Testing how different levels of entanglement affect predictability in practical setups

Abstract

The generation of series of random numbers is an important and difficult problem. Appropriate measurements on entangled states have been proposed as the definitive solution. In principle, this solution requires reaching the challenging “loophole-free” condition, which is unattainable in a practical situation nowadays. Yet, it is intuitive that randomness should gradually deteriorate as the setup deviates from that ideal condition. In order to test whether this trend exists or not, we prepare biphotons with three different levels of entanglement: moderately entangled ($S = 2.67$), marginally entangled ($S = 2.06$), and non-entangled ($S = 1.42$) in a setup that mimics a practical situation. The indicators of randomness we use here are: a battery of standard statistical tests, Hurst exponent, an evaluator of Kolmogorov complexity, Takens’ dimension of embedding, and augmented Dickey–Fuller and Kwiatkowski–Phillips–Schmidt–Shin to check stationarity. A nonparametrical statistical ANOVA (Kruskal–Wallis) analysis reveals a strong influence of the level of entanglement with randomness when measured with Kolmogorov complexity in three time series with P-values and strength factor $\epsilon ^2$: $P = 0.0015$, $\epsilon ^2 = 0.28$; $P = 4.5\times 10^{-4}$, $\epsilon ^2 = 0.67$ and $P = 5.6\times 10^{-4}$, $\epsilon ^2 = 0.16$. The setup is pulsed with time stamping, what allows generate different series applying different methods with the same data, even after the experimental run has ended, and to compare their raw randomness. It also allows the stroboscopic reconstruction of time variation of entanglement.

Appendices

Appendix I: Indicators of randomness (NIST, Hurst, complexity, Takens’ $d_{\mathrm{E}}$, ADF, and KPSS)

The battery developed by the NIST consists of 15 different tests. Not all of them can be applied in all cases, because of series’ length. The battery and its details are available in the NISTs page [1]. We just mention here that they essentially check Borel normality, hidden periodicities, and decay of mutual information.

Hurst exponent is related to autocorrelation decay rate. It is usually named H and normalized between 0 and 1. H $> \frac{1}{2}$ means the series has long range correlations, H $< \frac{1}{2}$ that it has strong fluctuations in the short term, while H $\approx \frac{1}{2}$ means that it fluctuates uniformly along its length.

Complexity (Kolmogorov) has advantages over other methods of detecting regular behavior. It does not need, in principle at least, to assume stationary probabilities (see below). It applies to series of any length. On the other hand, complexity cannot be actually computed; it can only be estimated from the rate of compressibility of the series using, f.ex., the algorithm devised by Lempel and Ziv [30]. Here we use the approach developed by Kaspar and Schuster [31] and implemented by Mihailovic [32] to estimate normalized complexity K. This value is designed to be near to 0 for a periodic or regular sequence, and near to 1 for a random one. For relatively short and strongly fluctuating series, values $K>1$ may occur. Note that this approach is ultimately statistical too.

Nonlinear analysis [33] provides a completely different approach. In a chaotic system, for example, few dynamical variables are linked through nonlinear equations in such a way that the evolution is apparently random. Nevertheless, the evolution involves few degrees of freedom and is partially predictable. This is a fundamental difference with “true” random evolution, which can be thought of as requiring a very high (eventually, infinite) number of degrees of freedom to be described (recall Kolmogorov’s complexity). Takens’ reconstruction theorem and related methods allow measuring the number of dimensions of the object in phase space within which the system evolves, and hence to discriminate chaos from randomness. That number is called dimension of embedding, $d_{\mathrm{E}}$. A definite value of $d_{\mathrm{E}}$ (which is always much smaller than the series’ length) indicates the series is non-random. In some cases, this approach allows the prediction of future elements of the series within a horizon of predictability. This horizon is roughly given by the inverse of the largest Lyapunov exponent. Predictability is the ultimate demonstration of not-randomness. This method was able to reveal the existence of regularities in one of the runs of the Innsbruck experiment [22, 29].

Excepting Takens’ $d_{\mathrm{E}}$, all the mentioned indicators are statistical and hence require the series to be stationary. There are two main types of non-stationarity. One: the series’ statistical parameters follow a continuous and slow evolution (trend-stationary). Deviations from the average trend vanish as the number of elements in the series increases. By identifying and correcting the trend, the series can be made stationary again. Two: a deviation affects the values of the statistical parameters in a permanent way through the series (unit-root). Standard tests for the two types of non-stationarity exist (KPSS and ADF), but, because of the very nature of the involved hypotheses and methods, they do not provide definitive conclusions. Used together, they indicate the most probable nature of the series. KPSS tests the null hypothesis that the series is trend-stationary, against the alternative of unit-root. Obtaining “0” (1) indicates that stationarity cannot (can) be discarded. ADF tests unit-root. Obtaining “0” (1) indicates that unit-root cannot (can) be discarded.

Appendix II: Statistical analysis

Statistical tests have been applied in order to support or not support the central hypothesis that there is an influence of entanglement on the randomness of the time series. To select the adequate statistical procedure, we test the normality of each time series. We perform the following tests in all Series I, II, and III with angles’ sets $\theta =8.6^{\circ }$ and $\theta =22.5^{\circ }$, measuring Kolmogorov complexity ($K_{\mathrm{c}}$), Hurst exponent (H), and NIST test battery: Kolmogorov–Smirnov, Stephens Method, Marsaglia Method, Lilliefors, Anderson–Darling, Cramer–Von Mises, Shapiro–Wilk, Shapiro–Francia, Jarque–Bera and D-Agostino and Pearson [34]. Only five time series show normality: all measuring H, Series III and Series II with both $\theta =8.6^{\circ }$ and $\theta =22.5^{\circ }$ and Series I with $\theta =22.5^{\circ }$. In addition, Lavene’s test of our data reported a P-value $< 0.005$ indicating a violation of the assumption of homoscedasticity. Consequently, a Kruskal–Wallis test or nonparametric one-way ANOVA was suitable to quantify the measured effect rather than ANOVA for normal distribution. In order to find how strong the influence is, it was used the method of the epsilon square ($\epsilon ^2$) to estimate of effect size [35]. This coefficient assumes the value from 0 (indicating no relationship) to 1 (indicating a perfect relationship). As there is no single method to quantify this effect size, $\epsilon ^2$ will be interpreted as r (Pearson correlation coefficient) or $R^2$ (coefficient of determination); a rule of thumb [36]: 0.01 $\sim$ small, 0.06 $\sim$ medium, $>0.14 \sim$ large.

Appendix III

Average values of Kolmogorov complexity and Hurst exponent for series generated by Methods I, II, and III with CHSH and informatic set of angles (Tables 4, 5, 6).

Table 4 Numerical average values of Kolmogorov complexity and Hurst exponent of Series I and Series II with $\theta =22.5^{\circ }$ and Series I, II, and III with $\theta =8.6^{\circ }$ and $S=2.67$

Table 5 Numerical average values of Kolmogorov complexity and Hurst exponent of Series I and II with $\theta =22.5^{\circ }$ and Series I, II, and III with $\theta =8.6^{\circ }$ and $S=2.06$

Table 6 Numerical average values of Kolmogorov complexity and Hurst exponent of Series I and II with $\theta =22.5^{\circ }$ and Series I, II, and III with $\theta =8.6^{\circ }$ and $S=1.42$