Parallel versus nested-adaptive superchannel discrimination

Problem

Can nested-adaptive strategies improve the Stein exponent for discriminating two finite-dimensional quantum superchannels? A superchannel maps channels \(\mathcal N:A\to B\) to channels \(C\to D\) and admits a realization

\begin{equation} \Theta(\mathcal N) =\mathcal D\circ(\mathcal N\otimes\operatorname{id}_S)\circ\mathcal E, \qquad \mathcal E:C\to A S, \quad \mathcal D:B S\to D. \tag{1} \end{equation}

The memory system \(S\) in Eq. (1) is internal to the superchannel. In a fully parallel \(n\)-use strategy, one applies \(\Theta_i^{\otimes n}\) to a joint \(n\)-partite inserted channel and then tests the resulting output on a joint input state. A nested-adaptive strategy may instead recursively insert the channel produced by one tested use into the channel slot of another, with arbitrary compatible CPTP maps between uses and a final binary measurement. For \(\mathsf S\in\{\mathrm{par},\mathrm{nest}\}\), let

\begin{equation} \begin{aligned} \beta_{\varepsilon,n}^{\mathsf S}(\Theta_1\|\Theta_2) &:=\inf\{\beta_n(P):P\in\mathsf S_n,\ \alpha_n(P)\leq\varepsilon\},\\ \zeta_{\mathsf S}(\Theta_1\|\Theta_2) &:=\lim_{\varepsilon\downarrow0}\liminf_{n\to\infty} -\frac1n\log_2 \beta_{\varepsilon,n}^{\mathsf S}(\Theta_1\|\Theta_2), \end{aligned} \tag{2} \end{equation}

where \(\alpha_n\) and \(\beta_n\) are the type-I and type-II errors. Is

\begin{equation} \zeta_{\mathrm{nest}}(\Theta_1\|\Theta_2) =\zeta_{\mathrm{par}}(\Theta_1\|\Theta_2) \tag{3} \end{equation}

for every pair \(\Theta_1,\Theta_2\)?

Source

Hirche explicitly conjectured the equality in Eq. (3) while developing the first asymptotic discrimination framework for quantum superchannels [Hir23].

Progress

Reports do not certify correctness or automatically change the problem's status. Progress policy.

The fully parallel Stein exponent in Eq. (2) is known exactly:

\begin{equation} \zeta_{\mathrm{par}}(\Theta_1\|\Theta_2) =D_{\rm sc}^{\infty}(\Theta_1\|\Theta_2), \tag{4} \end{equation}

In Eq. (4), \(D_{\rm sc}^{\infty}\) is the regularized superchannel relative entropy optimized over joint inserted channels and input states [Hir23].
Every parallel strategy can be embedded into a nested one. Hirche’s amortized-superchannel meta-converse therefore gives

\begin{equation} D_{\rm sc}^{\infty}(\Theta_1\|\Theta_2) \leq\zeta_{\mathrm{nest}}(\Theta_1\|\Theta_2) \leq D_{\rm sc}^{A}(\Theta_1\|\Theta_2), \tag{5} \end{equation}

with \(D_{\rm sc}^{A}\) the amortized superchannel relative entropy [Hir23].
The channel chain rule underlying Eq. (5) implies \(D_{\rm sc}^{A}=D_{\rm sc}^{\infty}\), and hence Eq. (3), conditional on the diamond-smoothed channel AEP in Problem . No unconditional proof or counterexample is known [Hir23].
For classical superchannels, all relevant relative-entropy amortizations collapse and product strategies already achieve the optimum; thus Eq. (3) holds in that special case [Hir23].
Reported progress: PR #81.

Comment

This problem compares two causal strategy classes. It is distinct from the problem on fully general superchannel-discrimination strategies, which allows arbitrary interleavings of the physical components of different superchannel uses.

References

[Hir23]: C. Hirche, “Quantum Network Discrimination,” Quantum 7, 1064 (2023).DOI arXiv

Parallel versus nested-adaptive superchannel discrimination

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