Measurement of time-dependent $\mathcal{CP}$ violation in charmless $B$ decays
at LHCb

Stefano Perazzini on behalf of the LHCb Collaboration
Istituto Nazionale di Fisica Nucleare (INFN)
Sezione di Bologna
Via Irnerio 46, 40126 Bologna, Italy

##  1 Abstract

In the following we present the measurements of time-dependent $\mathcal{CP}$
violation in charmless $B$ meson decays performed by LHCb analyzing the $p-p$
collision data collected at a center-of-mass energy of 7 TeV during the 2010
and 2011 LHC runs. In particular we will focus on the analysis of charmless
two-body $B$ decays where the direct and mixing-induced $CP$ asymmetry terms
of the $B^{0}\to\pi^{+}\pi^{-}$ and $B_{s}^{0}\to K^{+}K^{-}$ decays have been
measured using 0.69 fb-1 of data collected during 2011. The measurement of the
branching ratio of the $B_{s}^{0}\to K^{*0}\overline{K}^{*0}$ decay, using 35
pb-1 collected during 2010, is also reported. In the end we show the relative
branching ratios of all the decay modes of $B_{\left(s\right)}^{0}\to
K_{S}h^{+}h^{\prime-}$ decays (where $h^{\left(\prime\right)}=\pi,K$),
measured analyzing 1 fb-1 of data collected during 2011.

##  2 Introduction

In this paper we report measurements of time-dependent $\mathcal{CP}$
violation in charmless $B$ meson decays performed at LHCb [1]. We will also
briefly discuss the LHCb potential with other measurements in this sector. All
the results here shown are obtained analyzing the $p-p$ collision data
collected at a center-of-mass energy of 7 TeV during the 2010 and 2011 LHC
runs. The time-dependent $\mathcal{CP}$ asymmetry of either a $B^{0}$ or a
$B_{s}^{0}$ meson decaying into a $\mathcal{CP}$-eigenstate $f$ can be written
as:

$A_{\mathcal{CP}}\left(t\right)=\frac{\Gamma_{\overline{B}\to
f}\left(t\right)-\Gamma_{B\to f}\left(t\right)}{\Gamma_{\overline{B}\to
f}\left(t\right)+\Gamma_{B\to f}\left(t\right)}=\frac{A_{dir}\cos{\left(\Delta
mt\right)}+A_{mix}\sin{\left(\Delta
mt\right)}}{\cosh{\left(\frac{\Delta\Gamma}{2}t\right)}+A_{\Delta}\sinh{\left(\frac{\Delta\Gamma}{2}t\right)}},$
(1)

where $\Gamma\left(t\right)$ represents the time dependent decay rate of the
initial $B$ or $\overline{B}$ meson to the final state $f$, $\Delta m$ and
$\Delta\Gamma$ are the $B$ meson oscillation frequency and decay width
difference respectively, and where the relation
$A_{dir}^{2}+A_{mix}^{2}+A_{\Delta}^{2}=1$ holds. Within this
parameterization, $A_{dir}$ and $A_{mix}$ account for $\mathcal{CP}$ violation
in the decay and in the interference between mixing and decay, respectively.
Since the decay amplitudes of charmless $B$ decays receive important
contribution from penguin diagrams, such a class of decays represents a
powerful tool in the search for physics beyond the Standard Model. In fact new
particles may appear as virtual contributions inside the loop diagrams,
altering the Standard Model expectation for $A_{dir}$ and $A_{mix}$.

The LHCb experiment has a great potential in this sector thanks to its
capabilities of efficiently triggering and reconstructing charmless $B$
decays, to its excellent decay-time resolution allowing to follow the fast
$B_{s}^{0}$ oscillations and to the large cross section for $B$ hadron
production at LHC.

##  3 $B^{0}\to\pi^{+}\pi^{-}$ and $B_{s}^{0}\to K^{+}K^{-}$

The $B^{0}\to\pi^{+}\pi^{-}$ and $B_{s}^{0}\to K^{+}K^{-}$ decay amplitudes
receive contributions from tree, electroweak penguin, strong penguin and
annihilation topologies. The presence of penguin diagrams makes these decays
sensitive to New Physics, but with the drawback of theoretical uncertainties
in the determination of relevant hadronic parameters. In Refs. [3] strategies
are presented in order to use the $U$-spin symmetry (i.e. the invariance of
strong interaction dynamics under the quark exchange $d\leftrightarrow s$)
relating the $B^{0}\to\pi^{+}\pi^{-}$ and $B_{s}^{0}\to K^{+}K^{-}$ decays, in
order to constraint hadronic parameters and determine the CKM phase $\gamma$
and the $B_{s}^{0}$ mixing phase $\phi_{s}$.

