# Proposal for witnessing non-classical light with the human eye

A. Dodel  Department of Physics, University of Basel, Klingelbergstrasse 82,
4056 Basel, Switzerland  A. Mayinda  Department of Physics, University of
Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland  E. Oudot  Department of
Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
A. Martin  Group of Applied Physics, University of Geneva, Ch. de Pinchat 22,
1211 Geneva, Switzerland  P. Sekatski  Institut for Theoretische Physik,
Universitat of Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria  J.-D.
Bancal  Department of Physics, University of Basel, Klingelbergstrasse 82,
4056 Basel, Switzerland  N. Sangouard  Department of Physics, University of
Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland

(August 27, 2024)

###### Abstract

We give a complete proposal showing how to detect the non-classical nature of
photonic states with naked eyes as detectors. The enabling technology is a
sub-Poissonian photonic state that is obtained from single photons,
displacement operations in phase space and basic non-photon-number-resolving
detectors. We present a detailed statistical analysis of our proposal
including imperfect photon creation and detection and a realistic model of the
human eye. We conclude that a few tens of hours are sufficient to certify non-
classical light with the human eye with a p-value of 10%.

##  1 Introduction & motivations

Efforts have been recently devoted to the realization of quantum experiments
with the human eye. This endeavor is however challenging. The proposal of Ref.
[1] which uses many entangled photon pairs to realize a Bell test with the eye
does not allow one to violate a Bell inequality with a realistic model of the
eye. Refs. [2, 3] which propose to amplify entanglement of a photon pair
through a phase covariant cloning, can lead to entanglement detection with
eye-based detectors provided that strong assumptions are made on the source.
While no assumption is needed on the functioning of the eye, it is necessary
to assume that the source produces true single photons. From a practical point
of view, phase-covariant cloning is also difficult to implement. In
particular, cloning is inherently multimode when implemented with a non-linear
crystal as suggested in Ref. [2]. The undesired modes can be filtered out but
at the price of introducing substantial loss. Ref. [4] provides a technically
simpler solution by using displacement operations on single-photon
entanglement. This proposal allows one to detect entanglement with the eye
without assumption on the source but needs a precise description of the visual
system. Indeed, the entanglement witness proposed in Ref. [4] relies on a
well-defined model of the eye thus requiring a detailed characterization of
the human eye. Importantly, in both Ref. [2] and [4], entanglement is detected
before the amplification. That is, these proposals allow one to conclude that
few-photon entanglement can be detected by the human eye upgraded by phase-
covariant cloning and displacement operations respectively. The question we
address in this manuscript is how the quantum nature of light can be directly
detected with the eye.

The motivations are twofold. First, our proposal is a fascinating attempt to
get closer to the quantum world. Indeed, it is conceptually very different
from standard quantum optics experiments where measurements are done by photon
detectors and the sole role of experimentalists in the measurement process is
to analyse the experimental data stored on a computer. The envisioned
experiment is unitary until the eye, so if a collapse happens it does not
happen before the eye. Second, such an experiment interfaces quantum light and
biological systems. Inspired by the great success of quantum optics in
revolutionizing communications [5], metrology [6], sensing [7] or computing
[8], this experiment of a new kind may flourish with important applications
for biomedical research.

As stated before, the proposal of Ref. [4] is appealing as it uses simple
ingredients, namely single-photon entanglement and displacement operations. In
this manuscript, we derive a witness for non-classical states and we show how
the same ingredients allow one to reveal the non-classical nature of a
superposition state with the eye. Our witness needs no assumption on the
photon number produced by the source or on the precise modelling of the eye.
It simply relies on the assumption that the probability to detect light
increases with the photon number. While entanglement detection requires
measurements in different bases, the experiment that we propose is simpler as
it uses displacement operations with fixed amplitudes and phases. It does not
need interferometric stabilization of optical paths and is very robust against
loss. We show, through a detailed feasibility study including a realistic
model of the human eye with a reasonable recovery time as well as imperfect
photon creation and detection, that a few tens of hours are sufficient for our
witness to conclude about non-classicality with a p-value of 10%. Our results
point towards a concrete proposal for implementing the first experiment where
the quantum nature of light is revealed directly with the human eye.

##  2 Witnessing non-classicality with rudimentary detectors

Coherent states $\ket{\alpha}$ of a harmonic oscillator (or a mode of the
electromagnetic field) saturate the uncertainty relations for any pair of
quadratures as well as for amplitude and phase [9]. In addition, they are
eigenstates of the positive frequency part of the quantized field and vector
potential operators [10]. For these reasons, the set of coherent states is
thought as the most classical subset of all possible pure states of light. In
this context, a state which can be expressed as a mixture of coherent states
$\ket{\alpha}$

$\rho_{\text{class}}=\int\mathrm{d}^{2}\alpha\\-\
p(\alpha)|\alpha\rangle\langle\alpha|,\\-\ \text{with}\\-\ p(\alpha)\geq 0$
(1)

is considered classical, and any state which cannot be decomposed in this way
is then non-classical. It is easy to see that the convex combination of
coherent states in Eq. (1) satisfies
$\frac{\langle\hat{N}^{2}\rangle-\langle\hat{N}\rangle}{\langle\hat{N}\rangle^{2}}\geq
1$ with $\hat{N}$ the number operator [11]. Hence, a photon-counting detector
can be used to witness the non-classical nature of a light state. If the
photon-counting results reveal
$\frac{\langle\hat{N}^{2}\rangle-\langle\hat{N}\rangle}{\langle\hat{N}\rangle^{2}}<1,$
we can indeed conclude that the measured state is non-classical. Note that all
non-classical states lead to entanglement when combined with the vacuum on a
beamsplitter [12]. The link with entanglement helps clarifying the notion of
non-classical states.

