[Paper Review] Private-Capacity Bounds for Bosonic Wiretap Channels

Ligong Wang, Jeffrey H. Shapiro+2 more|arXiv (Cornell University)|Feb 6, 2012

Quantum Information and Cryptography8 references3 citations

TL;DR

This paper establishes tight upper and lower bounds on the private capacity of single-mode and multiple-mode bosonic wiretap channels, proving that the private capacity scales as $ g((2 heta - 1)ar{n}) $ in the low-photon-number regime. It further derives a private-capacity lower bound for atmospheric turbulence channels using only second moments of the channel matrix, enabling practical estimation under near-field conditions.

ABSTRACT

We prove an upper bound on the private capacity of the single-mode noiseless bosonic wiretap channel. Combined with a previous lower bound, we obtain the low photon-number asymptotic expression for the private capacity. We then show that the multiple-mode noiseless bosonic wiretap channel is equivalent to parallel single-mode channels, hence the single-mode bounds can be applied. Finally, we consider multiple-spatial-mode propagation through atmospheric turbulence, and derive a private-capacity lower bound that only requires second moments of the channel matrix.

Motivation & Objective

• To derive a tight upper bound on the private capacity of the single-mode noiseless bosonic wiretap channel.
• To establish the low-photon-number asymptotic behavior of the private capacity by combining upper and lower bounds.
• To extend single-mode results to multiple-mode channels by showing equivalence to parallel single-mode channels.
• To derive a private-capacity lower bound for multiple-spatial-mode propagation through atmospheric turbulence using only second moments of the channel matrix.
• To enable practical capacity estimation in near-field free-space optical communication under turbulence, with minimal channel state information.

Proposed method

• Proves an upper bound on the private capacity $ C_{ ext{P}}( heta,ar{n}) riangleq U( heta,ar{n}) = g((2 heta - 1)ar{n}) $ for $ heta > 1/2 $, using convexity and majorization arguments.
• Leverages the previously known lower bound $ L( heta,ar{n}) = g( hetaar{n}) - g((1- heta)ar{n}) $ for $ heta > 1/2 $, and combines it with the new upper bound to determine the asymptotic behavior at low $ ar{n} $.
• Demonstrates that the multiple-mode noiseless bosonic wiretap channel is equivalent to parallel single-mode channels, allowing direct application of single-mode bounds.
• Uses convexity and Schur-convexity of the lower bound function $ L^{ ext{M}}(oldsymbol{ heta},ar{n}) $ to derive a lower bound under turbulence.
• Applies majorization theory to show that the eigenvalues of $ ext{E}[oldsymbol{T}_{ab}^ op oldsymbol{T}_{ab}] $ majorize its diagonal elements, enabling a bound based only on second moments.
• Derives $ C_{ ext{P}}^{ ext{M}}(oldsymbol{T},ar{n}) riangleq ext{E}[L^{ ext{M}}(oldsymbol{H},ar{n})] riangleq ext{E}[L^{ ext{M}}(oldsymbol{M},ar{n})] riangleq L^{ ext{M}}(oldsymbol{ heta},ar{n}) $, where $ oldsymbol{ heta} $ are the eigenvalues of $ ext{E}[oldsymbol{T}_{ab}^ op oldsymbol{T}_{ab}] $.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the private capacity of the single-mode bosonic wiretap channel at low photon numbers?
• RQ2 Can a tight upper bound on the private capacity be derived for the single-mode noiseless bosonic wiretap channel?
• RQ3 How does the private capacity scale in multiple-mode bosonic wiretap channels under high spectral and photon efficiency?
• RQ4 What is the private capacity lower bound for multiple-spatial-mode free-space optical links affected by atmospheric turbulence?
• RQ5 Can the private capacity lower bound be expressed using only second-order statistics of the channel matrix?

Key findings

• The private capacity of the single-mode noiseless bosonic wiretap channel is bounded above by $ U( heta,ar{n}) = g((2 heta - 1)ar{n}) $ for $ heta > 1/2 $, and this bound is tight in the low-photon-number regime.
• The combination of the new upper bound and the known lower bound $ L( heta,ar{n}) $ yields the exact low-photon-number asymptotic private capacity: $ C_{ ext{P}}( heta,ar{n}) o g((2 heta - 1)ar{n}) $ as $ ar{n} o 0 $.
• The multiple-mode noiseless bosonic wiretap channel is equivalent to parallel single-mode channels, so the single-mode bounds apply directly to the multiple-mode case.
• For multiple-spatial-mode propagation through atmospheric turbulence, a private-capacity lower bound is derived that depends only on the eigenvalues of $ ext{E}[oldsymbol{T}_{ab}^ op oldsymbol{T}_{ab}] $, i.e., second moments of the channel matrix.
• The derived lower bound $ C_{ ext{P}}^{ ext{M}}(oldsymbol{T},ar{n}) riangleq L^{ ext{M}}(oldsymbol{ heta},ar{n}) $ is tightest when $ oldsymbol{ heta} $ are the eigenvalues of $ ext{E}[oldsymbol{T}_{ab}^ op oldsymbol{T}_{ab}] $, and can be evaluated for focused-beam, Hermite-Gaussian, or Laguerre-Gaussian modes.
• The lower bound is valid regardless of the basis used to represent the channel matrix, making it robust and applicable to various spatial-mode configurations in free-space optical communication.