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Showcase: Device-Independent Quantum Cryptography

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Device-Independent Quantum Information Processing

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

In this chapter we consider the showcase of device-independent quantum cryptography and show how the security proof of device-independent cryptographic protocols can be performed via a reduction to IID. We introduce a general framework for obtaining proofs of device-independent security for a broad range of cryptographic tasks. For the sake of explicitness, we focus in this chapter on the task of device-independent quantum key distribution (DIQKD).

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Notes

  1. 1.

    Since the initial announcement of our work  [1], our framework has already been applied to a variety of additional tasks, including conference key agreement  [2], randomness expansion  [1] and privatization  [3], as well as randomness generation with sub-linear quantum resources  [4].

  2. 2.

    This is crucial for experimental implementations of device-independent protocols. Our quantitive results have been applied to the analysis of the first experimental implementation of a protocol for randomness generation in the fully device-independent framework  [5].

  3. 3.

    From that point onward standard classical post-processing steps, e.g., error correction and privacy amplification, suffice to prove the security of the protocol; recall Sect. 4.2.3.

  4. 4.

    The security proof presented in  [20] is similar in spirit (but technically very different) to the one presented here. It bounds the total amount of smooth min-entropy generated in the protocol in a round-by-round fashion but the entropy accumulated in a single round is not the von Neumann entropy.

  5. 5.

    In particular, in a setting with two distinct parties, Alice and Bob, communication is required to actually implement Protocol 11.1. We ignore this as it is not relevant for the analysis.

  6. 6.

    See e.g. Fig. 11.4, where one can see that finite-size effects can play an important role up to even moderately large values of \(n\approx 10^{10}\).

  7. 7.

    Consider for example a device in which the initial state \(\rho _{Q_{A_1}Q_{B_1}Q_{A_2}Q_{B_2}} = |\varPhi \rangle \langle \varPhi |_{Q_{A_1}Q_{B_2}}\otimes |\varPhi \rangle \langle \varPhi |_{Q_{A_2}Q_{B_1}}\), i.e., the systems over \(Q_{A_1}\) and \(Q_{B_2}\) are entangled. Thus, \(A_1\) and \(B_2\) may be correlated even given \(B_1X_1Y_1T_1\). In this case the Markov-chain conditions do not hold since the side-information \(B_2\) reveals information regarding the past output \(A_1\).

  8. 8.

    A different model for the sequential process could have been one in which the initial quantum state itself includes the registers \(\textit{\textbf{X}}\) and \(\textit{\textbf{Y}}\) and the channel is defined such that a measurement is performed on those registers to get the inputs (and then use the device in the protocol). When starting with maximally mixed states over \(\textit{\textbf{X}}\textit{\textbf{Y}}\) the entire sequential process is exactly the same as the one described by our EAT channels. However, when coming to construct a min-tradeoff function with this (somewhat strange) alternative choice of channels, we see that the set \(\varSigma (p)\) can include states in which, e.g., \(X_i=0\) with probability 1 (since we need to consider all possible input states). In the context of Bell inequalities, this is similar to dropping the “free choice assumption”. Clearly, if this had been the case, the only min-tradeoff function one could construct is the constant function \(f_{\min }(p)=0\) for all p, which is trivial and useless.

  9. 9.

    Formally, we will need to extend the function to all probability distributions p (even those with \(p(\perp )\ne 1-\gamma \)). We can extended the function in any way we wish, while keeping it convex and differentiable.

  10. 10.

    We define the functions g and \(f_{min}\) only in the regime in which the protocol does not abort, i.e., \( p(1)/\gamma \ge 3/4\).

  11. 11.

    Alternatively, one could replace \(p(1)/\gamma \) with \(p(1)/(p(0)+p(1))\), which is more meaningful, in the definition of the function g. However, since the \(\mathrm {d}f_{\min }/\mathrm {d}p(1)\) will affect the final smooth min-entropy bound, using \(p(1)/\gamma \) leads to better quantitive results.

  12. 12.

    We assume that the value of \(T_i\) is exchanged over a classical authenticated channel to which the device D does not have access. In particular, Alice’s part of the device is independent from the value of \(T_i\) given \(X_i\).

  13. 13.

    The point \(p_{\mathrm {cut}}\) can later be chosen such that the derived smooth entropy bounds are optimised.

  14. 14.

    For any \(\omega _{\mathrm {exp}}\) there are many devices that fit this description; an explicit example can be found in Sect. 4.2.4.

  15. 15.

    O denotes the classical information sent from Alice to Bob during error correction; see Sect. 4.2.2.

  16. 16.

    This is why we made the distinction between \(B_i\) in the entropy accumulation protocol and \(\tilde{B}_i\) in the DIQKD protocol.

  17. 17.

    Here a slightly more general version of the EAT than the one given in Sect. 9.2.3 is needed, in which the event \(\varOmega \) can be defined via A, B, X, Y and not only C; see  [22] for the details.

  18. 18.

    The second order term of the smooth min-entropy rate given in Lemma 11.8 scales with \(\gamma \), roughly, as \(1/\gamma \), while in Appendix C.2 the dependency is roughly \(1/\sqrt{\gamma }\). The modified analysis can be seen as a “patch” used to overcome the non-optimal dependency of the EAT given in Theorem 9.3 on the testing probability in the considered protocols. This issue was overcome in a more recent version of the EAT  [26].

  19. 19.

    This should not be dismissed as can be seen from the following state of affairs. [33] reports an advantage in terms of the min-entropy when considering the full statistics instead of merely the violation if the CHSH inequality. Comparing the bound on the min-entropy from the full statistics to the bound on the von Neumann entropy from the violation alone, both evaluated on the quantum states produced by the honest implementations, we find that it is still better to use the bound on the von Neumann entropy as we do here. Thus, to truly see if an advantage can be gained by considering the full statistics, one should aim to a direct bound on the von Neumann entropy.

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Authors and Affiliations

  1. Senior Scientist, Physics of Complex Systems, Weizmann Institute of Science, Rechovot, Israel

    Rotem Arnon-Friedman

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Correspondence to Rotem Arnon-Friedman .

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Arnon-Friedman, R. (2020). Showcase: Device-Independent Quantum Cryptography. In: Device-Independent Quantum Information Processing. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-60231-4_11

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