Felix Leditzky

In this work we improve the quantum communication rates of various quantum channels of interest using permutation-invariant quantum codes. We focus in particular on parametrized families of quantum channels and aim to improve bounds on their quantum capacity threshold, defined as the lowest noise level at which the quantum capacity of the channel family vanishes. These thresholds are important quantities as they mark the noise level up to which faithful quantum communication is theoretically possible. Our method exploits the fact that independent and identically distributed quantum channels preserve any permutation symmetry present at the input. The resulting symmetric output states can be described succinctly using the representation theory of the symmetric and general linear groups, which we use to derive an efficient algorithm for computing the channel coherent information of a permutation-invariant code. Our approach allows us to evaluate coherent information values for a large number of channel copies, e.g., at least 100 channel copies for qubit channels. We apply this method to various physically relevant channel models, including general Pauli channels, the dephrasure channel, the generalized amplitude damping channel, and the damping-dephasing channel. For each channel family we obtain improved lower bounds on their quantum capacities. For example, for the 2-Pauli and BB84 channel families we significantly improve the best known quantum capacity thresholds derived in [Fern, Whaley 2008]. These threshold improvements are achieved using a repetition code-like input state with non-orthogonal code states, which we further analyze in our representation-theoretic framework.

Quantum teleportation is a quantum communication primitive that allows a long-distance quantum channel to be built using pre-shared entanglement and one-way classical communication. However, the quality of the established channel crucially depends on the quality of the pre-shared entanglement. In this work, we revisit the problem of using noisy entanglement for the task of teleportation. We first show how this problem can be rephrased as a state discrimination problem. In this picture, a quantitative duality between teleportation and dense coding emerges in which every Alice-to-Bob teleportation protocol can be repurposed as a Bob-to-Alice dense coding protocol, and the quality of each protocol can be measured by the success probability in the same state discrimination problem. One of our main results provides a complete characterization of the states that offer no advantage in one-way teleportation protocols over classical states, thereby offering a new and intriguing perspective on the long-standing open problem of identifying such states. This also yields a new proof of the known fact that bound entangled states cannot exceed the classical teleportation threshold. Moreover, our established duality between teleportation and dense coding can be used to show that the exact same states are unable to provide a non-classical advantage for dense coding as well. We also discuss the duality from a communication capacity point of view, deriving upper and lower bounds on the accessible information of a dense coding protocol in terms of the fidelity of its associated teleportation protocol. A corollary of this discussion is a simple proof of the previously established fact that bound entangled states do not provide any advantage in dense coding.

Communication networks have multiple users, each sending and receiving messages. A multiple access channel (MAC) models multiple senders transmitting to a single receiver, such as the uplink from many mobile phones to a single base station. The optimal performance of a MAC is quantified by a capacity region of simultaneously achievable communication rates. We study the two-sender classical MAC, the simplest and best-understood network, and find a surprising richness in both a classical and quantum context. First, we find that quantum entanglement shared between senders can substantially boost the capacity of a classical MAC. Second, we find that optimal performance of a MAC with bounded-size inputs may require unbounded amounts of entanglement. Third, determining whether a perfect communication rate is achievable using finite-dimensional entanglement is undecidable. Finally, we show that evaluating the capacity region of a two-sender classical MAC is in fact NP-hard.

With sensitive information encoded in data, it is important to ensure the privacy to those sensitive information while learning useful information about the data. Towards that, quantum versions of differential privacy frameworks have been introduced. Privatizing data often comes with a cost. Meaning that, there are perfectly private mechanisms one could use that may lead to no utility depending on the application. However, privacy-utility tradeoffs in the quantum setting are not extensively studied. In this work, we study optimal privacy-utility tradeoffs for both generic and application-specific utility metrics when privacy is quantified by $(\varepsilon,\delta)$-quantum local differential privacy. As generic measures, we focus on optimizing fidelity and trace distance between the original state and the privatized state, showing that the depolarizing mechanism achieves the optimal utility while obtaining analytical expressions for utility for all privacy parameter regimes. Next, we study a specific application where one needs to learn the expectation of an observable with respect to an input state (property of quantum data), given access to only privatized states. There, we obtain a lower bound on the number of samples of privatized data required to achieve a fixed accuracy guarantee with high probability by utilizing lower bounds on private quantum hypothesis testing. We also obtain private mechanisms that achieve order optimality with respect to the privacy parameters and accuracy parameters, showcasing how the awareness of the task can be utilized to improve the utility in contrast to using mechanisms that optimize generic utility metrics. Furthermore, we show that the number of samples required to privately learn the expectation values scales as $\Theta((\varepsilon \beta)^{-2})$, where $\varepsilon \in (0,1)$ is the privacy parameter and $\beta$ is the accuracy tolerance. We also study a private version of classical shadows, which may be useful for the private estimation of properties of quantum states and processes.

In this work, we study the sample complexity of two variants of product testing when restricted to single-copy measurements. In particular, we consider both bipartite product testing (i.e., does there exist at least one non-trivial cut across which the state is product) and multipartite product testing (i.e., is the state fully product across every cut). For the first variant, we prove an exponential lower bound on the sample complexity of any algorithm for this task which utilizes only single-copy measurements. When comparing this with known efficient algorithms that utilize multi-copy measurements, this establishes an exponential separation for this and several related entanglement learning tasks. For the second variant, we prove another sample lower bound that establishes a separation between single- and multi-copy strategies. To obtain our results, we prove a crucial technical lemma that gives a lower bound on the overlap between tensor products of permutation operators acting on subsystems of states that themselves carry a tensor structure. Finally, we provide an algorithm for multipartite product testing using only single-copy, local measurements, and we highlight several interesting open questions arising from this work.

We study the information-theoretic properties of the full two-parameter family of quantum channels covariant under the orthogonal group O(d), which we call Brauer channels. This family contains the depolarizing and generalized Werner-Holevo channels as one-parameter subfamilies, and its Choi matrix admits a three-block decomposition via Schur-Weyl duality for O(d). We derive closed-form expressions for the maximum output p-norms of Brauer channels and identify a region where two-copy multiplicativity of these norms fails. We prove that the minimum output Renyi 2-entropy is additive for two copies across the entire CPTP region, extending prior partial results. Using orthogonal covariance, we simplify several classical and quantum capacity upper bounds, including SDP-based, approximate entanglement-breaking, approximate covariance, Rains information, and additive-extension bounds, and compare them across the family of Brauer channels. Finally, we derive a closed-form expression for the coherent information of qudit repetition codes through multiple copies of any Brauer channel. We find that repetition codes exhibit positive coherent information beyond the hashing bound in a broad region of channels that includes the qudit depolarizing channel, demonstrating superadditivity of coherent information for Brauer channels.