Further author information: Elena Losero: E-mail<EMAIL_ADDRESS>
Giuseppe Ortolano: E-mail<EMAIL_ADDRESS>
# Quantum reading: the experimental set-up

Elena Losero  Laboratory of Quantum Nano-Optics, EPFL, 1015, Lausanne,
Switzerland  Giuseppe Ortolano  Quantum metrology and nano technologies
division, INRiM, Strada delle Cacce 91, 10135 Torino, Italy  Fabio Saccomandi
Quantum metrology and nano technologies division, INRiM, Strada delle Cacce
91, 10135 Torino, Italy  Ivano Ruo-Berchera  Quantum metrology and nano
technologies division, INRiM, Strada delle Cacce 91, 10135 Torino, Italy
Stefano Pirandola  Department of Computer Science, University of York, York
YO10 5GH, United Kingdom  Marco Genovese  Quantum metrology and nano
technologies division, INRiM, Strada delle Cacce 91, 10135 Torino, Italy

###### Abstract

The protocol of quantum reading refers to the quantum enhanced retrieval of
information from an optical memory, whose generic cell stores a bit of
information in two possible lossy channels. In the following we analyze the
case of a particular class of optical receiver, based on photon counting
measurement, since they can be particularly simple in view of real
applications. We show that a quantum advantage is achievable when a
transmitter based on two-mode squeezed vacuum (TMSV) states is combined with a
photon counting receiver, and we experimentally confirm it. In this paper,
after introducing some theoretical background, we focus on the experimental
realisation, describing the data collection and the data analysis in detail.

###### keywords:

quantum reading, quantum advantage, two-mode squeezed vacuum, photon counting

##  1 INTRODUCTION

A general strategy to read data a classical digital memory can be modeled as
follows [1]: a bipartite bosonic system, called transmitter $T(\rho,M,L)$, in
a state $\rho$ formed by a signal system with $M$ modes and an idler system
with $L$ modes, is used to discriminate between two values of transmittance
$\tau_{\mu}$ storing the value of a classical bit, $u={0,1}$. The signal
system probes a single cell of the memory while the idler system goes directly
to the receiver $R$, where an appropriate joint measurement of the final
bipartite state is performed to retrieve the value of the bit with a certain
probability of error $P_{err}$. Given this scenario the goal is to minimize
$P_{err}$ under the energy constraint where $N$ photons in total are addressed
to the memory cell. Assuming an optimal measurement at the receiver, the
minimization of $P_{err}$ over all the possible transmitters with fixed signal
energy is difficult to solve. However, in the relevant case of a classical
state $\rho$, having a positive Glauber–Sudarshan P representation, a lower
bound on the error probability can be found. In particular in Ref. [1] the
error probability for discrimination between the final states
$\mathcal{E}_{0}(\rho)$ and $\mathcal{E}_{1}(\rho)$, where $\mathcal{E}_{\mu}$
represents the action of the attenuation channel induced by $\tau_{\mu}$, is
found to be:

$P^{cla}_{err}\geq\mathcal{C}(N,\tau_{0},\tau_{1})=\frac{1-\sqrt{1-e^{-N(\sqrt{\tau_{1}}-\sqrt{\tau_{0}})^{2}}}}{2}.$
(1)

Here ${C}(N,\tau_{0},\tau_{1})$ sets an absolute limit on the performance that
can be achieved using any transmitter in a classical state. Note that it
depends only on the mean number of signaling photons $N$ and the values of the
transmittance $\tau_{0}$ and $\tau_{1}$. Note also that this bound is not
necessarily tight, i.e. it is not known if it exists a classical measurement
scheme able to saturate it. A probability of error $P_{err}$ means that on
average $1-H(P_{err})$ bits are recovered, where $H(\cdot)$ denotes the binary
Shannon entropy [2]. It can be formally proven [1] that this limit can be
beaten by a certain class of quantum transmitters, namely EPR correlated ones,
assuming an unspecified optimal measurement at the receiver. Denoting the
probability of error of a quantum transmitter as $P_{err}^{qua}$, we can
evaluate the gain in bits over the classical bound as:

$G_{a}(N,\tau_{0},\tau_{1})=1-H(P_{err}^{qua})-(1-H(\mathcal{C}(N,\tau_{0},\tau_{1}))).$
(2)

##  2 PHOTON COUNTING DISCRIMINATION STRATEGIES

The macroscopic effect of an attenuation channel (e.g. $\mathcal{E}_{0}$ and
$\mathcal{E}_{1}$) is a reduction of the main energy of the system. In
particular, considering an electromagnetic field, the mean number of photons
$N$ after a channel characterized by a transmission $0\leq\tau\leq 1$, will be
reduced to $\tau N$. From this consideration, a natural approach to
discriminate the channels is to perform a direct photon counting measurement.
In the case of parameter estimation, when the goal is to estimate the
continuous parameter $\tau$, it can be proven that a photon counting
measurement combined with Fock states or high number of TMSV replicas is
actually able to reach the ultimate quantum limit. For the discrete case, i.e.
the discrimination problem considered here, such a proof has not been given
before our work. Note that this class of discrimination strategies is
extremely relevant from the experimental point of view since they can be
easily implemented.

