Adaptive Quantum Channel Discrimination

• Adaptive quantum channel discrimination is the process of optimally distinguishing between quantum channels using feedback-assisted, sequential input strategies.
• It leverages mathematical tools like the geometric Rényi divergence and chain rule properties to equate adaptive protocols with parallel strategies.
• This approach enables converting adaptive protocols into parallel ones with practical error bounds, offering insights for infinite-dimensional quantum systems.

Adaptive quantum channel discrimination concerns the fundamental and applied problem of optimally distinguishing between two or more quantum channels when sequential, outcome-dependent input choices—that is, adaptive protocols—are permitted. It is central in quantum information theory, communication, quantum metrology, quantum machine learning, and quantum-enhanced sensing, particularly when the underlying quantum channels may be infinite-dimensional or structurally complex. The central technical question is whether adaptive (possibly feedback- and memory-assisted) protocols yield superior discrimination performance to non-adaptive (parallel, block) strategies in various operational regimes, and how their limitations can be rigorously characterized by quantum information-theoretic divergences.

1. Framework of Adaptive Quantum Channel Discrimination

In quantum channel discrimination, a decision-maker (the "tester") is provided with a black-box channel that is either $\mathcal{E}$ or $\mathcal{F}$ and is allowed $n$ uses of the channel with the goal of minimizing the error probability of mistaking one for the other. In adaptive strategies, the tester is permitted to interleave channel uses with arbitrary quantum operations and to choose each channel input and any intermediate operations as a function of all previous outputs—possibly using quantum memory between steps. This is in contrast to non-adaptive (parallel) protocols, where a possibly entangled input is distributed across $n$ channel uses, which are then measured at the end in a joint—but otherwise fixed—fashion.

The principal figure of merit is the asymptotic exponential decay rate of the type-II error probability, subject to constraints on the type-I error, as $n \to \infty$. Various divergences and error exponents—e.g., the Umegaki relative entropy, sandwiched/geometric Rényi divergences, and quantum hypothesis testing quantities—characterize the ultimate performance limits.

2. Main Results: Asymptotic Equivalence of Adaptive and Parallel Strategies

A core result established for infinite-dimensional channels in (Bergh et al., 2023) (extending finite-dimensional findings (Bergh et al., 2022)) is that, for the asymmetric Stein’s lemma regime, the asymptotic error exponent achieved by the optimal adaptive discrimination protocol coincides with that achievable by non-adaptive (parallel) strategies, provided the geometric Rényi divergence between channels is finite for some $\alpha > 1$. Specifically, the asymptotic rate $R$ of the (type-II) error probability is given by:

$$R = \lim_{n \to \infty} \frac{1}{n} D^{\text{reg}}(\mathcal{E} \Vert \mathcal{F}),$$

where $D^{\text{reg}}$ is the regularized divergence between channels (typically involving a supremum or infimum over input states and possibly ancilla).

Crucially, adaptive strategies—which allow for analytic and measurable adjustments of input states based on previous measurement outcomes—do not improve this exponent. The result is established for the geometric Rényi divergence, which is shown to possess both a chain rule and additivity property in infinite dimensions (see below).

3. Chain Rule, Additivity, and Implications for Infinite Dimensions

The geometric Rényi divergence, $D_\alpha^\text{geo}(\mathcal{E} \Vert \mathcal{F})$, is defined via minimal reverse tests and is constructed to be operationally meaningful. The proof of equivalence relies on two key properties:

• Chain rule: For any quantum channels $\mathcal{E}$ and $\mathcal{F}$ and any state preparation/measurement process, the divergence after the channel satisfies

$$D_\alpha^\text{geo}\big(\mathcal{E}(\rho) \Vert \mathcal{F}(\sigma)\big) \leq D_\alpha^\text{geo}(\rho \Vert \sigma) + D_\alpha^\text{reg}(\mathcal{E} \Vert \mathcal{F}),$$

where $D_\alpha^\text{reg}$ is the regularized divergence (the limiting asymptotic exponent).