Crucial aspects for measuring time dependent $\mathcal{CP}$ asymmetries in
$B^{0}\to\pi^{+}\pi^{-}$ and $B_{s}^{0}\to K^{+}K^{-}$ decays are the event
selection, the calibration of the particle identification (PID) and the
flavour tagging. These decays are mainly selected by the hadronic trigger of
LHCb, that exploits high transverse momentum and large impact parameter with
respect to primary vertices, typical of $B$ hadron decay products. Then the
sample is further refined applying a set of kinematic and topological
criteria. Finally the 8 final states ($\pi^{+}\pi^{-}$, $K^{+}K^{-}$,
$K^{+}\pi^{-}$, $\pi^{+}K^{-}$, $p\pi^{-}$, $\overline{p}\pi^{+}$, $pK^{-}$
and $\overline{p}K^{+}$) are separated into exclusive subsamples by means of
the PID information provided by the two ring-imaging Cherenkov detectors of
LHCb. The calibration of the PID observables is performed using large samples
of pions, kaons and protons selected from $D^{*+}\rightarrow
D^{0}(K^{-}\pi^{+})\pi^{+}$ and $\Lambda\rightarrow p\pi^{-}$ decays (and
their charge conjugates). As a demonstration of its PID capabilities, using
$0.37$ fb-1 of integrated luminosity collected during 2011, LHCb measured the
relative branching ratios of various two-body charmless $B$ decays [5]. The
measurements of the branching ratios of the $B_{s}^{0}\to K^{+}K^{-}$ and
$B_{s}^{0}\to\pi^{+}K^{-}$ decays are the most precise to date. In addition,
the $B_{s}^{0}\to\pi^{+}\pi^{-}$ decay is observed for the first time ever
with a significance of more than 5$\sigma$.

The determination of the initial flavour of the signal $B$ meson (the so-
called ”flavour tagging”) is obtained using a multivariate algorithm that
analyzes the decay products of the other $B$ hadron in the event [6]. The
response of the algorithm is calibrated by measuring the oscillation of the
flavour specific decay $B^{0}\rightarrow K^{+}\pi^{-}$, in which the amplitude
is related to the effective mistag rate. Using the measured mistag rate as an
external input to a two dimensional (invariant mass and decay time) maximum
likelihood fit of the $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ spectra, the direct
and mixing-induced $CP$ asymmetry terms $A_{\pi\pi}^{dir}$,
$A_{\pi\pi}^{mix}$, $A_{KK}^{dir}$ and $A_{KK}^{mix}$ have been measured. The
results, obtained using 0.69 fb-1 of data collected during 2011, are [7]:

$\displaystyle A_{\pi\pi}^{dir}=0.11\pm 0.21\mathrm{(stat.)}\pm
0.03\mathrm{(syst.)}$ (2) $\displaystyle A_{\pi\pi}^{mix}=-0.56\pm
0.17\mathrm{(stat.)}\pm 0.03\mathrm{(syst.)}$ (3) $\displaystyle
A_{KK}^{dir}=0.02\pm 0.18\mathrm{(stat.)}\pm 0.04\mathrm{(syst.)}$ (4)
$\displaystyle A_{KK}^{mix}=0.17\pm 0.18\mathrm{(stat.)}\pm
0.05\mathrm{(syst.)},$ (5)

with correlations $\rho\left(A_{\pi\pi}^{dir},A_{\pi\pi}^{mix}\right)=-0.34$
and $\rho\left(A_{KK}^{dir},A_{KK}^{mix}\right)=-0.10$. $A_{\pi\pi}^{dir}$ and
$A_{\pi\pi}^{mix}$ are measured for the first time at a hadronic machine and
are compatible with the previous results from $B$-Factories [8]. The time-
dependent $\mathcal{CP}$ asymmetry terms $A_{KK}^{dir}$ and $A_{KK}^{mix}$ are
measured for the first time ever and are compatible with theoretical
expectations [4].