###  2.1 Effect of the attenuation on the photon number distribution

The output of a photon counting measurement on a field $\hat{a}$ in a generic
state $\rho$, is a classical random variable $n$ distributed as
$P_{0}(n)=\langle n|\rho|n\rangle$, where $|n\rangle$ is the eigenstate with
eigenvalue $n$ of the number operator $\hat{n}=\hat{a}^{\dagger}\hat{a}$ of
the field. The effect of an attenuation channel $\mathcal{E}_{\tau}$ on a
field can be modeled using a Beam Splitter (BS) with transmission $\tau$ for
which the input-output relations are well known:
$\hat{a}_{t}=\sqrt{\tau}\hat{a}+i\sqrt{(1-\tau)}\hat{v}$ and
$\hat{a}_{r}=i\sqrt{(1-\tau)}\hat{a}+\sqrt{\tau}\hat{v}$. Here $\hat{a}$
denotes the input field, $\hat{a}_{t}$ and $\hat{a}_{r}$ the transmitted and
reflected fields, $\hat{v}$ is the field at the second port of the BS. Thermal
noise in the channel can be modeled taking $\hat{v}$ in a thermal state. In
the following however thermal noise will be neglected so that $\hat{v}$ will
be considered to be in a vacuum state $|0\rangle$.

The photon number distribution for the output state is found to be:

${{{\color[rgb]{0,0,0}\langle
N|\mathcal{E}_{\tau}(\rho)|N\rangle=\sum^{\infty}_{n=N}\langle n|\rho|n\rangle
B(n|m,\tau),}}}$ (3)

where $B(n|m,\tau)=\genfrac{(}{)}{0.0pt}{0}{n}{N}\tau^{N}(1-\tau)^{n-N}$ is a
binomial distribution with $n$ trials and probability of success $\tau$. The
process can be seen as each photon undergoing a Bernoulli trial with
probability of success $\tau$ leading to a binomial distribution with $n$
trials, $B(n,\tau)$. An arbitrary initial distribution $P_{0}(n)$ will then be
compounded, by $\mathcal{E}_{\tau}$, with a binomial distribution $B(n,\tau)$
in accordance with Eq.(3).

From here thereafter the quantity considered in the characterization of a
state will be its photon number distribution, in terms of which Eq.(3) can be
rewritten as:

$P(n)=\sum_{m=0}^{\infty}P_{0}(m)B(n|m,\tau)$ (4)

In the case of multivariate distributions, the case of interest when
considering a multipartite system, the relevant quantity is the joint photon
number distribution, $P(n_{1},..,n_{\mathcal{N}})$. Considering an initial
distribution $P_{0}(n_{1},..,n_{\mathcal{N}})$ going through $\mathcal{N}$
independent channels with attenuation
$\tau=\\{\tau_{1},...,\tau_{\mathcal{N}}\\}$ the final distribution will be,
similarly to the univariate case:

$P(n_{S},..,n_{\mathcal{N}})=\sum_{m_{1},...,m_{\mathcal{N}}=0}^{\infty}P_{0}(m_{1},..,m_{\mathcal{N}})\prod_{i}^{\mathcal{N}}B(n_{i}|m_{i},\tau_{i}).$
(5)

###  2.2 Bayesian discrimination

Suppose the pair of counts $\textbf{n}=(n_{S},n_{I})$ is the result of a
measurement, from the signal and idler channels respectively, after the signal
from a known source has passed trough a channel $\mathcal{E}_{i}$. If we
consider only two possible channels, $\mathcal{E}_{0}$ and $\mathcal{E}_{1}$
(corresponding to a transmittance $\tau_{0}$ and $\tau_{1}$ respectively), a
discrimination between the two hypothesis can be done considering the Bayesian
probability $P(\tau_{i}|\textbf{n})$ given by:

$\displaystyle P(\tau_{i}|\textbf{n})$
$\displaystyle=\frac{P(\textbf{n}|\tau_{i})P(\tau_{i})}{P(\textbf{n})}=$
$\displaystyle=\frac{P(\textbf{n}|\tau_{i})}{P(\textbf{n}|\tau_{0})+P(\textbf{n}|\tau_{1})},$
(6)

where we are considering the discrimination between two equi-probable
channels, i.e. $P(\tau_{i})=1/2$. Given the constant prior, the Bayesian
probability is proportional to the likelihood $P(\textbf{n}|\tau_{i})$.

