Mark M. Wilde

Many fundamental quantities in quantum information processing are instances of quantum divergences - functionals on quantum states that satisfy natural axioms grounded in information-theoretic principles. Recently, a new class of divergences - the f-divergences - has gained prominence in quantum information theory and received operational interpretations, while being long established in the classical setting. Furthermore, Frenkel showed that the Umegaki relative entropy is a special case of a quantum f-divergence for the function f(x) = x log x; building on this, Hirche et al. introduced a parameterized family of f-divergences that, in appropriate regimes, recovers the sandwiched and Petz relative entropies as regularizations. Taken together, these results reveal a tight link between the best-understood quantum divergences - the Umegaki, Petz, and sandwiched relative entropies - on a technical level and the general class of f-divergences, thereby strongly motivating a program that connects f-divergences to concrete quantum information tasks as already started by Cheng et al. In this contribution, we develop a variational formulation that approximates general quantum f-divergences to arbitrary precision. These approximations yield (i) efficient evaluation of the quantum relative entropy of channels and already used as the core numerical method in quantum many body physics and (ii) computation of asymptotic key rates in DIQKD in particular in the scenario of two switches in routed Bell scenarios.

The Barnum-Knill theorem states that the optimal success probability in the multiple state discrimination task is not more than the square root of the success probability when the pretty good or square-root measurement is used for this task. An assumption of the theorem is that the underlying ensemble consists of finitely many quantum states over a finite-dimensional quantum system. Motivated in part by the fact that the success probability is not a relevant metric for continuous ensembles, in this paper we provide a generalization of the notion of pretty good measurement and the Barnum-Knill theorem for general quantum ensembles, including those described by a continuous parameter space and an infinite-dimensional Hilbert space. To achieve this, we also design a general metric of performance for quantum measurements that generalizes the success probability, namely, the expected gain of the measurement with respect to a positive score function. A notable consequence of the main result is that, in a Bayesian estimation task, the mean square error of the pretty good measurement does not exceed twice the optimal mean square error.

One of the primary applications of classical Boltzmann machines is generative modeling, wherein the goal is to tune the parameters of a model distribution so that it closely approximates a target distribution. Training relies on estimating the gradient of the relative entropy between the target and model distributions, a task that is well understood when the classical Boltzmann machine has both visible and hidden units. For some years now, it has been an obstacle to generalize this finding to quantum state learning with quantum Boltzmann machines that have both visible and hidden units. In this paper, I derive an analytical expression for the gradient of the quantum relative entropy between a target quantum state and the reduced state of the visible units of a quantum Boltzmann machine. Crucially, this expression is amenable to estimation on a quantum computer, as it involves modular-flow-generated unitary rotations reminiscent of those appearing in my prior work on rotated Petz recovery maps. This leads to a quantum algorithm for gradient estimation in this setting. I then specialize the setting to quantum visible units and classical hidden units, and vice versa, and provide analytical expressions for the gradients, along with quantum algorithms for estimating them. Finally, I replace the quantum relative entropy objective function with the Petz--Tsallis relative entropy; here I develop an analytical expression for the gradient and sketch a quantum algorithm for estimating it, as an application of an independent derivation of a formula for the derivative of the matrix power function, which also involves modular-flow-generated unitary rotations. Ultimately, this paper demarcates progress in training quantum Boltzmann machines with visible and hidden units for generative modeling and quantum state learning.

xtendibility of bosonic Gaussian states is a key issue in continuous-variable quantum information. We show that a bosonic Gaussian state is $k$-extendible if and only if it has a Gaussian $k$-extension, and we derive a simple semidefinite program, whose size scales linearly with the number of local modes, to efficiently decide $k$-extendibility of any given bosonic Gaussian state. When the system to be extended comprises one mode only, we provide a closed-form solution. Implications of these results for the steerability of quantum states and for the extendibility of bosonic Gaussian channels are discussed. We then derive upper bounds on the distance of a $k$-extendible bosonic Gaussian state to the set of all separable states, in terms of trace norm and R\’enyi relative entropies. These bounds, which can be seen as “Gaussian de Finetti theorems,” exhibit a universal scaling in the total number of modes, independently of the mean energy of the state. Finally, we establish an upper bound on the entanglement of formation of Gaussian $k$-extendible states, which has no analogue in the finite-dimensional setting.

We introduce the genuine multipartite Rains entanglement (GMRE) as a measure of genuine multipartite entanglement that can be computed using semi-definite programming. Similar to the Rains relative entropy (its bipartite counterpart), the GMRE is monotone under selective quantum operations that completely preserve the positivity of the partial transpose, implying that it is a multipartite entanglement monotone. As a consequence, we show that the GMRE bounds both the one-shot standard and probabilistic approximate GHZ-distillable entanglement from above. We also develop a generalization of this quantity that incorporates other entropies, including quantum Rényi relative entropies.

In classical information theory, the Doeblin coefficient of a classical channel provides an efficiently computable upper bound on the total-variation contraction coefficient of the channel, leading to what is known as a strong data-processing inequality. Here, we investigate quantum Doeblin coefficients as a generalization of the classical concept. In particular, we define various new quantum Doeblin coefficients, one of which has several desirable properties, including concatenation and multiplicativity, in addition to being efficiently computable. We also develop various interpretations of two of the quantum Doeblin coefficients, including representations as minimal singlet fractions, exclusion values, reverse max-mutual and oveloH informations, reverse robustnesses, and hypothesis testing reverse mutual and oveloH informations. Our interpretations of quantum Doeblin coefficients as either entanglement-assisted or unassisted exclusion values are particularly appealing, indicating that they are proportional to the best possible error probabilities one could achieve in state-exclusion tasks by making use of the channel. We also outline various applications of quantum Doeblin coefficients, ranging from limitations on quantum machine learning algorithms that use parameterized quantum circuits (noise-induced barren plateaus), on error mitigation protocols, on the sample complexity of noisy quantum hypothesis testing, and on mixing, distinguishability, and decoupling times of time-varying channels. All of these applications make use of the fact that quantum Doeblin coefficients appear in upper bounds on various trace-distance contraction coefficients of a channel. Furthermore, in all of these applications, our analysis using Doeblin coefficients provides improvements of various kinds over contributions from prior literature, both in terms of generality and being efficiently computable.

