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| Monday | Tuesday | Wednesday | Thursday | Friday | |
|---|---|---|---|---|---|
| 9:30–10:30 | Tomamichel | Hayashi | Fawzi | Hiai | Sutter |
| 10:30–11:00 | Coffee break | Coffee break | Coffee break | Coffee break | Coffee break |
| 11:00–12:00 | Dalai | Cheng | Christandl | Hirche | Bergh |
| 12:00–13:30 | Lunch | Lunch | Lunch | Lunch | Lunch |
| 13:30–14:30 | Reguła | Berta | Rubboli | Mosonyi | |
| 14:30–15:00 | Coffee break | Coffee break | Free afternoon | Coffee break | Coffee break |
| 15:00–16:00 | Buscemi | Brandão | Renes | ||
| Evening | Reception (18:00) | Social dinner (map) |
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Why Petz?
An attempt to justify the occasion of our meeting.
Some Open Problems Related to List Decoding for Classical-Quantum Channels
I will discuss some recent results and some open problems in the study of the reliability function of classical-quantum channels, with a particular focus on the low-rate region. In this regime, list decoding is known to play an important role, and this creates a connection with a fundamental problem in multi-hypothesis testing that requires new tools. A number of results are available in the classical case that have not yet been extended to the classical-quantum setting. Recent results suggest that significant progress on these questions may be possible: quite surprisingly, some problems that are very difficult in the classical case turn out to admit rather simple answers for a large class of pure-state channels.
Rethinking quantum smooth entropies: Tight one-shot analysis of quantum privacy amplification
We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering the state-of-the-art asymptotic i.i.d. error exponents shown there. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.
Why Petz? Because Bayes!
Bayesian updating is usually introduced as the natural way of revising a prior in light of new information: change what must be changed, and no more. In classical probability theory, this principle (sometimes called "the minimum change principle") leads to Bayes' rule. In quantum theory, the same idea becomes more delicate, since noncommutativity leaves us without a canonical notion of joint distribution, conditional probability, or even what exactly should be updated. In this talk, I will discuss a formulation of quantum Bayesian updating based on a principle of minimum change. Rather than updating only marginal states, the construction treats input-output processes as the basic objects and asks for a retrodictive map that is consistent with the observed data while deviating from the prior description as little as possible. In many cases of practical interest, this map is precisely the Petz transpose map. This suggests a simple interpretation for the Petz map: it is not merely a useful recovery map appearing in quantum information theory, but the quantum analogue of Bayesian inversion. I will then describe applications to quantum measurement retrodiction, uncertainty relations, and (if time permits) the emergence of irreversibility. This is work done in collaboration with (in alphabetical order): Ge Bai, Kohtaro Kato, Jiaxi Kuang, Teruaki Nagasawa, Valerio Scarani, Kensei Torii, Eyuri Wakakuwa.
Operational interpretation of the reverse sandwiched Renyi divergences in composite quantum hypothesis testing
We study the Hoeffding regime of composite quantum hypothesis testing, in which each hypothesis is specified by a sequence of sets of quantum states. We establish quantum Hoeffding bounds under a set of structural assumptions, orthogonal to those of our previous framework. A notable consequence is the direct operational interpretation of the reverse sandwiched Rényi divergence for \(\alpha \in (0,1)\): for the task of discriminating a thermal equilibrium state from a probe state subject to unknown dephasing in the energy eigenbasis, with free Hamiltonian evolution as a special case, the optimal Hoeffding exponent is given exactly by this divergence evaluated on a single copy of the system. The same task in the Stein regime is governed by the reverse quantum relative entropy, providing its operational interpretation as well. This behavior contrasts both with the simple independent and identically distributed (i.i.d.) setting, where the Petz Rényi divergence and the Umegaki relative entropy govern the Hoeffding and Stein exponents, respectively, and with many composite settings, where only regularized many-copy formulas are available. This finding reveals that passing from simple to composite hypotheses can fundamentally change which quantum divergence determines the operational limits of discrimination, and suggests a new avenue for seeking operational interpretations of quantum divergences by lifting simple hypotheses to richer composite scenarios.
Multiple Quantum Hypothesis Testing: Pairwise Bounds, Harmonic-Mean Bounds, and Asymptotics
We consider Bayesian discrimination among multiple quantum states and establish a dimension-free one-shot upper bound on the minimum probability of error in terms of the sum of pairwise errors. This resolves a conjecture of Audenaert and Mosonyi [J. Math. Phys. 55 (2014)] and improves the multiple quantum Chernoff bound of Li [Ann. Statist. 44 (2016)] by removing its dimension-dependent prefactor. In the asymptotic many-copy regime, our bound proves the achievability of the multiple quantum Chernoff distance for arbitrary separable Hilbert spaces, thereby settling the previously open infinite-dimensional case, and further yields constant-factor sharp asymptotics for the optimal error probability.
Tight any-shot quantum decoupling
We prove a novel one-shot decoupling theorem formulated in terms of quantum relative entropy distance, with the decoupling error bounded by two sandwiched Rényi conditional entropies. In the asymptotic i.i.d. setting of standard information decoupling via partial trace, we show that this bound is ensemble-tight in quantum relative entropy distance and thereby yields a characterization of the associated decoupling error exponent in the low-cost-rate regime. As applications we show a single-letter expression for the exact error exponent of quantum state merging in terms of Petz-Rényi conditional entropies and (ii) novel entropic uncertainty relations for sets of multiple measurements.