After measurement, the transmittance $\tau_{i}$ can be chosen such that
$P(\tau_{i}|\textbf{n})\geq P(\tau_{j}|\textbf{n})$, that given the
proportionality implies an analogous condition on the likelihood, with
probability of success $P(\tau_{i}|\textbf{n})$. The probability of success in
function of the pair n can then be written as
$P_{s}(\textbf{n})=\max_{i=0,1}P(\tau_{i}|\textbf{n})$. Averaging
$P_{s}(\textbf{n})$ over the pair distribution
$P(\textbf{n})=\frac{1}{2}\big{(}P(\textbf{n}|\tau_{0})+P(\textbf{n}|\tau_{1})\big{)}$
yields the expected probability of success,$P_{s}(\tau_{0},\tau_{1})$, of the
recovery:

$\displaystyle P_{s}(\tau_{0},\tau_{1})$
$\displaystyle=\frac{1}{2}\sum_{\textbf{n}}\max_{i=0,1}P(\tau_{i}|\textbf{n})(P(\textbf{n}|\tau_{0})+P(\textbf{n}|\tau_{1}))=$
$\displaystyle=\frac{1}{2}\sum_{\textbf{n}}\max_{i=0,1}P(\textbf{n}|\tau_{i}).$
(7)

###  2.3 Photon-counting strategy for classical states

In this section we focus on the explicit discrimination strategy based on
photon counting presented in Sec. 2.2, while considering classical states. In
particular, we focus on the relevant case of a transmitter in a coherent case
$\rho_{coh}$, e.g. the state that can be easily obtained whit a laser. A
coherent state presents a Poissonian photon number distribution:
$P_{0}(n)=P_{\lambda}(n)$. An important property of $P_{\lambda}(n)$ is that
the attenuation process does not change the form of the distribution, i.e. the
distribution at the output of the attenuation channel $\mathcal{E}_{\tau}$ is
$P_{\lambda\tau}(n)$.

In the following we will consider a single mode coherent state signaling $N$
photons, $T(\rho_{coh},1,0)$. In the coherent case, given the independence of
the events described by a Poisson distribution, considering $M$ independent
modes rather than a single one would not change the results given that the $N$
photons in the single mode are already independent from each other.

Considering two channels of transmittance $\tau_{0}<\tau_{1}$ and a source
with initial distribution $P_{\lambda}(n)$, after the channel the distribution
will be either $P_{\lambda\tau_{0}}(n)$ or $P_{\lambda\tau_{1}}(n)$. As
described in section 2.2 we will chose $\tau_{0}$ for each value of $\bar{n}$
such that $P_{\lambda\tau_{0}}(\bar{n})\geq P_{\lambda\tau_{1}}(\bar{n})$.

Solving this condition for $\bar{n}$ yields the threshold:

$\bar{n}\leq\frac{\lambda(\tau_{1}-\tau_{0})}{\log(\tau_{1}/\tau_{0})}\doteq
n_{th}.$ (8)

The average probability of success in this case,
$P_{s}^{cla,pc}(\tau_{0},\tau_{1})$, will then be given by:

$\displaystyle P_{s}^{cla,pc}(\tau_{0},\tau_{1})$
$\displaystyle=\sum_{n=0}^{\infty}P_{s}(n)\frac{1}{2}\big{(}P_{\lambda\tau_{0}}(n)+P_{\lambda\tau_{1}}(n)\big{)}=$
$\displaystyle=\frac{1}{2}\sum_{n=0}^{n_{th}}P_{\lambda\tau_{0}}(n)+\frac{1}{2}\sum_{n=n_{th}}^{\infty}P_{\lambda\tau_{1}}(n)$
$\displaystyle=\frac{1}{2}\Big{(}1+\frac{\Gamma(\lfloor
n_{th}+1\rfloor,\lambda\tau_{0})-\Gamma(\lfloor
n_{th}+1\rfloor,\lambda\tau_{1})}{\lfloor n_{th}\rfloor!}\Big{)},$ (9)

where $\lfloor x\rfloor$ denotes the floor and $\Gamma(x,y)$ is the incomplete
gamma function. The probability of error for the photon counting strategy in
the case of a classical input state (i.e. a coherent state) can be computed
as:

