The active entropy photon-number inequality: exact minima, energy budgets and input rigidity

What this paper enables

For researchers analysing bosonic amplifiers and Gaussian optical channels, this paper connects three practical questions: What is the lowest possible output entropy? Which input reaches that limit with the available entropy and photon budget? What does an output close to the limit reveal about the source? It provides exact entropy benchmarks, a finite convex allocation problem for resource-constrained source design, and quantitative bounds on the structure of near-optimal inputs. The competing inputs may be non-Gaussian and entangled across modes.

An attainable entropy limit for amplification

Let \(S\) denote von Neumann entropy in nats, define \(g(r)=(r+1)\log(r+1)-r\log r\) with \(g(0)=0\), and write \(e=g^{-1}\). The entropy photon number of an \(n\)-mode state is \(\mathcal N_n(\rho)=e(S(\rho)/n)\). For any finite \(n\ge1\), two independent input systems \(A\) and \(B\), each with finite mean total photon number and arbitrary internal correlations, obey the following bound under amplification of gain \(G>1\):

\[ \mathcal N_n(\rho_C)\ge G\mathcal N_n(\rho_A)+(G-1)\bigl[\mathcal N_n(\rho_B)+1\bigr]. \]

Here \(\rho_C\) is the amplifier output. Products of identical thermal modes in each input attain this bound at prescribed input entropies. With an independent product thermal environment, it determines the exact minimum output entropy at every finite block length, including optimisation over entangled signal inputs. This removes that entropy optimisation from communication converses that require it; operational capacity still depends on the communication model and its resource constraint.

Source design with both entropy and photon constraints

For a finite product \(\Phi=\bigotimes_{j=1}^n\Phi_j\) of centered canonical phase-insensitive Gaussian channels with thermal action \(r\mapsto a_jr+b_j\), the paper solves the joint optimisation at finite input entropy \(T\ge0\) and finite actual mean total photon budget \(E\). The factors have \(a_j>0\), with \(b_j\ge\max\{0,a_j-1\}\) for covariant channels and \(b_j\ge a_j\) for contravariant channels. Writing \(N\) for the original total photon-number operator, the exact answer is

\[ \begin{aligned} M_\Phi(T,E)&:=\min_{\substack{S(\rho)=T\\\operatorname{Tr}(N\rho)\le E}} S(\Phi\rho)\\ &=\min_{\substack{s_j\ge0,\ \sum_j s_j=T\\\sum_j e(s_j)\le E}}\sum_j g\bigl(a_j e(s_j)+b_j\bigr), \qquad E\ge n e(T/n). \end{aligned} \]

The last condition is exactly the feasibility threshold. The scalar variables \(s_j\) are input entropies, and the programme is convex in those variables. An optimal allocation \(s_j^*\) gives the attaining input \(\sigma=\bigotimes_j\omega_{e(s_j^*)}\), where \(\omega_r\) is a one-mode thermal state of mean photon number \(r\). Its output is a thermal product with occupations \(a_je(s_j^*)+b_j\). Thus the calculation returns both the performance limit and a source that reaches it, over the full admissible joint-state class. A photon cap can activate a second mode even when the entropy-only optimum puts all entropy into one mode. The result includes zero allocations and the stated quantum-limited channel boundaries.

The physical-budget statement applies to the specified canonical class. The separate result for general one-mode Gaussian frames imposes an entropy constraint alone: input squeezing changes the physical photon cost.

Input structure and a reusable contraction bound

For the same canonical channel class and any product thermal reference \(\sigma=\bigotimes_j\omega_{r_j}\) with finite positive occupations \(r_j\), every finite-energy joint input satisfies the sharp comparison

\[ D(\Phi\rho\Vert\Phi\sigma)\le\kappa D(\rho\Vert\sigma),\qquad \kappa=\max_j\frac{a_j h(a_jr_j+b_j)}{h(r_j)},\qquad h(r)=\log(1+1/r). \]

Here \(D(\rho\Vert\sigma)=\operatorname{Tr}[\rho(\log\rho-\log\sigma)]\) is quantum relative entropy. The bound applies to correlated and entangled inputs and can be used independently of the allocation problem. Its constant is sharp over the full finite-energy input class; the displacement attaining that ratio need not obey an additional photon cap saturated by the reference.

At an allocation optimum with positive occupations and finite multipliers, a positive energy multiplier makes the thermal product unique and lets the output-entropy excess control \(D(\rho\Vert\sigma)\). For the two independent amplifier ports, exact equality instead identifies compatible Gaussian inputs with a shared symplectic shape reflected on the creation-operator port. At fixed mode number, gain and a common finite energy bound for each port, a power of the entropy-photon deficit controls full trace distance to that family. These are inferences from a known entropy deficit; estimating it from experimental data and determining the required sample size remain separate measurement and statistical tasks.