Aadil Oufkir

In their seminal work, Bennett et al. [IEEE Trans. Inf. Theory (2002)] showed that, with sufficient shared randomness, one noisy channel can simulate another at a rate equal to the ratio of their capacities. We establish that when coding above this channel interconversion capacity, the exact strong converse exponent is characterized by a simple optimization involving the difference of the corresponding Renyi channel capacities with Holder dual parameters. We extend this result to the entanglement-assisted interconversion of classical-quantum channels, showing that the strong converse exponent is likewise determined by differences of sandwiched Renyi channel capacities. The converse bound is obtained by relaxing to non-signaling assisted codes and applying Holder duality together with the data processing inequality for Renyi divergences. Achievability is proven by concatenating refined channel coding and simulation protocols that go beyond first-order capacities, achieving exponentially small conversion errors.

We study the quantum umlaut information, a correlation measure defined for bipartite quantum states as a reversed variant of the quantum mutual information. We show that it has an operational interpretation as the asymptotic error exponent in the hypothesis testing task of deciding whether a given bipartite state is product or not. We generalise the umlaut information to quantum channels, where it also extends the notion of `oveloh information' [Nuradha et al., arXiv:2404.16101]. We prove that channel umlaut information is additive for classical-quantum channels, while we observe additivity violations for fully quantum channels. Inspired by recent results in entanglement theory, we then show as our main result that the regularised umlaut information constitutes a fundamental measure of the quality of classical information transmission over a quantum channel - as opposed to the capacity, which quantifies the quantity of information that can be sent. This interpretation applies to coding assisted by activated non-signalling correlations, and the channel umlaut information is in general larger than the corresponding expression for unassisted communication as obtained by Dalai for the classical-quantum case [IEEE Trans. Inf. Theory 59, 8027 (2013)]. In the classical unassisted setting, the channel umlaut information has a further operational interpretation as the zero-rate error exponent of list decoding in the large list limit. Combined with prior works on non-signalling--assisted zero-error channel capacities, our findings imply a dichotomy between the settings of zero-rate error exponents and zero-error communication. While our results are single-letter only for classical-quantum channels, we also give a single-letter bound for fully quantum channels in terms of the `geometric' version of umlaut information.

We study the problems of certifying and learning local quantum Hamiltonians and their associated Gibbs states. We first address Hamiltonian certification given real-time access to the dynamics of an unknown k-local Hamiltonian. Given oracle access to its time-evolution operator and a fully specified target Hamiltonian, the task is to decide whether the two Hamiltonians are identical or differ by at least a prescribed accuracy in normalized Frobenius norm, while minimizing the total evolution time. We introduce the first certification protocol that achieves optimal performance for all constant-locality Hamiltonians. For general n-qubit, k-local, traceless Hamiltonians, our algorithm succeeds with high probability using total evolution time that scales inversely with the target accuracy, and for constant locality this matches the fundamental lower bound, achieving Heisenberg-limit scaling. In contrast to prior approaches, our method requires neither inverse evolution nor controlled operations, and relies only on forward real-time dynamics. We then turn to thermal states generated by local Hamiltonians. We develop algorithms for both learning and certifying Gibbs states that are fully sample-efficient in all relevant parameters. For polynomially bounded temperature, our methods achieve exponential improvements over general quantum state tomography. While the learning algorithm is inherently time-inefficient due to covering arguments, the certification algorithm is both sample- and time-efficient, resolving a previously open question on efficient Gibbs state testing. Together, these results establish optimal or near-optimal complexity bounds for characterizing local quantum systems in both dynamical and thermal regimes.