$P_{err}^{cla,pc}(\tau_{0},\tau_{1})=1-P_{s}^{cla,pc}(\tau_{0},\tau_{1}).$
(10)

This quantity can be compared with the theoretical classical limit in Eq. (1).
A comparison for $\tau_{0}=0.8$ and $\tau_{1}=1$ is shown in Fig. 1. The gain
in terms of bits of a quantum strategy having error probability
$P^{qua}_{err}$ over the classical photon counting bound is:

$G_{emp}(N,\tau_{0},\tau_{1})=1-H(P_{err}^{qua})-(1-H(P_{err}^{cla,pc}(\tau_{0},\tau_{1}))).$
(11)

Figure 1:  _Comparison of classical limits_. A comparison between the
theoretical classical limit, supposing an optimal measurement, reported in Eq.
(1), and the limit for classical states when an explicit photon counting
measurement is considered, reported in Eq. (10).

###  2.4 Photon-counting strategy for a two-mode squeezed vacuum state

A quantum state, commonly produced in optical laboratories, having
correlations that surpass the classical limit is the two-mode squeezed vacuum
(TMSV) state, $|\psi\rangle_{\text{TMSV}}$:

$|\psi\rangle_{\text{TMSV}}=\sum_{n=0}^{\infty}c_{N}(n)|n\rangle_{S}|n\rangle_{I},$
(12)

where $S$ and $I$ denote a signal and idler system and
$|c_{N}(n)|^{2}=\frac{N^{n}}{(1+N)^{n+1}}$ is a thermal distribution with mean
number of photons $N$. The distribution of the count pair
$\textbf{n}=(n_{S},n_{I})$ is then:

$P(\textbf{n})=|\langle
n_{S},n_{I}|\psi\rangle_{\text{TMSV}}|^{2}=|c_{N}(n)|^{2}\delta(n-n_{S})\delta(n-n_{I}).$
(13)

After the signal system passes through a channel $\tau$ this distribution is
mapped according to Eq. (5) into $P(\textbf{n}|\tau)$:

$P(\textbf{n}|\tau)=|c_{N}(n_{I})|^{2}B(n_{S}|n_{I},\tau).$ (14)

Once again, when discriminating between two channels of transmittance
$\tau_{0}<\tau_{1}$, the value chosen is the one for which the likelihood in
Eq. (14) is higher. Solving this condition for $n_{S}$ we get:

$n_{S}\leq\Big{(}\frac{\log(\tau_{1}/\tau_{0})}{\log((1-\tau_{0})/(1-\tau_{1}))}+1\Big{)}^{-1}n_{I}\doteq
n_{th}(n_{I}).$ (15)

Using Eq. (2.2), the probability of success for the TMSV-based transmitter and
photon-counting receiver is:

$P_{s}^{TMSV,pc}(\tau_{0},\tau_{1})=\frac{1}{2}\sum_{n_{I}=0}^{\infty}\sum_{n_{S}=0}^{n_{th}(n_{I})}P(\textbf{n}|\tau_{0})+\frac{1}{2}\sum_{n_{I}=0}^{\infty}\sum_{n_{S}=n_{th}(n_{I})}^{\infty}P(\textbf{n}|\tau_{1}).$
(16)

Remarkably, the numerical simulations of Ref. [3] demonstrate that this
approach offers a gain respect to the optimal classical bound in Eq. (1) and
the classical photon counting bound in Eq. (10). Note that the numerical
investigation shows a quantum advantage even with a single TMSV state but the
discrimination is more effective in presence of an high number of copies $M\gg
1$. This is the regime that we have in our experiment. Note that when the
number of modes $M$ is much higher than the mean number of photons $n$ and at
the same time $n\gg 1$, the photon number distributions are well approximated
by a multivariate normal distribution. We will use this approximation in our
analysis

###  2.5 Effect of quantum efficiency

A quantity characterizing every quantum optical experimental setup is the
quantum efficiency $0\leq\eta\leq 1$. This quantity expresses the fraction of
photons entering the experimental setup that are actually detected. Being
interested in the quantum reading experimental realisation this quantity is of
particular interest. It takes into account photon losses from interaction with
the environment and optical components as well as the intrinsic quantum
efficiency of the detector used and its electronic noise and, in case of
correlated beams, the efficiency in detecting correlated photons can be
accounted by this coefficient as well.

It is well known that quantum properties require a good quantum efficiency
$\eta$ of the setup to be revealed and used to get an advantage over classical
limits, and that those advantages are usually very sensitive to variations in
the value of $\eta$. An analysis of its effect is then necessary to obtain
accurate predictions of the experimental outcome.

