# Probing the structure of $\chi_{c1}(3872)$ with photon transition form
factors

Izabela Babiarz<EMAIL_ADDRESS>Institute of Nuclear Physics,
Polish Academy of Sciences, ul. Radzikowskiego 152, PL-31-342 Kraków, Poland
Roman Pasechnik<EMAIL_ADDRESS>Department of Physics, Lund
University, SE-223 62 Lund, Sweden  Wolfgang Schäfer
<EMAIL_ADDRESS>Institute of Nuclear Physics, Polish Academy of
Sciences, ul. Radzikowskiego 152, PL-31-342 Kraków, Poland  Antoni Szczurek
<EMAIL_ADDRESS>Institute of Nuclear Physics, Polish Academy of
Sciences, ul. Radzikowskiego 152, PL-31-342 Kraków, Poland  College of
Mathematics and Natural Sciences, University of Rzeszów, ul. Pigonia 1,
PL-35-310 Rzeszów, Poland

###### Abstract

We propose to study the structure of the enigmatic $\chi_{c1}(3872)$ axial
vector meson through its $\gamma^{*}_{L}\gamma\to\chi_{c1}(3872)$ transition
form factor. We derive a light-front wave function representation of the form
factor for the lowest $c\bar{c}$ Fock-state. We found that the reduced width
of the state is well within the current experimental bound recently published
by the Belle collaboration. This strongly suggests a crucial role of the
$c\bar{c}$ Fock-state in the photon-induced production. Our results for the
$Q^{2}$ dependence can be tested by future single tagged $e^{+}e^{-}$
experiments, giving further insights into the short-distance structure of this
meson.

###### pacs:

12.38.Bx, 13.85.Ni, 14.40.Pq

Introduction–Starting from the discovery of the $\chi_{c1}(3872)$ state by the
Belle collaboration Choi  _et al._ (2003), recent years have seen a surge of
discoveries of new hadronic states Lebed  _et al._ (2017); Olsen  _et al._
(2018); Chen  _et al._ (2023). The microscopic structure of $\chi_{c1}(3872)$
state is still under intense debate in the literature Karliner  _et al._
(2018); Brambilla  _et al._ (2020). While its quantum numbers $J^{\rm
PC}=1^{++}$ imply a possible $\chi_{c1}(2P)$ charmonium Achasov and Rogozina
(2015), its mass appears to be close to the $D^{0}\bar{D}^{*0}$ threshold
suggesting a picture based on a very weakly bound meson molecule Tornqvist
(2004) (for a review, see e.g. Ref. Guo  _et al._ (2018) and references
therein). The latter option also gives a natural explanation of the strong
isospin violation in its decays. However, such a violation can also be
accommodated in the structure of couplings to $c\bar{c}$ and meson-meson
channels Danilkin and Simonov (2010); Ferretti  _et al._ (2013); Bruschini and
González (2023); Coito  _et al._ (2013); Yamaguchi  _et al._ (2020), e.g. via
mixing of molecular and compact states studied in Ref. Kang and Oller (2017).
For a discussion of another possible picture of $\chi_{c1}(3872)$ as a compact
tetraquark, see Refs. Maiani  _et al._ (2005); Ali  _et al._ (2017).

There is an open question of to what extent production mechanisms of
$\chi_{c1}(3872)$ can shed light on its internal structure. The inclusive
$\chi_{c1}(3872)$ production cross section has been recently measured at the
LHC as a function of its transverse momentum Aaboud  _et al._ (2017);
Chatrchyan  _et al._ (2013); Aaij  _et al._ (2022). A striking similarity with
the production rate of $\psi(2S)$ points to the importance of a compact
component in the wave function. Indeed, the measured transverse momentum
distributions are well described assuming a large $c\bar{c}$ component Cisek
_et al._ (2022). Studying $\chi_{c1}(3872)$ production in a cleaner
environment than hadronic collisions would provide new opportunities to
understand its structure better.

