Abstract
In this chapter we consider the showcase of non-signalling parallel repetition, introduced in Sect. 4.1, and show how threshold theorems derived under the IID assumption can be extended to threshold theorems for general strategies, using a reduction to IID.
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Notes
- 1.
Most steps of our proof can be used as is when considering classical and quantum players as well. There is one lemma, however, which we do not know how to modify so it can capture the classical and quantum case. We explain the difficulty later on.
- 2.
When considering games with only two players, the requirement for complete support can be dropped but, even though we focus on two-player games, we do not discuss this here; see [4] for the details.
- 3.
See Theorem 10.11 for the formal statement.
- 4.
- 5.
As discussed in Sect. 8.4, this is inevitable.
- 6.
For the formal statement see Theorem 10.10.
- 7.
For simplicity we assume n is even; otherwise replace n/2 by \(\lceil n/2 \rceil \) and modify everything else accordingly.
- 8.
More commonly in the literature, one considers a definition in which \(\mathbb {E}_{(x,y)}\) is replaced by \(\max _{x,y}\). We use Definition 10.4 since it allows us to apply Sanov’s theorem later on.
- 9.
If \(\mathsf {data}_1\) does not include an index in which the inputs are (x, y) then the test \(\mathcal {T}^{(A\rightarrow B,x,y,b)}\) rejects by definition (recall Definition 10.3).
- 10.
This motivates our signalling measure given in Definition 10.3.
- 11.
Note that while Bob’s inputs, \(\varvec{{y}}_{\mathsf {data}_1}= y_1,\dots ,y_{n/2}\), are fixed in a specific instance of the guessing game, Alice’s inputs are still distributed according to the prior \(\mathrm {Q}_{XY}(x|y)\).
- 12.
To see this note that since the box is non-signalling between Alice and Bob, Bob can check in which copy the test passes even before Alice uses her input. Therefore, the probability to pass the test is independent of Alice’s inputs and hence must be non-zero for any of them.
- 13.
- 14.
- 15.
This is not to say that all strategies are permutation invariant but only that the optimal strategy can be assumed to be permutation invariant. It is perhaps interesting to note that, more commonly, the optimal strategies are taken to be, without loss of generality, deterministic in proofs of classical parallel repetition and pure in proofs of quantum parallel repetition. Here we are choosing to focus on permutation invariant strategies instead.
- 16.
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Arnon-Friedman, R. (2020). Showcase: Non-signalling Parallel Repetition. In: Device-Independent Quantum Information Processing. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-60231-4_10
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