The imperfect experimental setup acts as an amplitude attenuation channel,
with transmission $\eta$, on the input quantum state. The effect of an
imperfect quantum efficiency is therefore indistinguishable from the
attenuation coming from the memory cell. From the point of view of the photon
number distribution this is stated by the composition property of the binomial
distribution characterizing the process:

$\displaystyle\sum_{m=n}^{N}B(m|N,\tau)B(n|m,\eta)$
$\displaystyle=\sum_{m=n}^{N}\genfrac{(}{)}{0.0pt}{0}{N}{m}\genfrac{(}{)}{0.0pt}{0}{m}{n}\tau^{m}(1-\tau)^{N-m}\eta^{n}(1-\eta)^{m-n}=$
$\displaystyle=\genfrac{(}{)}{0.0pt}{0}{n}{N}\sum_{m=n}^{N}\genfrac{(}{)}{0.0pt}{0}{N-n}{m-n}\tau^{m}(1-\tau)^{N-m}\eta^{n}(1-\eta)^{m-n}=$
$\displaystyle=\genfrac{(}{)}{0.0pt}{0}{n}{N}(\tau\eta)^{n}(1-\tau)^{N-n}\sum_{\alpha=0}^{N-n}\genfrac{(}{)}{0.0pt}{0}{N-n}{\alpha}\Big{(}\frac{\tau(1-\eta)}{1-\tau}\Big{)}^{\alpha}=$
$\displaystyle=\genfrac{(}{)}{0.0pt}{0}{n}{N}(\tau\eta)^{n}(1-\tau)^{N-n}\Big{(}1+\frac{\tau(1-\eta)}{1-\tau}\Big{)}^{N-n}=$
$\displaystyle=\genfrac{(}{)}{0.0pt}{0}{n}{N}(\tau\eta)^{n}(1-\tau\eta)^{N-n}=$
$\displaystyle=\sum_{m=n}^{N}B(m|N,\eta)B(n|m,\tau)=B(n|N,\tau\eta).$ (17)

In view of this property a generalization of the result of the previous
section to a _signal system_ with an efficiency $\eta_{s}<1$ is readily found
by performing the substitution:

$\tau_{\alpha}\rightarrow\eta_{s}\tau_{\alpha}.$ (18)

As said above, this is due to the indistinguishability of the two attenuation
processes, $\eta_{s}$ and $\tau_{\alpha}$, that the quantum state undergoes;
it shows that the discrimination of two quantum channels $\tau_{0}$ and
$\tau_{1}$ with a signal quantum efficiency $\eta_{s}$ is equivalent to the
discrimination between two channels $\eta_{s}\tau_{0}$ and $\eta_{s}\tau_{1}$
with perfect quantum efficiency. This argument does not depend on the
measurement performed, so it is valid also in the case of the classical
theoretical bound in Eq. (1), i.e. when computing it the effective channels to
be considered are $\eta_{s}\tau_{0}$ and $\eta_{s}\tau_{1}$.

##  3 EXPERIMENTAL REALIZATION

###  3.1 Experimental set-up

A scheme of the experimental set-up is reported in Fig. 2(b). Comparing it
with Fig. 2(a) it can be appreciated how the different elements involved in
the quantum reading model are experimentally implemented.

Figure 2: (a) Model of the reading of a classical digital memory (b)
Simplified schematic of the experimental setup. In the BBO crystal the multi-
mode squeezed vacuum state is generated. The probe (signal) beam passes
through the memory cell realized by an absorbing layer with two value of
transmittance, $\tau_{0}$ or $\tau_{1}$ and is addressed to a region of the
the CCD camera chip. The reference (idler) beam goes directly to an other
region the chip, without interacting with the cell. The integrated signals
from the two regions, $n_{S}$ and $N_{R}$ respectively, are recorded in each
run. BBO: Type-II-Beta-Barium-Borate non linear crystal. IF: interferential
filter ($800\pm 20$)nm. CCD: charge-coupled device camera.

