Abstract
The study of quantum information unveils new possibilities for remarkable forms of computation, communication, and cryptography by investigating different ways of manipulating quantum states. Crucially, the analysis of quantum information processing tasks must be based, in one way or another, on the actual physical processes used to implement the considered task; the physical processes must be inherently quantum as otherwise no advantage can be gained compared to classical information processing. In most applications, the starting point of the analysis is an explicit and exact characterisation of the quantum apparatus, or device, used to implement the task of interest.
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Notes
- 1.
Clearly, one cannot perform any cryptographic task if the device includes a transmitter that just sends all the information to the adversary. Few minimal assumptions regarding the device will be needed; see Sect. 3.3. Depending on the considered task, some of the assumptions can be enforced in practice while others may require some minimal level of trust.
- 2.
Notice that even if Alice and Bob did have some information about the physical apparatus, the device-independent framework does not allow them to take advantage of this information in the analysis. For example, Alice and Bob may be able to distinguish a device that uses the polarisation of a photon to encode a qubit from one based on superconducting qubits (even the author is able to do that). Yet, this information is not to be used when treating the device as a black box.
- 3.
Consider for example the case of device-independent QKD. Classical devices can always be pre-programmed by the adversary to output a fixed key of her choice.
- 4.
The formal definitions of parallel and sequential devices are given in Chap. 6.
- 5.
Perhaps surprisingly, as far as the author is aware the idea of a “reduction to IID” does not appear or used in classical information processing and cryptography.
- 6.
In the context of QKD, security under the IID assumption is called security against collective attacks.
- 7.
The reductions themselves are not necessarily simple, but that is fine. They are technical tools that are only proved once and can then be used to simplify many other proofs. The researcher using the reduction does not need to reprove anything.
- 8.
This is in agreement with Occam’s razor; while there is no notion of the “right proof” out of several possible proofs (assuming they are all mathematically correct), the simplest proof usually turns out to be the most useful and insightful one.
- 9.
A commonly used example is that of “data compression”. There, one would like to encode an n bit string using less bits. If we allow for some small error when decoding the data, the smooth max-entropy roughly describes the number of bits needed. However, for a large enough number of independent repetitions, less bits suffice and the exact amount is governed by the Shannon entropy.
- 10.
To be more precise, some requirements regarding the process, or protocol, in which the sequential device is to be used must hold. This is explained in details in Chap. 9.
- 11.
This is actually a generalisation of the more commonly known parallel repetition question, in which one wishes to upper-bound the probability of winning all the n games.
- 12.
When first encountering the question of parallel repetition it may seem surprising that the players can do better using a parallel device, but this is indeed the case; see Sect. 4.1.2 a concrete example.
- 13.
We are jumping ahead now with the aim of being able to explain Theorem 1.2 to readers who are already somewhat familiar with device-independent information processing and non-signalling systems. For a reader unfamiliar with these topics, the mathematical statements may seem puzzling without further explanations. We will get back to the discussed theorem in Chap. 10, after giving all the preparatory information throughout the thesis. A reader unfamiliar with the used terminology can therefore skip the current discussion without the risk of missing out.
- 14.
In other words, the local strategy of each player does require “communication between the games”: In order to (locally) answer the i’th question received from the referee, the player needs to know his j’th question (with \(i\ne j\)).
- 15.
An IID device is illustrated in the bottom of Fig. 1.2. We can then think of each copy \(\mathrm {O}_{AB|XY}\) as describing a single copy of the smaller boxes in the figure, while \(\mathrm {P}_{\varvec{A}\varvec{B}|\varvec{X}\varvec{Y}} = \mathrm {O}_{AB|XY}^{\otimes n}\) described the device including all the copies together.
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Arnon-Friedman, R. (2020). Introduction. In: Device-Independent Quantum Information Processing. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-60231-4_1
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