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.

We study the noise threshold for positive quantum capacity for the qubit depolarizing channel. We explore analytically the action of the qubit depolarizing channel on the symmetric subspaces of the input qubits, in the limit of asymptotically many uses of the channel. We observe the emergence of a bosonic Gaussian channel. Furthermore, the codes previously developed for the depolarizing channel can be translated to codes for the emergent Gaussian channel, and it is easier to further optimize these codes for the simpler emergent Gaussian channel. Translating these codes back to the depolarizing channel leads to extremely good input states for the coherent information of the depolarizing channel producing new lower bounds on the noise threshold for positive capacity. In addition to improved lower bounds on the threshold, this newly found link between depolarizing noise and Gaussian channels offers a novel perspective contributing to our understanding of these symmetric codes.

Random unitaries are useful in quantum information and related fields, but hard to generate with limited resources. An approximate unitary k-design is an ensemble of unitaries with an underlying measure over which the average is close to a Haar random ensemble up to the first k moments. A particularly strong notion of approximation bounds the distance from Haar randomness in relative error. Such relative-error approximate designs are secure against queries by an adaptive adversary trying to distinguish it from a Haar ensemble. We construct relative-error approximate unitary k-design ensembles for which communication between subsystems is O(1) in the system size. These constructions use the alternating projection method to analyze overlapping Haar averages, giving a bound on the convergence speed to the full averaging with respect to the 2-norm. Using von Neumann subalgebra indices to replace system dimension, the 2-norm distance converts to relative error without introducing any additional dimension dependence. We use these constructions as the building blocks of a two-step protocol that achieves a relative-error design in O((logm+log(1/ϵ)+klogk)kpolylog(k)) depth, where m is the number of qudits in the complete system and ϵ the approximation error. This sublinear depth construction answers a variant of [Harrow and Mehraban 2023, Section 1.5, Open Questions 1 and 7]. Moreover, entanglement generated by the sublinear depth scheme follows area laws on spatial lattices up to corrections logarithmic in the full system size.

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.