Moreover for few photon states,
$\langle\hat{N}^{2}\rangle-\langle\hat{N}\rangle$ can be approximated by $\sim
2\langle|2\rangle\langle 2|\rangle$ and $\langle\hat{N}\rangle^{2}$ by
$\sim\langle|1\rangle\langle 1|\rangle^{2}.$ Hence, one can use a 50/50
beamsplitter and two non-photon-number-resolving detectors to witness the non-
classical nature of few photon states by checking that the two-fold
coincidences ($\sim\langle|2\rangle\langle 2|\rangle/2$) are smaller than the
product of singles ($\sim\langle|1\rangle\langle 1|\rangle^{2}/4$), cf. [13]
for a proper derivation. Can one still use this criterion in presence of other
kinds of detectors? We now address the question of the conditions required to
witness the non-classical nature of a light source with a 50/50 beamsplitter
and two detectors.

Let us consider an arbitrary detector with a binary outcome, one corresponding
to click, the other one to no-click. We label $p_{s}(\alpha)$ the probability
to get a click when a coherent state $\ket{\alpha}$ impinges on such a
detector. In a scenario where two of these detectors are placed after a 50/50
beamsplitter, the probability to get a twofold coincidence with any classical
state is given by $p_{c}(\rho_{\text{class}})=\int\mathrm{d}^{2}\alpha\\-\
p(\alpha)p_{s}(\alpha/\sqrt{2})^{2}$ whereas the probability of a single
detection is given by $p_{s}(\rho_{\text{class}})=\int\mathrm{d}^{2}\alpha\\-\
p(\alpha)p_{s}(\alpha/\sqrt{2})$. This simply comes from the fact that a
coherent state splits into two similar coherent states on a beamsplitter
$\ket{\alpha}\overset{\text{BS}}{\longrightarrow}\ket{\frac{\alpha}{\sqrt{2}}}_{t}\otimes\ket{\frac{\alpha}{\sqrt{2}}}_{r}$.
The Cauchy-Schwarz inequality $\int f(\mu)^{2}\mathrm{d}\mu\int
g(\mu)^{2}\mathrm{d}\mu\geq\left(\int f(\mu)g(\mu)\mathrm{d}\mu\right)^{2}$
for $f=1$, $g=p_{s}(\alpha/\sqrt{2})$ and
$\mathrm{d}\mu=p(\alpha)\mathrm{d}^{2}\alpha$ then implies

$\frac{p_{c}(\rho_{\text{class}})}{p_{s}(\rho_{\text{class}})^{2}}\geq 1.$ (2)

In other words, any detector can be used to witness non-classicality as long
as one has two copies of this particular detector. It suffices to place these
detectors after a 50/50 beamsplitter and to record the number of singles and
coincidences. If the ratio between the probability of having a coincidence and
the square of the probability of singles is smaller than one, we can safely
conclude that the measured state is non-classical. We show in the appendix A
that the ratio between the coincidence and the product of singles is a witness
for non-classicality even if the two detectors after the beamsplitter are not
identical and the beamsplitter is not balanced, as long as $p_{s}(\alpha)$ is
an increasing function of the photon number $|\alpha|^{2}$ for both detectors.
These results are used in the next section to show how to detect non-classical
states with the human eye.

##  3 Witnessing non-classicality with the human eye

Let us start this section by recalling how to model the response of the human
eye to weak light stimuli. In a landmark experiment Hecht, Shlaer and Pirenne
tested the capability of the human eye to detect light pulses containing only
a few photons [14], see also [15]. In their experiment, an observer was
presented with a series of multimode thermal light pulses and asked to report
when the light is seen. Similar results have been obtained much more recently
with coherent light pulses (monomode light also having a Poissonian photon-
number distribution) [16], thus indicating that the response of the eye does
not depend on the number of modes. Interestingly, the results of both
experiments are very well reproduced by a model in which coherent states are
sent onto a threshold detector preceded by loss. In particular, the
experimental data of Ref. [14] is compatible with a threshold at $\theta=7$
photons and an efficiency of $\eta_{e}=8\%,$ see Fig. 1 in Ref. [4]. Note that
these numbers depend on the psychophysics, i.e. the dark adaptation, the
choice of dead-times and methods for eliciting responses from the observer
about his experience of light stimuli. In particular, the recent results
reported in Ref. [16] are compatible with lower thresholds and several
references [17, 18] suggest higher efficiencies. In the remainder of the
paper, we keep the model of the eye with parameters associated to the seminal
work of Hecht and co-workers ($\theta=7$ and $\eta_{e}=8\%$). We show that
these parameters are conservative, i.e. higher efficiencies or lower
thresholds reduce the number of experimental runs that are needed to conclude
about non-classicality.

Figure 1: Result of an auto-correlation $(g^{(2)}(0))$ measurement in which
two eyes are placed after a 50/50 beamsplitter. The ratio between the
probability to see light with both eyes and the square of the probability to
see light with one eye is recorded for an input state
$\mathcal{D(\alpha)}(\ket{0}+\ket{1})/\sqrt{2},$ considering real $\alpha.$ We
here show this ratio as a function of $\alpha.$ Ratios smaller than one (red
dashed line) witness the non-classical nature of the state.

