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Classical Capacity of Single-Mode Bosonic Gaussian Channels

This paper proves that for all single-mode bosonic Gaussian channels, the classical capacity is achieved by independently prepared Gaussian states with Gaussian-distributed displacements, thereby confirming the additivity of the energy-constrained Holevo information and characterizing the precise asymptotic trade-off between communication rate and decoding error.

Original authors: Gerardo Adesso, Ludovico Lami, Vittorio Giovannetti

Published 2026-10-16

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CC BY 4.0

Original authors: Gerardo Adesso, Ludovico Lami, Vittorio Giovannetti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Light carries our world's most vital information, from the data streaming through fiber-optic cables to the signals beaming between satellites. Yet, this flow is never perfect. As light travels, it encounters noise from the environment and loses energy, much like a whisper fading in a crowded room. The fundamental question for engineers and physicists has long been how much information can be pushed through these noisy channels before errors become unavoidable. This limit, known as the classical capacity, depends on how we encode the message into the light itself. We can vary the brightness, the timing, or the phase of the waves. But because light is made of particles called photons that follow the strange rules of quantum mechanics, the way we prepare these particles matters immensely. For decades, scientists have wondered if the best way to send information is to use simple, independent pulses of light, or if we could do better by linking the pulses together in complex, entangled patterns that span across many uses of the channel.

A team of researchers has now settled this debate for the most common type of optical link: the single-mode bosonic Gaussian channel. This model describes standard fiber-optic and free-space communication where noise and loss are present. The researchers proved that the most efficient strategy is far simpler than many had hoped. They demonstrated that the maximum rate of reliable communication is achieved not by using complex, entangled signals or exotic non-Gaussian states, but by using independent, prepared pulses of light that follow a specific statistical pattern. Specifically, the optimal method involves taking a standard state of light and shifting its position randomly in a way that follows a bell-curve distribution. This approach works perfectly across all energy levels, whether the signal is very weak or quite strong.

The study also resolved a subtle trade-off that arises when the available energy is low. In these conditions, one might try to "squeeze" the light, a quantum technique that reduces uncertainty in one property of the wave at the expense of another, to match the noise of the channel. The researchers showed exactly how to balance this squeezing against the random shifting of the signal. They found that below a certain energy threshold, the optimal strategy changes: the light should be squeezed in only one direction and shifted randomly in that same direction, rather than in both. Above this threshold, the strategy becomes more flexible, allowing for shifts in both directions. Crucially, they proved that no amount of entanglement between successive uses of the channel can beat this simple, independent strategy. The best way to send a message is to treat each pulse as a separate event, prepared with the right amount of squeezing and random displacement.

Beyond finding the maximum speed, the paper also maps out what happens when we try to push faster than that limit. If we allow for a small, acceptable rate of errors, the researchers derived an exact formula showing how much faster we can go. They found that the capacity with a small error rate is mathematically equivalent to the error-free capacity calculated at a slightly higher energy level. This means that by accepting a few mistakes, we can effectively use more energy to send more data. However, the story changes if we impose a strict limit on the maximum number of photons allowed in any single message. Under this stricter rule, trying to send data faster than the capacity leads to a rapid collapse in performance. The probability of correctly decoding the message drops exponentially as the rate increases, ensuring that there is no way to sneak extra information through by gambling on rare, high-energy events.

These findings provide a complete and exact map for the ultimate limits of classical communication over optical links. They confirm that for these fundamental channels, the complex quantum correlations that often promise advantages in other scenarios offer no benefit here. The optimal transmitter is one that uses independent, Gaussian-distributed signals, and the optimal receiver can be designed to match this simplicity. The results hold true whether the noise is random and independent or correlated across time, provided the noise can be broken down into independent modes. By establishing these limits with mathematical certainty, the work gives engineers a clear target for designing future communication systems, showing exactly how much information can be extracted from light before the noise swallows it whole.

Technical Summary: Classical Capacity of Single-Mode Bosonic Gaussian Channels

Problem Statement
The paper addresses the fundamental problem of determining the classical capacity of single-mode bosonic Gaussian channels under a mean input energy constraint. The classical capacity is defined as the maximum reliable communication rate achievable with vanishing decoding error, allowing for arbitrary quantum signal states and collective measurements. A central open question in this domain has been whether entanglement between successive channel uses or the use of non-Gaussian signal states can outperform the standard strategy of using independently prepared Gaussian states with Gaussian-distributed random displacements. Specifically, the authors investigate whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and whether it coincides with its Gaussian counterpart for all input energies, including the low-energy regime where signal squeezing competes with displacement encoding.

