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Bures--Kuratowski metrics and simplicial complexes for completely bounded maps

Remus Floricel, Sarah Plosker, Avner Sadikov

Abstract

Let $A$ be a unital $C^*$-algebra and $H$ a Hilbert space. The cone $\CP(A,B(H))$ of completely positive maps carries the Bures metric $β$, closely related to the cb-norm. We introduce a family of Bures--Kuratowski (BK) metrics on $\CB(A,B(H))$ that extend $β$ exactly on $\CP(A,B(H))$. The construction combines a Kuratowski embedding of the Bures cone, based at an anchor $θ\in\CP(A,B(H))$, with a regular-representation Hausdorff coordinate arising from universal regular models. Each BK metric admits an $\ell^p$-wedge decomposition, splitting $\CB(A,B(H))$ into the Bures cone and a non-CP component attached at $θ$. We then study Vietoris--Rips and Čech complexes of BK metric spaces. The wedge formula yields explicit criteria for mixed simplices, a join-type description of the mixed Rips complex, and ball-intersection criteria for mixed Čech simplices. For finite point clouds, this makes the mixed simplicial geometry computable from the two component metrics and reveals new homological features arising from the interaction between the CP and non-CP sectors.

Bures--Kuratowski metrics and simplicial complexes for completely bounded maps

Abstract

Let be a unital -algebra and a Hilbert space. The cone of completely positive maps carries the Bures metric , closely related to the cb-norm. We introduce a family of Bures--Kuratowski (BK) metrics on that extend exactly on . The construction combines a Kuratowski embedding of the Bures cone, based at an anchor , with a regular-representation Hausdorff coordinate arising from universal regular models. Each BK metric admits an -wedge decomposition, splitting into the Bures cone and a non-CP component attached at . We then study Vietoris--Rips and Čech complexes of BK metric spaces. The wedge formula yields explicit criteria for mixed simplices, a join-type description of the mixed Rips complex, and ball-intersection criteria for mixed Čech simplices. For finite point clouds, this makes the mixed simplicial geometry computable from the two component metrics and reveals new homological features arising from the interaction between the CP and non-CP sectors.

Paper Structure

This paper contains 10 sections, 32 theorems, 113 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background: Bures distance and universal regular models
  3. Bures--Kuratowski metrics
  4. The Bures--Kuratowski metric bundle
  5. An lp wedge decomposition for BK metrics
  6. Vietoris--Rips and Čech complexes for BK metric spaces
  7. Vietoris--Rips complexes
  8. A finite-dimensional CP-side computation: $A=\mathbb C^n$, $H=\mathbb C$
  9. A finite-dimensional computation on a mixed cloud: $A=M_n$, $H=\mathbb C^m$
  10. Čech complexes

Key Result

Theorem 2.3

For every $\phi\in \mathrm{CB}(A,B(H))$ there exist a Hilbert space $K$, a regular homomorphism $\tau:A\to B(K)$, and an operator $W\in B(H,K)$ such that $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Corollary \ref{['ex:universal_loop']}: the BK complex at scale $t$ has 1-skeleton equal to the complete bipartite graph $K_{2,2}$, with bipartition $\{x_1,x_2\}\subset \mathcal{C}$ and $\{y_1,y_2\}\subset \mathcal{Y}$. Since there are no within-side edges, there are no triangles, and hence the flag (clique) complex adds no $2$-simplices. Therefore $\mathrm{VR}_t(S,d_\theta)$ is the graph itself, namely a $4$-cycle, and has homotopy type $S^1$.
  • Figure 2: The mixed point cloud $S=\{x_0,x_4,y_+,y_-\}$ from Example \ref{['ex:go2']}. The CP vertices $x_0,x_4$ are at Bures distance $2$, while the non-CP vertices $y_+,y_-$ are at $d_{\mathrm{reg}}$-distance $D$. Each cross distance is $\max\{1,r_\pm\}$. Under the condition $\max\{1,r_+,r_-\}\le t<\min\{2,D\}$, all cross edges are present and no within-side edges occur. Thus the Vietoris--Rips complex $\mathrm{VR}_t(S,d_\theta)$ has 1-skeleton $K_{2,2}$ and is homotopy equivalent to $S^1$ (Theorem \ref{['thm:bk-rips-loop']}).
  • Figure 3: A schematic comparison between BK Vietoris--Rips and ambient BK Čech complexes at the same scale $t$. Left: in the Rips complex, only the mixed edges are present, so the 1-skeleton is $K_{2,2}$ and the complex is a $4$-cycle. Right: in the ambient Čech complex, the glued basepoint $\theta\sim\ast$ may lie in all ambient balls without being a vertex of the cloud; this hidden witness fills in all simplices, so the complex becomes contractible. This illustrates the basepoint-driven cone effect from Corollary \ref{['cor:cech_cone_effect']} and the contrast with Corollary \ref{['ex:universal_loop']}.
  • Figure 4: Example \ref{['prop:cp-cech']}. Intrinsic and ambient Čech filtrations behave differently even for a simple three-point CP cloud. The intrinsic complex jumps directly from discrete to a simplex, while the ambient complex exhibits an intermediate path stage due to intersections occurring in the ambient Bures ray.

Theorems & Definitions (95)

  • Definition 2.1: Bures distance on $\mathrm{CP}(A,B(H))$ KSW
  • Definition 2.2: cf. BMS2017
  • Theorem 2.3: Regular representation theorem BMS2017
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6: BMS2017
  • Remark 2.7
  • Lemma 3.1
  • proof
  • Definition 3.2
  • ...and 85 more