Given the witness for non-classical states presented in the previous section,
we envision an experiment where two eyes are placed after a beamsplitter. The
event “click” corresponds to the case where the observer sees light, “no-
click” where no light is seen. The experiment is repeated several times to
access the probability to see light with one of the two eyes as well as the
joint probability to see light with both eyes. The ratio between the
coincidences and the product of singles is then used to reveal non-
classicality. This ratio is labelled $g^{(2)}(0)$ in analogy to the standard
autocorrelation measurement.

To make a complete proposal, we still need to find a quantum state for which
the non-classical nature can be revealed in such a setup. Note that sub-
Poissonian states, i.e. states for which the distribution in photon-number
space is narrower than the one of a coherent state with the same mean photon
number, are natural candidates for achieving $g^{(2)}(0)<1$ with threshold
detectors such as the human eye. This is because there is a regime where, for
the same probability of singles, the narrow photon-number distribution of a
sub-Poissonian state yields a lower coincidence probability than the one of
the corresponding coherent state. As an illustration, consider an ideal
threshold detector and a Fock state that has enough photons to eventually make
one of the detectors click, but not enough to give a coincidence.
While Fock states with large photon numbers are challenging to produce, a sub-
Poissonian state can be obtained in practice by displacing a superposition of
vacuum and single-photon Fock state in phase space. The resulting state
$\mathcal{D}(\alpha)\left|\frac{1}{\sqrt{2}}\left(0+1\right)\right\rangle$,
where $\mathcal{D}(\alpha)$ stands for a displacement operation, indeed has a
variance in photon-number space that is
$\frac{1+8|\alpha|^{2}-4Re(\alpha)^{2}}{2+4|\alpha|^{2}+4Re(\alpha)}$ times
that of a coherent state with the same mean photon number. This ratio admits
values that are below one, and interestingly, for a given strength of the
displacement $|\alpha|^{2}$, it is minimal and always inferior to unity when
$\alpha$ is real. Consequently, from here on we will only consider real
displacements.

Fig. 1 shows the value of $g^{(2)}(0)$ obtained when sending such a state on a
50/50 beamsplitter followed by two eyes as a function of the amplitude of
$\alpha.$ We see that the non-classical nature of
$\mathcal{D}(\alpha)\left|\frac{1}{\sqrt{2}}\left(0+1\right)\right\rangle$ can
be detected with the human eye as long $\alpha\leq 13.3.$ For larger $\alpha,$
the two eyes always see light and the ratio between coincidences and singles
tends to one. However, in the range of displacement values $\alpha\sim 10,$
one can expect non-negligible occurrence frequency for the event “seen” for
both eyes. These encouraging estimations compel us to make a detailed
feasibility study, i.e. to propose a practical way to create a single photon
superposed with vacuum, to account for imperfect generation efficiency,
channel loss, limited detection efficiencies and to conclude about the
statistics that is required to witness non-classicality with the human eye.

##  4 Proposed experiment

The experiment we envision is shown in Fig. 2. A source based on spontaneous
parametric down-conversion is used to create photon pairs, the detection (on
detector $D_{h}$ in Fig. 2) of one photon from a given pair serving to herald
the presence of its twin. The latter is then sent into a $50/50$ beamsplitter
to create path-entanglement, i.e. entanglement of the form
$(\ket{0}_{t}\ket{1}_{r}-\ket{1}_{t}\ket{0}_{r})/\sqrt{2}$ between the
transmitted and reflected modes of the beamsplitter which share a single
photon. The reflected mode is subsequently detected with a non-photon-number-
resolving detector (detector $D_{g}$ in Fig. 2) preceded by a displacement in
phase space $\mathcal{D}(\beta)$. With the appropriate displacement amplitude,
such a measurement performs a pretty good measurement along the $x$ direction
of the Bloch sphere having $\ket{0}$ and $\ket{1}$ as its north and south pole
respectively [19]. In other words, with the appropriate displacement, a
detection click projects the transmitted mode into a state close to
$(\ket{0}_{t}+\ket{1}_{t})/\sqrt{2}.$ Such a state is then displaced in phase
space, split using a 50/50 beamsplitter and sent to human observers. The
single and coincidence events are recorded and the experiment is repeated
until the observers can conclude about the non-classical nature of the
superposition state with enough statistical confidence. As it is not clear
what psychophysical test would allow to distinguish a dim flash of light
occurring in the left vs. the right eye and a temporal discrimination with a
single observer would require unrealistic delays, we envision an experiment
with two observers, each reporting on whether he/she sees light each time a
detection click is obtained on $D_{g}.$ We show below how to get a triggering
rate compatible with a synchronization of the two observers’ answers.

Figure 2: Schematic representation of the experiment envisioned to witness the
non-classical nature of a superposition state
$\mathcal{D}(\alpha)(\ket{0}_{t}+\ket{1}_{t})/\sqrt{2}$ with the human eye.
The superposition $(\ket{0}_{t}+\ket{1}_{t})/\sqrt{2}$ is prepared by first
sending a single photon into an unbalanced beamsplitter and by subsequent
detection of the reflected mode with a photon detector preceded by a
displacement operation. For displacements with a small enough amplitude, this
projects the transmitted mode into a state close to the desired superposition.
This superposition state is then displaced to produce the non-classical state
of interest. A 50/50 beamsplitter and two eyes are then used to analyse this
state with a measurement analogous to an auto-correlation measurement.