In this Letter, we work under the basic assumption of a $c\bar{c}$ $2P$-state
and suggest probing the $c\bar{c}$ component through the production of
$\chi_{c1}(3872)$ in the (virtual) photon-photon channel in single-tagged
$e^{+}e^{-}$ collisions. A first attempt to perform such a measurement has
been made by the Belle collaboration providing a bound on the reduced width
Teramoto  _et al._ (2021). A recent result of Ref. Li  _et al._ (2022) found
the reduced width significantly overshooting the Belle bound, hence,
concluding that the $c\bar{c}$ component is not relevant for the structure of
$\chi_{c1}(3872)$ as probed by photons. This represents a controversy with
recent studies in hadronic reactions Cisek  _et al._ (2022), where the
$c\bar{c}$ component almost explains experimental data.

Transition form factor–Due to the Landau-Yang theorem, at least one off-shell
photon is required for $\chi_{c1}$ production in the photon-photon fusion
channel. Here, we study the corresponding form factor for one longitudinal and
one transverse photon representing the amplitude of a longitudinal photon to
an axial $c\bar{c}$ meson transition in an external electromagnetic field
approaching the forward amplitude. For this purpose, we utilize the Light-
Front (LF) approach to transition form factors in the Drell-Yan frame Brodsky
_et al._ (1998). Here, the longitudinal photon with spacelike virtuality
$Q_{1}^{2}\equiv-q_{1}^{2}$ carries four-momentum
$q_{1}=(q_{1+},q_{1-},\bf{0}_{\perp})$, with $q_{1-}=-Q_{1}^{2}/(2q_{1+})$ and
polarization vector
$\varepsilon_{L}=1/Q_{1}\,(q_{1+},-q_{1-},\bf{0}_{\perp})$, while the plus-
momentum of the second photon vanishes $q_{2}^{+}=0$, such that
$q_{2}=(0,q_{2-},\mbox{\boldmath$q$}_{2})$. In the real photon limit,
$Q_{2}^{2}\equiv-q_{2}^{2}=\mbox{\boldmath$q$}_{2}^{2}\to 0$, i.e. when its
transverse momentum $\mbox{\boldmath$q$}_{2}\to 0$, the transition amplitude
vanishes linearly with $\mbox{\boldmath$q$}_{2}$ enabling us to extract the
relevant form factor. Indeed, in the considered frame, the LF plus component
of the electromagnetic current is free from parton number changing and
instantaneous fermion exchange contributions Brodsky and Hwang (1999), and can
be written in terms of the lowest $c\bar{c}$ Fock-state LF wave functions
(LFWFs) (from now on, we define $Q^{2}\equiv Q_{1}^{2}$):

$\langle{\chi_{c1}(\lambda_{A})}|J_{+}(0)|{\gamma^{*}_{L}(Q^{2})}\rangle=2q_{1+}\,\sqrt{N_{c}}\int\frac{dzd^{2}\mbox{\boldmath$k$}}{z(1-z)16\pi^{3}}\\\
\times\sum_{\lambda,\bar{\lambda}}\Psi^{(\lambda_{A})*}_{\lambda\bar{\lambda}}(z,\mbox{\boldmath$k$})\,(\mbox{\boldmath$q$}_{2}\cdot\nabla_{\mbox{\boldmath$k$}})\Psi^{\gamma_{L}}_{\lambda\bar{\lambda}}(z,\mbox{\boldmath$k$},Q^{2})\,.$
(1)

Above, the summation over the (anti)quark color indices has been performed,
$N_{c}=3$ is the number of colors in QCD, and we introduced the LF helicity
$\lambda_{A}=\pm 1,0$ of the axial meson $\chi_{c1}$, as well as
$c\bar{c}\to\chi_{c1}$ and $\gamma^{*}_{L}\to c\bar{c}$ LFWFs,
$\Psi^{(\lambda_{A})*}_{\lambda\bar{\lambda}}$ and
$\Psi^{\gamma_{L}}_{\lambda\bar{\lambda}}$, respectively. The integration is
over the internal LF momenta of quark $(c)$ and antiquark $(\bar{c})$, namely,
the LF momentum fraction $z=k_{c+}/q_{1+}$ of the $c$-quark and its transverse
momentum $k$ as illustrated in Fig. 1. The (anti)quark coupling to the
external field conserves the LF helicities $\lambda,\bar{\lambda}$ of quark
and antiquark, whose $\pm 1/2$ values are denoted by $\uparrow$ and
$\downarrow$, respectively.

