# Optimal photon budget allocation in E91 protocol

Melis Pahalı<EMAIL_ADDRESS>Electrical and Electronics Engineering
Department, Özyeğin University, Istanbul 34794, Turkey  Kadir Durak
<EMAIL_ADDRESS>Electrical and Electronics Engineering Department,
Özyeğin University, Istanbul 34794, Turkey  Utku Tefek  u.tefek@adsc-
create.edu.sg Advanced Digital Sciences Center, Singapore 138602, Singapore

###### Abstract

In order for the deployment of quantum communication technologies in a global
scale, it is necessary to meet data exchange demand of various size
establishments per certain time intervals and it is important to make them
cost effective and feasible. In order to meet these requirements, a lot of
effort has been put into increasing key rate and having optimized systems. In
this study, we focus on the improvement of raw key rate in a standardized E91
QKD system without compromising its security. Our method is to optimize photon
budget allocation among three types of bits used in the system for different
purposes or occurred unavoidably. These three types of bits are key bits,
Bell’s inequality bits and discarded bits and their ratios were $25\%$, $50\%$
and $25\%$, respectively, in the proof-of-principle experiment. On the other
hand, we present $226.22\%$ increase in raw key rate with $83\%$, $10\%$ and
$7\%$ allocation of photon budget among three types of bits. These ratios are
achieved by replacing $50:50$ beam splitters with a $90:10$ beam splitter at
one communicating side and a $93:7$ beam splitter at the other communicating
side. Additionally, we demonstrate that the optimum beam splitting ratios can
vary depending on photon budget.

optimization, Ekert 1991, discrete-variable quantum key distribution.

††preprint: APS/123-QED

##  I INTRODUCTION

In quantum key distribution (QKD) literature, various methods were studied to
increase key rate. For example, encryption and decryption in various degrees
of freedoms [1] were put into use, brightness of entanglement source was
increased [2] via fidelity and repetition rate increase, error correction with
high speed [3] and varied schemes [4] were performed, high dimensional QKD
protocols [5, 6] were generated, alternative QKD schemes like twin-field QKD
[7, 8, 9, 10] were created and achievable distances were improved [11, 12,
13].

In discrete-variable QKD, there are two entanglement-based protocols E91 [14]
and BBM92 [15], and there are different protocols which use entangled photons
[16, 17]. The studies performed related to E91 are security analysis [18, 19],
examination of protocol under a certain noise model [19], employing
optoelectronic polarization rotators (OPRs) in the implementation of violation
of Bell’s inequality [20] or throughout the QKD system [21] as dynamic
components instead of employing beam splitters and wave plates [22] as the
conventional static components, usage of entanglement between different
degrees of freedoms (hybrid entanglement) [23], efficiency analysis and
improvement [24], feasibility study on satellite-based communication [25],
side channel analysis [26], and eavesdropping attack [27].

Here, we studied E91 protocol. We focused on the key rate after the steps of
sifting and parameter estimation, before the steps of error correction and
privacy amplification, which can also be called raw key rate, and aimed at
increasing it.

In the entanglement-based QKD modified protocol of Ekert which is studied in
Ref. [22], there are three types of bits whose ratios are determined according
to the beam splitting ratios of beam splitters at Alice’s and Bob’s sides. The
three types of bits are key bits, Bell’s inequality bits and discarded bits
whose ratios are $25\%$, $50\%$ and $25\%$, respectively. These ratios are not
optimum and our method of increasing the key rate is to optimize these ratios.

Photon budget defines the number of photons entering the detection module of a
communicating side in a certain time interval. While the photon budget can
take various values, a typical value is $10^{6}/$\mathrm{s}$$, as also assumed
and used in our study.

For this study, our motivation is to increase the ratio of key bits in order
to maximize key rate given photon budget without compromising the security of
key. This can be achieved by decreasing the ratios of discarded bits and
Bell’s inequality bits. We examined various beam splitting ratios of beam
splitters at Alice’s and Bob’s sides, and determined the optimum ratio of key
bits yielding the maximum key rate. However, this optimization study can also
be implemented to the QKD systems employing OPRs which are used for the same
function as beam splitters and wave plates. This would be done by determining
OPRs’ optic axes’ directions and the speed of change of the optic axes.
Furthermore, this optimization study can also be implemented to the QKD
systems which would use metamaterials having quantum phase transition
materials for polarization change to be used in faster cryptography. The
security of raw key relies on quantum bit error rate and Bell’s inequality
parameter. Quantum bit error rate will be a threshold determining the
continuation or termination of the communication system. Bell’s inequality
parameter which depends on the amount of Bell’s inequality bits will quantify
how secure the system would be.

