Testing randomness with photons by direct characterization of optical $t$-designs

Generating and characterizing randomness is fundamentally important in both classical and quantum information science. Here we report the experimental demonstration of ensembles of pseudorandom optical processes comprising what are known as $t$-designs. We show that in practical scenarios, certain finite ensembles of two-mode transformations—1- and 2-designs—are indistinguishable from truly random operations for 1- and 2-photon quantum interference, but they fail to mimic randomness for 2- and 3-photon cases, respectively. We make use of the fact that $t$-photon behavior is governed by degree-$2t$ polynomials (in the parameters of the optical process), to experimentally verify the ensembles' behavior for complete bases of polynomials, ensuring that average outputs will be uniform for arbitrary configurations. It is in this sense that a $t$-design is deemed to be a potentially useful pseudorandom resource.

Figure 1

(a) $t$-photon interference is described by degree-$2t$ polynomials. (b) Our setup samples complete sets of independent degree-2 (degree-4) polynomials in the matrix elements of $U$ using interference of 1-photon (2-photon) states, generated from spontaneous parametric down-conversion (SPDC) in type-1 BiBO. Polarized photon pairs generated in paths $a$ and $b$ are combined onto $d$ in the photon-number state ${|1\rangle }_H{|1\rangle }_V$, using a half-wave plate (HWP) and a polarizing beam splitter (PBS). Degree-6 polynomials were sampled with the 3-photon state ${|2\rangle }_H{|1\rangle }_V$ in path $d$, observed by postselecting from the 4-photon term of pulsed SPDC state: 3-photons detected across single photon avalanche diodes (SPADs) 1–15 and one photon heralded at SPAPD 16 (see Supplemental Material for further details [15]). The quantum process $T=Q_{\mathrm{out}}U{Q}_{\mathrm{in}}$ was implemented using wave plates (QWP, HWP, dashed boxes). Photon-number-resolving detection [16] uses spatially multiplexed (SMUX) SPADs (1–15), used in conjunction with a commercial 16-channel time-correlated single photon counting system (TCSPC).

Figure 2

Polynomials $p_1,...,p_9$ extracted from the measured normalized single photon intensity $|T_{1,1}{|}^2$, averaged over settings corresponding to (a) ${\mathfrak{D}}_1$ and (b) ${\mathfrak{D}}_2$. Measured distributions (solid color) are plotted with ideal theoretical values (empty boxes). Dashed lines represent the ideal Haar value $P_H=1/2$, which is always uniform for normalized probability distributions.

Figure 3

Polynomials $q_1,...,q_{25}$ extracted from experiment by measuring the 2-photon correlation $|T_{1,1}{T}_{2,2}+T_{1,2}{T}_{2,1}{|}^2$ and averaging over settings that correspond to realizing (a) ${\mathfrak{D}}_1$, (b) ${\mathfrak{D}}_2$, and for comparison (c) 12 unitary operators chosen randomly from the Haar distribution that do not form a 2-design. Photon distinguishability alters the realized polynomials from their ideal values and the average they would converge to when sampling $U$ over a Haar distribution (green line). This is characterized in the experiment and the ideal theoretical polynomials (empty boxes) and the Haar average (dashed line) are corrected accordingly (see Supplemental Material for details [15]). The average statistical fidelity, ${\sum }_i\sqrt{p_i{q}_i}$, between the probability distributions extracted from our experiment and the ideal distributions is $99.26\pm 0.02\%$ for the 600 2-photon experiments used to generate these plots.

Figure 4

Detection outcomes for 3-photon polynomials $r_i$, averaged over ${\mathfrak{D}}_2$. Solid color represents measured data, empty boxes represent theory, and error bars are computed from assuming Poisson-distributed detection noise. Haar averages for perfect interference and characterized photon distinguishability are marked by the black dashed and green lines, respectively. The average statistical fidelity between measured probability distributions and theory is $97.55\pm 0.03\%$ for the seventy-two 3-photon experiments.

Figure 5

Uniformity of each data set, plotted in terms of (a) the variance over the set of polynomials measured and (b) the maximum deviation of the average probability, as a percentage, from the expected Haar average (black dashed line in Fig. 2 and green solid lines in Figs. 3 and 4).

Figure 6

Left-hand axis: Running average of $P_{\text{est}}={|T_{1,1}{T}_{2,2}+T_{1,2}{T}_{2,1}|}^2$ estimated from each of 64 independent 2-photon experiments (overlaid orange points) as $U$ implements uniformly at random elements of ${\mathfrak{D}}_1$ for polynomial $q_{19}$. The blue line marks the value that $P_{\text{set}}$ should ideally converge to, taking into account characterized indistinguishability of photon pairs (see Supplemental Material [15]). Green and black dashed lines mark the values that $P_{\text{est}}$ would converge to when sampling uniformly from the Haar measure, for perfect and characterized indistinguishability, respectively. Right-hand axis, purple line: The number of standard deviations that $P_{\text{est}}$ differs from the Haar average $P_H=1/3$, quantifying confidence of directly discriminating a non-Haar ensemble.