Amortization collapse for superchannel divergences

Problem

Does amortization collapse for the geometric Rényi divergence of arbitrary finite-dimensional quantum superchannels? For compatible states, let

\begin{equation} \begin{aligned} D_{\max}(\rho\|\sigma) &:=\inf\{\lambda:\rho\leq2^\lambda\sigma\},\\ \widehat D_\alpha(\rho\|\sigma) &:=\frac1{\alpha-1}\log_2\operatorname{Tr}\!\left[ \sigma\bigl(\sigma^{-1/2}\rho\sigma^{-1/2}\bigr)^\alpha \right],\qquad 1<\alpha\leq2, \end{aligned} \tag{1} \end{equation}

with the standard support conventions. For either divergence \(\mathbf D\in\{D_{\max},\widehat D_\alpha\}\), define its channel extension and channel-amortized extension by

\begin{equation} \begin{aligned} \mathbf D_{\rm ch}(\mathcal N\|\mathcal M) &:=\sup_{\rho_{RA}} \mathbf D(\mathcal N(\rho)\|\mathcal M(\rho)),\\ \mathbf D_{\rm ch}^{A}(\mathcal N\|\mathcal M) &:=\sup_{\rho_{RA},\sigma_{RA}} \{\mathbf D(\mathcal N(\rho)\|\mathcal M(\sigma)) -\mathbf D(\rho\|\sigma)\}, \end{aligned} \tag{2} \end{equation}

In Eq. (2), identity maps on \(R\) are implicit, and the optimizations allow an arbitrary reference of sufficient finite dimension. For superchannels \(\Theta_1,\Theta_2\), set

\begin{equation} \begin{aligned} \mathbf D_{\rm sc}(\Theta_1\|\Theta_2) &:=\sup_{\mathcal N} \mathbf D_{\rm ch}(\Theta_1(\mathcal N)\|\Theta_2(\mathcal N)),\\ \mathbf D_{\rm sc}^{A}(\Theta_1\|\Theta_2) &:=\sup_{\mathcal N,\mathcal M} \{\mathbf D_{\rm ch}^{A} (\Theta_1(\mathcal N)\|\Theta_2(\mathcal M)) -\mathbf D_{\rm ch}^{A}(\mathcal N\|\mathcal M)\}. \end{aligned} \tag{3} \end{equation}

Is

\begin{equation} \mathbf D_{\rm sc}^{A}(\Theta_1\|\Theta_2) =\mathbf D_{\rm sc}(\Theta_1\|\Theta_2) \tag{4} \end{equation}

for \(\mathbf D=\widehat D_\alpha\) in Eq. (1), \(1<\alpha\leq2\), and all superchannel pairs? The definitions also include \(D_{\max}\) to state its settled subcase below. The amortized suprema use pairs with finite subtracted divergence; other infinite values have the standard support convention.

Source

Hirche’s Remark 5.4, after Eqs. (75)–(77), leaves amortization collapse open in general superchannel discrimination settings [Hir23]. For the nested definition in Eq. (3), the max-relative-entropy case follows directly from CP order, as shown below; the remaining target here is the geometric Rényi case.

Progress

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For point-to-point channels, amortization of the max-relative entropy collapses:

\begin{equation} D_{\max,{\rm ch}}^{A}(\mathcal N\|\mathcal M) =D_{\max,{\rm ch}}(\mathcal N\|\mathcal M). \tag{5} \end{equation}

The proof of Eq. (5) uses the CP-order structure of channel max-relative entropy [WBHK20].
Fang and Fawzi proved the analogous point-to-point channel collapse

\begin{equation} \widehat D_{\alpha,{\rm ch}}^{A}(\mathcal N\|\mathcal M) =\widehat D_{\alpha,{\rm ch}}(\mathcal N\|\mathcal M), \qquad 1<\alpha\leq2, \tag{6} \end{equation}

from a chain rule for the geometric Rényi divergence [FF21]. Equation (6) concerns point-to-point channels; it does not settle the superchannel claim.
For the nested superchannel definition, the max-relative-entropy subcase also collapses. Indeed, let \(t=2^{D_{\max,\mathrm{ch}}(\mathcal N\|\mathcal M)}\) and \(s=2^{D_{\max,\mathrm{sc}}(\Theta_1\|\Theta_2)}\) be finite. Channel max-relative entropy is characterized by CP order, and physical superchannels preserve that order. Thus

\begin{equation} \mathcal N\leq_{\mathrm{CP}}t\mathcal M \quad\Longrightarrow\quad \Theta_1(\mathcal N)\leq_{\mathrm{CP}}t\Theta_1(\mathcal M) \leq_{\mathrm{CP}}ts\Theta_2(\mathcal M), \tag{7} \end{equation}

where \(\mathcal A\leq_{\mathrm{CP}}\mathcal B\) means \(\mathcal B-\mathcal A\) is completely positive. Taking logarithms in Eq. (7) and using Eq. (5) bounds each term of the amortized supremum by \(\log_2s\). Choosing \(\mathcal N=\mathcal M\) gives the reverse inequality, proving Eq. (4) for \(\mathbf D=D_{\max}\). If the unamortized divergence is infinite, the same reverse inequality already proves equality in the extended sense. This is a direct consequence of the cited channel CP-order characterization [WBHK20]; it does not prove collapse of Hirche’s larger fully amortized quantities [Hir23].

Comment

The remaining displayed question is geometric Rényi amortization collapse for nested-adaptive superchannel discrimination. The max-relative-entropy subcase is settled by Eq. (7). Whether the larger fully amortized divergences controlling braided and fully general strategies collapse is a further, stronger question.

References

[WBHK20]: M. M. Wilde, M. Berta, C. Hirche, and E. Kaur, “Amortized Channel Divergence for Asymptotic Quantum Channel Discrimination,” Letters in Mathematical Physics 110, 2277–2336 (2020).DOI arXiv
[FF21]: K. Fang and H. Fawzi, “Geometric Rényi Divergence and its Applications in Quantum Channel Capacities,” Communications in Mathematical Physics 384, 1615–1677 (2021).DOI arXiv
[Hir23]: C. Hirche, “Quantum Network Discrimination,” Quantum 7, 1064 (2023).DOI arXiv

Amortization collapse for superchannel divergences

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