Methodology
The authors employ a rigorous information-theoretic approach grounded in entropy inequalities and the properties of Gaussian states. Their methodology proceeds through several key steps:

Canonical Form Reduction: They consider single-mode Gaussian channels in a canonical form characterized by parameters $k$ (transmissivity/amplification), $y$ (noise strength), and $\xi$ (phase sensitivity). This reduction preserves photon number and Holevo information.
Sharp Output Entropy Bound (Theorem 1): The core technical innovation is the derivation of a sharp, additive lower bound on the output entropy of parallel uses of Gaussian channels. This bound depends solely on the second moments (quadrature variances) of the input state. The proof utilizes an affine entanglement inequality (Proposition 8) and a specific dilation of the channel (Lemma 9) involving symplectic unitaries and auxiliary vacuum modes. This bound is tight for products of squeezed vacuum states.
Moment-Resolved Optimality (Theorem 2): By combining the lower bound on individual output entropies with an upper bound on the entropy of the average output (derived via subadditivity and the Gaussian maximum entropy property), the authors establish that the Holevo information for any ensemble is bounded by a function of the input second moments. This function is shown to be jointly concave.
Optimization and Additivity: The concavity of the objective function allows the reduction of the multi-mode optimization problem to a single-mode optimization over the allocation of the total energy budget. This leads to a "water-filling" solution for correlated noise models where the noise matrices commute.
Rate-Error Trade-off Analysis: The paper extends the capacity analysis to regimes allowing a non-zero average decoding error and scenarios with strict photon-number cutoffs, utilizing relative entropy arguments and energy-filtering techniques to establish strong converse properties.

Key Contributions and Results

Proof of Gaussian Optimality for All Energies: The primary result is the proof that for every single-mode bosonic Gaussian channel and every input energy, the classical capacity is achieved by independently prepared Gaussian states with Gaussian-distributed random displacements. This resolves the additivity conjecture for this class of channels in the affirmative.
- Specifically, neither non-Gaussian inputs nor entanglement between channel uses can increase the communication rate.
- This result holds even in the low-energy regime ( $N < N_{th}$ ), where the optimal Gaussian ensemble differs from the high-energy case: the optimal seed is a squeezed vacuum displaced only in one quadrature, rather than a thermal state with displacements in both.
Exact Classical Capacity Formula: The authors provide an exact expression for the capacity $C(\Phi, N)$ , which coincides with the single-use Gaussian Holevo information $\chi_G(\Phi, N)$ .
- For energies above a threshold $N_{th}$ , the capacity is given by the difference between the entropy of the average output and the minimum output entropy.
- For energies below $N_{th}$ , the capacity is determined by a transcendental optimization involving a squeezed seed displaced in a single quadrature.
Strong Additivity and Energy Allocation: The paper establishes strong additivity for products of single-mode Gaussian channels. The optimal strategy for a collection of channels (or a channel with memory that can be diagonalized) involves dividing the total energy budget among the modes according to a water-filling principle, optimizing each mode independently.
Rate-Error Trade-off (Theorem 6): Under the mean-energy constraint, the paper derives an exact formula for the capacity allowing a specific average decoding error $\epsilon$ . The capacity $C_\epsilon(\Phi, N)$ is exactly equal to the vanishing-error capacity at a rescaled energy: $C(\Phi, N/(1-\epsilon))$ . This determines the entire asymptotic rate-error trade-off for a fixed average error constraint.
Strong Converse under Photon-Number Cutoff (Proposition 7): While a mean-energy constraint alone does not guarantee a strong converse (success probability does not necessarily vanish exponentially above capacity), the authors prove that under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially for any rate above the capacity. They provide explicit bounds on the decay exponent.
Correlated Noise Models: The results are extended to channels with correlated Gaussian noise (memory channels). If the position and momentum noise matrices commute, the channel decomposes into independent normal modes, and the capacity is determined by the water-filling formula over these modes.

Significance and Claims
The paper claims to identify the "ultimate limits of classical communication" for fundamental models of optical communication. By proving that Gaussian encodings are optimal across the entire energy spectrum, the work settles a long-standing question regarding the necessity of non-Gaussian strategies or entangled inputs for bosonic Gaussian channels.

The authors emphasize that their results:

Unify the understanding of capacity across high and low energy regimes, clarifying the transition in optimal signal structure (from thermal-like to single-quadrature squeezed).
Resolve the additivity problem for single-mode Gaussian channels, confirming that the regularized capacity equals the single-use Gaussian capacity.
Characterize the reliability of communication beyond capacity, providing exact rate-error trade-offs for a fixed average error and establishing strong converse properties under physically motivated constraints (photon-number cutoffs).
Provide a benchmark for assessing the efficiency of optical transmitters and the cost of reliability in optical links.

The paper does not propose new experimental setups or applications but rather provides the theoretical foundation and exact limits for existing optical communication models, including fiber and free-space links. It explicitly notes that while the capacity is now known, the problem of determining sharp converse exponents (the optimal decay rate of error) remains an open quantitative problem.

Technical Summary: Classical Capacity of Single-Mode Bosonic Gaussian Channels

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