Note that in this setup, one can tune the transmission coefficient of the
first beamsplitter along with the displacement amplitude $\beta$, effectively
modifying the input state for the autocorrelation measurement. Finally, we
observed that the closest state to
$\mathcal{D}(\alpha)\left|\frac{1}{\sqrt{2}}\left(0+1\right)\right\rangle$ is
obtained by choosing a highly unbalanced beamsplitter with transmission $t\sim
1$ and using a displacement $\mathcal{D}(\beta)$ with almost zero amplitude.
In this case, we get a very partially entangled state and maximum coherence of
the conditional state $(\ket{0}_{t}+\ket{1}_{t})/\sqrt{2}$ is restored by
measuring the reflected mode almost along the $z$ direction and post-selecting
the case where a click is obtained. This favors larger fidelities of the
conditional state because the measurement noise is reduced when it gets closer
to the z direction [19]. However, the probability to get a click drops when
the transmission of the beamsplitter increases. There is thus a trade-off
between the “quality” of the states produced by the source and the rate at
which they are produced. The parameters $\beta$ and $t$ have to be optimized
in view of the statistics needed to witness non-classicality, cf. below.

Several requirements need to be satisfied for implementing the experiment
proposed in Fig. 2. (i) The efficient generation of pure, indistinguishable
and narrowband single photons is the first one. A straightforward way to
create photons with these properties from spontaneous parametric down-
conversion is to combine short, Fourier-limited pump pulses with a narrow-band
filtering of the heralding photons. This results in Fourier-limited heralded
photons with the spectrum of the pump [20]. To ensure a high coupling
efficiency of these heralded photons into an optical fiber, a plane wave pump
is required and the heralding photons need to be spatially filtered with a
single mode fiber before being detected. This projects the heralded photons
into the fundamental spatial mode of the fiber, and hence allows one to reach
very high coupling efficiencies [21]. (ii) The photons need to have a color
that can be seen by the human eye. This can be fulfilled with a pump at
$405$nm down-converted into non-degenerate photon pairs at $1536$ and $550$nm.
The advantage is threefold. $550$nm is very well suited for the human eye and
the photons in the telecom band can be efficiently filtered both spatially and
in frequency. The telecom mode can also be seeded with a stable cw telecom
laser to generate the coherent states that are needed for the displacement
operations, cf. below. (iii) The click rate on the detector $D_{g}$ in Fig. 2
needs to be adapted to the timescale of the response of the human eye as it
sets a start for the observers. This can be done by reducing the repetition
rate of the pump laser with an optical chopper. The heralding rate on $D_{h}$
and thus on $D_{g},$ can then be easily set by tuning the laser intensity and
the duty cycle of the optical chopper, c.f. below. (iv) To implement the
displacement operations, we need an unbalanced beamsplitter and coherent
pulses with Poissonian photon distribution that are indistinguishable from the
photons at 550 nm in all degrees of freedom. This can be done using difference
frequency generation. More precisely, we propose to use a second non-linear
crystal, identical to the first one and pumped by the same laser but with a
narrow seed of the telecom mode. In contrast to spontaneous parametric down-
conversion, the seed results in coherent states at 550 nm with the
characteristics of the pump laser, i.e. Fourier-limited coherent states with
the spectrum of the pump [22]. Since the coherent states created in this way
and the single photons at 550 nm are generated from the same pump, their
indistinguishability is insensitive to the pump fluctuations. Note also that
with a ps pump, the effect of frequency fluctuations of the telecom laser is
negligible. The slow fluctuations in intensity of the latter can be recorded
and taken into account once the measurements are done. In the worst case, they
can be monitored and corrected with a feedback loop. Albeit with different
wavelengths, the proposed technique has already been used successfully in
various experiments [22, 23].

Figure 3: Schematic of the setup to produce superposition states close to
$\mathcal{D}(\alpha)(\ket{0}_{t}+\ket{1}_{t})/\sqrt{2}$ and to detect their
quantum nature with the human eye. Star: laser, $\chi^{2}:$ non-linear
crystal, DM: dichroic mirror, f: filters, WP: wave-plates, PBS: polarizing
beamsplitter, BS: beamsplitter. See text for details.