Organization of the paper is the following. In Sec. II, we investigated the
uncertainty of the violation of Bell’s inequality dependence to the number of
photons used in the calculation of it and to the beam splitting ratios. In
Sec. III, we examined the effect of dead time of a detector in photon counting
and examined the concept on which photon budget allocation in the E91 protocol
[22] is based. In Sec. IV, we introduced key rate expression that we use in
our study and introduced the optimized photon budget. In Sec. V, we presented
the effects of photon detection efficiency and dark count of a detector on
photon counting, explicitly. Lastly, in Sec. VI, we summarize the topic by
giving our results and mention the importance of the study.

##  II Dependencies of UNCERTAINTY OF VIOLATION OF BELL’S INEQUALITY

In the process of obtaining a final raw key to be used in the encryption and
decryption, key bits and Bell’s inequality bits are collected for a period of
time. At the end of each period, the security of key bits is quantified with a
parameter, which is generated by using Bell’s inequality bits. If this
parameter, so called Bell’s inequality parameter, is below a threshold, that
block of key bits cannot contribute to final raw key and is discarded. Bell’s
inequality parameter is calculated frequently within the process of key
distribution until final raw key could be obtained. Whenever the parameter is
above the threshold, the related block of key bits can be used to contribute
final raw key. In this way, the blocks of key bits are collected and final raw
key is obtained.

The Bell’s inequality parameter we used throughout the chapter is given in Eq.
(1).

$S=E(a_{0},b_{0})+E(a_{0},b_{1})-E(a_{1},b_{0})+E(a_{1},b_{1})$ (1)

where $a_{0}$($b_{0}$) and $a_{1}$($b_{1}$) are two sets of detectors located
at Alice’s (Bob’s) detection module. Each set of detectors consists of two
detectors. In other words, each of $a_{0}$($b_{0}$) and $a_{1}$($b_{1}$) can
be called a basis set and consists of two orthogonal basis. Thus each detector
is related to a basis and is dedicated to measure a polarization direction
which is shown in Fig. 1a [22]. $E(a_{i},b_{i})$ is a correlation coefficient
related to measurement results obtained from basis sets $a_{i}$ and $b_{i}$.
Lastly, $S$ is the combined correlation coefficient used to quantify the
quantumness a QKD system has. While the Bell’s inequality satisfies $|S|\leq
2$, which we call classical region, its violation is realized when $2<|S|\leq
2\sqrt{2}$.

In a pair of measurement among Alice and Bob, the coincidence of two of the
basis in Fig. 1a (a) results in a Bell’s inequality bit being obtained in the
QKD system. The reflected outputs of B1 and B3 in Fig. 1a (b) represent the
same basis set, thus the coincidence of these basis sets results in a key bit
being obtained in the system. Lastly, the coincidence of a measurement at the
reflected output of B1 and a measurement at the basis set labeled with
$\\{3^{\prime},4^{\prime}\\}$ results in a discarded bit being obtained in the
system. Namely, the type of a bit is determined by which basis sets coincide.

(

(a)

(

(b)  Figure 1: (a) Measurement basis sets used in the calculation of the
amount of violation of Bell’s inequality at Alice’s and Bob’s sides. The four
basis sets and their components are as follows: $a_{0}\equiv\\{3,4\\}$,
$a_{1}\equiv\\{5,6\\}$, $b_{0}\equiv\\{1^{\prime},2^{\prime}\\}$,
$b_{1}\equiv\\{3^{\prime},4^{\prime}\\}$. (b) B1, B2, B3: Beam splitters. PBS:
Polarizing beam splitter. WP: Wave plate. The numbers indicate detectors, in
other words, the basis sets’ components. Figure on the left and right
represent the detection modules at Alice’s and Bob’s sides, respectively.
Reflected outputs of B1 and B3 constitute key bit detection parts of the
detection modules.

a) $5($$)$$3($$)$$4($$)$$6($$)$Alice
$3^{\prime}($$)$$4^{\prime}($$)$$2^{\prime}(H)$$1^{\prime}(V)$Bob b)

The total number of photons which is used in the calculation of $S$ is
allocated to the components of $S$, $n_{kl}$. The mathematical expression of
it is given in Eq. (2).

