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Thermal balance and the amplifier entropy photon-number inequality

Harjas Sandhu

Published Oct 21, 2026• 0 citations
Publisher

Abstract

What does an amplifier produce when there is nothing to amplify? Give a quantum amplifier two empty inputs and the output you keep is no longer empty. The energy comes from the pump. The extra contribution has a quantum origin: an amplifier needs a second input to preserve the rules that its output must obey, and that input matters even when it starts in the vacuum. For thermal inputs, the answer can be calculated exactly. At gain 2, an input with one photon per mode and a second input in the vacuum produce a thermal output with three photons per mode. But this calculation leaves a more interesting question: if we keep the input entropies fixed and allow more complicated quantum states, can we make the output entropy smaller? The amplifier entropy photon-number inequality, conjectured by De Palma, Mari and Giovannetti, says that thermal inputs already give the lower bound. Entropy photon number expresses entropy on a thermal scale: it is the occupation a thermal state would need to have the same entropy per mode. It need not equal the actual photon number of the state being studied. This manuscript presents a proof candidate for the inequality with two independent, finite-energy input groups. The modes within either group may be correlated or entangled. The proposed bound is sharp, with equality attained by thermal inputs. The argument brings together two descriptions of the same amplifier. One comes from the geometry of entropy, the other from relaxation towards thermal equilibrium. The amplifier’s second input changes the signs in both. At a regularised entropy minimum, the proposed argument pairs those changes so that the energy corrections cancel; a trace-norm thermal Stein estimate then helps close the finite-energy argument. The paper also presents a Gaussian spectral refinement and equality analysis, and a bound on correlations between the two outputs conditional on the general inequality. The Gaussian-input inequality itself is established prior work. The variational framework and four-weight interpolation are adapted from OpenAI’s beam-splitter argument.

DOI
10.5281/zenodo.23526643

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What does an amplifier produce when there is nothing to amplify? Give a quantum amplifier two empty inputs and the output you keep is no longer empty. The energy comes from the pump. The extra contribution has a quantum origin: an amplifier needs a second input to preserve the rules that its output must obey, and that input matters even when it starts in the vacuum. For thermal inputs, the answer can be calculated exactly. At gain 2, an input with one photon per mode and a second input in the vacuum produce a thermal output with three photons per mode. But this calculation leaves a more interesting question: if we keep the input entropies fixed and allow more complicated quantum states, can we make the output entropy smaller? The amplifier entropy photon-number inequality, conjectured by De Palma, Mari and Giovannetti, says that thermal inputs already give the lower bound. Entropy photon number expresses entropy on a thermal scale: it is the occupation a thermal state would need to have the same entropy per mode. It need not equal the actual photon number of the state being studied. This manuscript presents a proof candidate for the inequality with two independent, finite-energy input groups. The modes within either group may be correlated or entangled. The proposed bound is sharp, with equality attained by thermal inputs. The argument brings together two descriptions of the same amplifier. One comes from the geometry of entropy, the other from relaxation towards thermal equilibrium. The amplifier’s second input changes the signs in both. At a regularised entropy minimum, the proposed argument pairs those changes so that the energy corrections cancel; a trace-norm thermal Stein estimate then helps close the finite-energy argument. The paper also presents a Gaussian spectral refinement and equality analysis, and a bound on correlations between the two outputs conditional on the general inequality. The Gaussian-input inequality itself is established prior work. The variational framework and four-weight interpolation are adapted from OpenAI’s beam-splitter argument.
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.
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.
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.
Recently de Palma et al. [IEEE Trans. Inf. Theory 63, 728 (2017)] proved---using Lagrange multiplier techniques---that under a non-zero input entropy constraint, a thermal state input minimizes the output entropy of a pure-loss bosonic channel. In this note, we present our attempt to generalize this result to all single-mode gauge-covariant Gaussian channels by using similar techniques. Unlike the case of the pure-loss channel, we cannot prove that the thermal input state is the only local extremum of the optimization problem. %It is unclear to us why the same techniques do not lead to a proof for amplifier channels. However, we do prove that, if the conjecture holds for gauge-covariant Gaussian channels, it would also hold for gauge-contravariant Gaussian channels. The truth of the latter leads to a solution of the triple trade-off and broadcast capacities of quantum-limited amplifier channels. We note that de Palma et al. [Phys. Rev. Lett. 118, 160503 (2017)] have now proven the conjecture for all single-mode gauge-covariant Gaussian channels by employing a different approach from what we outline here. Proving a multi-mode generalization of de Palma et al.'s above mentioned result---i.e., given a lower bound on the von Neumann entropy of the input to an nnn-mode lossy thermal-noise bosonic channel, an nnn-mode product thermal state input minimizes the output entropy---will establish an important special case of the conjectured Entropy Photon-number Inequality (EPnI). The EPnI, if proven true, would take on a role analogous to Shannon's EPI in proving coding theorem converses involving quantum limits of classical communication over bosonic channels.
Abstract: Two-way classical assistance does not increase the energy-constrained quantum or private capacity of the quantum-limited bosonic amplifier. We establish this optimality through a matching converse for adaptive protocols with arbitrary separable quantum memories. For amplification gain κ>1\kappa>1κ>1 and an unconditional mean input photon number bounded by NNN, quantum transmission, entanglement distribution, private communication, and secret-key agreement share the capacityg(κN+κ−1)−g((κ−1)(N+1))g(\kappa N+\kappa-1)-g((\kappa-1)(N+1))g(κN+κ−1)−g((κ−1)(N+1)), where ggg denotes the entropy of a thermal mode. The proof extends sector teleportation simulation from pure loss to amplification. A correspondence between their conditional resources provides the entanglement estimate needed to retain the transmitted-energy budget throughout the adaptive converse, despite the amplifier's unbounded output photon number. We also determine the strong-converse thresholds under the same averaged energy constraint. For every N>0N>0N>0, these equal log⁡2[κ/(κ−1)]\log_2[\kappa/(\kappa-1)]log2​[κ/(κ−1)], separating the energy-dependent vanishing-error capacities from an energy-independent strong-converse threshold.
We prove the long-standing conjecture stating that Gaussian thermal input states minimize the output von Neumann entropy of one-mode phase-covariant quantum Gaussian channels among all the input states with a given entropy. Phase-covariant quantum Gaussian channels model the attenuation and the noise that affect any electromagnetic signal in the quantum regime. Our result is crucial to prove the converse theorems for both the triple trade-off region and the capacity region for broadcast communication of the Gaussian quantum-limited amplifier. Our result extends to the quantum regime the entropy power inequality that plays a key role in classical information theory. Our proof exploits a completely new technique based on the recent determination of the p→q norms of the quantum-limited amplifier [De Palma et al., arXiv:1610.09967]. This technique can be applied to any quantum channel.