Concretely, we envision an experiment where a Ti-Sa laser is doubled to create
$2-3$ps pulses at $405$nm with a repetition rate of $80$ MHz, see Fig. 3.
These pulses are then used to pump two crystals in order to be down-converted
to $1536$ and $550$nm respectively. The first crystal will be used to create
pure single photons at $550$nm by picking up a single spatial and frequency
mode of the photons at $1536$nm with a monomode fiber and a narrowband
spectral filter. Coherent states that are indistinguishable from the photons
at $550$ nm are generated by seeding the second crystal with a pulsed telecom
laser. Let us emphasize that the critical point of this experimental
implementation is the noise. In standard experiment, the noise is filtered out
by analyzing the detection times to discriminate between true and false
events. As the response of the human eye is not fast enough for such a
temporal discrimination, we need to be sure that a limited number of undesired
photons can reach the eye of the observer. First, we propose to decrease the
repetition rate of the pump laser to 1.6 MHz using an optical chopper with a
duty cycle of 0.02. By tuning the pump intensity to get a pair emission
probability of $0.8\times 10^{-3}$ and considering a global detection
efficiency of 0.08 for $D_{h}$ (i.e. a coupling efficiency of 0.8, a filter
transmission of 0.4 and a raw detection efficiency of 0.25), we get a
heralding rate on $D_{h}$ of $\sim 100$ Hz. Moreover, we consider a coupling
efficiency of the heralded photon at $550$nm of $\eta_{c}=0.8$ in agreement
with the experimental results reported e.g. in Ref. [21]. The detection
efficiency of the visible detector in the upper arm of Fig. 2 is assumed to be
$\eta_{d}=0.5$ which is realistic even when including the transmission loss
from the source to the detector and the inefficiencies of linear optical
elements. We neglect mismatches in the indistinguishability of the photons and
coherent states at $550$nm, which is well justified given the results of Ref.
[22] where the visibility of the Hong-Ou-Mandel interference between a single
photon and a coherent state created via identical crystals as described before
was only limited by the statistics of the coherent state. We set the
transmission $t=98\%$ which, together with the value of the displacement
$\beta\sim 0.08$ chosen to minimize the total number of experimental runs (cf.
below), ensures that $1\%$ of the heralds on $D_{h}$ lead to a click on
$D_{g}$. Meanwhile the conditional state generated on the lower arm shows a
near maximal $95\%$ fidelity with respect to
$\mathcal{D}(\alpha)\ket{\frac{1}{\sqrt{2}}(0+1)}$.

The dominant noise in this scenario comes from the coherent states that are
used for the displacement operations. We propose to trigger the seed that is
used to generate these coherent states on detections in $D_{h}$. In this case,
the noise is $\sim 100$ times greater than the signal. To reduce it further, a
pulse picker is placed in front of the eyes which is triggered by detections
on $D_{g}.$ Considering an extinction ratio of 1:2000, we get a signal-to-
noise ratio of $\sim 20,$ which should be more than enough to perform the
proposed measurement. Note that the pulse picker also filters out other
sources of noise, including the spontaneous emission of the crystal used to
generate single photons at 550 nm (that is negligible with respect to the
noise due to coherent states). Note also that $\sim 100$ns are needed to
trigger the pulse picker on detections by $D_{g},$ which requires a delay line
of 20m of fiber, representing negligible loss for typical attenuation
$<12$dB/km at $550$nm.

##  5 Statistics

To conclude the feasibility analysis of the proposed experiment, we now turn
to the question of statistics, and determine the number of runs needed to
exclude the possibility that the observed finite statistics are the result of
measurements on a classical state. This is a particularly relevant question in
our case, as the repetition rates that can be attained with the human eye are
much lower than the slowest commercial detectors. The statistical study that
we describe in this section aims at estimating the time-resource that an
experimenter would have to allocate to such an experiment for the efficiencies
discussed in the previous section, depending on the accuracy he wants to
achieve.

The statistical issue is essentially an estimation of the odds of having
$g^{(2)}(0)<1$ from a classical photon-number distribution. To answer this we
consider the multinomial joint probability

$\begin{split}P(N_{s},N_{c})=&p_{c}^{N_{c}}(p_{s}-p_{c})^{N_{s}-N_{c}}(1-p_{s})^{N-N_{s}}\\\
&\times{N\choose N_{c},N_{s}-N_{c},N-N_{s}}\end{split}$ (3)

of obtaining $N_{s}$ singles and $N_{c}$ coincidences out of $N$ experimental
runs, from the knowledge of the single and coincidence probabilities in one
round $\\{p_{s},p_{c}\\}$. Note that we assume here that the single
probability on each eye is identical, and that the runs are independently and
identically distributed (i.i.d.). Further note that the form of the above
distribution, whose natural variables are $N_{c}$ and $N_{s}-N_{c}$, stresses
the dependence of the events “single” and “coincidence”. Indeed we have
defined a single in one arm regardless of the situation in the other arm,
hence a coincidence is counted as a single as well. The outcome “single only”
has an occurrence probability $p_{s}-p_{c}$ as can be seen in the multinomial
expression. Both the quantum scenario presented before, with
$\\{p_{s}(\rho_{\text{q}}),p_{c}(\rho_{\text{q}})\\}$ depending on the non-
classical state $\rho_{\text{q}}$, and the classical one with
$\\{p_{s}(\rho_{\text{c}}),p_{c}(\rho_{\text{c}})\\}$ such that
$p_{c}(\rho_{\text{c}})\geq p_{s}^{2}(\rho_{\text{c}})$ give rise to a
probability distribution that we label respectively by $P^{q}(N_{s},N_{c})$
and $P^{c}(N_{s},N_{c}).$

We then choose an estimator $\chi$ which is a function of the total number of
singles $N_{s}$ and coincidences $N_{c}$ observed in $N$ rounds of the
experiment, cf. below. For a given $N$, this estimator takes the value
$\chi(N_{s},N_{c})$ with probabilities $P^{q}(N_{s},N_{c})$ and
$P^{c}(N_{s},N_{c})$ in the quantum and classical scenarios. The probability
of observing a value of $\chi$ smaller than a given value $\chi_{0}$ in the
quantum (classical) case after $N$ rounds is thus given by