##  4 Other charmless decays

Using other charmless $B$ decays where time-dependent $\mathcal{CP}$
asymmetries can be measured, LHCb already produced results. Using just 35 pb-1
collected during 2010, LHCb observed for the first time the $B_{s}^{0}\to
K^{*0}\overline{K}^{*0}$ decay [9]. This decay is governed by pure penguin
topologies offering an interesting potential in the quest for New Physics. It
has been pointed out [10] that the time dependent $\mathcal{CP}$ asymmetries
in this decay can be used to measure the CKM phase $\gamma$ and the
$B_{s}^{0}$ mixing phase $\phi_{s}$ owing to the information provided by the
$U$-spin related decay $B^{0}\to K^{*0}\overline{K}^{*0}$. The signal yield
obtained from the fit to the invariant mass distribution is equal to $49.8\pm
7.5$ candidates and the corresponding branching ratio is
$\mathcal{BR}\left(B_{s}^{0}\to K^{*0}\overline{K}^{*0}\right)=\left(2.81\pm
0.46\mathrm{(stat.)}\pm 0.45\mathrm{(syst.)}\pm
0.34\left(f_{s}/f_{d}\right)\right)\times 10^{-5}$, where the last error comes
from the knowledge of the $b$-quark hadronization probabilities.

Decay | $\mathcal{BR}\times 10^{-6}$
---|---
$\mathcal{BR}\left(B^{0}\to K_{S}K^{\pm}\pi^{\mp}\right)$ | $5.8\pm 0.9\pm 0.9\pm 0.2$
$\mathcal{BR}\left(B^{0}\to K_{S}K^{+}K^{-}\right)$ | $26.3\pm 2.0\pm 2.0\pm 1.1$
$\mathcal{BR}\left(B_{s}^{0}\to K_{S}\pi^{+}\pi^{-}\right)$ | $11.9\pm 3.0\pm 2.0\pm 0.5$
$\mathcal{BR}\left(B_{s}^{0}\to K_{S}K^{\pm}\pi^{\mp}\right)$ | $97\pm 7\pm 10\pm 4$
$\mathcal{BR}\left(B_{s}^{0}\to K_{S}K^{+}K^{-}\right)$ | $4.2\pm 1.5\pm 0.9\pm 0.2$
Table 1: Branching ratios of the $B_{\left(s\right)}^{0}\to
K_{S}h^{+}h^{\prime-}$ decays measured by LHCb. The first error is
statistical, the second is systematic and the third is due to the uncertainty
on $\mathcal{BR}\left(B^{0}\to K_{s}\pi^{+}\pi^{-}\right)$ that is used as a
normalization.

Another important result is the measurement of the relative branching ratios
of all the decay modes of $B_{\left(s\right)}^{0}\to K_{S}h^{+}h^{\prime-}$
decays (where $h^{\left(\prime\right)}=\pi,K$) [11]. The dominant topologies
governing these decays are $b\rightarrow q\overline{q}s$ loop transitions
(with $q=u,d,s$), where new particles in several extensions of the Standard
Model may appear as virtual contributions. Several strategies have been
proposed in order to extract information on the CKM phase $\gamma$ and also on
the $B^{0}$ and $B_{s}^{0}$ mixing phases $\phi_{d}$ and $\phi_{s}$ [12] from
the time-dependent analysis of these decays in the Dalitz plane. As a first
step, using the full 1 fb-1 collected during the 2011, LHCb measured the
relative branching ratios of all the $B_{\left(s\right)}^{0}\to
K_{S}h^{+}h^{\prime-}$ modes with respect to the well established
$\mathcal{BR}\left(B^{0}\to K_{s}\pi^{+}\pi^{-}\right)$. The results are
reported in Tab. 1 where the first error is statistical, the second is
systematic and the third is due to the uncertainty on
$\mathcal{BR}\left(B^{0}\to K_{s}\pi^{+}\pi^{-}\right)$. First evidence of the
two decays $B_{s}^{0}\to K_{S}\pi^{+}\pi^{-}$ and $B_{s}^{0}\to
K_{S}K^{+}K^{-}$ is obtained with a significance of 4.3$\sigma$ and
3.3$\sigma$ respectively. Furthermore, LHCb observed for the first time the
$B_{s}^{0}\to K_{S}K^{\pm}\pi^{\mp}$ decay.

