Abstract
In this thesis, we are interested in analysing the behaviour of multi-round boxes when such boxes are used to play many non-local games, e.g., while running a cryptographic protocol. In the previous chapter we discussed the different models of multi-round boxes (the parallel and sequential ones). As we saw, their behaviour can be quite complex.
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Notes
- 1.
As always, a box is a conditional probability distribution and its definition is therefore independent of the distribution of the inputs, \(\textit{\textbf{x}}\) and \(\textit{\textbf{y}}\), which can be arbitrary (depending on how the box is being used). It is perhaps helpful to note that the idea here is that, while the inputs of the different rounds may be correlated in general (i.e., not IID), the box itself does not “create” further correlations between the rounds (in contrast to parallel and sequential boxes). In any case, in most scenarios the inputs are usually taken to be IID random variables as well.
- 2.
Recall that in the device-independent setting we assume that the adversary is the one constructing the box. Device-independent protocols are expected to abort, with high probability, when an adversarial device is detected.
- 3.
An example of the analysis of the von Neumann entropy for single-round boxes was presented in Sect. 5.2. The AEP motivates the analysis done in that section when working under the IID assumption.
- 4.
- 5.
Note that this theorem is actually a non-asymptotic version of the AEP, as it describes also the convergence rate for finite n (i.e., it includes also the second order term). The limit, stated as Eq. (7.5) below, follows trivially from the presented theorem.
- 6.
For the time being we are not interested in the explicit form of \(\delta \); this will be discussed when relevant.
- 7.
The above only (roughly) explains why the smooth entropies are considered, without addressing the second order term of the AEP. The second order term does not come from the law of large numbers but its refinement—the central limit theorem.
- 8.
The converse bound roughly follows from the monotonicity of the so called \(\alpha \)-entropies. For details see [6, Sect. 6.4].
- 9.
The classical AEP, given as Theorem 7.2, can be easily written also in terms of conditional entropies if the conditioning is done on classical systems (then one can directly define the probability distribution of A as the conditional one). This is not the case when conditioning on quantum systems. That is to say that the statement of the theorem which includes conditional entropies does not follow directly from a “non-conditional” variant.
- 10.
- 11.
Hoeffding’s inequality tells us even more; it says that when using the optimal IID strategy the probability of winning less than \(1-\alpha -\beta \) fraction of the games is also decreasing exponentially fast.
- 12.
It is the equivalence of all purifications that allows us to go from an IID assumption regarding Alice and Bob’s state \(\rho _{\varvec{Q_A}\varvec{Q_B}}\) to an IID assumption regarding the state \(\rho _{\varvec{Q_A}\varvec{Q_B}\textit{\textbf{E}}}\), which also includes Eve. Interestingly, the same thing cannot be done when considering non-signalling boxes and adversaries. It follows from [20] that the extension of a non-signalling IID box to the adversary does not necessarily have an IID structure as well. (See also [20], where the box itself is assumed to have a subsystem structure similar to that of an IID box while the structure of the adversary’s system is unrestricted).
- 13.
We previously wrote \(\sigma \) as the tripartite state \(\sigma _{Q_AQ_BE}\) while here we are referring also to the classical register A. What is meant by this notation is that \(\sigma \) is a state which can lead to winning probability \(\omega \) when measured with some given measurements \(\left\{ M_a^x\right\} \) and \(\left\{ M_b^y\right\} \). The result of measuring \(Q_A\) with \(\left\{ M_a^x\right\} \) defines the RV A.
- 14.
If the box is malicious or simply noisier than we wished for, we anyhow expect the protocol to abort. Thus, we only ask that the error correction protocol does not abort with high probability when the honest implementation of the devices is used since, otherwise, it will affect the completeness of the protocol.
References
Holenstein T, Renner R (2011) On the randomness of independent experiments. IEEE Trans Inf Theory 57(4):1865–1871
Renner R (2008) Security of quantum key distribution. Int J Quantum Inf 6(01):1–127
Tomamichel M, Colbeck R, Renner R (2009) A fully quantum asymptotic equipartition property. IEEE Trans Inf Theory 55(12):5840–5847
Tomamichel M (2012) A framework for non-asymptotic quantum information theory. arXiv:1203.2142
Cover TM, Thomas JA (2012) Elements of information theory. Wiley, New York
Tomamichel M (2015) Quantum information processing with finite resources: mathematical foundations, vol 5. Springer, Berlin
Renner R, Wolf S (2004) Smooth rényi entropy and applications. In: International symposium on information theory, 2004. ISIT 2004. Proceedings, p. 233. IEEE
Shannon CE (1948) A mathematical theory of communication. Bell Syst Techn J 27
Tomamichel M, Hayashi M (2013) A hierarchy of information quantities for finite block length analysis of quantum tasks. IEEE Trans Inf Theory 59(11):7693–7710
Barrett J, Hardy L, Kent A (2005) No signaling and quantum key distribution. Phys Rev Lett 95(1):010503
Acín A, Gisin N, Masanes L (2006) From Bell’s theorem to secure quantum key distribution. Phys Rev Lett 97(12):120405
Acín A, Massar S, Pironio S (2006) Efficient quantum key distribution secure against no-signalling eavesdroppers. New J Phys 8(8):126
Scarani V, Gisin N, Brunner N, Masanes L, Pino S, Acín A (2006) Secrecy extraction from no-signaling correlations. Phys Rev A 74(4):042339
Acín A, Brunner N, Gisin N, Massar S, Pironio S, Scarani V (2007) Device-independent security of quantum cryptography against collective attacks. Phys Rev Lett 98(23):230501
Masanes L (2009) Universally composable privacy amplification from causality constraints. Phys Rev Lett 102(14):140501
Pironio S, Acín A, Brunner N, Gisin N, Massar S, Scarani V (2009) Device-independent quantum key distribution secure against collective attacks. New J Phys 11(4):045021
Hänggi E, Renner R, Wolf S (2010) Efficient device-independent quantum key distribution. Advances in Cryptology-EUROCRYPT 2010. Springer, Berlin, pp 216–234
Hänggi E, Renner R (2010) Device-independent quantum key distribution with commuting measurements. arXiv:1009.1833
Masanes L, Pironio S, Acín A (2011) Secure device-independent quantum key distribution with causally independent measurement devices. Nature Commun 2:238
Masanes L, Renner R, Christandl M, Winter A, Barrett J (2014) Full security of quantum key distribution from no-signaling constraints. IEEE Trans Inf Theory 60(8):4973–4986
Arnon-Friedman R, Ta-Shma A (2012) Limits of privacy amplification against nonsignaling memory attacks. Phys Rev A 86(6):062333
Renner R, Wolf S (2005) Simple and tight bounds for information reconciliation and privacy amplification. Advances in cryptology-ASIACRYPT 2005. Springer, Berlin, pp 199–216
Devetak, I. and Winter, A. (2005). Distillation of secret key and entanglement from quantum states. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, volume 461, pages 207–235. The Royal Society
Arnon-Friedman R, Renner R (2015) de Finetti reductions for correlations. J Math Phys 56(5):052203
Dupuis F, Fawzi O, Renner R (2016) Entropy accumulation. arXiv:1607.01796
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Arnon-Friedman, R. (2020). Working Under the IID Assumption. In: Device-Independent Quantum Information Processing. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-60231-4_7
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