[Paper Review] Optimization of Quantum Information Processing Maximizing Mutual Information

V. P. Belavkin, R. L. Stratonovich|ArXiv.org|Nov 4, 2005

Quantum Information and Cryptography3 citations

TL;DR

This paper derives the optimal quantum decoding strategy for Gaussian quantum channels by maximizing classical mutual information through quasi-measurements. It proves that coherent state representations form the optimal basis for indirect (heterodyne) measurement, achieving the information capacity $\mathsf{I} = \mathrm{Sp}\ln\big{[}1 + S/(N+1)\big{]}$, which matches the classical Gaussian channel capacity with effective noise $N+I$. This establishes the optimality of heterodyne detection in quantum communication systems with additive noise.

ABSTRACT

A model of quantum noisy channel with input encoding by a classical random vector is described. An equation of optimality is derived to determine a complete set of wave functions describing quantum decodings based on quasi-measurements maximizing the classical amount of transmitted information. A solution of this equation is found for the Gaussian multimode case with input Gaussian distribution. It is described by the overcomplete family of coherent vectors describing an optimal quasimeasurement of the canonical annihilation amplitudes in the output Hilbert space. It is found that the optimal decoding in this case realizes the maximum amount I=Spln[1+S/(N+1)] of the classical information as transmitted via the classical Gaussian channel with the effective noise covariance matrix N+I. A physical realization of optimal quasi-measurement based on an indirect (heterodyne) observation of the canonical operators is suggested.

Motivation & Objective

• To determine the optimal quantum decoding strategy that maximizes classical information transmission in noisy quantum channels.
• To resolve whether coherent state-based quasi-measurements are optimal for decoding coherently modulated signals.
• To derive a general equation of optimality for non-orthogonal quantum measurements in the context of classical-quantum channels.
• To provide a physical realization of optimal decoding using indirect (heterodyne) measurements of canonical operators.
• To establish the information capacity of Gaussian quantum channels under optimal decoding, comparing it with classical limits.

Proposed method

• Derives an optimality equation for quantum decoding based on quasi-measurements using operator-valued positive measures $\Pi(\mathrm{d}\beta)$.
• Represents the quasi-measurement as $\Pi(\mathrm{d}\beta) = \varphi_\beta \varphi_\beta^* \mu(\mathrm{d}\beta)$, where $\{\varphi_\beta\}$ are non-orthogonal wave functions in Hilbert space.
• Applies Bayesian risk criteria to formulate optimal information processing in classical-quantum channels with additive noise.
• Solves the optimality equation for the Gaussian multimode case with input Gaussian distribution, yielding coherent states as the optimal basis.
• Demonstrates that optimal decoding corresponds to indirect measurement of canonical annihilation amplitudes via heterodyne detection.
• Validates the solution using Gaussian integral identities and trace operations, confirming consistency with known information capacity formulas.

Experimental results

Research questions

• RQ1 Is the decoding based on coherent state quasi-measurements optimal for Gaussian quantum channels?
• RQ2 What is the complete set of wave functions that maximizes mutual information in a quantum noisy channel with classical input encoding?
• RQ3 Can the optimal quasi-measurement be physically realized through indirect (heterodyne) observation of non-commuting observables?
• RQ4 What is the maximum classical information capacity achievable in a Gaussian multimode quantum channel under optimal decoding?
• RQ5 How does the quantum information capacity compare to the classical Gaussian channel capacity in the limit of low or high signal-to-noise ratios?

Key findings

• The optimal decoding for Gaussian multimode quantum channels is achieved by an overcomplete family of coherent states $|\beta\rangle$, which realize an ideal quasi-measurement.
• The maximum mutual information transmitted is $\mathsf{I} = \mathrm{Sp}\ln\big{[}1 + S/(N+1)\big{]}$, equivalent to the classical Gaussian channel with effective noise covariance matrix $N+I$.
• The optimal quasi-measurement is physically realizable via indirect heterodyne detection of canonical annihilation amplitudes in the output Hilbert space.
• The solution confirms the optimality of heterodyne detection in quantum communication, resolving prior uncertainty about its performance relative to theoretical limits.
• In the classical limit ($h\nu/\theta \ll 1$), the quantum capacity reduces to the classical expression $\mathsf{I} = \int \ln(1 + \sigma_\nu^2 / \theta) \, \mathrm{d}\nu$, validating consistency with classical information theory.
• For weak signals ($s_\nu \ll 1$), quantum corrections become significant, and the capacity scales as $\mathsf{I} \approx \int (1 - e^{-h\nu/\theta}) s_\nu \, \mathrm{d}\nu$, showing non-classical behavior at low temperatures.