## References

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# Kinship and antagonism of chains in entangled biphoton controlled synthesis

Y.I. Ozhigov1,2, I.R. Pluzhnikov1
1.Lomonosov Moscow State University,
faculty of computational mathematics and Cybernetics,
2\. K. A. Valiev Russian Academy of Sciences, Moscow, Russia

###### Abstract

The synthesis of 3 and 4 abstract polymer chains divided into two sexes is
considered, where the degree of kinship of the chains is determined by their
overlap. It is shown that the use of some types of entangled bi-photon in one-
way control gives a difference in the degree of kinship between the legal and
nonlegal pairs that is unattainable with classical control. This example
demonstrates the quantum superiority in distributed computing, coming from the
violation of the Bell inequality. It may be of interest for revealing the
quantum mechanisms of synthesis of real biopolymers with directional
properties.

The synthesis of polymer chains with special bonding properties between them
is an urgent task that finds numerous applications, because it determines the
property of material made of such polymers . The gluing of polymers can have,
for example, the form of photocrosslinking - such polymers are used in
biological applications (see [1]); the scheme of this process in simplified
form is represented at the picture 1. Random arrays of peptides is used in
testing vaccines; the immunosignature effect withe them is considered in [2].
The task of synthesis biopolymers with anti bacterial properties is actual
since the past century (from the plenty of recent results see, for example,
[3]). Assembly is often carried out from a pair of types of monomers - a
typical example is styrene-butadiene rubber, the polymer of which is created
from two types of monomers (see figure2).

##  1 Introduction

Figure 1: The simplest form of gluing: fibrin gel preparation and drug/growth
factor entrapment. Picture adapted from [1]

.

However, the properties of polymer chain arrays depend on the type of their
interaction, which in turn, is determined at the stage of their synthesis - as
the subtle properties of the connection of monoblocks that determine their
spatial location. In the simplest case, the interaction can be represented as
the gluing of polymers in separate sections. In this article, we will consider
the quantum control of the synthesis of abstract polymer chains from two
monomeric links. This control, carried out through the entangled state of a
pair of photons, allows us to achieve a degree of differentiation of the
degree of gluing of different chains, which is unattainable with classical
control.

Figure 2: Styrene - butadiene polymer

.

The possibility of such a subtle distinction between the structure of polymer
chains of the same nature is based on the property of quantum non locality of
photons that control the synthesis of chains, namely, a violation of the Bell
inequality for bi photons in the EPR state.

The quantum non locality resulting from the experimentally confirmed violation
of Bell’s inequality is the most striking manifestation of the counter-
intuitive nature of the microcosm. An experiment invented by Bell in the 60s
of the twentieth century and implemented 20 years later ([4],[5]), showed the
fundamental nature of quantum theory and the priority of its spirit over the
classical understanding of nature. Discussions about the nature of quantum non
locality have been going on for more than half a century ([6]), approaches to
understanding this nature are also offered through various information
exchange protocols, for example, through non-local games ([7]). Photon
entanglement experiments involve obtaining multidimensional entanglement from
the angular momentum modes of bi-photon pairs ([8], as well as four photons
([9]).

Non-locality formally follows from the existence of entangled multiparticle
states, the simplest of which is the two-qubit state
$\frac{1}{\sqrt{2}}(|01\rangle\pm|10\rangle)$, where the $+$ sign gives a
triplet $|t\rangle$, and the $-$ singlet $|s\rangle$. Both of these states
have a maximum degree of entanglement, which is defined as entropy
$tr(\rho_{1}\ ln(\rho_{1}))$ the relative density matrix of the first qubit.
Entanglement also underlies the so-called quantum acceleration of classical
computing ([10]). However, if the superiority of a scalable quantum computer
requires the full-scale operation of an ensemble of hundreds of qubits, then
on more limited models of computing this superiority can be demonstrated with
a few qubits, which lies within the current capabilities of experimental
physics. This paper is devoted to such a limited model-distributed computing
with one-way control.

Finding out the role of quantum non locality in the management of distributed
computing will allow us to better understand the nature of this phenomenon,
but also promises to give a practical effect, allowing us to find more
effective forms of such control. For bi-photon control, an example of such
phenomenon is given in the paper [11]. In this paper, we will strengthen this
result by showing that bi-photonic control can also give the effect of
differentiation between legal and nonlegal pairing of the chains assembled in
the different locations.

