We investigate the performance of parallel and adaptive quantum channel discrimination strategies for a finite number of channel uses. It has recently been shown that, in the asymmetric setting with ...asymptotically vanishing type I error probability, adaptive strategies are asymptotically not more powerful than parallel ones. We extend this result to the non-asymptotic regime with finitely many channel uses, by explicitly constructing a parallel strategy for any given adaptive strategy, and bounding the difference in their performances, measured in terms of the decay rate of the type II error probability per channel use. We further show that all parallel strategies can be optimized over in time polynomial in the number of channel uses, and hence our result can also be used to obtain a poly-time-computable asymptotically tight upper bound on the performance of general adaptive strategies.
Typically, non-life insurance claims data is studied in claims development triangles which display the two time axes accident years and development years. Most stochastic claims reserving models ...assume independence between different accident years. Therefore, such models fail to model claims inflation appropriately, because claims inflation acts on all accident years simultaneously. We introduce a Bayes chain ladder reserving model which enables us to model claims inflation. In this model we derive analytical formulas for the posterior distribution, the claims reserves and their prediction uncertainty.
We study asymmetric binary channel discrimination, for qantum channels acting on separable Hilbert spaces. We establish quantum Stein's lemma for channels for both adaptive and parallel strategies, ...and show that under finiteness of the geometric Rényi divergence between the two channels for some \(\alpha > 1\), adaptive strategies offer no asymptotic advantage over parallel ones. One major step in our argument is to demonstrate that the geometric Rényi divergence satisfies a chain rule and is additive for channels also in infinite dimensions. These results may be of independent interest. Furthermore, we not only show asymptotic equivalence of parallel and adaptive strategies, but explicitly construct a parallel strategy which approximates a given adaptive \(n\)-shot strategy, and give an explicit bound on the difference between the discrimination errors for these two strategies. This extends the finite dimensional result from B. Bergh et al., arxiv:2206.08350. Finally, this also allows us to conclude, that the chain rule for the Umegaki relative entropy in infinite dimensions, recently shown in O. Fawzi, L. Gao, and M. Rahaman, arxiv:2212.14700v2 given finiteness of the max divergence between the two channels, also holds under the weaker condition of finiteness of the geometric Rényi divergence. We give explicit examples of channels which show that these two finiteness conditions are not equivalent.
We study convergence rates of the Trotter-Kato splitting \(e^{A+L} = \lim_{n \to \infty} (e^{L/n} e^{A/n})^n\) in the strong operator topology. In the first part, we use complex interpolation theory ...to treat generators \(L\) and \(A\) of contraction semigroups on Banach spaces, with \(L\) relatively \(A\)-bounded. In the second part, we study unitary dynamics on Hilbert spaces and develop a new technique based on the concept of energy constraints. Our results provide a complete picture of the convergence rates for the Trotter splitting for all common types of Schr\"odinger and Dirac operators, including singular, confining and magnetic vector potentials, as well as molecular many-body Hamiltonians in dimension \(d=3\). Using the Brezis-Mironescu inequality, we derive convergence rates for the Schr\"odinger operator with \(V(x)=\pm |x|^{-a}\) potential. In each case, our conditions are fully explicit.
J. Math. Phys. 62, 092205 (2021) Fawzi and Fawzi recently defined the sharp R\'enyi divergence, $D_\alpha^\#$,
for $\alpha \in (1, \infty)$, as an additional quantum R\'enyi divergence with
nice ...mathematical properties and applications in quantum channel discrimination
and quantum communication. One of their open questions was the limit ${\alpha}
\to 1$ of this divergence. By finding a new expression of the sharp divergence
in terms of a minimization of the geometric R\'enyi divergence, we show that
this limit is equal to the Belavkin-Staszewski relative entropy. Analogous
minimizations of arbitrary generalized divergences lead to a new family of
generalized divergences that we call kringel divergences, and for which we
prove various properties including the data-processing inequality.