Figure 1: An illustration of the meson production mechanism in photon-photon
fusion in the light-cone dipole picture, with relevant ingredients and
kinematics.

Furthermore, it is instructive to utilize the general covariant
parametrization of the $\gamma^{*}\gamma^{*}\to\chi_{c1}$ amplitude of Ref.
Babiarz  _et al._ (2022), which is similar to the one found in Ref. Poppe
(1986) and is based on $\gamma^{*}\gamma^{*}$ c.m. frame helicity amplitudes.
We notice that only one term contributes to the transition amplitude of
interest in the limit $Q_{2}^{2}\to 0$,

$\varepsilon_{L}^{\mu}n^{-\nu}{\mathcal{M}}_{\mu\nu\rho}E^{*\rho}\rightarrow
4\pi\alpha_{\rm em}\,\tilde{G}_{\nu\rho}n_{-}^{\nu}E^{*\rho}\,\frac{F_{\rm
LT}(Q^{2},0)}{q_{1}\cdot q_{2}}\,,$ (2)

where $\alpha_{\rm em}=e^{2}/4\pi$ is the fine structure constant,
$E=E(\lambda_{A})$ is the polarization vector of the axial meson,
$n_{-}=(0,1,\bf{0}_{\perp})$, and
$\tilde{G}_{\nu\rho}\equiv\varepsilon_{\nu\rho\alpha\beta}q_{1}^{\alpha}q_{2}^{\beta}$.
We choose the LF spin projection $\lambda_{A}=+1$, and obtain:

$\langle{\chi_{c1}(+1)}|J_{+}(0)|{\gamma^{*}_{L}(Q^{2})}\rangle=\,2q_{1+}\,\frac{q_{2x}-iq_{2y}}{\sqrt{2}}\\\
\times\frac{\sqrt{4\pi\alpha_{\rm em}}F_{\rm
LT}(Q^{2},0)}{Q^{2}+M_{\chi}^{2}}\,,$ (3)

in terms of the considered $\chi_{c1}$ meson mass $M_{\chi}=(3871.65\pm 0.06)$
MeV Workman  _et al._ (2022) and the photon-meson transition form factor
$F_{\rm LT}(Q^{2},0)$. Combining this expression with Eq. (1) and using the
well-known expression for the perturbative LF wave function of the
longitudinal photon’s $c\bar{c}$ component (see e.g. Ref. Kovchegov and Levin
(2013)),

$\displaystyle\Psi^{\gamma_{L}}_{\lambda\bar{\lambda}}(z,\mbox{\boldmath$k$},Q^{2})=ee_{c}\sqrt{z(1-z)}\,{2z(1-z)Q\over\mbox{\boldmath$k$}^{2}+\epsilon^{2}}\,\delta_{\lambda,-\bar{\lambda}}\,,$
(4)

with $\epsilon^{2}=m_{c}^{2}+z(1-z)Q^{2}$, charm (anti)quark mass $m_{c}$, and
the electric charge of the charm quark is $e_{c}=2/3$, one arrives at the LFWF
representation of the transition form factor:

