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Angshuman Bhattacharya

Publications and source records attributed to Angshuman Bhattacharya.

12 recordsLinked to original sources
arXiv2026

A multigraph approach to confusability in quantum channels

We introduce a new approach to confusability in a quantum channel, namely quantum confusability multigraph, which incorporates the output information into the graphical structure. By``counting" the edges between two vertices of this confusability multigraph, one recovers the traditional confusability ``single-edged" graph of the channel. With this physical motivation, we therefore develop a theory of quantum multigraphs from Weaver's quantum relations point of view and explore its quantum graph theoretic properties. Finally, we provide a necessary and sufficient condition characterizing those quantum multigraphs that arise as quantum confusability multigraphs.

math.QA↗
arXiv2024

Ergodic decomposition in the space of unital completely positive maps

The classical decomposition theory for states on a C*-algebra that are invariant under a group action has been studied by using the theory of orthogonal measures on the state space \cite{BR1}. In \cite{BK3}, we introduced the notion of \textit{generalized orthogonal measures} on the space of unital completely positive (UCP) maps from a C*-algebra $A$ into $B(H)$. In this article, we consider a group $G$ that acts on a C*-algebra $A$, and the collection of $G$-invariant UCP maps from $A$ into $B(H)$. This article examines a $G$-invariant decomposition of UCP maps by using the theory of generalized orthogonal measures on the space of UCP maps, developed in \cite{BK3}. Further, the set of all $G$-invariant UCP maps is a compact and convex subset of a topological vector space. Hence, by characterizing the extreme points of this set, we complete the picture of barycentric decomposition in the space of $G$-invariant UCP maps. We establish this theory in Stinespring and Paschke dilations of completely positive maps. We end this note by mentioning some examples of UCP maps admitting a decomposition into $G$-invariant UCP maps.

math.OA↗
arXiv2023

Generalized Orthogonal Measures on the Space of Unital Completely Positive Maps

A classical result by Effros connects the barycentric decomposition of a state on a C*-algebra to the disintegration of the GNS representation of the state with respect to an orthogonal measure on the state space of the C*-algebra. In this note, we take this approach to the space of unital completely positive maps on a C*-algebra with values in B(H), connecting the barycentric decomposition of the unital completely positive map and the dis-integration of the minimal Stinespring dilation of the same. This generalizes Effros' work in the non-commutative setting. We do this by introducing a special class of barycentric measures which we call generalized orthogonal measures. We end this note by mentioning some examples of generalized orthogonal measures.

math.OA↗
arXiv2023

Barycentric decompositions in the space of weak expectations

The space of weak expectations for a given representation of a (unital) separable C*-algebra is a compact convex set of (unital) completely positive maps in the BW topology, when it is non-empty. An application of the classical Choquet theory gives a barycentric decomposition of a weak expectation in that set. However, to complete the barycentric picture, one needs to know the extreme points of the compact convex set in question. In this article, we explicitly identify the set of extreme points of the space of weak expectations for a given representation, using operator theoretic techniques.

math.OA↗
arXiv2022

Decomposable Weak Expectations

In this article we define a special class of weak expectations for a representation of a separable unital C*-algebra, called decomposable weak expectation. We give necessary and sufficient conditions for such kind of weak expectations to exist for a given representation. Then we define decomposable measures on the state space of a C*-algebra and show that the GNS representation of a state admits a decomposable weak expectation if and only if there is a decomposable measure on the state space. Further we give an example of a decomposable weak expectation.

math.OA↗
arXiv2020

Property (T), property (F) and residual finiteness for discrete quantum groups

We investigate connections between various rigidity and softness properties for discrete quantum groups. After introducing a notion of residual finiteness, we show that it implies the Kirchberg factorization property for the discrete quantum group in question. We also prove the analogue of Kirchberg's theorem, to the effect that conversely, the factorization property and property (T) jointly imply residual finiteness. We also apply these results to certain classes of discrete quantum groups obtained by means of bicrossed product constructions and study the preservation of the properties (factorization, residual finiteness, property (T)) under extensions of discrete quantum groups.

math.QA↗
arXiv2014

Relative weak injectivity of operator system pairs

The concept of a relatively weakly injective pair of operator systems is introduced and studied in this paper, motivated by relative weak injectivity in the C*-algebra category. E. Kirchberg \cite{Kr} proved that the C*-algebra $C^*(\mathbb{f}_\infty)$ of the free group $\mathbb{f}_\infty$ on countably many generators characterizes relative weak injectivity for pairs of C*-algebras by means of the maximal tensor product. One of the main results of this paper shows that $C^*(\mathbb{f}_\infty)$ also characterises relative weak injectivity in the operator system category. A key tool is the theory of operator system tensor products \cite{KP1,KP2}.

math.OA↗
arXiv2013

Coincidence of Schur multipliers of the Drury-Arveson space

In a purely multi-variable setting (i.e., the issues discussed in this note are not interesting in the single variable operator theory setting), we show that the coincidence of two operator valued Schur class multipliers of a certain kind on the Drury-Arveson space is characterized by the fact that the associated colligations (or a variant, obtained canonically) are `unitarily coincident' in a sense to be made precise in the last section of this article.

math.FA↗
arXiv2011

Complete Pick Positivity and Unitary Invariance

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel $k_S(z,w) = (1 - z\ow)^{-1}$ for $|z|, |w| < 1$, by means of $(1/k_S)(T,T^*) \ge 0$, we consider an arbitrary open connected domain $Ω$ in $\BC^n$, a complete Nevanilinna-Pick kernel $k$ on $Ω$ and a tuple $T = (T_1, ..., T_n)$ of commuting bounded operators on a complex separable Hilbert space $\clh$ such that $(1/k)(T,T^*) \ge 0$. For a complete Pick kernel the $1/k$ functional calculus makes sense in a beautiful way. It turns out that the model theory works very well and a characteristic function can be associated with $T$. Moreover, the characteristic function then is a complete unitary invariant for a suitable class of tuples $T$.

math.FA↗