Locality is a fundamental feature of many physical time evolutions. Assumptions on locality and related structural properties also underlie recently proposed procedures for learning an unknown Hamiltonian from access to the induced time evolution. However, no protocols to rigorously test whether an unknown Hamiltonian is in fact local were known. We investigate Hamiltonian locality testing as a property testing problem, where the task is to determine whether an unknown Hamiltonian H is k-local or epsilon-far from all k-local Hamiltonians, given access to the time evolution along H. First, we emphasize the importance of the chosen distance measure: With respect to the operator norm, a worst-case distance measure, incoherent quantum locality testers require at least order 2^n many time evolution queries and an expected total evolution time of order 2^n/epsilon, and even coherent testers need at least order 2^(n/2) many queries and order 2^(n/2)/epsilon total evolution time. In contrast, when distances are measured according to the normalized Frobenius norm, corresponding to an average-case distance, we give a sample-, time-, and computationally efficient incoherent Hamiltonian locality testing algorithm based on randomized measurements. In fact, our procedure can be used to simultaneously test a wide class of Hamiltonian properties beyond locality. Finally, we prove that learning a general Hamiltonian remains exponentially hard with this average-case distance, thereby establishing an exponential separation between Hamiltonian testing and learning. Our work initiates the study of property testing for quantum Hamiltonians, demonstrating that a broad class of Hamiltonian properties is efficiently testable even with limited quantum capabilities, and positioning Hamiltonian testing as an independent area of research alongside Hamiltonian learning.

How many copies of a quantum process are necessary and sufficient to construct an approximate classical description of it? We extend the result of Surawy-Stepney, Kahn, Kueng, and Guta (2022) to show that tildemathcalO(din^3dout^3/ε^2) copies are sufficient to learn any quantum channel mathdsC^dintimes dinrightarrowmathdsC^douttimes dout to within ε in diamond norm. Moreover, we show that Ømega(din^3dout^3/ε^2) copies are necessary for any strategy using incoherent non-adaptive measurements. This lower bound applies even for ancilla-assisted strategies.

Learning about quantum states is a fundamental problem in physics and quantum computing. As people are often interested in certain properties of quantum states instead of a complete description, shadow tomography has gained significant attention, where the goal is to learn the expectation values of $M$ observables $O_1, \ldots, O_M$ with $\varepsilon$ accuracy. In near-term devices, however, noise is prevalent and often unexpected. Thus, it is crucial to design algorithms that work well in the worst case. We study the practical single-copy setting and assume $\gamma$-fraction of the \emph{outcomes} can be arbitrarily corrupted by an adversary. We show that all non-adaptive shadow tomography algorithms must incur an error of $\varepsilon=\tilde{\Omega}(\gamma\min\{\sqrt{M}, \sqrt{d}\})$ for some choice of observables, even with unlimited copies. Unfortunately, the classical shadows algorithm by \cite{huang2020predicting} and naive algorithms that directly measure each observable suffer even more. We design an algorithm that achieves an error of $\varepsilon=\tilde{O}(\gamma\max_{i\in[M]}\|O_i\|_{HS})$, which nearly matches our worst-case error lower bound for $M\ge d$ and guarantees better accuracy when the observables have stronger structure. Remarkably, the algorithm only needs $n=\frac{1}{\gamma^2}\log(M/\delta)$ copies to achieve that error with probability at least $1-\delta$, matching the sample complexity of the classical shadows algorithm that achieves the same error without corrupted measurement outcomes. Our algorithm is conceptually simple and easy to implement. Classical simulation for fidelity estimation shows that our algorithm enjoys much stronger robustness than~\cite{huang2020predicting} under adversarial noise. Finally, based on a reduction from full-state tomography to shadow tomography, we prove that for rank $r$ states, both the near-optimal asymptotic error of $\eps=\tilde{O}(\gamma\sqrt{r})$ \emph{and} copy complexity $\tilde{O}(dr^2/\eps^2)=\tilde{O}(dr/\gamma^2)$ can be achieved for adversarially robust state tomography, closing the large gap in \cite{AliakbarpourBCL2025robustquantum} where optimal error can only be achieved using pseudo-polynomial number of copies in $d$.