$P(\chi^{q/c}\leq\chi_{0})=\sum_{N_{s},N_{c}|\chi(N_{s},N_{c})\leq\chi_{0}}P^{q/c}(N_{s},N_{c}).$
(4)

On one side, the quantum distribution tells us what is the probability with
which we can expect to observe (in a quantum experiment) a value of $\chi$
smaller or equal to some value $\chi_{0}$. We write this probability

$P_{\text{stop}}=P(\chi^{q}\leq\chi_{0}).$ (5)

On the other side, the classical distribution allows us to define the p-value
$\epsilon$ associated with the rejection of the null hypothesis “the state is
classical” once a value $\chi_{0}$ is observed. This p-value is given by

$\epsilon=\max_{p_{c}\geq p_{s}^{2}}P(\chi^{c}\leq\chi_{0})$ (6)

where the maximum is taken over all classical scenarios satisfying
$p_{c}(\rho_{\text{c}})\geq p_{s}^{2}(\rho_{\text{c}}).$ Alternatively, we can
read the relation (6) as a definition of the critical value of the estimator
$\chi_{0}$ which needs to be obtained in order to rule out all classical
states with a confidence of $1-\epsilon$. Choosing first the p-value, Eq. (6)
gives $\chi_{0}$ which can then be used to get the probability to stop at the
$N^{th}$ run using Eq. (5). The average number of runs that is needed to rule
out classical states can finally be estimated as (cf. Appendix B)

$\langle N\rangle\simeq\sum_{j\geq
0}\frac{n(2j+1)}{2}\left(P_{\text{stop}}(n(j+1))-P_{\text{stop}}(nj)\right)$
(7)

where $n$ is a coarse-graining parameter used to make the computation faster.

The question at this stage is what is a good choice for the estimator. Let us
consider the space of frequencies defined by
$(f_{s}^{2},f_{c})\equiv\left(\left(\frac{N_{s}}{N}\right)^{2},\frac{N_{c}}{N}\right).$
We choose a set of coordinates $\\{x,y\\}$ to cancel the covariance and to
equal the variances of $P^{q}(N_{s},N_{c})$ in the x and y directions at first
order in $\frac{1}{N}$. This is achieved by setting

$\begin{cases}x=\sqrt{\frac{c}{b}}f_{s}^{2}+\frac{d}{\sqrt{cb}}f_{c}\\\
y=\sqrt{\frac{b}{c}}f_{c}\end{cases}$ (8)

with

$\begin{cases}b=\sqrt{\frac{(1-p_{c}(\rho_{\text{q}}))p_{s}(\rho_{\text{q}})}{(1-p_{s}(\rho_{\text{q}}))p_{c}(\rho_{\text{q}})}-1}\\\
c=\frac{1-p_{c}(\rho_{\text{q}})}{2p_{s}(\rho_{\text{q}})(1-p_{s}(\rho_{\text{q}}))}\\\
d=-1\end{cases}.$

The projection of $P^{q}(N_{s},N_{c})$ in the $x-y$ plane hence defines
circular isolines, cf. Fig. 4 red isolines. The dashed black line in Fig. 4
distinguishes the frequencies coming from classical and non-classical states.
In particular, the distributions with mean values lying on this boundary come
from states with $p_{c}(\rho_{\text{c}})=p_{s}^{2}(\rho_{\text{c}}),$ i.e.
coherent states with various $p_{s}.$ The classical scenario that best
reproduce the quantum statistics is quite clearly a coherent state which
minimizes the Euclidean distance to the quantum distribution, i.e. centered on
the orthogonal projection of the quantum distribution onto the dashed black
line of Fig. 4. Such a coherent states is associated with

$p_{s}(\rho_{\text{c}})=\sqrt{\frac{c(c+d)p_{s}(\rho_{\text{q}})^{2}+(d(c+d)+b^{2})p_{c}(\rho_{\text{q}})}{b^{2}+(c+d)^{2}}}.$
(9)

Calling $(x^{\prime}_{0},y^{\prime}_{0})$ the center of the corresponding
distribution $P^{c}(N_{s},N_{c}),$ an estimator of the form

$\chi=y^{\prime}-y^{\prime}_{0}+a(x^{\prime}-x^{\prime}_{0})^{2},$ (10)

where $x^{\prime}=\cos(\phi)\,x+\sin(\phi)\,y$,
$y^{\prime}=\cos(\phi)\,y-\sin(\phi)\,x$ and
$\phi=\arccos{\frac{c+d}{\sqrt{b^{2}+(c+d)^{2}}}}$ is intuitively minimized by
the coherent state satisfying (9) for appropriate $a$, as $\phi$ is such that
the axis of the parabola is orthogonal to the classical/non-classical
boundary.

Figure 4: Projections in the modified frequency plane $\\{x,y\\}$ defined in
Eq. (8) of the probability distributions $P^{q}(N_{s},N_{c})$ for the quantum
scenario presented in Fig. 2 (red isolines) and $P^{c}(N_{s},N_{c})$ for the
coherent state defined in (9) (green isoline). The blueish contour line is the
estimator given in Eq. (10). The dashed black line separates the mean values
of quantum and classical states as witnessed by a $g^{(2)}(0)$ measurement. In
particular, the shaded area includes all states with $g^{(2)}(0)\geq 1$.

