Delayed-onset additivity violation for minimum output Rényi entropy

Unsolved ID op_c0b1045a614d2353 Last edited 4 September 2026

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Field: Quantum Communication
Topics: Additivity and regularization
Matrix and entropy inequalities
Quantum channel structure

Problem

Does there exist a finite-dimensional quantum channel whose minimum output Rényi entropy is additive for every tensor power below some order and first becomes strictly subadditive at that order? Let \(\Phi:\mathcal L(A)\to\mathcal L(B)\) be completely positive and trace preserving. For \(p>0\), define the Rényi entropy and the corresponding minimum output entropy by

\begin{equation} S_p(\sigma) :=\begin{cases} \displaystyle\frac{1}{1-p}\log_2\operatorname{Tr}(\sigma^p), &p\neq1,\\[2mm] -\operatorname{Tr}(\sigma\log_2\sigma),&p=1, \end{cases} \qquad S_{p,\min}(\Phi) :=\min_{\substack{\rho\succeq0\\\operatorname{Tr}\rho=1}} S_p\!\left(\Phi(\rho)\right). \tag{1} \end{equation}

With the quantities in Eq. (1), determine whether there are \(p>0\), an integer \(m\geq3\), and a channel \(\Phi\) such that

\begin{equation} S_{p,\min}(\Phi^{\otimes n}) =nS_{p,\min}(\Phi) \quad\text{for every }1\leq n<m, \qquad S_{p,\min}(\Phi^{\otimes m}) <mS_{p,\min}(\Phi). \tag{2} \end{equation}

The restriction \(m\geq3\) in Eq. (2) makes the lower-power requirement nontrivial: the equality at \(n=1\) is automatic, whereas equality at \(n=2\) is required.

Source

Ruskai explicitly asks whether additivity can hold for all tensor powers below an integer \(m\) and fail at the \(m\)th power [Rus07]. Equation (2) uses the intended delayed-onset formulation \(m\geq3\), excluding the tautological one-copy-only condition obtained when \(m=2\).

Progress

Derksen and Lovitz construct explicit finite-dimensional self-channel violations \(S_{p,\min}(\Phi^{\otimes2})<2S_{p,\min}(\Phi)\) for every \(p>1\). These examples resolve the literal \(m=2\) reading, but fail the required two-copy equality in Eq. (2) for every \(m\geq3\) [DL26].
Hastings disproved minimum-output von Neumann entropy additivity at \(p=1\) using finite-dimensional random channels. The violation is already witnessed at a two-channel tensor product and therefore does not provide a channel that remains additive through every lower nontrivial power in Eq. (2) [Has09].

Comment

No channel is known to satisfy Eq. (2), and no general theorem is known that promotes two-copy additivity to additivity of all self-tensor powers. The unresolved issue is therefore the existence or impossibility of a genuinely delayed first violation at some \(m\geq3\).

References

[Rus07]: M. B. Ruskai, “Open Problems in Quantum Information Theory,” arXiv preprint (2007), Problem 19, p. 16.DOI arXiv
[DL26]: H. Derksen and B. Lovitz, “Constructive Counterexamples to the Additivity of Minimum Output Rényi Entropy of Quantum Channels for All \(p>1\),” arXiv preprint (2026), version 2.arXiv
[Has09]: M. B. Hastings, “Superadditivity of Communication Capacity Using Entangled Inputs,” Nature Physics 5, 255–257 (2009).DOI arXiv

Delayed-onset additivity violation for minimum output Rényi entropy

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