• Additivity: The chain rule implies that the geometric Rényi divergence is additive under the tensor product of channels, even in infinite dimensions:

$$D_\alpha^\text{geo}(\mathcal{E}^{\otimes n} \Vert \mathcal{F}^{\otimes n}) = n D_\alpha^\text{geo}(\mathcal{E} \Vert \mathcal{F}).$$

This enables one to analyze asymptotic discrimination rates and convert adaptive protocols into (approximately) equivalent parallel protocols.

An important implication is that the chain rule for the Umegaki relative entropy (the usual quantum relative entropy) in infinite dimensions, previously shown under the strong condition of finite max-relative divergence [(Fawzi et al., 2022)v2], actually holds under the strictly weaker condition of finite geometric Rényi divergence.

4. Explicit Construction: Conversion and Error Bounds

The equivalence is established constructively: for any adaptive $n$-shot strategy, there exists a parallel strategy that approximates its performance with rigorous (quantified) bounds on the difference in discrimination errors. The explicit bound (see [(Bergh et al., 2023), main results]) takes the form:

$$\frac{1}{n} D_H^{\alpha_a}(\mathcal{E}(\rho_n) \Vert \mathcal{F}(\sigma_n)) \leq \frac{1}{1-\alpha_a} \left[ \frac{1}{m} D_H^{\alpha_p}\left(\mathcal{E}^{\otimes m}(\nu) \Vert \mathcal{F}^{\otimes m}(\nu)\right) + \text{error term} \right]$$

for appropriate choices of parameters $\alpha_a,\alpha_p$ and $m$ large relative to $n$. The error term decays with $m,n$, so for large enough $m$ the parallel strategy is arbitrarily close in performance to the adaptive one. This generalizes finite-dimensional results (Bergh et al., 2022) to infinite-dimensional settings.

5. Separation of Finiteness Conditions and Explicit Examples

The paper also demonstrates that the conditions for finiteness of the geometric Rényi divergence and of the max-relative divergence between channels are not equivalent in infinite dimensions. Explicit channel constructions (e.g., generalized depolarizing channels based on classical distributions with different tail behaviors) illustrate that geometric Rényi divergence may be finite while the max-divergence is infinite.

This mathematical distinction is operationally significant: it shows that the asymptotic equivalence and the strong chain rule for the relative entropy can be ensured under weaker conditions than previously thought, thereby broadening the applicability of these results.

6. Operational and Theoretical Implications

• Generalization: The approach unifies the treatment of the discriminability of quantum channels in both finite and infinite dimensions. It confirms that even in complex, potentially unbounded Hilbert spaces, the essential operational properties of the geometric Rényi divergence (chain rule, additivity) hold and govern the performance of discrimination strategies.
• Practicality: By enabling the explicit conversion of adaptive discrimination protocols into parallel implementations, the work provides tools for practical evaluation and optimization of discrimination strategies in both theory and experiment.
• Foundations: These results strengthen the conclusion that, at least when divergences are well-behaved, the added complexity of adaptivity provides no improvement in the leading-order discrimination exponent. This suggests that resources spent on building adaptive protocols can, in many cases, be reallocated without loss to optimized parallel schemes.

7. Future Directions and Open Problems

While the equivalence of adaptive and parallel strategies is established here for the asymptotic exponential error rate under geometric Rényi divergence finiteness, it remains an open problem to characterize the minimal conditions on channel pairs and divergences needed for this result in even more general settings. Further, the quantification of finite-size effects and the development of methods for discriminating channels with more exotic structures or input constraints are promising directions.

Finally, the demonstrated non-equivalence of the finiteness conditions for different divergences calls for a deeper investigation into how various quantum information-theoretic measures influence the capability and resource requirements of channel discrimination protocols in infinite-dimensional quantum systems.