Aadil Oufkir

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.

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.

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.