[Paper Review] Classical Information Capacity of the Bosonic Broadcast Channel

Saikat Guha, Jeffrey H. Shapiro|ArXiv.org|Apr 15, 2007

Wireless Communication Security Techniques8 references4 citations

TL;DR

This paper establishes that the classical capacity region of the lossless Bosonic broadcast channel is achieved by coherent-state encoding with coherent detection, reducing it to a classical degraded Gaussian broadcast channel. Under a minimum output-entropy conjecture (Strong Conjecture 2), coherent-state modulation achieves the ultimate classical information capacity, with evidence supporting the conjecture via Gaussian, Wehrl, and Rényi entropy analyses.

ABSTRACT

We show that when coherent-state encoding is employed in conjunction with coherent detection, the Bosonic broadcast channel is equivalent to a classical degraded Gaussian broadcast channel whose capacity region is dual to that of the classical Gaussian multiple-access channel. We further show that if a minimum output-entropy conjecture holds true, then the ultimate classical information capacity of the Bosonic broadcast channel can be achieved by a coherent-state encoding. We provide some evidence in support of the conjecture.

Motivation & Objective

• To determine the classical information capacity region of the Bosonic broadcast channel under coherent-state encoding and coherent detection.
• To establish a duality between the Bosonic broadcast channel and classical Gaussian broadcast channels.
• To prove that the ultimate classical capacity is achievable with coherent-state modulation if a minimum output-entropy conjecture holds.
• To provide analytical and numerical evidence supporting the conjecture, particularly for Gaussian and Wehrl entropy cases.

Proposed method

• The authors model the lossless Bosonic broadcast channel using a beam splitter with transmissivity η, mapping input modes to output modes.
• They show that coherent-state encoding with coherent detection maps the quantum broadcast channel to a classical degraded Gaussian broadcast channel.
• The capacity region is derived under the assumption of Strong Conjecture 2, which posits that thermal input states minimize output von Neumann entropy.
• The analysis uses von Neumann entropy, Holevo information, and duality principles from classical multiple-access and broadcast channels.
• The authors provide evidence for the conjecture using Gaussian states, Wehrl entropy, and Rényi entropy of integer order.
• Numerical comparisons are made between coherent detection (homodyne/heterodyne) and optimum reception, showing coherence-based schemes approach the ultimate capacity.

Experimental results

Research questions

• RQ1 Can the classical capacity region of the Bosonic broadcast channel be fully characterized under coherent-state encoding and coherent detection?
• RQ2 Is there a duality between the Bosonic broadcast channel and classical Gaussian broadcast channels similar to that in classical information theory?
• RQ3 Does the minimum output-entropy conjecture (Strong Conjecture 2) hold for non-Gaussian and general quantum states?
• RQ4 Can coherent-state modulation achieve the ultimate classical capacity of the Bosonic broadcast channel if the conjecture is true?
• RQ5 What is the performance gap between coherent detection and optimum reception in terms of achievable rates?

Key findings

• The classical capacity region of the Bosonic broadcast channel is equivalent to that of a classical degraded Gaussian broadcast channel when coherent-state encoding and coherent detection are used.
• Under Strong Conjecture 2, the ultimate classical information capacity of the Bosonic broadcast channel is achievable with coherent-state modulation.
• Homodyne detection outperforms heterodyne detection at low photon numbers due to lower noise, while heterodyne performs better at high photon numbers due to its two-fold bandwidth advantage.
• Numerical results show that coherent-state encoding with homodyne or heterodyne detection approaches the conjectured ultimate capacity region, especially as photon number increases.
• Evidence supports Strong Conjecture 2 for Gaussian states, Wehrl entropy, and Rényi entropy of integer order, with additional support for Poisson, Binomial, and Bose-Einstein input distributions.
• The duality between classical multiple-access and broadcast channels does not extend to Bosonic channels, indicating a fundamental difference in quantum versus classical capacity regions.