Abstract
Multi-round parallel boxes, discussed in Sect. 6.1, can display an almost arbitrary behaviour and hence are complicated to analyse.
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Notes
- 1.
Depending on the context, the term system may refer to a probability distribution, a quantum state, or a box.
- 2.
The mentioned theorems rely on some initial subsystem structure and/or a bound on the dimension of the subsystems. In the device-independent setting one cannot start with such assumptions regarding the considered boxes in general.
- 3.
The definition and the derived theorem are independent of the nature of the box, i.e., if it is classical, quantum, non-signalling, or even signalling. This will be addressed in Sect. 8.2.
- 4.
In a quantum de Finetti statement a permutation takes a state \( \left| {\phi _1} \right\rangle \otimes \dots \left| {\phi _n} \right\rangle \) to \( \left| {\phi _{\pi ^{-1}(1)}} \right\rangle \otimes \dots \left| {\phi _{\pi ^{-1}(n)}} \right\rangle \). That is, the quantum states themselves are being permuted.
- 5.
Depending on the considered scenario, the application of the permutation may be a purely theoretical step or needs to be done in practice.
- 6.
As previously mentioned, we focus on the case of two parties. The definition extends to any number of parties trivially.
- 7.
In [13], a more general version of Theorem 8.3 was proven, in which further symmetries of \(\mathrm {P}_{\textit{\textbf{A}}\textit{\textbf{B}}|\textit{\textbf{X}}\textit{\textbf{Y}}}\) (on top of permutation invariance) can be exploited to construct more structured de Finetti boxes and prove de Finetti reductions with improved parameters. Theorem 8.3 was then derived as a corollary. To keep things (relatively) concise, we present in this thesis a direct proof of Theorem 8.3.
- 8.
- 9.
Note that \(\tau _{\textit{\textbf{A}}\textit{\textbf{B}}|\textit{\textbf{X}}\textit{\textbf{Y}}}\) may be signalling, as in our previous statements. The fact that we are considering non-signalling extensions only means that the marginals \(\tau _{\textit{\textbf{A}}\textit{\textbf{B}}|\textit{\textbf{X}}\textit{\textbf{Y}}}\) and \(\tau _{C|Z}\) of \(\tau _{\textit{\textbf{A}}\textit{\textbf{B}}C|\textit{\textbf{X}}\textit{\textbf{Y}}Z}\) are well defined.
- 10.
Linearity refers here to the linearity of the test in the box \(\mathrm {P}_{\textit{\textbf{A}}\textit{\textbf{B}}|\textit{\textbf{X}}\textit{\textbf{Y}}}\), which follows from the fact that the test interacts only once with \(\mathrm {P}_{\textit{\textbf{A}}\textit{\textbf{B}}|\textit{\textbf{X}}\textit{\textbf{Y}}}\) (or, in other words, the test gets only a single copy of the box).
- 11.
Let us briefly explain why the notation of a test considered in Sect. 8.3.1 is not appropriate in the cryptographic setting. When considering tests, we were interested in events defined over \(\mathcal {X}^n\times \mathcal {Y}^n\times \mathcal {A}^n\times \mathcal {B}^n\). Whether an output of a protocol (a key, for example) is secure to use cannot be defined as an event. Security depends on the process of producing the key rather on the specific data that was produced during the run of the protocol.
- 12.
- 13.
Note, however, that the extension \(\tau _{\textit{\textbf{A}}\textit{\textbf{B}}C|\textit{\textbf{X}}\textit{\textbf{Y}}Z}\) itself cannot be written as a convex combination of IID boxes, only its marginal \(\tau _{\textit{\textbf{A}}\textit{\textbf{B}}|\textit{\textbf{X}}\textit{\textbf{Y}}}\) is a de Finetti box. Furthermore, \(\tau _{\textit{\textbf{A}}\textit{\textbf{B}}|\textit{\textbf{X}}\textit{\textbf{Y}}}\) may be signalling in general, as before.
- 14.
Though this does not make them useless; see Chap. 10.
- 15.
Weaker statements, e.g., with a pre-factor sub-exponential in n, may also be of interest in certain applications.
- 16.
The hope here is that by adding the additional weight on non-quantum or signalling boxes one could account for the “gap” between Eq. (8.6) and the known parallel repetition results.
- 17.
We present only the bipartite case; [20, Theorem 4.3] is stated for an arbitrary number of parties.
- 18.
Reference [7] presented the first de Finetti reduction, i.e., an inequality relation between permutation invariant systems and de Finetti systems (all previous de Finetti-type theorems gave other types of relations between the two systems). The term “de Finetti reduction” was not used at that time and the authors chose the name “post-selection technique” as they first proved the quantum analogue of Lemma 8.6.
- 19.
In the presence of certain types of symmetries (in addition to permutation invariance) one can derive such de Finetti reductions; see [13].
- 20.
In this language, the original result of de Finetti [1] stated that all infinitely-exchangeable distributions (i.e., distributions that are n-exchangeable for any \(n\ge k\)) are equal to distributions of the form of a convex combination of IID distributions.
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Arnon-Friedman, R. (2020). Reductions to IID: Parallel Interaction. In: Device-Independent Quantum Information Processing. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-60231-4_8
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