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Hilbert space-valued Gaussian processes, and quantum states

Palle E. T. Jorgensen, James Tian

TL;DR

The paper develops a universal factorization framework for operator-valued PD kernels, linking Hilbert-space representations, dilations, and RKHS methods, with applications to Hilbert-space-valued Gaussian processes. It shows that K(s,t) can be realized as K(s,t) = V_s^* V_t through a concrete RKHS H_tildeK, embedding dilation theory (including Stinespring) into a function-space setting. The authors apply this framework to POVMs and quantum information, formulating completeness and inverse-problem perspectives for state inference from measurement data, and providing finite-dimensional kernel-approximation schemes with PVM dilations. The approach yields explicit, computable realizations that unify classical operator-dilation theory with modern quantum-measurement tasks and covariance analyses in Gaussian processes.

Abstract

We offer new results and new directions in the study of operator-valued kernels and their factorizations. Our approach provides both more explicit realizations and new results, as well as new applications. These include: (i) an explicit covariance analysis for Hilbert space-valued Gaussian processes, (ii) optimization results for quantum gates (from quantum information), (iii) new results for positive operator-valued measures (POVMs), and (iv) a new approach/result in inverse problems for quantum measurements.

Hilbert space-valued Gaussian processes, and quantum states

TL;DR

The paper develops a universal factorization framework for operator-valued PD kernels, linking Hilbert-space representations, dilations, and RKHS methods, with applications to Hilbert-space-valued Gaussian processes. It shows that K(s,t) can be realized as K(s,t) = V_s^* V_t through a concrete RKHS H_tildeK, embedding dilation theory (including Stinespring) into a function-space setting. The authors apply this framework to POVMs and quantum information, formulating completeness and inverse-problem perspectives for state inference from measurement data, and providing finite-dimensional kernel-approximation schemes with PVM dilations. The approach yields explicit, computable realizations that unify classical operator-dilation theory with modern quantum-measurement tasks and covariance analyses in Gaussian processes.

Abstract

We offer new results and new directions in the study of operator-valued kernels and their factorizations. Our approach provides both more explicit realizations and new results, as well as new applications. These include: (i) an explicit covariance analysis for Hilbert space-valued Gaussian processes, (ii) optimization results for quantum gates (from quantum information), (iii) new results for positive operator-valued measures (POVMs), and (iv) a new approach/result in inverse problems for quantum measurements.
Paper Structure (6 sections, 13 theorems, 73 equations)

This paper contains 6 sections, 13 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Factorization of $\mathcal{L}\left(H\right)$-valued p.d. kernels
  3. Relation to classical dilation theory
  4. $H$-valued Gaussian processes
  5. Completeness, POVMs, quantum states, and quantum information
  6. An inverse problem

Key Result

Theorem 2.1

Let $K:S\times S\rightarrow\mathcal{L}\left(H\right)$ be a p.d. kernel. Then there exists a RKHS $H_{\tilde{K}}$, and a family of operators $V_{s}:H\rightarrow H_{\tilde{K}}$, $s\in S$, such that and Conversely, if there is a Hilbert space $L$ and operators $V_{s}:H\rightarrow L$, $s\in S$, such that and eq:b2 holds, then $L\simeq H_{\tilde{K}}$.

Theorems & Definitions (31)

  • Theorem 2.1: Universal Factorization
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 3.1
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • ...and 21 more