$\eta=\sum_{k}\sum_{l}n_{kl}$ (2)

where $\eta$ is the total number of photons used in the calculation of $S$,
$n_{kl}$ is the components of $S$ with $k\equiv\\{3,4,5,6\\}$ and
$l\equiv\\{1^{\prime},2^{\prime},3^{\prime},4^{\prime}\\}$ shown in Fig. 1a
(b). $n_{kl}$ can be written in terms of two independent parameters, which are
$\eta$ and the reflectance of the beam splitter B3 in Fig. 1a (b). Thus, it
can be interpreted as $S$ is a function of independent and random variables.
Propagation of uncertainty can be applied to $S$ and the uncertainty of $S$
can be expressed as given in Eq. (3) [28, 29].

$\delta S=\sqrt{\sum_{k}\sum_{l}\Big{(}\frac{\partial S}{\partial
n_{k,l}}\delta n_{k,l}\Big{)}^{2}}$ (3)

where $\delta$ stands for uncertainty. The definition of uncertainty of
coincidence counts is given in Eq. (4). Uncertainty of $S$, $\delta S$, can
give an idea about how reliable an $S$ would be relatively. Lastly, $\partial$
stands for partial derivative.

$\delta n_{k,l}=\sqrt{n_{k,l}}$ (4)

A mixed state consisting of singlet state
$\ket{\psi^{-}}=\frac{1}{\sqrt{2}}(\ket{01}-\ket{10})$ having visibility
$0.95$ and white noise, which is also called Werner state, was used in the
quantum channel. For the mixed state $\rho$, the uncertainty of $S$ depending
on the various numbers of photons and various reflectance of beam splitter B3
was computed and shown in Fig. 2.

Figure 2: The graph of the uncertainty of $S$ dependence to the number of
photons and the reflectance of beam splitter B3. Color bar represents the
uncertainty of $S$.

We considered different numbers of photons as independent experiments and
modeled the probability distribution of $S$ in each case with a Gaussian
distribution approximation whose standard deviation is the uncertainty of $S$
in Fig. 2. We called the region $|S|\leq 2$ as classical region, computed the
areas under the distributions which fall into the classical region and showed
the dependence of the probability to the number of photons and reflectance of
beams splitter B3 in Fig.3.

Figure 3: The graph of the dependence of the probability to the number of
photons and the reflectance of beam splitter B3. Color bar represents the
probability of obtaining a value from the classical region for $S$.

The probability shows the percentage of all the blocks of key bits collected
in a key distribution process which cannot contribute to the final raw key. In
other words, it represents the ratio of key rate which will be discarded in
unit time interval. Even if the quantum channel is secure, communicating sides
can obtain an $S$ value from the classical region at that probability. We will
call the key rate remains after the decrease at that probability as average
key rate in the rest of the paper.

##  III DEAD TIME EFFECT IN PHOTON COUNTING

Dead time is the bottleneck which limits photon counting in a detector. The
number of photons observed at a detector may differ from the number of photons
reaching the detector. The relation between them is given in Eq. (5).

$N=\frac{n}{n\cdot t\cdot 10^{-12}+1}$ (5)

where $n$ is the number of photons reaching detector per second, $t$ is the
dead time of detector in terms of $\mathrm{ps}$ and takes value within the
range of $1$\mathrm{ps}$\leq t\leq 10^{12}$\mathrm{ps}$$, and $N$ is the
number of photons observed at detector per second. Dependence of $N$ to $t$ is
shown in Fig. 4. Dead time changes within the range of
$t=[1$\mathrm{ns}$,10$\mathrm{\SIUnitSymbolMicro s}$]$ because of presenting
the most common values encountered in active and passive quenching detectors.

Figure 4: The graph of the number of photons observed at a detector per second
dependence to the dead time of detector. The legend represents $n$, namely,
the number of photons reaching detector per second. The curves from bottom to
top correspond to $n$ from $1e+06/$\mathrm{s}$$ to $1e+12/$\mathrm{s}$$.