We experimentally produce a multi-mode squeezed vacuum state of light
exploiting the spontaneous parametric down conversion process in a non linear
crystal. More specifically, we pump a $(1\mathrm{cm})^{3}$ type-II-Beta-
Barium-Borate crystal with a CW laser of $\lambda_{p}=405$nm and power of
$100$mW. The laser emission is triggered by a digital signal coming from the
detector, i.e. the laser emits only in correspondence of the acquisition time.
An interferential filter at $(800\pm 20)$nm performs a spectral selection of
the down-converted photons allowing only the photons around the degenerate
frequency ($\lambda_{d}=2\lambda_{p}=810$nm) to reach the detector. Using a
system of two lenses, the correlation in momentum of two down-converted
photons is mapped into spatial correlations at the detection plane. In
particular, the far-field plane of the emission, where spatial correlation
occurs, is firstly realized in the focal plane of the far-field first lens,
having focal length $f_{FF}=1$cm. Using a second system of lenses with
equivalent focal length of about $f_{IM}=1.6$cm, named as imaging lens in the
figure, the far-field plane is imaged to the detection plane, with a
magnification factor of $M=7.8$. The detector is a charge-coupled-device (CCD)
camera (Princeton Instrument Pixis 400BR Excelon), working in linear mode,
with high quantum efficiency (nominally $>95\%$ at $810$nm), $100\%$ fill
factor and low electronic noise. We work at the 100kHz digitization rate, to
keep the noise as low as possible. The physical pixels of the camera measure
$13\mu$m, nevertheless in this experiment, where spatial resolution is not
crucial, we group them by $12\times 12$ hardware binning, thus lowering the
acquisition time and the read-out noise.

Figure 3: Typical frame acquired at the CCD camera. $S_{S}$ and $S_{I}$
indicate the two detection areas considered: being symmetric respect to the
center of symmetry (CS), they present quantum photon number correlation. On
the contrary, $S_{S}$ and $S^{\prime}_{I}$ are only classically correlated.
The average number of counts per pixel in these regions is of the order of
$10^{4}$, integrating on the areas considered the values $n_{S}$ and $n_{I}$
are obtained and used in subsequent elaboration. $A_{SL}$ is the region used
to evaluate the straylight. The different dotted cicles indicate the different
wavelength of the photons in their correspondence. Due to the interferential
filter used, photon of $780\mathrm{nm}\leq\lambda\leq 810\mathrm{nm}$ are
detected. The degeneracy frequency is
$\lambda_{d}=2\lambda_{pump}=810\mathrm{nm}$.

In Fig. 3 a typical frame acquired by the CCD is reported: $S_{S}$ and
$S_{I}$, symmetric respect to the center of symmetry, indicate the two
detection areas considered, corresponding to the signal and idler beams
respectively. In our case these areas contain $\sim 50$ macro-pixels and
collect $M_{sp}=8\cdot 10^{2}$ spatial modes, where the linear size of a mode
has been evaluated by an intensity spatial correlation measurement to be of
about 5$\mu$m. For the further data elaboration we integrate the signals in
$S_{S}$ and $S_{I}$ and we name as $n_{S}$ and $n_{I}$ the number of photons
detected in the two regions respectively. To estimate the number of temporal
modes collected in a single frame we can compare its acquisition time, $\sim
20$ms, with the coherence time of the SPDC process, $10^{-12}$s. It results
$M_{t}=2\cdot 10^{10}$. The favourable condition of working with an high
number of modes $M=M_{sp}\cdot M_{t}\gg 1$ is thus satisfied. Considering that
we typically work with $\langle n_{S}\rangle\sim 3\cdot 10^{5}$, we can
conclude that the number of photons per spatio-temporal mode is
$\nu=\frac{\langle n_{S}\rangle}{M_{sp}\cdot M_{t}}\sim 2\cdot 10^{-8}\ll 1$.

To reproduce the quantum-reading scheme presented in Fig. 2(a) we insert in
the focal plane of the first lens a coated glass-slide, playing the role of
the digital memory support. It presents a deposition of variable transmission
$\tau_{0}$ ($0.990<\tau_{0}<1$), while outside the deposition the transmission
is $\tau_{1}=1$. The probe beam interception with the glass slide represents
the single cell memory. Its transmittance can assume two values, $\tau_{0}$ or
$\tau_{1}$, depending on the presence or absence of the deposition
respectively, and therefore can be used to store a bit of information. In
order to have the same optical path of the probe beam, also the reference beam
passes through the glass, but never intercepts the deposition.

The two beams of a two mode squeezed vacuum state present non-classical photon
number correlation given by the entangled nature of the state. This is
demonstrated from the squeezed shape of the joint distributions in Fig. 4 (a),
where it is reported $n_{I}$ as function of $n_{S}$, for 1000 frames. Blue
data correspond to $\tau_{0}=0.993$ while red data correspond to $\tau_{1}=1$.
For comparison, Fig. 4 (b) shows $n_{I}$ as function of $n_{S}$, for 1000
frames, in absence of quantum correlation (e.g. what we could obtain splitting
a classical beam with a 50% beam splitter). Since both signal and idler beams
are already shot noise limited from the practical viewpoint (more precisely
they follow a multi-thermal statistics with a number of photons per mode
$\nu\ll 1$, which approaches a Poisson’s distribution), further squeezing of
the distribution does not occur if quantum correlation are not present. To
experimentally reproduce this scenario with our set-up we consider two regions
on the detector, $S_{S}$ and $S^{\prime}_{I}$, not symmetric respect the
center of symmetry (see the dashed square in Fig. 3).

