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Interpolation for completely positive maps: numerical solutions

Calin-Grigore Ambrozie, Aurelian Gheondea

TL;DR

This work addresses the problem of interpolating completely positive maps $\varphi\in\mathrm{CP}(M_n,M_k)$ that satisfy prescribed data $\varphi(A_\nu)=B_\nu$ by recasting the search in terms of the Choi matrix $\Phi_\varphi$. It develops two computational routes: (i) semidefinite programming on a PSD variable $X$ with linear trace constraints, and (ii) a convex minimization framework using $V(x)=\mathrm{tr}(e^{\sum x_\iota C(\iota)})-\sum x_\iota b_\iota$ whose critical point yields a positive solution; a numerical example demonstrates the approach. The paper also provides a linear-functional characterization of solvability and shows a commutative-data case reduces to linear programming, linking to prior results. Collectively, the methods offer practical tools to construct CP maps (and trace-preserving channels) matching prescribed data, with implications for quantum channel interpolation and design.

Abstract

We present certain techniques to find completely positive maps between matrix algebras that take prescribed values on given data. To this aim we describe a semidefinite programming approach and another convex minimization method supported by a numerical example.

Interpolation for completely positive maps: numerical solutions

TL;DR

This work addresses the problem of interpolating completely positive maps that satisfy prescribed data by recasting the search in terms of the Choi matrix . It develops two computational routes: (i) semidefinite programming on a PSD variable with linear trace constraints, and (ii) a convex minimization framework using whose critical point yields a positive solution; a numerical example demonstrates the approach. The paper also provides a linear-functional characterization of solvability and shows a commutative-data case reduces to linear programming, linking to prior results. Collectively, the methods offer practical tools to construct CP maps (and trace-preserving channels) matching prescribed data, with implications for quantum channel interpolation and design.

Abstract

We present certain techniques to find completely positive maps between matrix algebras that take prescribed values on given data. To this aim we describe a semidefinite programming approach and another convex minimization method supported by a numerical example.

Paper Structure

This paper contains 9 sections, 3 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The Choi matrix.
  4. Equivalent setting of the problem.
  5. Solutions by means of semidefinite programming.
  6. Solutions via a convex minimization technique.
  7. A Numerical Example.
  8. Characterization in terms of linear functionals.
  9. The case of commutative data

Key Result

Theorem 2.1

ca. The system of equations (transl) admits solutions $X>0$ if and only if the function $V$ defined by (vv) has a critical point (of minimum), that is, if and only if $\lim_{\| x\| \to \infty }V(x)=+\infty$. In this case, (transl) has also the (positive) particular solution where $x^0 =(x_{\iota}^0 )_\iota$ is the critical point of $V$.

Theorems & Definitions (8)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof