[Paper Review] Interstellar Communication. VIII. Hard limits on the number of bits per photon

Michael Hippke|arXiv (Cornell University)|Jan 18, 2018

Dark Matter and Cosmic Phenomena5 references3 citations

TL;DR

This paper establishes fundamental quantum limits on information encoding per photon in interstellar communication using Heisenberg’s uncertainty principle and Holevo’s bound. It finds that practical systems are limited to ~59 bits per photon due to physical constraints, rising to ~171 bits per photon at Planck energy, with trade-offs between information efficiency and data rate.

ABSTRACT

A photon can encode several bits of information based on an alphabet of its time of arrival, energy, and polarization. Heisenberg's uncertainty principle places a limit on measuring pairs of physical properties of a particle, limiting the maximal information efficiency to <59 bits per photon in practice, and <171 bits per photon at Planck energy, at a data rate of one photon per second.

Motivation & Objective

• To determine the ultimate physical limits on information encoding per photon in interstellar communication.
• To analyze how quantum mechanics, particularly Heisenberg’s uncertainty principle, constrains photon information efficiency.
• To evaluate the impact of optical system limits (e.g., surface smoothness, diffraction) and interstellar propagation effects on achievable data rates.
• To quantify the trade-off between information efficiency (bits per photon) and dimensional information efficiency (bits per mode) for data rate optimization.
• To assess the role of quantum measurement limits, such as the Helstrom bound, in determining minimum bit error rates.

Proposed method

• Applies Heisenberg’s uncertainty principle (ΔEΔt ≥ ħ/2) to derive the minimum temporal width Δt_min of a photon pulse based on its spectral bandwidth Δλ and central wavelength λ₀.
• Uses the Fourier transform relationship for Gaussian pulses to model Δt_min ≈ 0.5λ₀² / (Δλ c), with K ≈ 0.5 for practical estimation.
• Derives the number of temporal modes M per photon using M = λ₀ / (2c t_dura), where t_dura is the pulse duration.
• Applies the Holevo bound for quantum information efficiency (PIE = g(ηM)), approximating PIE ≈ log₂(2c t_dura / λ₀) for high M and ideal conditions.
• Incorporates polarization as an additional degree of freedom, increasing PIE by 1 bit per photon when two states are used.
• Evaluates bit error rates using the Helstrom bound, P_Helstrom = ½(1 − √(1 − e^−4/PIE)), to determine the quantum limit on error performance.

Experimental results

Research questions

• RQ1 What is the maximum number of bits that can be encoded per photon under the constraints of quantum mechanics and physical optics?
• RQ2 How do the Heisenberg uncertainty principle and Planck-scale limits affect the minimum pulse width and thus information capacity per photon?
• RQ3 What is the trade-off between information efficiency (bits per photon) and dimensional information efficiency (bits per mode) in interstellar communication systems?
• RQ4 How do interstellar propagation effects such as dispersion and scattering limit usable bandwidth and pulse duration?
• RQ5 What is the fundamental lower bound on bit error rate for photon-based interstellar communication, and how close can practical receivers get to this limit?

Key findings

• The maximum practical information efficiency is limited to approximately 59 bits per photon due to quantum and physical constraints, assuming one photon per second and λ₀ = 1 nm.
• At Planck energy (λ₀ ≈ 10⁻³⁴ m), the theoretical upper bound reaches ~171 bits per photon, though such conditions are physically unattainable with current technology.
• The use of polarization as an additional degree of freedom increases PIE by 1 bit per photon, doubling the number of distinguishable states.
• Pulse duration and wavelength are critical: for a 1-second pulse at 1 nm wavelength, PIE ≈ 59 bits/photon, derived from Δt_min and M = λ₀ / (2c t_dura).
• Interstellar dispersion limits pulse width to Δt_disp > 10⁻⁸ s at meter wavelengths, which is stricter than the uncertainty limit (≈10⁻¹⁰ s), thus dominating bandwidth constraints.
• The Helstrom bound sets a fundamental lower limit on bit error rate, with practical receiver implementations converging toward this limit, and forward error correction adding only a few percent overhead.