The probability that enough statistics is obtained after N runs to exclude the
classical distribution $P^{c}(N_{s},N_{c})$ with the estimator given in Eq.
(10) can be computed numerically as a function of the steepness of the
parabola $a$ and the amplitude of displacement operations $\alpha,\beta$.
After checking that the considered classical strategy is indeed optimal for
the estimator (10), we obtained the optimal values $a=40$ and
$(\alpha,\beta)\simeq(10.99,0.08)$ for the efficiencies discussed in the
previous section and the model of the eye matching the data of Hecht and co-
workers $(\theta=7,\eta_{e}=8\%)$. The results are shown in Fig. 5 for
p-values of $1\%$ and $10\%$. We see for example that after $350000$ runs, we
have more than $50\%$ chance of being able to rule out classical states with a
confidence of $1-\epsilon=99\%$. For $n=12500,$ we find $\langle
N\rangle\simeq 402964$ for a confidence of $99\%.$ Note that to perform
$403000$ runs with a repetition rate of $1$Hz takes about $112$ hours. The
latter provides an upper bound on the timescale of the proposed experiment to
get a p-value of $1\%.$ A similar analysis for a p-value of $10\%$ shows that
$46$ hours are likely to be enough to detect the non-classical nature of a
single photon superposed with vacuum using the human eye. This goes down to 35
hours when considering a threshold at 3 photons while keeping 8% efficiency
and to 29 hours for an efficiency of 10% and a threshold at 7 photons.

Figure 5: Probability to get enough statistics to conclude about non-
classicality as a function of the number of runs N for a p-value of $1\%$
(blue dotted line) and $10\%$ (red dashed line).

##  6 Conclusion

We have presented a concrete proposal for a quantum experiment with the human
eye, including the full analysis of the measurement statistics. It uses simple
components, namely path-entanglement, displacement operations in phase space
and non-photon-number-resolving detectors, to certify with naked eyes the non-
classical nature of a state of light. We have given a detailed recipe using
parametric conversions and photon-counting techniques only, i.e. commercially
available devices working at room temperature that are routinely used in
practice. We have shown that the statistics obtained in a few tens of hours
would be sufficient to certify non-classicality with a p-value of $10\%.$ This
was obtained with realistic models of the human eye and taking loss and non-
unit efficiencies of photon detectors into account. We believe that these
timescales are well within reach in practice primarily because the data do not
need to be taken in a row. Following in particular the implementation proposed
in Fig. 3 where a single photon and a coherent state with different
polarizations impinge on a polarizing beamsplitter to follow the same optical
path and where a set of wave plates and a polarizing beamsplitter are used to
make the displacement operations, we can certify from our past experiment [23]
that the setup is extremely stable even without active stabilization of
relative path-length fluctuations. It is thus very likely that the data
acquisition can be stopped and started again later for several tens of hours
without problem. Despite many preconceptions, we expect the response of the
eye to be consistent over long minutes after appropriate dark adaptation. Slow
threshold or efficiency drifts can be taken into account easily by periodic
re-calibration of the amplitude of displacement operations. We thus see our
work as a concrete and realistic proposal to realize the first experiment
where the non-classical nature of light is detected directly with the human
eye.

##  7 Acknowledgements

We thank V. Caprara-Vivoli, M. Munsch, Botond Roska, Hendrik Scholl and R.
Warburton for valuable discussions. This work was supported by the Swiss
National Science Foundation (SNSF), through the NCCR QSIT and the Grant number
PP00P2-150579, the John Templeton foundation and the Austrian Science Fund
(FWF), Grant number J3462 and P24273-N16.

##  8 Appendices

###  8.1 Autocorrelation with different arbitrary detectors

Let us recall the definition of a classical state as given in the main text:
$\rho_{cl}=\int\mathrm{d}^{2}\alpha\\-\ p(\alpha)|\alpha\rangle\langle\alpha|$
with $p(\alpha)\geq 0$. We now relax the constraint on the symmetry between
the two arms in the autocorrelation measurement, and label $(1,2)$
respectively the reflected and transmitted beams. Each of those beams is sent
to a detector which can be different from the other one and the beamsplitter
prior to detection is allowed to be unbalanced with coefficients $r/t$. Using
the transformation rules for a coherent state on a beamsplitter, it is
straightforward to express the probabilities of interest as an integral of the
probabilities of singles for appropriate coherent states

$\displaystyle P_{s_{1}}(\rho_{cl})=\int
p(\alpha)P_{s_{1}}\left(\sqrt{r}\alpha\right)\mathrm{d}^{2}\alpha$
$\displaystyle P_{s_{2}}(\rho_{cl})=\int
p(\alpha)P_{s_{2}}\left(\sqrt{t}\alpha\right)\mathrm{d}^{2}\alpha$
$\displaystyle P_{c}(\rho_{cl})=\int
p(\alpha)P_{s_{1}}\left(\sqrt{r}\alpha\right)P_{s_{2}}\left(\sqrt{t}\alpha\right)\mathrm{d}^{2}\alpha.$

Instead of the autocorrelation which is a ratio of two quantities, we focus on
the difference

$\displaystyle
D(\rho_{cl})=P_{c}(\rho_{cl})-P_{s_{1}}(\rho_{cl})P_{s_{2}}(\rho_{cl})$
$\displaystyle\quad=\int
p(\alpha)P_{s_{1}}\left(\sqrt{r}\alpha\right)P_{s_{2}}\left(\sqrt{t}\alpha\right)\mathrm{d}^{2}\alpha$
$\displaystyle\qquad-\int
p(\alpha)P_{s_{1}}\left(\sqrt{r}\alpha\right)\mathrm{d}^{2}\alpha\int
p(\alpha)P_{s_{2}}\left(\sqrt{t}\alpha\right)\mathrm{d}^{2}\alpha,$