There are three types of $n$ with which we are dealing. Photon budget is
allocated among three types of $n$. They are the numbers of photons reaching
detectors or detectors’ groups which are dedicated to obtain three types of
bits. The numbers of photons reaching these detectors are determined by the
ratios of three types of bits. $N_{key}$ will the number of photons observed
per second and will be used in the calculation of key rate. In the rest of the
paper, dead time was be taken as $1$\mathrm{ps}$$. Given the dead time, we
continued our study with the numbers of photons observed in three types of
bits.

##  IV AVERAGE KEY RATE

The key rate formula we used is that of device-independent QKD which is based
on collective attacks scenario [30]. It is the first multiplier in Eq. (6)
whose unit is bits/symbol.

$\Bigg{(}1-h(Q)-h\Big{(}\frac{1+\sqrt{((S+\delta
S)/2)^{2}-1}}{2}\Big{)}\Bigg{)}\cdot N_{key}\cdot(1-P)$ (6)

where $h(Q)$ is the binary entropy of quantum bit error rate $Q$,
$h\Big{(}\frac{1+\sqrt{((S+\delta S)/2)^{2}-1}}{2}\Big{)}$ is the accessible
information about the quantum channel by an eavesdropper where $S$ is the mean
and $\delta S$ is the uncertainty of Bell’s inequality parameter. In the
comparison of different distributions of $S$ in the experiments using
different total number of photons, the range of $\mu\pm\sigma$, where $\mu$ is
the mean and $\sigma$ is the standard deviation of the distribution, can be
used. Thus from the range of $S\pm\delta S$, we chose $S+\delta S$ because of
being closer to the classical region to distinguish the effect of different
distributions of $S$ as the worst case scenario that can be encountered in an
experiment. $N_{key}$ is as defined in Sec. III and is the conversion
parameter from bits/symbol to bps which would be the unit of the key rate used
in the rest of the paper since each symbol or photon carries one bit of
information. $P$ is the probability of obtaining an $S$ from the classical
region and with the use of it the key rate formula turns into average key
rate.

The average key rates that can be obtained depending on the beam splitting
ratios, and the maximized key rate can be seen in Fig. 5. While average key
rate is $0.1331$Mbps in the standard configuration which is the employment of
$50:50$ beam splitters [22] and is shown as a red circle in the figure, it
becomes $0.4342$Mbps in the employment of $90:10$ beam splitter at Alice’s
side and $97:3$ beam splitter at Bob’s side and shown as a blue circle in the
figure. Hence our method presents an increase in the average key rate up to
$226.22\%$. Also, while the ratios of key bits, Bell’s inequality bits and
discarded bits are $25\%$, $50\%$ and $25\%$ in the standard configuration,
their ratios in the maximized key rate configuration is $87\%$, $10\%$ and
$3\%$. Additionally, the number of photons used in the calculation of $S$
becomes $10^{5}/$\mathrm{s}$$ with the $10\%$ allocation. $S$ with its
uncertainty corresponding to that much of photons is $-2.6870\pm 0.0255$.
Bell’s inequality is violated with $26.95$ standard deviations (SD), it can be
seen in Fig. 6. In the literature, there are various numbers of standard
deviations in the violations of Bell’s inequality, such as $6$SD, $12$SD [31],
$8.5$SD, $9.3$SD [32], $10$SD [33, 34], $24$SD [35, 34], $203$SD [36], and
there is no threshold other than the trivial conclusion that the number of
standard deviations $>0$ must be satisfied. Thus $26.95$SD is within the
acceptable range of the numbers of standard deviations of the violation, and
$10\%$ allocation instead of $50\%$ to the Bell’s inequality bits does not
expose a compromise in the security of the QKD system which is based on $S$.

Figure 5: The graph of the average key rate dependence to the beam splitting
ratio for key at Bob’s side, namely, dependence to the beam splitting ratio at
the reflected output of B3 in Fig. 1a (b). The legend represents the beam
splitting ratio for key at Alice’s side, namely, the beam splitting ratio at
the reflected output of B1 in Fig. 1a (b). The curves from bottom to top
correspond to the beam splitting ratio for key at Alice’s side from $0.1$ to
$0.9$. Figure 6: The graph of the number of standard deviations of the
violation of Bell’s inequality dependence to the beam splitting ratios for
key. The legend represents the beam splitting ratio for key at Alice’s side.

