Quantum Biometrics with Retinal Photon Counting

It is known that the eye’s scotopic photodetectors, rhodopsin molecules, and their associated phototransduction mechanism leading to light perception, are efficient single-photon counters. We here use the photon-counting principles of human rod vision to propose a secure quantum biometric identification based on the quantum-statistical properties of retinal photon detection. The photon path along the human eye until its detection by rod cells is modeled as a filter having a specific transmission coefficient. Precisely determining its value from the photodetection statistics registered by the conscious observer is a quantum parameter estimation problem that leads to a quantum secure identification method. The probabilities for false-positive and false-negative identification of this biometric technique can readily approach ${10}^{-10}$ and ${10}^{-4}$, respectively. The security of the biometric method can be further quantified by the physics of quantum measurements. An impostor must be able to perform quantum thermometry and quantum magnetometry with energy resolution better than ${10}^{-9}\hslash$, in order to foil the device by noninvasively monitoring the biometric activity of a user.

Figure 1

(a) Equivalence of the proposed biometric measurement with an optical setup consisting of a filter, the transmission coefficient $\alpha$, of which is to be estimated, followed by an ideal (unit quantum efficiency) single-photon counter (SPC), having a threshold equal to the brain’s light perception threshold of at least $K=6$ photons illuminating a ganglion-receptive field. The output of the SPC is then registered by the conscious observer. The transmissivity of the filter captures the photon losses along the particular beam path ending on a particular retinal spot, including the detection efficiency of the illuminated rods. (b) Probability to see the flash of coherent light having an average photon number $\tilde{I}$ for three indicative values of the parameter $\alpha$.

Figure 2

Random walks of $\ln (R_n/R_0)$ (Alice, blue and upwards; Eve, red and downwards) and distribution of interrogation times $T$. (a),(b) Illumination of spots with (a) ${\alpha }_L=0.05$ and ${\alpha }_H=0.15$. For these values of ${\alpha }_L$ and ${\alpha }_H$ it is $q=0.096$ and $\tilde{I}=62.4$. (c),(d) Illumination of spots with low $\alpha$ and high $\alpha$ uniformly distributed in [0.02,${\alpha }_L$] and [${\alpha }_H$,0.18], respectively. If $\ln (R_n/R_0)$ becomes higher (lower) than the upper (lower) threshold defined by $-\ln (p_{\mathrm{FP}})$ shown by the black solid line [$\ln (p_{\text{FN}})$ shown by the dashed black line] in (a),(c), the interrogation stops and Alice is identified (Eve is rejected). The obtained mean values of $T$ shown in (b) and (d) are consistent with the bounds (9). In (a) and (c) we plot just 100 walks, the distributions (b) and (d) are obtained from 5000 walks.

Figure 3

Biometric strategy. (a) We suppose Alice’s retina consists of $100\times 100$ nonoverlapping ganglion-receptive cells that can be individually addressed with a laser pulse. The light path ending in each one of those is described by a characteristic $\alpha$ value, which here takes a random value between ${\alpha }_{\min }=0.02$ and ${\alpha }_{\max }=0.2$ (we ignore the spatial dependence of $\alpha$, i.e., in a real retina the average $\alpha$ decreases going from the center to the periphery). The color coding in (a) is white for high-$\alpha$ values, i.e., $0.16\le \alpha \le 0.2$, gray for $0.04\le \alpha \le 0.16$, and black for low-$\alpha$ values, i.e., $0.02\le \alpha \le 0.04$. We can easily find a number of white pixels forming a pattern, for example, the number 2 (upper yellow box) or the letter $Y$ (lower yellow box). To the set of pixels forming the pattern we add another set of low-$\alpha$ pixels so that in the combined set there is no discernible pattern. However, Alice would ideally (only high-$\alpha$ pixels fire) just see the pattern, as shown in (b) and (d). In contrast, even if Eve is equipped with an ideal photodetector, all she would see is the combined set of pixels, shown in (c) and (e). Realistically, Alice can miss some high-$\alpha$ pixels and see noise from some firing low-$\alpha$ pixels. A few realizations of a Poisson simulation [a pixel $(i,j)$ is lighted if a Poisson random variable with average ${\alpha }_{ij}\tilde{I}$ exceeds the detection threshold $K=6$, where we used $\tilde{I}=72$] are shown in (f1)–(f6) for the case of 2. (g1)–(g18) Example for realizing $p_{\mathrm{FP}}\approx {10}^{-10}$ with $m=8$ interrogations. Among the illuminated spots of (c) we can form at least 18 patterns, in particular 2, 4, 6, $S$, $v$, 7, $x$, $b$, $f$, 3, $h$, $t$, $q$, $d$, $Z$, $L$, %, $U$. The tested subject is offered these 18 patterns from which she is supposed to pick her response. While Alice would most probably perceive 2, since 2 is the pattern formed by Alice’s high-$\alpha$ spots, Eve’s only option would be to randomly pick her response among the 18 choices, thus after eight interrogations one could achieve $(1/18{)}^8\approx {10}^{-10}$ for the false-positive probability.

Figure 4

(a) The desired level of precision in the estimate of the true $\alpha$ sets a range of acceptable values around the true $\alpha$, given by ${\alpha }_L$ and ${\alpha }_R$, defining the $\alpha$ acceptance region, which lies within the possible $\alpha$ values, ranging from ${\alpha }_{\min }$ to ${\alpha }_{\max }$. The $\alpha$ axis is mapped into the corresponding probability space by the function $G_K(x)$, which gives the probability to see a flash when $x$ photons on average are detected by the tested spot of the retina. The inverse map leads from the measured $\mathbb{P}[\mathrm{see}]$ to $\alpha$. (b) Number of interrogations per spot $\nu$ required to achieve a false-positive probability $p_{\mathrm{FP}}$ for a test involving $\mu$ retinal spots, for various values of the false-negative probability $p_{\mathrm{FN}}$. For this plot we took $\mu =50$.