We prove the entropy photon-number inequality for two independent bosonic inputs with finite mean energy, for every finite number of modes. This resolves the entropy photon-number conjecture in this finite-energy setting while allowing arbitrary entanglement among the modes within either input. As a consequence, we determine the exact minimum output entropy at fixed input entropy for every finite tensor power of an identical thermal attenuator, over finite-energy inputs that may be entangled across modes. We also obtain the exact classical capacity region of the degraded two-receiver pure-loss bosonic broadcast channel under a mean photon constraint.
We prove the entropy photon-number inequality for two independent bosonic inputs with finite mean energy, for every finite number of modes. This resolves the entropy photon-number conjecture in this finite-energy setting while allowing arbitrary entanglement among the modes within either input. As a consequence, we determine the exact minimum output entropy at fixed input entropy for every finite tensor power of an identical thermal attenuator, over finite-energy inputs that may be entangled across modes. We also obtain the exact classical capacity region of the degraded two-receiver pure-loss bosonic broadcast channel under a mean photon constraint.
Abstract How much does a nearly minimal output entropy reveal about the quantum states entering an amplifier? We prove the active entropy photon-number inequality for two independent multimode inputs, each with arbitrary internal correlations, and characterise every equality case. Writing 𝒩n(ρ) = g−1(S(ρ)/n), where g(x) = (x + 1) log(x + 1) − x log x, the output obeys 𝒩n(C) ≥ G𝒩n(A) + (G − 1)[𝒩n(B) + 1]. Equality requires Gaussian inputs with one common symplectic shape, reflected on the creation-operator port. At fixed finite mode number and input energy, a power of the actual output gap bounds the full trace distance to a compatible Gaussian pair with the original means and entropies. For a pure one-mode signal and a specified Gaussian idler, a Wehrl-entropy bridge gives a square-root bound. The inequality also fixes the exact constrained output-entropy minimum of every finite block of thermal amplifiers. For unequal one-mode Gaussian channels, the optimisation over all entangled and non-Gaussian block inputs reduces to scalar entropy allocation, with a sharp threshold at which a second mode becomes active. These results determine both the optimal output and the input structure that can approach it, turning an entropy benchmark into a quantitative test of source models. Version 1.2 — 8 October 2026 This revision develops the input-rigidity theory: near saturation of the active EPnI now constrains both independent inputs jointly. Complete equality classification. For two independent finite-energy multimode inputs, equality is characterised by a common symplectic Gaussian shape, with reflection on the creation-operator port. Each input may contain arbitrary internal correlations. Quantitative stability for arbitrary inputs. At fixed finite mode number, gain and input-energy bound, a power-law estimate controls the full trace distance to one compatible Gaussian pair matching the original means and entropies. The theorem includes pure and nonfaithful states and states with infinite Fock support. A stronger one-mode bound. For a pure signal and a specified Gaussian idler, which may be mixed, a Wehrl-entropy bridge gives a square-root estimate and an explicit sufficient tolerance for the input error. Expanded proofs and figures. New appendices provide the complete arguments and explicit constants. A new vector figure connects the equality geometry with the quantitative input-error bound. The active EPnI, exact finite-block thermal-amplifier minima and exact scalar entropy allocation for heterogeneous Gaussian channels are retained. The 49-page PDF contains the complete manuscript, proofs, four figures and references.
Determining the ultimate classical information carrying capacity of electromagnetic waves requires quantum-mechanical analysis to properly account for the bosonic nature of these waves. Recent work has established capacity theorems for bosonic single-user and broadcast channels, under the presumption of two minimum output entropy conjectures. Despite considerable accumulated evidence that supports the validity of these conjectures, they have yet to be proven. In this paper, it is shown that the second conjecture suffices to prove the classical capacity of the bosonic wiretap channel, which in turn would also prove the quantum capacity of the lossy bosonic channel. The preceding minimum output entropy conjectures are then shown to be simple consequences of an entropy photon-number inequality (EPnl), which is a conjectured quantum-mechanical analog of the entropy power inequality (EPI) from classical information theory.
The bosonic quantum entropy power inequality bounds the entropy of a beam-splitter output in terms of the input entropies. We determine its complete equality class among independent inputs of finite mean photon number: for any number of modes and any transmissivity strictly between zero and one, equality in either the linear or the exponential form holds if and only if the inputs are Gaussian with the same covariance matrix, allowing arbitrary displacements. The main step shows that equality for one output forces the two outputs to be independent: a thermal auxiliary converts equality into preservation of mutual information by a fixed noisy channel, and a two-outcome instrument shows that preservation of mutual information requires a product state with the reference. The quantum Darmois--Skitovich theorem then gives Gaussianity, while a central-limit argument reduces exponential equality to linear equality.
Determining the ultimate classical information carrying capacity of electromagnetic waves requires quantum-mechanical analysis to properly account for the bosonic nature of these waves. Recent work has established capacity theorems for bosonic single-user, broadcast, and wiretap channels, under the presumption of two minimum output entropy conjectures. Despite considerable accumulated evidence that supports the validity of these conjectures, they have yet to be proven. Here we show that the preceding minimum output entropy conjectures are simple consequences of an entropy photon-number inequality, which is a conjectured quantum-mechanical analog of the entropy power inequality from classical information theory.
Previous work on the classical information capacities of bosonic channels has established the capacity of the single-user pure-loss channel, bounded the capacity of the single-user thermal-noise channel, and bounded the capacity region of the multiple-access channel. The latter is a multiple-user scenario in which several transmitters seek to simultaneously and independently communicate to a single receiver. We study the capacity region of the bosonic broadcast channel, in which a single transmitter seeks to simultaneously and independently communicate to two different receivers. It is known that the tightest available lower bound on the capacity of the single-user thermal-noise channel is that channel's capacity if, as conjectured, the minimum von Neumann entropy at the output of a bosonic channel with additive thermal noise occurs for coherent-state inputs. Evidence in support of this minimum output entropy conjecture has been accumulated, but a rigorous proof has not been obtained. We propose a minimum output entropy conjecture that, if proved to be correct, will establish that the capacity region of the bosonic broadcast channel equals the inner bound achieved using a coherent-state encoding and optimum detection. We provide some evidence that supports this conjecture, but again a full proof is not available.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number N and allowed average decoding error ε equals the vanishing-error capacity at mean photon number N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.