Incremental steps of the beam splitting ratio at Alice’s side is $0.1$ and at
Bob’s side is $0.001$. More precise optimum values can be obtained by reducing
the incremental steps of the beam splitting ratios in the simulations for the
QKD systems which enable more sensitive implementations. However, the change
in the beam splitting ratios from $50:50$ to roughly $90:10$ would give a
considerable increase in the key rate as shown in this study.

Fig. 7 is plotted to show at which beam splitting ratios a maximized average
key rate can be achieved. In this figure, the beam splitting ratios at both
sides are chosen as the same as indicated in the label of x-axis. Photon
budget changes within the range of $[10^{3},10^{9}]/$\mathrm{s}$$. Color bar
shows normalized average key rate in which average key rates at various beam
splitting ratios for a photon budget is normalized according to the maximized
average key rate. After $n=10^{6}/$\mathrm{s}$$, maximized average key rate
seems to happen when beam splitting ratios for key $=0.9$. It is suggested to
look at average key rate change in a graph of the type of Fig. 5. If the
global maximum average key rate cannot be seen in such a graph, the conclusion
that this experimental setup for higher photon budgets may not be effective
can be inferred.

Figure 7: Normalized average key rate dependence to photon budget and beam
splitting ratios for key at both sides.

##  V Photon Detection Efficiency and Dark Count Effects in Photon Counting

Besides dead time, photon detection efficiency (PDE) and dark count (DC) also
take part in the relation between the numbers of photons reaching a detector
and observed at the detector per second. They can be incorporated as given in
Eq.(7).

$N=\frac{n\cdot PDE+DC}{n\cdot PDE\cdot t\cdot 10^{-12}+1}$ (7)

where $n$, $t$ and $N$ are as defined in Eq. 5. In this study, PDE and CD are
taken as $1$ and $0/$\mathrm{s}$$, respectively, for the simplification. The
effect of PDE can be seen in the left hand side of Fig. 8a, which is a
vertical shift in the linear region. And the effect of DC can be seen in the
right hand side of Fig. 8a, which shows itself as noise while working with
relatively lower number of photons. And it shows the performance of a detector
while working with relatively low number of photons. In this figure, dead time
was taken as $22$\mathrm{ns}$$ for all the curves and substituted into Eq.7 in
terms of $\mathrm{ps}$ as compatible with Eq.5. Alternative to Eq.(7), a
formula which characterizes a detector should be used in a specific
implementation in order to obtain average key rates achievable.

(

(a)

(

(b)  Figure 8: The graphs of the number of photons observed vs the number of
photons reaching detector. The effect of photon detection efficiency and dark
count to the number of photons observed.

a) b)

##  VI CONCLUSION

In summary, we focused on a standardized E91 QKD protocol, in which $50:50$
beam splitters are employed at both Alice’s and Bob’s sides. Depending on the
positioning of measurement basis sets with respect to the beam splitters,
there was the allocation of $25\%$, $50\%$ and $25\%$ of a photon budget among
three types of bits, which are key bits, Bell’s inequality bits and discarded
bits. We calculated raw key rate for the $50:50$ beam splitters and for the
standard allocation as a reference point. Then, we examined the value of the
key rate for various beam splitting ratios and for various allocations of the
photon budget. We searched the optimum beam splitting ratios and allocations
yielding the global maximum key rate, and showed that $226.22\%$ increase in
key rate can be achievable with $90:10$ and $97:3$ beam splitting ratios, and
with the allocation of $87\%$, $10\%$ and $3\%$. Additionally, we showed that
optimum beam splitting ratios can vary for different photon budgets in a
constant configuration of experimental setup by defining a parameter called
normalized average key rate and by using it. An E91 QKD protocol may have
various configurations such as not including all the basis sets used in this
study or establishing symmetrical detection modules in terms of basis sets at
both communicating sides or the realization of basis sets by using OPRs
instead of beam splitters and wave plates. The optimization of photon budget
allocation should be applied to all the experimental setups of the similar
kind for the best use of resources and for the maximization of key rate.

## ACKNOWLEDGMENT

This study was supported by the Scientific and Technological Research Council
of Turkey (TÜBİTAK) Academic Research Funding Programs Directorate (ARDEB),
Primary Subjects R&D Funding Program (Program No. 1003), Project No. 118E991.

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