The criterion of quantum superiority over classical calculations in
distributed computing is given not in terms of the calculation time, but in
terms of the quality of the calculation result. The quality is detected after
comparing the results in various remote nodes of the computing system, which
requires communication between these nodes at light speed, so such protocols
cannot exceed the light threshold for transmitting the information intended by
the user. Nevertheless, it is precisely the non-locality that creates the
qualitative superiority of the quantum model over classical distributed
computing. It is interesting that the use of photonic quartets does not give
advantage over the effect of bi-photons for this task.

The differentiation of legal and nonlegal pairs of chains may play a role in
the synthesis of bio polymers in living organism, namely for the immune
response to the antigen.

In conclusion, we will give an interpretation of quantum non locality in the
form of pre-programmed propagation of the wave function of a multiphoton
system, which can intuitively clarify the mechanism of action of quantum non
locality in distributed computing.

##  2 Violation of Bell’s inequality

The experiment proposed by Bell ([12]) consists of the following. Two persons-
Alice and Bob are given one qubit from a pair of qubits, after which each of
them measures his qubit in a randomly selected one with probability $1/2$ the
basis of two: for Alice, the choice is between the basis of the eigenvectors
of the operators $\sigma_{x}$ and $\sigma_{z}$, for Bob - from the
eigenvectors of the operators $\frac{1}{\sqrt{2}}(\sigma_{x}+\sigma_{z})$ and
$\frac{1}{\sqrt{2}}(\sigma_{x}-\sigma_{z})$. The result of the measurement is
considered to be the eigenvalue corresponding to the measurement - in all
cases it will be $\pm 1$.

Figure 3: Bell experiment. Alice chooses between $\sigma_{x}$ and $\sigma_{z}$
, Bob - between $\frac{1}{\sqrt{2}}(\sigma_{x}+\sigma_{z})$ and
$\frac{1}{\sqrt{2}}(\sigma_{x}-\sigma_{z})$

.

Denote by $a$ and $b$ results of Alice’s measurement when selecting the basis
corresponding to the operators $\sigma_{x}$ and $\sigma_{z}$ accordingly, and
through $X$ and $Y$ \- results of the Bean measurement when selecting the
basis corresponding to the operators
$\frac{1}{\sqrt{2}}(\sigma_{x}+\sigma_{z})$ and
$\frac{1}{\sqrt{2}}(\sigma_{x}-\sigma_{z})$. After the measurement, we make a
random variable $\xi$ equal to $aX$, $aY$, $bX$ \- when Alice and Bob choose
the corresponding bases, and $-bY$ \- when choosing the bases corresponding to
$b$ and $Y$.

It is easy to see that if the measurement result of any participant does not
depend on the orientation of the detector of the other participant, the
mathematical expectation of the value $\xi$ cannot be greater than $1/2$,
since in this case we can write $M\xi=M\frac{1}{4}(aX+aY+bX-
bY)=\frac{1}{4}M(a(X+Y)+b(X-Y))\leq 1/2$, where all occurrences of $a$, $b$,
$X$, and $Y$ are the same numbers, and, consequently, one of the brackets is
zero, and the other is modulo 2.

However, suppose that the pair of photons distributed to Alice and Bob is in
the $|t\rangle$ state. Then the quantum calculation of the mathematical
expectation by the formula $M_{\psi}(A)=tr(A|\psi\rangle\langle\psi|)$ gives
$M\xi=1/\sqrt{2}$. This means that the measurement result of one of the
participants depends on the orientation of the detector of the other
participant. This is quantum non locality. It proves the existence of
instantaneous action at a distance, since the choice of the detector
orientation can be made at the moment when the photon flies into Alice’s
laboratory, and nothing moving at light speed can reach Bob’s laboratory to
affect its result.

"Knowing" about the orientation of Alice’s detector "reaches" Bob’s lab at a
"speed" many times faster than light, which is contrary to the spirit of
relativism. What is surprising is that there is no contradiction to the letter
of the theory of relativity, since this channel cannot be used to transmit the
information conceived by Alice to Bob at superluminal speed; to fix the
violation of Bell’s inequality, it is necessary to bring together the results
of Alice’s and Bob’s measurements, which can only be done at the speed of
light.

