Reductions to IID in Device-independent Quantum Information Processing

Abstract

The field of device-independent quantum information processing concerns itself with devising and analysing protocols, such as quantum key distribution and quantum tomography, without referring to the quality of the physical devices utilised to execute the protocols. Instead, the analysis is based on the observed correlations that arise during a repeated interaction with the devices and, in particular, their ability to violate the so called Bell inequalities. Since the analysis of device-independent protocols holds irrespectively of the underlying physical device, it implies that any device can be used to execute the protocols: If the apparatus is of poor quality, the users of the protocol will detect it and abort; otherwise, they will accomplish their goal. This strong statement comes at a price — the analysis of device-independent protocols is, a priori, extremely challenging. Having good techniques at hand is thus crucial. The thesis presents an approach that can be taken to simplify the analysis of device-independent information processing protocols. The idea is the following: Instead of analysing the most general device leading to the observed correlations, one should first analyse a significantly simpler device that, in each interaction with the user, behaves in an identical way, independently of all other interactions. We call such a device an independently and identically distributed (IID) device. As the next step, special techniques are used to prove that, without loss of generality, the analysis of the IID device implies similar results for the most general device. Such techniques reduce the problem of analysing the general scenario to that of analysing an IID one and, hence, we term them reductions to IID. We present two mathematical techniques that can be used as reductions to IID in the device-independent setting: de Finetti reductions for correlations and the entropy accumulation theorem. Each technique is accompanied by a showcase-application that exemplifies the reduction’s usage and benefits. Specifically, we use our de Finetti reduction to prove a non-signalling (super-quantum) parallel repetition theorem, belonging to a family of theorems discussed in theoretical computer science. The entropy accumulation theorem is used to prove the security of device-independent quantum cryptographic protocols. Performing the analysis via a reduction to IID instead of directly analysing the most general scenarios leads to simpler proofs and significant quantitive improvements, matching the tight results proven when analysing IID devices. In particular, our analysis of device-independent quantum key distribution protocols produces essentially optimal key rates and noise tolerance, crucial for all future experimental implementations of device-independent cryptography.