We prove the quantum Zeno effect in open quantum systems whose evolution, governed by quantum dynamical semigroups, is repeatedly and frequently interrupted by the action of a quantum operation. For ...the case of a quantum dynamical semigroup with a bounded generator, our analysis leads to a refinement of existing results and extends them to a larger class of quantum operations. We also prove the existence of a novel strong quantum Zeno limit for quantum operations for which a certain spectral gap assumption, which all previous results relied on, is lifted. The quantum operations are instead required to satisfy a weaker property of strong power-convergence. In addition, we establish, for the first time, the existence of a quantum Zeno limit for the case of unbounded generators. We also provide a variety of physically interesting examples of quantum operations to which our results apply.
We derive the 3D quintic NLS as the mean field limit of a Bose gas with
three-body interactions. The quintic NLS is energy-critical, leading to several
new difficulties in comparison with the cubic ...NLS which emerges from Bose gases
with pair-interactions. Our method is based on Bogoliubov's approximation,
which also provides the information on the fluctuations around the condensate
in terms of a norm approximation for the N-body wave function.
Quantum hypothesis testing (QHT) has been traditionally studied from the information-theoretic perspective, wherein one is interested in the optimal decay rate of error probabilities as a function of ...the number of samples of an unknown state. In this paper, we study the sample complexity of QHT, wherein the goal is to determine the minimum number of samples needed to reach a desired error probability. By making use of the wealth of knowledge that already exists in the literature on QHT, we characterize the sample complexity of binary QHT in the symmetric and asymmetric settings, and we provide bounds on the sample complexity of multiple QHT. In more detail, we prove that the sample complexity of symmetric binary QHT depends logarithmically on the inverse error probability and inversely on the negative logarithm of the fidelity. As a counterpart of the quantum Stein's lemma, we also find that the sample complexity of asymmetric binary QHT depends logarithmically on the inverse type II error probability and inversely on the quantum relative entropy, provided that the type II error probability is sufficiently small. We then provide lower and upper bounds on the sample complexity of multiple QHT, with it remaining an intriguing open question to improve these bounds. The final part of our paper outlines and reviews how sample complexity of QHT is relevant to a broad swathe of research areas and can enhance understanding of many fundamental concepts, including quantum algorithms for simulation and search, quantum learning and classification, and foundations of quantum mechanics. As such, we view our paper as an invitation to researchers coming from different communities to study and contribute to the problem of sample complexity of QHT, and we outline a number of open directions for future research.
We develop a resource theory of symmetric distinguishability, the fundamental objects of which are elementary quantum information sources, i.e., sources that emit one of two possible quantum states ...with given prior probabilities. Such a source can be represented by a classical-quantum state of a composite system \(XA\), corresponding to an ensemble of two quantum states, with \(X\) being classical and \(A\) being quantum. We study the resource theory for two different classes of free operations: \((i)\) \({\rm{CPTP}}_A\), which consists of quantum channels acting only on \(A\), and \((ii)\) conditional doubly stochastic (CDS) maps acting on \(XA\). We introduce the notion of symmetric distinguishability of an elementary source and prove that it is a monotone under both these classes of free operations. We study the tasks of distillation and dilution of symmetric distinguishability, both in the one-shot and asymptotic regimes. We prove that in the asymptotic regime, the optimal rate of converting one elementary source to another is equal to the ratio of their quantum Chernoff divergences, under both these classes of free operations. This imparts a new operational interpretation to the quantum Chernoff divergence. We also obtain interesting operational interpretations of the Thompson metric, in the context of the dilution of symmetric distinguishability.
Fawzi and Fawzi recently defined the sharp Rényi divergence, \(D_\alpha^\#\), for \(\alpha \in (1, \infty)\), as an additional quantum Rényi divergence with nice mathematical properties and ...applications in quantum channel discrimination and quantum communication. One of their open questions was the limit \({\alpha} \to 1\) of this divergence. By finding a new expression of the sharp divergence in terms of a minimization of the geometric Rényi divergence, we show that this limit is equal to the Belavkin-Staszewski relative entropy. Analogous minimizations of arbitrary generalized divergences lead to a new family of generalized divergences that we call kringel divergences, and for which we prove various properties including the data-processing inequality.
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