Abstract
The goal of this chapter is to present the basic information needed while reading the thesis. It is by no means a comprehensive review of the topic of device-independent information processing. A reader completely unfamiliar with the concepts of non-locality and device-independent protocols is encouraged to read the survey [1] and book [2].
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Notes
- 1.
We distinguish the quantum state from the correlations throughout the thesis: \(Q_A\) and \(Q_B\) denote quantum registers belonging to Alice and Bob while A and B denote their classical outputs.
- 2.
The definition of a quantum box over a bipartite Hilbert space \(\mathscr {H}_{Q_A} \otimes \mathscr {H}_{Q_B}\) is the standard one in the context of non-relativistic quantum mechanics. When studying relativist quantum mechanics one considers a single Hilbert space \(\mathscr {H}\) and two commuting measurements acting on it (instead of tensor product measurements). The two definitions coincide when restricting the attention to finite dimensional Hilbert spaces but otherwise different in general [3].
- 3.
Though common, this is a rather confusing and unjustified terminology. As clear from Eq. (3.3), quantum correlations are also local, in the sense that each component performs a local operation on its part of the state.
- 4.
Separating quantum boxes from non-signalling ones is a far more complicated task; see, e.g., [4].
- 5.
Other hyperplanes represent the trivial conditions of positivity of normalisation of the conditional probability distributions which are relevant for all sets.
- 6.
Notice that the statement that some quantum states violate Bell inequalities is independent from the statement that classical boxes cannot violate the inequality; it could have been the case that no box is able to violated such inequalities. This would have implied that all quantum correlations can be written in the form of Eq. (3.4) and, hence, can be described as arising from some shared randomness, or an “hidden variable”, \(\lambda \).
- 7.
Depending on the context, one can further restrict the allowed strategies by considering classical or quantum boxes.
- 8.
Notice the notation: w denotes a winning condition (function) while \(\omega \) is the winning probability (a number). W will be used to denote the random variable describing whether a game is won or lost. In any case, the difference between these three objects is always clear from the text.
- 9.
For the inputs \((x,y)=(1,2)\) one can set either \(w_{\text {CHSH}}=1\) or 0 (it is not relevant later on); for completeness we choose \(w_{\text {CHSH}}=1\) in this case, following previous works.
- 10.
A purification \(\rho _{Q_AQ_BE}\) is the most general extension of a quantum state \(\rho _{Q_AQ_B}\), in the sense that it gives Eve the maximal amount of information regarding Alice and Bob’s marginal state. Hence, in the cryptographic setting we always say that Eve holds the purifying system E, without loss of generality—any adversary holding a system \(E'\) which is not the purifying system E can only be weaker than that holding E.
- 11.
We emphasise again that Eve is not required to measure her quantum state at any particular point.
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Arnon-Friedman, R. (2020). Preliminaries: Device-Independent Concepts. In: Device-Independent Quantum Information Processing. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-60231-4_3
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