Figure 4: (a) $n_{I}$ in function of $n_{S}$, for 1000 frames. In this case
the regions $S_{I}$ and $S_{S}$, symmetric respect to the center of symmetry,
are considered. Blue data correspond to $\tau_{0}\sim 0.993$, while red data
corresponds to $\tau_{1}=1$. (b) $n^{\prime}_{I}$ in function of $n_{S}$, for
1000 frames. In this case the regions $S^{\prime}_{I}$ and $S_{S}$ are
considered. Blue dots correspond to $\tau_{0}\sim 0.993$, while red dots
corresponds to $\tau_{1}=1$.(c) $n_{S}$ relative frequency distribution for
$\tau_{0}\sim 0.993$ (blue histogram) and $\tau_{1}=1$ (red histogram).

In Fig. 4 (c) it is reported the frequency histogram of $n_{S}$, for
$\tau_{0}=0.993$ (blue data) and $\tau_{1}=1$ (red data). This is the
information we have access if we consider only the signal beam, which alone
does not present any quantum feature. While in Fig. 4 (a) the two
distributions, for $\tau_{0}$ and $\tau_{1}$ respectively, are well separated,
in Fig. 4 (c) they significantly overlap. Intuitively, this difference is at
the basis of the quantum enhancement offered by using the TMSV state rather
than a single classical beam.

We implement the readout memory strategy exploiting two mode squeezed vacuum
intensity correlations presented in Sec. 2.4 and we experimentally evaluate
its information gain respect to both the classical bound in Eq. (1), $G_{a}$,
and the classical single beam photon-counting strategy discussed in Sec. 2.3
($G_{emp}$). Different number of photons $N$ probing the memory cell and
different values of $\tau_{0}$ are considered. To change $\tau_{0}$ we simply
consider depositions of different absorption by moving the glass slide
appropriately. For changing the number of probing photons we simply change the
dimension of the $S_{S}$ and $S_{I}$ regions, avoiding the necessity of
acquiring new data with a different laser power. In a calibration phase, we
estimate all the parameters necessary in the subsequent analysis: $N$,
$\tau_{0}$, $\eta_{S}$, $\eta_{I}$, being $\eta_{S}$ and $\eta_{I}$ the
channel efficiency on the signal and idler beam respectively. The method used
to estimate these efficiencies exploits the peculiarity of the two-mode
squeezed vacuum state and is discussed in Sec. 3.1.1. The values of the
experimental parameters estimated in the calibration phase are used to compute
the theoretical classical bounds and to compute the probabilities
$P_{\eta_{S},\eta_{I}}(\textbf{n}|\tau)$ necessary for applying the Bayesian
decision strategy presented in Sec. 2.2

For the experimental estimation the error probability $P_{err}$ while using
the two mode squeezed vacuum state we proceed as follows: we consider two sets
of frames (in our case 10000 frames per set are acquired), one for
$\tau=\tau_{0}$ and one for $\tau=\tau_{1}$. For each frame, we estimate the
value of the memory cell according to the measured ($n_{S},n_{I}$) and the
strategy discussed in Sec. 2.4. Repeating this procedure for all the frames in
both sets and considering the ratio between the number of incorrect choices
and the total number of frames considered, we estimate the error probability
$P_{err}$. A similar approach can be adopted for evaluating the error
probability in the single beam classical strategy, simply considering only
$n_{S}$ and the strategy presented in Sec. 2.3.

From $P_{err}$ we can straightforward calculate the associated binary entropy
$H(P_{err})$ and the information gain of the two mode squeezed vacuum strategy
over both the optimal classical bound ($G_{a}$, see Eq. (2)) and the single-
beam classical approach ($G_{emp}$, see Eq. (11)).

The experimental results are reported in Fig. 5.

(a) $N=1.15\cdot 10^{5}$

(b) $N=3.1\cdot 10^{5}$

(c) $N=5.2\cdot 10^{5}$

Figure 5: Experimental gain $G$ of quantum reading (bits) as a function of the
lower transmissivity $\tau_{0}$. The three panels are obtained for different
mean photon number in the signal beam: (a) $N=1.15\cdot 10^{5}$, (b) $3.1\cdot
10^{5}$, and (c) $5.2\cdot 10^{5}$. Blue data refers to the gain with respect
to the classical optimal bound in Eq. (1). Red data refers to the gain with
respect to the classical photon-counting bound given in Eq. (10), obtained
from the marginal distribution of the signal. The error bars represent the
experimental uncertainty in the estimated gains at $1\sigma$. The mean
detection efficiency of signal and idler channels $\eta_{S}$ and $\eta_{I}$
are $\eta_{S}=0.78$ and $\eta_{I}=0.77$. The electronic noise is estimated to
be $\nu_{e}\sim 10^{4}$.