Lower bounds to conditional mutual information
N.A.
Multi-index Schatten Norms
Multi-index Schatten norms are (quasi-)norms on operators on a tensor product of Hibert spaces. I will discuss such norms, the link to conditional entropies and their applications to their additivity properties. Based on joint works with Jan Kochanowski, Cambyse Rouzé and Thomas van Himbeeck https://arxiv.org/abs/2502.01611 and https://arxiv.org/abs/2604.14055.
Fault-tolerant quantum I/O
Usual scenarios of fault-tolerant computation are concerned with the fault-tolerant realization of quantum algorithms that compute classical functions, such as Shor's algorithm for factoring. In particular, this means that input and output to the quantum algorithm are classical. In contrast to stand-alone single-core quantum computers, in many distributed scenarios, quantum information might have to be passed on from one quantum information processing system to another one, possibly via noisy quantum communication channels with noise levels above fault-tolerant thresholds. In such situations, quantum information processing devices will have quantum inputs, quantum outputs or even both, which pass qubits among each other.
Hockey stick \(f\)-divergences in von Neumann algebras
A new notion of quantum \(f\)-divergences called hockey stick \(f\)-divergences has recently been developed by several authors. We consider the notion in the general von Neumann algebra setting, thus extending simultaneously the classical \(f\)-divergences and the hockey stick \(f\)-divergences in the finite dimensional case. In this talk, after giving the definition in the general setting, we discuss their basic properties such as differentiability of the hockey stick divergences, representation from Neyman-Pearson tests, joint lower semicontinuity, martingale convergences, and sufficiency/reversibility. This is a part of joint ongoing work with Milan Mosonyi and Marco Tomamichel.
New Frontiers for Contraction Coefficients
Contraction coefficients give improvements for the data processing inequality and are fundamental to many applications in quantum computing. Essentially, they quantify the loss of distinguishability between two quantum states stemming from a (completely) positive map. While a lot of progress has been made recently towards understanding the structure of these coefficients, data processing inequalities can cover much more general scenarios. In this talk I will discuss recent progress and ongoing work, aiming to generalize contraction coefficients beyond the current framework.
From classical to quantum conditional entropies
We present the most general form of classical conditional entropies singled out by a set of operational axioms. We then discuss how these quantities can be extended from classical to quantum states, and illustrate their relevance in applications.
Dualities in CQ problems
It is reasonably well-known that the properties of suitably-chosen pairs of classical-quantum states (or pairs of channels) are tightly constrained by entropy dualities and similar relations. This phenomenon underpins the security of quantum key distribution and has found use in characterizing CQ channel coding error exponents, constructing lossy compression tasks, and bounding properties of iterative decoding algorithms. In this talk I want to argue that this area is still largely unexplored and perhaps has significant further applications. One reason for optimism is that there are much more general entropy duality statements known even in the very general setting of algebraic quantum field theory. Perhaps other information processing protocols can be profitably understood via duality. However, the main reason for optimism is that the Regev reduction, as used in the decoded quantum interferometry (DQI) optimization algorithm, is very closely related to CQ duality. This suggests that duality may have further applications in understanding the performance of quantum algorithms.
Robust generalized quantum Stein's lemma
We study the problem of distinguishing almost independent and identically distributed states from separable states. We prove that the regularized relative entropy of entanglement of the associated iid state quantifies the optimal error exponent for asymmetric quantum hypothesis testing in this setting. This establishes a robust version of the generalized quantum Stein's lemma of Brandao and Plenio, which assumes an exact iid structure. In particular, our result shows that the original argument of Brandao and Plenio, which contains a logical gap, can be made rigorous.
Generalized Quantum Stein’s Lemma and Reversibility of Quantum Resource Theories for Classical-Quantum Channels
We prove a Generalized Quantum Stein’s Lemma for classical-quantum (c-q) channels, which characterizes the composite hypothesis testing problem of testing a c-q channel against a sequence of sets of c-q channels (satisfying certain natural assumptions), under parallel strategies. For this, we prove that the optimal asymptotic asymmetric error exponent is given by the regularization of the Umegaki channel divergence, minimized over the sets. This then allows us to prove the reversibility of resource theories of classical-quantum channels in a natural framework where the set of free operations is the set of all asymptotically resource non-generating superchannels. The talk will also highlight the difficulties still present in generalizing this approach to general quantum channels and draw connections to other open problems in quantum channel discrimination.
State discrimination error exponents for quasifree states on a fermionic lattice
The trade-off relations between the two types of error probabilities in binary i.i.d. quantum state discrimination can be expressed by single-copy formulas in terms of the Petz-type and the sandwiched Rényi divergences of the two states representing the two hypotheses. In the non-i.i.d. setting, the error exponents can usually be expressed in terms of regularized Rényi divergences, which do not admit explicit formulas in general. Here, we consider a class of states, quasifree states on fermionic lattices, and give explicit formulas for a wide range of regularized Rényi divergences, including alpha-z, log-Euclidean, maximal, measured, and the recently introduced integral Rényi divergences. We show that the case where there is a single mode at each lattice site becomes asymptotically classical, with all the different types of regularized Rényi-divergences being equal, while in the case of multiple modes per site, non-commutativity persists under regularization, and all the different types of Rényi divergences give different regularized values in general. We also provide examples of states where the discrimination error goes to zero with a speed that is super-exponential in the system size.
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