Multimode constrained output entropy of a pure-loss channel

Problem

Is Guha’s multimode Strong Conjecture 2 true for every correlated input state? Fix \(n\geq1\), \(K\geq0\), and \(0<\eta<1\). Let the modes \(A_1,\ldots,A_n\) be in the vacuum and let \(\rho_{B^n}\) be an arbitrary joint \(n\)-mode state. Apply identical beam splitters whose output annihilation operators satisfy

\begin{equation} \hat c_i=\sqrt{\eta}\,\hat a_i +\sqrt{1-\eta}\,\hat b_i, \qquad i\in\{1,\ldots,n\}, \tag{1} \end{equation}

and denote the resulting joint state of \(C_1,\ldots,C_n\) by \(\rho_{C^n}\). In the convention of Eq. (1), \(\eta\) multiplies the vacuum \(A\) port, so the attenuator from the information-bearing \(B\) port to \(C\) has transmissivity \(1-\eta\). Define the thermal entropy function by

\begin{equation} g(x):=(x+1)\log_2(x+1)-x\log_2x, \qquad x\geq0, \tag{2} \end{equation}

with \(0\log_2 0:=0\). Using the function in Eq. (2), impose the sole input-entropy constraint \(S(\rho_{B^n})=ng(K)\). Does the state produced in Eq. (1) always obey

\begin{equation} S(\rho_{C^n})\geq ng((1-\eta)K)? \tag{3} \end{equation}

Equality in Eq. (3) is attained when \(\rho_{B^n}\) is a tensor product of \(n\) thermal states of mean photon number \(K\); the conjecture asserts that arbitrary correlations cannot lower the output entropy further.

Source

Guha, Shapiro, and Erkmen explicitly formulated Eq. (3) as Strong Conjecture 2 [GSE07]. Wilde records the same multimode statement as the hypothesis needed for the converse in the pure-loss dynamic-capacity theorem [Wil17].

Progress

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De Palma, Trevisan, and Giovannetti proved Eq. (3) for one mode and every input state of fixed entropy. Their theorem leaves the arbitrary correlated case \(n\geq2\) open [DTG17].
For arbitrary \(n\), thermal product states are proved optimal when the input is restricted to states diagonal in some product basis [DTG17b]. Related unrestricted-input multimode optimality is known for entanglement-breaking Gaussian attenuators, but those noise regimes do not settle the nontrivial quantum-limited attenuator in Eq. (1) [DP19].
Falco and De Palma proved a multimode conditional quantum entropy power inequality for general linear mixing. Its resulting exponential- entropy lower bound is weaker than the thermal \(g\)-function bound in Eq. (3), so it does not close Strong Conjecture 2 [FDP25].

Comment

The unresolved regime is \(n\geq2\) with an arbitrary, possibly entangled and non-Gaussian, joint input of fixed entropy. Unconstrained Gaussian minimum-output-entropy theorems and one-mode constrained theorems are not equivalent to Eq. (3).

References

[GSE07]: S. Guha, J. H. Shapiro, and B. I. Erkmen, “Classical Capacity of Bosonic Broadcast Communication and a New Minimum Output Entropy Conjecture,” Physical Review A 76, 032303 (2007).DOI arXiv
[Wil17]: M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press (2017), Sec. 25.5.DOI arXiv
[DTG17]: G. De Palma, D. Trevisan, and V. Giovannetti, “Gaussian States Minimize the Output Entropy of the One-Mode Quantum Attenuator,” IEEE Transactions on Information Theory 63, 728–737 (2017).DOI arXiv
[DTG17b]: G. De Palma, D. Trevisan, and V. Giovannetti, “Multimode Gaussian Optimizers for the Wehrl Entropy and Quantum Gaussian Channels,” arXiv preprint (2017).arXiv
[DP19]: G. De Palma, “New Lower Bounds to the Output Entropy of Multi-Mode Quantum Gaussian Channels,” IEEE Transactions on Information Theory 65, 5959–5968 (2019).DOI arXiv
[FDP25]: A. Falco and G. De Palma, “The Multimode Conditional Quantum Entropy Power Inequality and the Squashed Entanglement of the Extreme Multimode Bosonic Gaussian Channels,” IEEE Transactions on Information Theory 71, 6075–6104 (2025).DOI arXiv

Multimode constrained output entropy of a pure-loss channel

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