Cheat E Coin

[Consuelo Dular]

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As in modern communication networks, the security of quantum networks will rely on complex cryptographic tasks that are based on a handful of fundamental primitives. Weak coin flipping (WCF) is a significant such primitive which allows two mistrustful parties to agree on a random bit while they favor opposite outcomes. Remarkably, perfect information-theoretic security can be achieved in principle for quantum WCF. Here, we overcome conceptual and practical issues that have prevented the experimental demonstration of this primitive to date, and demonstrate how quantum resources can provide cheat sensitivity, whereby each party can detect a cheating opponent, and an honest party is never sanctioned. Such a property is not known to be classically achievable with information-theoretic security. Our experiment implements a refined, loss-tolerant version of a recently proposed theoretical protocol and exploits heralded single photons generated by spontaneous parametric down conversion, a carefully optimized linear optical interferometer including beam splitters with variable reflectivities and a fast optical switch for the verification step. High values of our protocol benchmarks are maintained for attenuation corresponding to several kilometers of telecom optical fiber.

In classical communication networks, there exist no secure SCF and WCF protocols without computational assumptions or trusting a third party2,5,6,7. Although accepting a nonzero abort probability allows for information-theoretically secure classical schemes to exist8, such schemes cannot detect malicious behaviours deviating from the original protocol. On the other hand, cheat-sensitive coin flipping becomes possible when using quantum properties. Quantum SCF protocols have in fact been shown to display a fundamental lower bound on their bias9, but quantum WCF may achieve biases arbitrarily close to zero10,11,12. Interestingly, quantum WCF can also be used for the construction of optimal quantum SCF and quantum bit commitment schemes13,14,15.

While quantum SCF protocols have been experimentally demonstrated16,17,18, along with other quantum two-party computations19,20,21,22,23, the implementation of quantum WCF has remained elusive so far, because of the absence of protocols bringing together the use of practical states and measurements with tolerance to losses. Recently, a linear optical implementation, exploiting photon-number encoding, was proposed in24, but the quantum advantage it can provide in terms of bias is very sensitive to losses: a dishonest party may always declare an abort when they are not satisfied with the outcome of the coin flip.

Here, we provide an experimental demonstration for quantum WCF. Our demonstration relies on the generation of heralded single photons by spontaneous parametric down conversion (SPDC), which are effectively entangled with the vacuum on a beam splitter of variable reflectivity. The outcome of the coin flip is then provided by the detection or absence of a photon. Our protocol is a refined version of the theoretical protocol from24, which provides a new desirable property in the presence of losses that relates to cheat sensitivity rather than bias: by dropping the condition from24 that both parties have equal probabilities of winning when cheating, our protocol allows them to detect whether their opponent is cheating during a verification step, and does not sanction an honest party, while retaining security in terms of bias. There are no known classical protocols that achieve such cheat sensitivity25,26. In order to emphasize the robustness of our protocol to losses, we show that it remains secure over an attenuation that corresponds to several kilometers of telecom optical fiber.

Note that contrary to the previous protocol24, we drop the balancing condition, which states that Alice and Bob should have equal probabilities of winning when using an optimal cheating strategy, as it cannot be satisfied together with the correctness condition in presence of experimental imperfections. Consequently, a practical balanced protocol would sanction an honest Alice for cheating, with non-zero probability. This impacts the cheat sensitivity, as one cannot trust the verification step if it sanctions honest parties (see Supplementary Note 1 for details on the protocol and the chosen conditions).

The preparation, decision and verification steps, along with the role of measurement outcome b and beam splitter reflectivities x, y, z, are detailed in Box 1. The sources for Alice and Bob representations can be found at Woman icons created by Freepik - Flaticon and Man icons created by Freepik - Flaticon, respectively.

