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What is a Gaussian channel, and when is it physically implementable using a multiport interferometer?

Repana Devendra, Tiju Cherian John, K. Sumesh

TL;DR

This work provides a rigorous unification of three common definitions of quantum Gaussian channels and derives a precise parameterization by matrices $X$ and $Y$, subject to the positivity condition $Y+i(J_{2d}-X^{T}J_{2d}X)\ge 0$, thereby linking abstract continuous-variable channel theory to physical implementations. It proves the equivalence of multiple characterizations of Gaussian channels, including action on Gaussian states, dual action on Weyl operators, and Gaussian dilation via unitary coupling to an environment, with a Stinespring-style realization. The paper then characterizes when such channels can be implemented using linear-optical multiport interferometers, giving necessary and sufficient conditions in terms of orthosymplectic dilations and specific structure of the $X,Y$ parameters, and resolving questions posed by Parthasarathy. Collectively, these results ground Gaussian-channel theory in concrete optical architectures, guiding design and analysis of continuous-variable quantum information processing.

Abstract

Quantum Gaussian channels are fundamental models for communication and information processing in continuous-variable quantum systems. This work addresses both foundational aspects and physical implementation pathways for these channels. Firstly, we provide a rigorous, unified framework by formally proving the equivalence of three principal definitions of quantum Gaussian channels prevalent in the literature, consolidating theoretical understanding. Secondly, we investigate the physical realization of these channels using multiport interferometers, a key platform in quantum optics. The central research contribution is a precise characterization of the channel parameters that correspond to Gaussian channels physically implementable via linear optical multiport interferometers. This characterization bridges the abstract mathematical description with concrete physical architectures. Along the way, we also resolve some questions posed by Parthasarathy (Indian J. Pure Appl. Math. 46, (2015)).

What is a Gaussian channel, and when is it physically implementable using a multiport interferometer?

TL;DR

This work provides a rigorous unification of three common definitions of quantum Gaussian channels and derives a precise parameterization by matrices and , subject to the positivity condition , thereby linking abstract continuous-variable channel theory to physical implementations. It proves the equivalence of multiple characterizations of Gaussian channels, including action on Gaussian states, dual action on Weyl operators, and Gaussian dilation via unitary coupling to an environment, with a Stinespring-style realization. The paper then characterizes when such channels can be implemented using linear-optical multiport interferometers, giving necessary and sufficient conditions in terms of orthosymplectic dilations and specific structure of the parameters, and resolving questions posed by Parthasarathy. Collectively, these results ground Gaussian-channel theory in concrete optical architectures, guiding design and analysis of continuous-variable quantum information processing.

Abstract

Quantum Gaussian channels are fundamental models for communication and information processing in continuous-variable quantum systems. This work addresses both foundational aspects and physical implementation pathways for these channels. Firstly, we provide a rigorous, unified framework by formally proving the equivalence of three principal definitions of quantum Gaussian channels prevalent in the literature, consolidating theoretical understanding. Secondly, we investigate the physical realization of these channels using multiport interferometers, a key platform in quantum optics. The central research contribution is a precise characterization of the channel parameters that correspond to Gaussian channels physically implementable via linear optical multiport interferometers. This characterization bridges the abstract mathematical description with concrete physical architectures. Along the way, we also resolve some questions posed by Parthasarathy (Indian J. Pure Appl. Math. 46, (2015)).
Paper Structure (9 sections, 5 theorems, 76 equations)

This paper contains 9 sections, 5 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symplectic and Orthosymplectic Matrices
  4. Weyl Representation and Bogoliubov Transformations
  5. Gaussian States
  6. Gaussian Channels
  7. Characterization of Linear Optical Channels
  8. Conclusion
  9. Supplementary Analysis of Existing Literature

Key Result

Proposition 2.2

Let $\rho = \rho_{(\mathbf{m}, S)}$ be a Gaussian state on $\Gamma(\mathbb{C}^{d})$. If $\mathbf{u}\in \mathbb{C}^{d}$ and $L$ is a symplectic transformation of $\mathbb{C}^d$, then In particular, if $L$ is a unitary operator then

Theorems & Definitions (22)

  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5: KRP12john2018infinite
  • proof
  • proof
  • Corollary 3.1
  • proof
  • Proposition 3.2
  • ...and 12 more