrices $F=(F_1,F_2,F_3)$ is a convex and compact subset in $\mathbb{R}^3$. Generically we find that $W(F)$ is a three-dimensional oval. Assuming $\dim(W(F))=3$, every one- or two-dimensional face of $W(F)$ is a segment or a filled ellipse. We prove that only ten configurations of thes...

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

The maximum-entropy inference assigns to the mean values with respect to a fixed set of observables the unique density matrix, which is consistent with the mean values and which maximizes the von Neumann entropy. A discontinuity was recently found in this inference method for three-level quantum systems. For arbitrary finite-level quantum systems,...

We define an information topology (I-topology) and a reverse information topology (rI-topology) on the state space of a C*-subalgebra of Mat(n,C). These topologies arise from sequential convergence with respect to the relative entropy. We prove that open disks, with respect to the relative entropy, define a base for them, while Csiszar has shown in...

We revisit the maximum-entropy inference of the state of a finite-level quantum system under linear constraints. The constraints are specified by the expected values of a set of fixed observables. We point out the existence of discontinuities in this inference method. This is a pure quantum phenomenon since the maximum-entropy inference is continuo...

The set of quantum states consists of density matrices of order $N$, which are hermitian, positive and normalized by the trace condition. We analyze the structure of this set in the framework of the Euclidean geometry naturally arising in the space of hermitian matrices. For N=2 this set is the Bloch ball, embedded in $\mathbb R^3$. For $N \geq 3$...

We study touching cones of a (not necessarily closed) convex set in a finitedimensional real Euclidean vector space and we draw relationships to other concepts in Convex Geometry. Exposed faces correspond to normal cones by an antitone lattice isomorphism. Poonems generalize the former to faces and the latter to touching cones, these extensions are...

Given any polar pair of convex bodies we study its conjugate face maps and we characterize conjugate faces of non-exposed faces in terms of normal cones. The analysis is carried out using the positive hull operator which defines lattice isomorphisms linking three Galois connections. One of them assigns conjugate faces between the convex bodies. The...

Convex support, the mean values of a set of random variables, is central in information theory and statistics. Equally central in quantum information theory are mean values of a set of observables in a finite-dimensional C*-algebra A, which we call (quantum) convex support. The convex support can be viewed as a projection of the state space of A an...

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

This work generalizes the interaction measure of multi-information known in probability theory to finite-level quantum systems. This is done in the more general context of the entropy distance from an exponential family. One of the most well-known measures of stochastic dependence is multi-information, applied in various fields including Neuroscien...