Advantage of fully general superchannel-discrimination strategies
- Field
- Topics
Problem
Can a fully general adaptive strategy attain a larger Stein exponent than every nested-adaptive strategy for discriminating two quantum superchannels? Fix finite-dimensional physical realizations
The systems \(S_i\) in Eq. (1) are internal memories. A nested strategy recursively places complete uses of \(\Theta_i\) inside one another. A fully general strategy may interleave the preprocessing and postprocessing components of different uses in any causally valid order, provided each \(\mathcal E_i\) precedes its matched \(\mathcal D_i\) and the tester cannot access the internal memory \(S_i\). Define the vanishing-type-I-error Stein exponent for a strategy class \(\mathsf S\) by
The definition in Eq. (2) uses the type-I and type-II errors \(\alpha_n(P)\) and \(\beta_n(P)\) of protocol \(P\). Does there exist a pair of realizations for which
where \(\mathrm{fg}\) denotes fully general strategies? If not, prove equality in Eq. (3) with \(>\) replaced by \(=\) for all superchannel pairs.
Source
Hirche explicitly identified the achievability and optimality of fully general adaptive strategies as the principal unresolved problem in quantum-network discrimination [Hir23].
Progress
Fully parallel strategies are exactly characterized by the regularized superchannel relative entropy. Since they are contained in the fully general class, Hirche’s meta-converse yields
\begin{equation} D_{\rm sc}^{\infty}(\Theta_1\|\Theta_2) \leq\zeta_{\mathrm{fg}}(\Theta_1\|\Theta_2) \leq D_{\rm sc}^{A*}(\Theta_1\|\Theta_2), \tag{4} \end{equation}where \(D_{\rm sc}^{A*}\) is a fully amortized superchannel relative entropy that accounts for the exposed preprocessing and postprocessing components [Hir23].
A braided strategy gives a concrete fully general ordering that need not be nested. No pair is known for which such an ordering makes the first inequality in Eq. (4) strict, and no protocol is known to attain \(D_{\rm sc}^{A*}\) in general [Hir23].
For classical superchannels, the fully amortized relative entropy collapses to the ordinary superchannel relative entropy. Product strategies are therefore optimal, ruling out Eq. (3) in the classical special case [Hir23].
Comment
The open alternatives are operationally different: either construct a strict advantage as in Eq. (3), prove a protocol that attains the upper bound in Eq. (4), or sharpen that converse until it meets an achievable nested or parallel rate.