##  3 The task of polymer chain synthesis

The quantum advantage is based on entanglement, and the most explicit
demonstration of it is given with the explicit use of non-locality. To this
end, we will consider the task of distributed computing with one-way control.
Here, the computing system consists of a mass of nodes distributed in space,
whose operation is controlled by signals from a single central processor,
which can send either classical states of bits - one for each node, or qubits
that are in an entangled state, which we denote by $|\Psi\rangle$.

Figure 4: Simultaneous synthesis of a polymer pair. The sign is determined by
measuring the photon for Alice (index 1) and Bob (index 2) in the observables:

$\begin{array}[]{lll}&a_{1}=\sigma_{x},&b_{1}=\sigma_{z},\\\
&a_{2}=\frac{1}{\sqrt{2}}(\sigma_{z}-\sigma_{x}),&b_{2}=-\frac{1}{\sqrt{2}}(\sigma_{x}+\sigma_{z})\end{array}$

In our task, we will talk about the simultaneous synthesis of abstract polymer
chains at 4 points far from each other. The synthesis consists in joining a
new link to the final link of the chain - a monomer taken at random from the
reservoir associated with this chain. In this case, the connection of a new
link is regulated by the result of measuring a photon that enters the detector
connected to this chain. The measurement result is one of the numbers $\pm 1$,
and this number determines how to connect a new link to the chain: $+1$ means
a connection with a forward shift, $-1$ \- with a backward shift.

The result of the synthesis of 4 chains is determined by their overlapping
each other in the form of an index of quality bonding: applying the better,
the higher the index. Monomers - separate chain links, will be of two types,
denoted by $a$ and $b$. Their structure is schematically shown in the figure.

The chain overlap index $Cr$ is the number of bonding monomers. Each monoblock
has an outer (convex) and inner (concave) surface, where the latter is
provided with a special ball located in its center. In a fixed position, two
monoblock can be glued together in one of the following cases: 1) their
surfaces or half of the surfaces are completely aligned by vertical
displacement, or 2) their central balls are at the same point when this shift
occurs, as shown in the figure reffig:polimers.

The physical structure of the polymer on which the bonding depends is
determined not only by the sequence of monoblocks in the chain; the bonding
also depends on an additional option: their exact location relative to each
other in the chain. Adjacent monoblocks in the polymer are connected by a
flexible bond that can either shrink by $dx$, which is a quarter of the length
of the monobloc, or stretch by the same length. In these cases, we will say
that the monoblock is shifted back or forward, respectively, relative to the
equilibrium position of the connection. During the synthesis, the monoblocks
are set with these restrictions and their positions are fixed. Then the two
chains are superimposed on each other and for each pair of overlapping
monoblocks, the presence of gluing is established. From the accepted
restriction, it follows that if in such a pair of overlapping monoblocks they
were shifted in one direction, they are glued together in the same way as if
there were no shifts; and if in different directions, the resulting shift is
half the length of the monoblock.

Chain synthesis occurs as a sequential attachment to each of the existing
chains of a new monoblock - the one that first appeared at the assembly point
of one and the other chains. Monoblocks are taken from the environment
surrounding the growth points, where they are in chaotic motion and both types
are equally distributed. In this case, you can move the newly attached
monoblock either backward or forward by a distance of $dx$. We will denote the
forward shift by $+$, and the backward shift by $-$. Each $j$ \- th pair of
monoblocks in both chains, superimposed on each other after synthesis,
correspond, thus, to the four $c_{j}^{1}c_{j}^{2}s_{j}^{1}s_{j}^{2}$, where
the last two terms are shifts of $s_{j}^{1,2}\in\\{+,-\\}$.

From our rules (see Figure 5), it follows that the gluing corresponds to pairs
of superimposed monoblocks of the form: $aa++(--),\
ab++(--),bb++(--),ba+-(-+)$, while pairs of a different type:
$aa+-(-+),ab+-(-+),bb+-(-+),ab++(--)$ do not give gluing. Note the asymmetric
behavior of monoblocks of the type $a$ and $b$: the pairs $ab$ and $ba$ are
glued together differently at the same shifts. This asymmetry looks like an
asymmetry in the Bell inequality, which will give us an improvement in the
quality of the resulting gluing under bi-photon control compared to classical
control.