The study of feedback control inspired by Maxwell's demon is central to the understanding of the relationship between thermodynamics and information. In this paper, we establish fundamental lower limits on the work costs of system conversion with quantum feedback, where quantum side information acquired in advance can be fed back to the system coherently by a controller. From two basic operational principles that every physically admissible feedback-control scheme should satisfy, we derive the tightest possible bounds on the single-shot work of formation and extractable work of an arbitrary quantum system given arbitrary quantum side information held by the controller. These bounds are expressed in terms of information measures generalizing both mutual informations and relative entropies. In the asymptotic limit, they lead to a generalized second law of thermodynamics with quantum feedback, featuring a conditional Helmholtz free energy. Our findings provide precise thermodynamic meanings for the negativity of single-shot conditional entropies and resolve an open problem in the axiomatic reconstruction of such conditional entropies.

The notions of privacy tests and k-extendible states have both been instrumental in quantum information theory, particularly in understanding the limits of secure communication. In this paper, we determine the maximum probability with which an arbitrary k-extendible state can pass a privacy test, and we prove that it is equal to the maximum fidelity between an arbitrary k-extendible state and the standard maximally entangled state. Our findings, coupled with the resource theory of k-unextendibility, lead to an efficiently computable upper bound on the one-shot, one-way distillable key of a bipartite state, and we prove that it is equal to the best-known efficiently computable upper bound on the one-shot, one-way distillable entanglement. We also establish efficiently computable upper bounds on the one-shot, forward-assisted private capacity of channels. Extending our formalism to the independent and identically distributed setting, we obtain single-letter efficiently computable bounds on the n-shot, one-way distillable key of a state and the n-shot, forward-assisted private capacity of a channel. For some key examples of interest, our bounds are significantly tighter than other known efficiently computable bounds.

Simulating open quantum systems is essential for modeling realistic dynamics beyond closed-system Hamiltonian evolution. Such dynamics are described by the Lindblad master equation for Markovian systems and arise in fields ranging from condensed matter and quantum chemistry to quantum optics and noise analysis in quantum devices. Existing algorithms typically rely on sparse-access or linear-combination-of-unitaries input models. We propose an alternative framework, Wave Matrix Lindbladization, inspired by density matrix exponentiation for sample-based Hamiltonian simulation. In this model, Hamiltonians and Lindblad operators are encoded directly into program states, enabling sample-based Lindbladian simulation. We present algorithms for this task, analyze their sample and gate complexities, and demonstrate both efficiency and optimality. We also show that our algorithms achieve better sample complexity than any tomographic strategy for Lindbladian simulation. This further suggests a form of quantum copy-protection, where program states allow Lindbladian simulation without revealing the operators they encode.

We study the capacity of a quantum channel for retrocausal communication, where messages are transmitted backward in time, from a sender in the future to a receiver in the past, through a noisy postselected closed timelike curve (P-CTC) mathematically represented by the channel. We completely characterize the one-shot retrocausal quantum and classical capacities, and we show that the corresponding asymptotic capacities are equal to the average and sum, respectively, of the channel's max-information and its regularized Doeblin information. This endows these information measures with a novel operational interpretation. Furthermore, our characterization can be generalized beyond quantum channels to all completely positive maps. This imposes information-theoretic limits on transmitting messages via postselected-teleportation-like mechanisms with arbitrary initial- and final-state boundary conditions, including those considered in various black-hole final-state models.

Distinguishing quantum states and channels lies at the core of quantum information processing. Quantum hypothesis testing and channel discrimination have been traditionally studied in the asymptotic setting, assuming access to an infinite number of samples of data or channel uses. However, in practice, finite resources are the regime of interest and demand a non-asymptotic approach. In this work, we study the non-asymptotic regime of the discrimination setting to obtain the optimal sample complexity for the task of the binary state discrimination problem, where our lower and upper bounds differ only by a constant factor of four. We also extend the non-asymptotic study (finite channel uses n) of channel discrimination and determine the optimal query complexity for discrimination of classical–quantum channels, thus also solving this dynamical generalization of the sample complexity question.

In this paper, we develop the resource theory of quantum secret key. Operating under the assumption that entangled states with zero distillable key do not exist, we define the key cost of a quantum state, and device. We study its properties through the lens of a quantity that we call the key of formation. The main result of our paper is that the regularized key of formation is an upper bound on the key cost of a quantum state. The core protocol underlying this result is privacy dilution, which converts states containing ideal privacy into ones with diluted privacy. Next, we show that the key cost is bounded from below by the regularized relative entropy of entanglement, which implies the irreversibility of the privacy creation-distillation process for a specific class of states. We further focus on mixed-state analogues of pure quantum states in the domain of privacy, and we prove that a number of entanglement measures are equal to each other for these states, similar to the case of pure entangled states. The privacy cost and distillable key in the single-shot regime exhibit a yield-cost relation, and basic consequences for quantum devices are also provided.