A Representation of Quantum Measurement in Order-Unit Spaces

Bernard d’Espagnat

Grigori Gennadievich Amosov, Айрат Мидхатович Бикчентаев, Vsevolod Zhanovich Sakbaev

We obtain a representation of the set of quantum states in terms of barycenters of nonnegative normalized finitely additive measures on the unit sphere $$S_1(\mathcal H)$$ of a Hilbert space $$\mathcal H$$ . For a measure on $$S_1(\mathcal H)$$ , we find conditions in terms of its properties under which the barycenter of this measure belongs to the set of extreme points of the family of quantum states and to the set of normal states. The unitary elements of a unital $$\mathrm C^*$$ -algebra are characterized in terms of extreme points. We also study extreme points $$\mathrm{extr}(\mathcal E^1)$$ of the unit ball $$\mathcal E^1$$ of a normed ideal operator space $$\langle\mathcal E,\|\kern1pt{\cdot}\kern1pt\|_{\mathcal E}\rangle$$ on $$\mathcal H$$ . If $$U\in\mathrm{extr}(\mathcal E^1)$$ for some unitary operator $$U\in\mathcal{B}(\mathcal H)$$ , then $$V\in\mathrm{extr}(\mathcal E^1)$$ for all unitary operators $$V\in\mathcal{B}(\mathcal H)$$ . In addition, we construct quantum correlations corresponding to singular states on the algebra of all bounded operators in a Hilbert space.

H. D. Zeh

Chao Lei, Apostol Vourdas, Alfred Wünsche

A contour representation in the unit disk based on SU(1,1) coherent states is introduced. The scalar product is given by a contour integral. The regions of convergence of the functions representing ket and bra states are studied. An analytic representation in the unit disk is also considered, where the scalar product is represented by an integral over the unit disk, with the Lobachevsky measure. Various relations which connect these analytic functions with other phase-space quantities are derived.

Debopam Ghosh

The article presents a mathematical framework to associate a Matrix Shell system (even type or of the odd type) with a unit trace, hermitian, positive definite or positive semi-definite matrix of order ‘2’ (Density matrix representation of single qubit quantum states). This framework therefore, allows any evolution scheme defined on the fundamental matrix space associated with the Matrix Shell system and consistent with the Matrix shell model formalism to be mapped to an evolution on or inside of the unit solid sphere, centered at the origin, in the three dimensional space( Bloch sphere representation of single qubit quantum states) Demonstration of the proposed mathematical framework is presented through its application on certain subsets of the Complex Matrix spaces M2×2(C) and M4×4(C) and on certain numerical examples from M3×3(C).

Fritz Haake, Wolfgang Weidlich