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.
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.
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.
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.
When does finite phase-space entropy stop guaranteeing proximity to coherent light? This article determines sharp thresholds for two broad optical source classes when the number of modes may grow. It separates the roles of entropy, mode number and photon energy in approximating an optical state by a single pure coherent state. For an m-mode state ρ, write Qρ(α)=⟨α∣ρ∣α⟩, with dμm=π−md2mα, and define M(ρ)=2m∫Qρ(α)2dμm(α),s=−lnM(ρ),D(ρ)=infα∥ρ−∣α⟩⟨α∣∥1. Here s is the second-order Wehrl entropy excess and D is the full trace-norm distance, whose maximum is 2. The collision quantity M is also the output purity after half loss and can be accessed using interference of two independent, identically prepared copies. Exact source-class thresholds. The two classes are states diagonal in a common passive-mode number basis, and arbitrary states supported on the vacuum together with configurations whose photons occupy only one mode at a time in a common passive basis. The latter class retains arbitrary complex coherences, including vacuum coherences. For either class X, with FX(s)=supρ∈X,−lnM(ρ)=sD(ρ)2 and the supremum over all finite mode numbers, FX(s)=4⟺s≥ln4. The plateau is approached through growing-mode sequences, not attained by a finite-mode state. At its onset, vanishing coherent overlap and M→1/4 force the one-photon probability to approach one, without an energy-bound assumption. This probability rigidity does not assert convergence of the mean energy. Sharp boundary geometry and energy. For the full number-diagonal class D, the approach from below has the exact corner lims↑ln4ln4−s4−FD(s)=e4. A small vacuum component changes the optimal coherent peak and produces the sharp slope. Separate all-state energy bounds show that vanishing coherent overlap requires at least one photon asymptotically, or two photons for pure states. Explicit sources attain these limiting energy costs, and exact-entropy constructions include their additional photon costs. The proofs retain arbitrary photon tails and constants uniform in the mode number. The earliest-threshold and corner results apply to the source classes stated above. The integrated article contains the complete proofs and two figure groups.
When arbitrary finite local dimensions are allowed, bosonic Gaussian states and nondegenerate classical Gaussian block vectors satisfy exactly the same unconditional balanced linear entropy inequalities. We establish this equivalence for von Neumann entropy, extending the known Rényi-2 correspondence. Spectral entropy integrals and compatible realification transfer classical determinant inequalities to bosonic states and, in the forward direction, to even fermionic Gaussian states under complete-mode canonical anticommutation relation (CAR) restrictions. A high-temperature limit gives the bosonic converse. Both transfers include pure modes and imply the Zhang-Yeung non-Shannon inequality. We also derive quantitative independence bounds from conditional mutual information ε = I(A:C | B). For bosons, bounded covariance or total photon number gives a trace-distance bound C√ε for ρAC relative to ρA ⊗ ρC, with spectral and matrix Gram refinements of the prefactor. For fermions, an explicit square-root bound requires no energy or spectral gap assumption, and a complementary bound of order √[ε log(1/ε)] is uniform in the number of conditioning modes with A, C fixed. Known Gaussian continuity theorems convert the new covariance estimates to state distance. The fixed-mode square-root exponents are optimal. Combining fermionic rigidity with the established entropy-deficit identity yields an operational lower bound on non-Gaussianity from CMI and a probability difference for an even measurement event on AC. Version 1.0. Complete manuscript with Appendices A–G.
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.
How much classical information can a noisy optical channel carry when both signal energy and preshared entanglement are limited? This work gives an exact capacity frontier for this joint resource problem. It allows one to determine the best asymptotic communication rate, price the benefit of additional entanglement, and distinguish the effect of tolerating errors from that of changing how entanglement is supplied. Exact capacity under two resource constraints The setting is any fixed bosonic Gaussian channel Φ with finitely many input and output modes. Signal energy is measured by a positive quadratic Hamiltonian HG=21RTGR, with G>0 and ground energy E0; R collects the canonical quadratures. The budgets satisfy E≥E0 and P≥0. For an n-use code, the mean total signal energy is at most nE, while the same initially shared maximally entangled state has Schmidt rank Kn satisfying logKn≤nP. Thus P is a strictly supplied entanglement rate. Encoders may be arbitrary joint quantum operations, and decoding may be collective. The classical capacity is C(E,P)=maxU≥Vphysical21Tr(GU)≤E,SG(V)≤P[SG(ΦU)+SG(V)−SG(ΦcV)]. Here V is a Gaussian seed covariance, U is the average-input covariance, and U−V is the displacement-modulation covariance. A covariance is physical when V+iΩ/2≥0, where Ω is the canonical commutation matrix and the vacuum covariance is I/2. The notation SG denotes Gaussian von Neumann entropy, and Φc is the full complementary channel, including any environment purification. All logarithms are natural, so rates are in nats per channel use. This formula replaces optimization over arbitrary quantum block codes by a finite-dimensional covariance optimization. It supplies both an achievable rate and a converse against non-Gaussian, correlated encodings. At P=0, it also determines the unassisted capacity of general fixed finite-mode Gaussian channels, including channels without a passive decomposition into independent modes. The reduction specifies the exact capacity value; it does not require a closed-form optimizer for every device. Energy and strict entanglement supply respond differently to errors For a fixed allowed average error 0≤ε<1, the exact capacity under the mean-energy constraint is Cε(E,P)=C(E0+1−εE−E0,P). Allowing failure rescales the mean energy above the ground state, but leaves the strict initial entanglement budget unchanged. The proof uses an exponential strong converse for codes whose inputs obey a hard total-energy cutoff. In a different model, where a readable flag permits random entanglement supply constrained only on average, the work establishes an achievable bound and a strict small-supply advantage for thermal attenuation at positive energy and fixed nonzero error. That comparison identifies the operational importance of the supply rule; it is not a complete capacity characterization of the flagged model. The communication value of entanglement For a thermal attenuator, let 0<η<1 be the transmissivity, b>0 the output mean photon number for vacuum input, and N the input mean photon number. Write its capacity as CN(P), and define g(N)=(N+1)log(N+1)−NlogN. The gain from entanglement obeys the sharp bound 0≤CN(P)−CN(0)≤1+bηP. For every fixed N>0, the coefficient η/(1+b) is attained as the initial right derivative at P=0. In the low-photon limit, the stronger resource law is CN(P)−CN(0)=1+bηmin{P,g(N)}+O(N),N↓0. The remainder is uniform over P≥0 at fixed η and positive b. Noise therefore determines the conversion efficiency, while g(N)∼Nlog(1/N) sets the entanglement scale sufficient for saturation. This uniform limit concerns the entanglement budget, not a simultaneous limit of vanishing thermal noise. Multimode resource allocation and scope A coupled two-mode example demonstrates how the frontier can guide resource allocation. A global resource-price certificate proves that a legal covariance pair exceeds the entire common-passive-frame Gaussian profile hull by at least 10−5 nat per two-mode use at specified budgets. The comparison includes all common passive frames, local squeezing and average-resource mixtures. The mechanism is the freedom to choose different mode allocations for the seed and average input; the benchmark is this precisely defined Gaussian class, rather than all unentangled encodings. The universal scope is the stated family of fixed finite-mode Gaussian channels, positive quadratic signal costs and strict initial maximally entangled Schmidt-rank supply, with unrestricted joint encoding and collective decoding. The model has no feedback or credit for returned entanglement. The capacity frontier and its resource consequences provide a common way to assess optical communication designs under these operational constraints.