PhD Defense Mads Friis Frand-Madsen
Title: Connections between Quantum Key Distribution and Quantum Data Hiding
We consider a classical communication setup with two spatially separated parties sharinga quantum system. In such a setup, entanglement is considered a valuable resource useful for a plethora of undertakings; for one, the generation of secret key, which is the primary concern of our work. In 2004, Horodecki et al., showed that the amount of secret key that
can be extracted from noisy bipartite quantum states may exceed the amount of distillable maximally entangled quantum bits. Their work was based on the intuition of quantum data hiding, but did not offer a quantitative relationship to this phenomenon. More recently, the connection between quantum key distribution and quantum data hiding was further developed upon by Christandl and Ferrara , who also used the connection as a tool to bound the rate at which secret key can be distributed in a network scenario. In this thesis, we consider the question of distinguishability of quantum states given an imperfect quantum memory. We define a rate at which secure classical data can be extracted from partially secure data with respect to an eavesdropper with imperfect quantum memory. We prove an upper bound and a lower bound on this rate under general assumptions in terms of entropic quantities. Finally, we introduce a rate at which classical data can be hidden from an eavesdropper with imperfect quantum memory, and we relate this back to the notion of quantum data hiding.
In a one-way classical communication scenario for quantum key distribution, we prove a lower bound on the distillable key of a generic state in terms of correlation and orthogonality of related quantum states. We elaborate on this by showing that the security of a key can be understood in terms of orthogonality of related quantum states. Finally, we also consider the problem of extending the distance of quantum key distribution through intermediate stations, a setting referred to as a quantum key repeater. Here, we exhibit situations, where we can lower bound the performance of the optimal key repeater strategy in terms of local state discrimination. This strategy outperforms the naive approach of first distilling maximally entangled states between the individual parties followed by entanglement swapping in certain special cases. Finally, we consider the current practical implications of the gap between distinguishabiliity of quantum states given perfect or imperfect quantum memory. In 2006, a large gap between the theoretical performance of an eavesdropper with classical or quantum side
information was shown in a randomness extraction protocol . Using current commercially available quantum hardware, we were, however, not able to achieve an advantage using a quantum protocol when comparing with an error-free classical protocol in this scenario.
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Link to Zoom: https://ucph-ku.zoom.us/j/7577657414
Matthias Christandl, Department of Mathematical Sciences, University of Copenhagen
Laura Mancinska, Department of Mathematical Sciences, University of Copenhagen
Karol Horodecki, Institute of Informatics, University of Gdansk
Eric Chitambar, Electrical and Computer Engineering, University of Illinois