Marcus Huber

High-dimensional (HD) entanglement promises both enhanced key rates and overcoming obstacles faced by modern-day quantum communication. However, modern convex optimization-based security arguments are limited by computational constraints; thus, accessible dimensions are far exceeded by progress in HD photonics, bringing forth a need for efficient methods to compute key rates for large encoding dimensions. In response to this problem, we present a flexible analytic framework facilitated by the dual of a semi-definite program and diagonalizing operators inspired by entanglement-witness theory, enabling the efficient computation of key rates in high-dimensional systems. To facilitate the latter, we show how matrix completion techniques can be incorporated to effectively yield improved, computable bounds on the key rate in paradigmatic high-dimensional systems of time- or frequency-bin entangled photons and beyond, revealing the potential for very high dimensions to surpass low dimensional protocols already with existing technology. In our accompanying work, we show how our findings can be used to establish finite-size security against coherent attacks for general HD-QKD protocols both in the fixed- and variable-length scenario and we examine the performance under realistic conditions. Detailed manuscripts for Refs. [1] and [2] can be found attached.

Entanglement based quantum communication offers an increased level of security in practical secret shared key distribution. One of the fundamental principles enabling this security -- the fact that interfering with one photon will destroy entanglement and thus be detectable -- is also the greatest obstacle. Random encounters of traveling photons, losses and technical imperfections make noise an inevitable part of any quantum communication scheme, severely limiting distance, key rate and environmental conditions in which QKD can be employed. Using photons entangled in their spatial degree of freedom, we show that the increased noise resistance of high-dimensional entanglement, can indeed be harnessed for practical key distribution schemes. We perform quantum key distribution in eight entangled paths at various levels of environmental noise and show key rates that, even after error correction and privacy amplification, still exceed 1 bit per photon pair and furthermore certify a secure key at noise levels that would prohibit comparable qubit based schemes from working.

High-dimensional entanglement promises not only increased key rates but overcoming some of the obstacles faced by modern-day quantum communication. Typically, key rates are computed via convex optimization procedures, which inherently limits the dimensionality one can analyze through computational constraints. Recent progress in high-dimensional photonics far exceeds these limitations and brings forth a need for (semi-)analytic methods to compute key rates in the regime of large encoding dimensions. We present a flexible analytic framework facilitated by the dual of a semi-definite program, enabling the computation of key rates in high-dimensional systems. This method, whether purely analytical or semi-numerical, hinges on diagonalizing specific operators influenced by entanglement witnesses and efficiently solving an optimization problem. To facilitate the latter, we show how matrix completion techniques can be incorporated to yield effective and computable bounds on the key rate in paradigmatic high-dimensional systems of time- or frequency-bin entangled photons and beyond.

In our work, we provide a clean security analysis of a new high-dimensional QKD setup with a Franson interferometer in the asymptotic limit and calculate secure key rates using a recent method developed. We argue that our new protocol is not only experimentally easier, as it does not require tomography of the polarization degree of freedom, but also allows for a clean security analysis without assumptions that were implicitly hidden in earlier analyses of similar and related protocols. We build a realistic noise model that takes environmental photons, dark counts, channel losses and non-unit detection efficiency into account and show that our new protocol allows secure key rates for twice as many environmental photons than comparable protocols available in literature. We want to highlight that while the security analysis of our protocol is rigorous and clean, the compared key rates for the compared protocol are actually only an upper bound (due to the assumptions implicitly hidden described earlier), so our new protocol outperforms previous settings by at least a factor of 2. Current free-space QKD implementations are only operable during night when environmental photons are low, but fail to provide secure keys during twilight and daytime, which is a major obstacle towards broad practical usage. Thus, doubling the robustness against environmental photons marks an important step forwards towards daylight-independent Quantum Key Distribution implementations.

High-dimensional entanglement promises to increase the information capacity of photons and isnow routinely generated exploiting spatio-temporal degrees of freedom of single photons. A curiousfeature of these systems is the possibility to certify entanglement despite strong noise in the data.We show that it is also possible to exploit this noisy entanglement by introducing a protocol thatuses mutliple subspaces of the high-dimensional system simultaneously. Our protocol can be used toincrease key rates in realistic conditions. To that end, we conduct two simulations of our protocol fornoise models that apply to the two most commonly used sources of high-dimensional entanglement:time-bins and spatial modes.