We assume that the growth of the polymer $C_{1}$ occurs at one point, and the
growth of $C_{2}$ occurs at another, and these points are separated by a large
distance (for example, they occur in different countries). The task is to
organize this synthesis so that the number of non-glued pairs of superimposed
monoblocks is minimal, or, in other words, that the number of glues is
maximum.

Figure 5: Superposition of two polymers. The arrows indicate the direction of
bond stretching (red) between adjacent monoblocks during polymer synthesis.
Overlays of the form $aa++(--),ab++(--),bb++(--),ba+-(-+)$ give gluing, the
rest of the gluing does not give. At the bottom, all pairs give glues.

In the paper [11], it is shown that the control of chain synthesis using bi-
photon states of the form $|t\rangle$ gives a gain in the overlap index of
more than $12\%$ compared to the classical control. This is shown by reducing
it to Bell’s inequality. The overlap index of two polymers can be calculated
using the formula $Cr=\frac{1}{2}(1+NonCr)$, where $NonCr$ is the number of
overlays with good bonding. And the random variable $NonCr$, by virtue of our
definitions, coincides with $\xi$, so that its mathematical expectation is
$1/\sqrt{2}$, which gives the desired result.

##  4 Case of three participants

We assume the order of qubits: Alice, Natalia, Bob, and let the pair Alice,
Bob be legitimate pair. Maximum the differentiation of gluing indices for the
three participants is 0.35 (0.5 for an illegitimate pair (which is the minimum
possible value) and 0.85 for a legitimate pair (which is the maximum possible
value)), where the legitimate pair has photons entangled with each other, and
the illegitimate one does not. This value is achieved for the following
conditions:

$\frac{1}{2}(|000\rangle+|010\rangle+|101\rangle+|111\rangle)=\frac{1}{2}(|0*0\rangle+1*1\rangle)(|*0*\rangle+|*1*\rangle)$

$\frac{1}{2}(|001\rangle+|010\rangle+|101\rangle+|110\rangle)$

$\frac{1}{\sqrt{2}}(|000\rangle+|011\rangle)$

$\frac{1}{\sqrt{2}}(|001\rangle+|010\rangle)$

$\frac{1}{\sqrt{2}}(|010\rangle+|111\rangle)$

$\frac{1}{\sqrt{2}}(|011\rangle+|110\rangle)$

For the case of four syllables, it is possible to recognize a legitimate
couple - knowing that the 3rd participant is included in it always, and the
state itself is received as tensor product of 2 states entangled pair and two
states an illegitimate participant, you can see which positions (1 or 2) in
the final state is 1 and 0 so, in order to "bind" with the 3 state to blow or
two combinations 00 and two 11 or two combinations 01 and two 10. For the case
of three syllables, a similar method is suitable.

The maximum level of gluing for a pair is achieved only under the above states
of a legitimate pair. Thus, it is possible to obtain all the listed states by
enumerating various values of the state of the not entangled with others
participant.

the maximum sum E(NonCr) for all opposite-sex couples of three people is
1.157, which is less than the same value for classical management (0.75 * 2).

##  5 Photon quartets for 4-chain synthesis

We generalize the result of [11] to 4 polymer chains divided into pairs by
gender. Let’s say we have 4 participants: Alice, Natasha, Bob, and Ivan, and
each of them synthesizes their polymer chain according to the same rules as in
the previous task, so that Alice and Natasha choose the observables between
$\sigma_{x}$ and $\sigma_{z}$, and Bob and Ivan choose between
$\frac{1}{\sqrt{2}}(\sigma_{x}+\sigma_{z})$ and
$\frac{1}{\sqrt{2}}(\sigma_{x}-\sigma_{z})$ respectively, regardless of each
other. If you divide the participants into fixed pairs, for example, Alice-
Bob, Natasha-Ivan, and give each pair a bifoton, as in the previous paragraph,
the winnings will be the same as before. But if this control calculates the
index of gluing an alternative pair, for example, Natasha and Bob, we will get
a lower index then in the classical control, since there is no coherence
between different bi-photons.