Under these conditions, the thermally-induced fluctuations are slow enough such that we can easily post-select the protocol runs in which there was no phase difference between the two arms of the interferometer. This post-selection does not threaten the protocol security, as the parties could monitor the interference before performing the coin flip, and agree on starting the protocol only when the phase difference is null. Single photons are detected with threshold superconducting nanowire single-photon detectors (SNSPDs) in order to maximize the detection efficiency. Finally, to simulate communication distance between Alice and Bob, and the corresponding losses induced by the photon storage that is necessary in this case, we use variable optical attenuators (VOAs).

The reader can refer to Supplementary Note 1 for the detailed proof and Supplementary Note 2 for the values used in our implementation. Then, as long as the parties are honest, we obtain the following probabilities for significant events:

Note here the importance of maximizing the interference visibility v so that Alice is not sanctioned while being honest. The above expressions also provide a systematic way to optimize the reflectivities for honest parties, which does not require their direct measurement (see Methods for details).

The abort probability is shown on the right axis, in magenta. The lines represent the theoretical evolution of probabilities, calculated via Eqs. (5) to (10), with efficiencies given in Table 1. The error bars are mainly due to error propagation on these efficiencies.

We notice that the abort probability takes relatively high values, even when we trivially set the communication distance to L = 0 km. This has to do with important losses, particularly in mating sleeves connecting the numerous optical fiber components, the delay line, or in crystalline components such as the PBSs or the optical switch. Significant improvements could be made by fusing optical components for instance. Other critical features are the single-photon coupling and SNSPDs efficiencies. Both of these aspects are being actively studied27,28,29,30,31,32 and could see significant improvement in the near future. We also notice that the winning probabilities of Alice and Bob are indeed very close and the probability of an honest party to be sanctioned is minimized.

Only one set of points is shown for the two axes, as these two events are complementary. The line is plotted from Eqs. (11) and (12), with \(\eta _A^s\) given in Table 1. The error bars are mainly due to error propagation on this efficiency. The observed deviation from the theory is linked to systematic errors when setting the reflectivities, which is discussed in Supplementary Note 2.

After refining a previous theoretical proposal for a practical quantum weak coin flipping protocol24, we were able to perform an implementation of this protocol by generating a heralded single photon, and entangling it effectively with the vacuum. Thanks to the use of low dark counts SNSPDs, tunable beam splitters and a fast optical switch, while keeping a high visibility in our fibered interferometer, we demonstrated a fair and cheat-sensitive protocol. Importantly, this last property allows to detect a cheating party with non-negligible probability.

Note that in order to sanction a dishonest party with high probability, one could systematically sanction the winning party, regardless of their honesty. Thus, in order to display genuine cheat sensitivity, we highlight the primary importance of the correctness condition, which ensures an honest party is never sanctioned for cheating. This forced us to ignore the balancing of the benefit gained by each party when adopting an optimal cheating strategy, which was previously assessed as a necessary condition for a weak coin-flipping protocol24. Still, we propose a way of restoring this balance, by using the deterrent factor and interest function introduced in the previous paragraphs.

S.N., M.B. and E.D. designed and S.N. developed the experimental setup. S.N. and V.Y. performed the protocol implementation and processed the data. S.N., V.Y., U.C. and M.B. performed the protocol analysis. All authors discussed the analysis of the data, and contributed to writing or proofreading the manuscript. I.K. and E.D. supervised the project.

Here's a single page pdf to save or print of the "Straightedge / Coin" head tensioning method that's sometimes suggested as a way to initially set head tension on a banjo. I don't use the method, but it is useful for someone new to setting tension when they have that "How tight is enough and how tight is too tight?" question.

Glad you liked it, Bart. It should be pretty easy to find when doing a search. I tried to incorporate the right words in the title that would match a lot of searches when someone is looking for that information.

Looking at that photo, Rudy, I don't see any information or explanation about what those numbers (I assume the thicknesses of the coins?) mean for head tension. In other words, I see no practical way to use that photo, without some added information along the lines of, "If you want head tension x, use coin y at distance z from the center of the head" or something like that."

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