Note that $D<0$ implies $g^{(2)}(0)<1.$ Upon inserting $\int
p(\beta)\mathrm{d}^{2}\beta=1$ in $P_{c}(\rho_{cl})$ and relabelling the dummy
variable $\alpha\leftrightarrow\beta$ in some of the terms we get

$\displaystyle D$
$\displaystyle(\rho_{cl})=\frac{1}{2}\int\mathrm{d}^{2}\alpha\\-\
p(\alpha)\int\mathrm{d}^{2}\beta\\-\ p(\beta)$
$\displaystyle\left(P_{s_{1}}\left(\sqrt{r}\alpha\right)-P_{s_{1}}\left(\sqrt{r}\beta\right)\right)\left(P_{s_{2}}\left(\sqrt{t}\alpha\right)-P_{s_{2}}\left(\sqrt{t}\beta\right)\right)$

We thus obtain that if the functions $P_{s_{1/2}}(\alpha)$ are increasing with
$|\alpha|^{2}$, then $D(\rho_{cl})\geq 0\Leftrightarrow
g^{(2)}(0)_{\rho_{cl}}\geq 1$, which entails the validity of our witness even
in the non-symmetrical case.

###  8.2 On the estimation of the average number of runs

We introduce a formalism to deal with the issue of finding a proper
probability distribution for the number of runs. We write the sequence of
measurements as a list of zeros and ones, binary stochastic results
corresponding respectively to $\chi_{mes}>\chi_{0}(N)$ and
$\chi_{mes}\leq\chi_{0}(N)$. It illustrates the situation where an
experimenter computes $\chi$ after each measurement (or alternatively after
each set of $m$ measurements) and decides if he carries on with the measures
(“0”) or stops because the results are already satisfactory (“1”). Ideally
what we would like to have is the probability
$P(n)=P(\underset{n}{\underbrace{0,0,...,1}})$ to reach the required
statistics after exactly $n$ runs. Unfortunately, obtaining this “true”
probability numerically represents a computational challenge. What we output
from our simulation $P_{\text{stop}}(N)$ is the probability to get a one at
$N^{th}$ position regardless of the preceding sequence. Let’s compare the
“cumulative distributions”

$\displaystyle\sum_{n\leq N}P(n)$
$\displaystyle=P(1)+P(0,1)+...+P(\underset{N-1}{\underbrace{0,...,0}},1)$
$\displaystyle P(1)$
$\displaystyle=\sum_{i_{2},...,i_{N}}P(1,i_{2},...,i_{N})\text{\hskip
8.53581ptwhere $i_{k}\in\\{0,1\\}$}$
$\displaystyle=P_{\text{stop}}(N)-\sum_{i_{2},...,i_{N-1}}P(0,i_{2},...,i_{N-1},1)$
$\displaystyle+\sum_{i_{2},...,i_{N-1}}P(1,i_{2},...,i_{N-1},0)$
$\displaystyle P(0,1)-\sum_{i_{2},...,i_{N-1}}P(0,i_{2},...,i_{N-1},1)$
$\displaystyle=-\sum_{i_{3},...,i_{N-1}}P(0,0,i_{3},...,i_{N-1},1)$
$\displaystyle+\sum_{i_{3},...,i_{N-1}}P(0,1,i_{3},...,i_{N-1},0)$
$\displaystyle\vdots$ $\displaystyle\sum_{n\leq
N}P(n)=P_{\text{stop}}(N)+\sum_{n=0}^{N-2}$ $\displaystyle
P(\underset{n}{\underbrace{0,...,0}},1,i_{n+2},...,i_{N-1},0).$

Therefore $P_{\text{stop}}(N)\leq\sum_{n\leq N}P(n)$. We would like to
translate it into an information on the expectation values. Let us switch to a
continuous viewpoint and introduce functions $f$ and $g$ standing for the
cumulative distributions, with $\forall x\hskip 2.84526ptg(x)<f(x)$ (thus $g$
and $f$ replace the $P_{\text{stop}}$ and $\sum_{n\leq N}P(n)$ of the previous
paragraph). We write the expectation values difference and integrate by part

$\displaystyle\int_{0}^{M}xf^{\prime}(x)\mathrm{d}x-\int_{0}^{M}xg^{\prime}(x)\mathrm{d}x$
$\displaystyle=M[f(M)-g(M)]$
$\displaystyle-\int_{0}^{M}\underset{>0}{\underbrace{(f(x)-g(x))}}\mathrm{d}x.$

We need to know how the first term behaves when $M\rightarrow\infty$. We
haven’t find a rigorous way to prove that it vanishes but we notice
$M[f(M)-g(M)]<M[1-g(M)]$, which we reasonably assume stays finite based upon
the numerical simulations. The latter indeed reveals that $N\longmapsto
N\left(1-P_{\text{stop}}(N)\right)$ shows a decreasing tendency after a given
$N$. From this we deduce
$\left<N\right>\leq\sum_{n}n\dfrac{\mathrm{d}P_{\text{stop}}}{\mathrm{d}n}$.

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