Conditions with maximum gluing differentiation "legitimate - illegitimate"
couples: There are 4 states that give a good overlap for Natalia-Bob and
Alice-Ivan pairs (0.85) and for Natalia-Ivan, Alice-Bob pairs, bad ( 0.5):
these are states

$\frac{1}{2}(|0011\rangle+|0110\rangle+|1001\rangle+|1100\rangle)$

$\frac{1}{2}(|0010\rangle+|0111\rangle+|1000\rangle+|1101\rangle)$

$\frac{1}{2}(|0001\rangle+|0100\rangle+|1011\rangle+|1110\rangle)$

$\frac{1}{2}(|0000\rangle+|0101\rangle+|1010\rangle+|1111\rangle)$

There are also 4 states that give bad overlap for Natalia-Bob and Alice-Ivan
pairs and good overlap for Natalia-Ivan and Alice-Bob pairs:

$\frac{1}{2}(|0010\rangle+|0100\rangle+|1011\rangle+|1101\rangle)$

$\frac{1}{2}(|0001\rangle+|0111\rangle+|1000\rangle+|1110\rangle)$

$\frac{1}{2}(|0000\rangle+|0110\rangle+|1001\rangle+|1111\rangle)$

$\frac{1}{2}(|0011\rangle+|0101\rangle+|1010\rangle+|1100\rangle)$

Also, the maximum difference in the values of the gluing indices is achieved
only when entangled bi-photons are distributed to legitimate pairs.

The maximum sum E(NonCr) for all 4 opposite-sex couples will be
2.7071067811865475, which means that for each pair it is impossible to win
over the classical control (the sum should have been> 0.75 * 4 = 3).

Figure 6: Synthesis of 4 chains. The dotted line shows all heterosexual pairs
of polymers, the superposition of which at $|\Psi\rangle\in{(1)}$ has a higher
quality than when using bi-photon control.

.

##  6 Minimal overlap

Also, for 3 and 4 participants, there are states where E(NonCr) for each
heterosexual pair becomes the minimum possible - 0.5 (including less than with
classical control)

$\frac{1}{2}(|000\rangle+|001\rangle+|110\rangle+|111\rangle)$

$\frac{1}{2}(|010\rangle+|011\rangle+|100\rangle+|101\rangle)$

$\frac{1}{2}(|0000\rangle+|0011\rangle+|1100\rangle+|1111\rangle)$

$\frac{1}{2}(|0100\rangle+|0111\rangle+|1000\rangle+|1011\rangle)$

For example, consider the state
$\frac{1}{2}(|0000\rangle+|0011\rangle+|1100\rangle+|1111\rangle)$ for Alice,
Nataly, Bob and Ivan. Let’s build a reduced density matrix for each
heterosexual pair. We will look at the values at the corresponding places of
the density matrix (we restrict ourselves only to those values for which at
least one in a pair has a nonzero amplitude value in the initial state)

Therefore, for any pair and any choice of observables (A), it will be true
that $tr(A\rho)$ = 0. Therefore, E(NonCr) = $\frac{1}{2}$(1+0) = $\frac{1}{2}$
for all opposite-sex couples.

##  7 Conclusion

We have consider the task of the comparison on gluing index in all
heterosexual pairs of chains obtained by simultaneous synthesis of 3 or 4
polymer chains at the remote locations under one-way control. We found that
the result in the different formulations strongly depends on the controlling
signals. The use of EPR biphoton pairs allow to reach the maximal difference
of gluing degree (kinship) between legal and nonlegal pairs and the minimal
gluing degree (antagonism) in all heterosexual pairs - both results are
unattainable for the classical control.

If the aim is to increase the kinship of legal pair and at the same time to
increase the antagonism for the illegal pair, then we must distribute the EPR
pair between the legal participants and keep illegal one disentangled with
them. The difference in the index of gluing between the legal and non legal
pairs will be then $0.35$.

If the aim is to reach the minimal possible index of gluing in all
heterosexual pair (total antagonism) we must distribute bi-photonic signals to
homosexual pairs and keep the different sexes non entangled.

This non symmetric behavior between the ways of success in the formally
opposite tasks can be expressed shortly. To get the maximal antagonism between
sexes we must entangle pairs with the same sex; to get the maximal difference
between legal and nonlegal heterosexual pair we must entangle the legal
heterosexual pair exclusively.

This conclusion is made only for our artificially build example. But this
example directly rests on the main feature of entanglement - the violation of
Bell inequality, hence the conclusion applies to the entangled control for the
more wide class of chains assembled in the remote locations that may have
interest for quantum biology.

##  8  Acknowledgements

The paper was prepared in the Moscow Center for Fundamental and Applied
Mathematics.

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