[Paper Review] On extreme Bosonic linear channels

A. S. Holevo|arXiv (Cornell University)|Nov 15, 2011

Spectral Theory in Mathematical Physics4 citations

TL;DR

This paper establishes that a Bosonic linear (quasi-free) channel is extreme if and only if its environment is pure, under nondegeneracy conditions. The key result is that a Gaussian channel between finite-mode Bosonic systems is extreme precisely when it has minimal noise, generalizing a one-mode conjecture by Ivan, Sabapathy, and Simon.

ABSTRACT

The set of all channels with fixed input and output is convex. We first give a convenient formulation of necessary and sufficient condition for a channel to be extreme point of this set in terms of complementary channel, a notion of big importance in quantum information theory. This formulation is based on the general approach to extremality of completely positive maps in an operator algebra due to Arveson. We then apply this formulation to prove the main result of this note: under certain nondegeneracy conditions, purity of the environment is necessary and sufficient for extremality of Bosonic linear (quasi-free) channel. It follows that Gaussian channel between finite-mode Bosonic systems is extreme if and only if it has minimal noise.

Motivation & Objective

• To characterize extreme points in the convex set of Bosonic linear channels using the complementary channel formalism.
• To extend the extremality criterion from general completely positive maps (via Arveson's theory) to the specific case of Bosonic Gaussian channels.
• To resolve a conjecture that one-mode Gaussian channels are extreme if and only if they have minimal noise.
• To establish a necessary and sufficient condition for extremality of Bosonic linear channels in terms of environment purity and noise minimality.

Proposed method

• Utilizes Arveson's characterization of extremality for completely positive maps in a C*-algebra framework.
• Applies the concept of complementary channels via Stinespring dilation, defining a channel and its complementary map in terms of an isometry $ V $.
• Establishes that a channel $ ilde{ ho} $ is extreme if and only if its complementary channel $ ilde{ ilde{ ho}} $ has trivial kernel, i.e., $ \mathrm{Ker}\tilde{\tilde{\rho}} = 0 $.
• Analyzes Gaussian channels via their characteristic functions and Weyl operators, expressing the channel as $ \Phi_{K,l,\mu}[W_B(z_B)] = W(Kz_B) \exp[il^T z_B - \frac{1}{2}z_B^T \mu z_B] $.
• Uses the covariance matrix formalism and Williamson's canonical form to analyze purity of Gaussian states and minimality of noise.
• Applies the duality relation $ \hat{K} = K^{-1}, \hat{l} = -(K^{-1})^T l, \hat{\mu} = (K^{-1})^T \mu K^{-1} $ to relate channel representations under invertible $ K $.

Experimental results

Research questions

• RQ1 Under what conditions is a Bosonic linear channel an extreme point in the convex set of all such channels?
• RQ2 Is purity of the environment necessary and sufficient for extremality of a Bosonic linear channel?
• RQ3 Does a Gaussian channel between finite-mode Bosonic systems have minimal noise if and only if it is extreme?
• RQ4 Can the one-mode conjecture by Ivan, Sabapathy, and Simon be generalized to multi-mode systems?
• RQ5 How does the structure of the complementary channel relate to extremality in the context of Gaussian channels?

Key findings

• A Bosonic linear channel is extreme if and only if its environment is pure, provided nondegeneracy conditions hold.
• For Gaussian channels between finite-mode Bosonic systems, extremality is equivalent to having minimal noise, i.e., $ \mu $ being a minimal solution to the positivity inequality $ \mu \geq \frac{i}{2}[\Delta_B - K^T \Delta_A K] $.
• The minimal noise condition implies that the corresponding dual state in the environment has a covariance matrix $ \alpha_D $ that is minimal in the sense of the partial order on symmetric matrices.
• A Gaussian state is pure if and only if its covariance matrix $ \alpha $ satisfies $ \alpha \geq \frac{i}{2}\Delta $ with equality in the minimal solution sense, i.e., all symplectic eigenvalues are $ \frac{1}{2} $.
• The duality relation $ (\hat{K}, \hat{l}, \hat{\mu}) = (K^{-1}, -(K^{-1})^T l, (K^{-1})^T \mu K^{-1}) $ generalizes the known duality for one-mode channels.
• The extremality criterion via kernel triviality of the complementary channel $ \tilde{\Phi} $, i.e., $ \mathrm{Ker}\tilde{\Phi} = 0 $, is equivalent to the channel being extreme.