The three figures are obtained for a different number of photons in the signal
beam: $\langle N\rangle\sim 1.15\cdot 10^{5},3.1\cdot 10^{5}$ and $5.2\cdot
10^{5}$ respectively. Blue data represents $G_{a}$, while red data $G_{emp}$.
The error bars at $1\sigma$ are obtained dividing the data into 10 sets and
independently evaluating the gains in each set. The theoretical curves
obtained using the theoretical model discussed above are also reported.

We can conclude that a quantum advantage in the regime under consideration is
present. In particular, we experimentally demonstrate a maximum quantum gain
of larger than 0.1 bit per cell over the classical optimal bound, and of $\sim
0.3$ over the classical photon counting strategy. These results paves the way
for future experimental applications.

####  3.1.1 Efficiency estimation

Among other quantities, the estimation of the quantum efficiency is
particularly crucial. To estimate it, we use an extension of the Klishko
method [4], extensively discussed in Ref. [5]. The key idea of this method is
that the detected photon number correlation between signal and idler while
considering a two-mode squuezed vacuum state is not perfect because $\eta$ is
not unitary: from the degradation of the the photon number correlation we can
infer the efficiency of the sistem. In the following we summarize the
operative procedure used to estimate $\eta_{S}$ and $\eta_{I}$, i.e. the
quantum efficiency for the signal and idler modes respectively.

Said $S_{S}$ and $S_{I}$ two conjugated regions at the detector, in the idler
and signal beam respectevely, we introduce the quantity:

$\gamma=\frac{\langle n_{S}\rangle}{\langle
n_{I}\rangle}=\frac{\eta_{S}}{\eta_{I}},$ (19)

where $\langle n_{S}\rangle$ and $\langle n_{I}\rangle$ are the mean number of
photons detected in $S_{S}$ and $S_{I}$ respectively. This quantity, which
represents the unbalancement between the two channels, is proportional to the
ratio of the two efficiencies. $\gamma$ can be easily experimentally estimated
and allows balancing the photon counts of the two channels. In addition to
losses, there are two main other sources of noise:

  * •

Stray-light, $\langle N_{SL}\rangle$: it is due to fluorescence of the laser
pump in the BBO crystal and in the interference filter. It can be estimated
considering the mean number of photons detected per pixel in a ”dark” region,
as $A_{SL}$ in Fig. 3.

  * •

Electronic noise, $\Delta^{2}_{el}$: it can be estimated from frames acquired
with the shutter closed, where the only noise source is the electronic noise.

We can now define $\sigma$ which quantifies the degree of correlation between
signal and idler, taking into account the efficiency unbalancement and the
presence of noise:

$\sigma_{\gamma,B}=\frac{\Delta^{2}(n_{S}-\gamma n_{I})}{\langle n_{S}+\gamma
n_{I}\rangle}\cdot\frac{\langle n_{S}\rangle}{\langle
n_{S}-N_{SL}\rangle}-\frac{\Delta^{2}_{el}+\langle N_{SL}\rangle}{\langle
n_{S}-N_{SL}\rangle}.$ (20)

It turns out that, considering $S_{S}$ and $S_{I}$ sufficiently big respect to
the coherence area of the spontaneus parametric down conversion process,
$\sigma_{\gamma,B}$ can be written in terms of $\eta_{S}$ as:

$\sigma_{\gamma,B}=\frac{1+\gamma}{2}-\eta_{S}.$ (21)

Inverting the expression in Eq. (21) and using the definition of $\gamma$, the
two efficiencies can thus be estimated as:

$\eta_{S}=\frac{1+\gamma}{2}-\sigma_{\gamma,B},$ (22)
$\eta_{I}=\frac{\eta_{S}}{\gamma}.$ (23)

The uncertainty on this estimate can be experimentally evaluated repeating the
measurement several times and then considering the standard deviation on the
values obtained.

###### Acknowledgements.

This work has been sponsored by the EU Horizon 2020 FET-Open project: “Quantum
readout techniques and technologies” (QUARTET, Grant agreement No 862644).

## References

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  * [3] Ortolano, G., Losero, E., Ruo-Berchera, I., Pirandola, S., and Genovese, M., “Experimental quantum reading with photon counting,” Science Advances 7, 4, eabc7796 (2021).
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