$\frac{f_{\rm
LT}(Q^{2})}{Q^{2}+M_{\chi}^{2}}=-2\sqrt{2N_{c}}\,e_{c}\int\frac{dzd^{2}\mbox{\boldmath$k$}}{16\pi^{3}}\,\frac{k_{x}+ik_{y}}{[\mbox{\boldmath$k$}^{2}+\epsilon^{2}]^{2}}\\\
\times\sqrt{z(1-z)}\Big{\\{}\Psi^{(+1)*}_{\uparrow\downarrow}(z,\mbox{\boldmath$k$})+\Psi^{(+1)*}_{\downarrow\uparrow}(z,\mbox{\boldmath$k$})\Big{\\}}\,.$
(5)

The dimensionless transition from factor $f_{\rm LT}(Q^{2})\equiv F_{\rm
LT}(Q^{2},0)/Q$ takes a finite value in the limit $Q^{2}\to 0$. The
representation of Eq. (5) can also be derived from more general expressions
for the transition form factors for two spacelike virtual photons found
earlier in Ref. Babiarz  _et al._ (2022).

Figure 2: The dimensionless $\gamma^{*}_{L}\gamma\to\chi_{c1}(2P)$ transition
form factor $f_{\rm LT}(Q^{2})$ found in Eq. (5).

A brief comment on the considered LFWFs is in order. In our analysis, we adopt
two different approaches. The first one is based on $c\bar{c}$ radial wave
functions in the $c\bar{c}$-pair rest frame obtained by solving the
Schrödinger equation for a variety of phenomenologically viable potential
models. Then, one appropriately transforms both the resulting radial wave
functions Terentev (1976), and their spin-orbital components Melosh (1974) in
order to describe the boosted meson states in the Drell-Yan frame. For further
details of this procedure, an overview of the potential models and related
theoretical uncertainties, see Refs. Krelina  _et al._ (2019); Cepila  _et
al._ (2019) and Babiarz  _et al._ (2022) for photoproduction of vector and
axial mesons, respectively. In the considered case of the first radial
excitation of the $\chi_{c1}$ meson – the $2P$ state – the relevant
combination of LFWFs takes the form:

$\sqrt{z(1-z)}\,\Big{\\{}\Psi^{(+1)*}_{\uparrow\downarrow}(z,\mbox{\boldmath$k$})+\Psi^{(+1)*}_{\downarrow\uparrow}(z,\mbox{\boldmath$k$})\Big{\\}}\\\
=(k_{x}-ik_{y})\,\sqrt{\frac{3}{2}}\frac{\pi\sqrt{M_{c\bar{c}}}}{2}\frac{u_{\rm
2P}(k)}{k^{2}}\,,$ (6)

where $u_{\rm 2P}(k)$ is the radial wave function in the $c\bar{c}$-pair rest
frame found for a given interquark interaction potential as a function of the
relative $c$ and $\bar{c}$ three-momentum,
$k=\sqrt{M_{c\bar{c}}^{2}-4m_{c}^{2}}/2$ Babiarz  _et al._ (2022), with
$M^{2}_{c\bar{c}}=(\mbox{\boldmath$k$}^{2}+m_{c}^{2})/z(1-z)$. The second
approach is based on the Basis Light Front Quantization (BLFQ) of Refs. Li
_et al._ (2016, 2017, 2022), where the LFWFs are obtained without referring to
non-relativistic $c\bar{c}$ interaction potentials and factorisation of radial
and spin-orbit components applicable only in the $c\bar{c}$-pair rest frame.
Instead, one formulates an LF-Hamiltonian problem

$\displaystyle H_{\rm
eff}|{\chi_{c};\lambda_{A},P_{+},\mbox{\boldmath$P$}}\rangle=M^{2}|{\chi_{c};\lambda_{A},P_{+},\mbox{\boldmath$P$}}\rangle\,,$
(7)

where the effective Hamiltonian used in Li  _et al._ (2016, 2017) contains a
term motivated by a “soft-wall” confinement from LF-holography, as well as a
longitudinal confinement potential supplemented by one gluon exchange
including the full spin-structure. The LFWFs from the Fock state decomposition

$|{\chi_{c};\lambda_{A},P_{+},\mbox{\boldmath$P$}}\rangle=\sum_{\lambda\bar{\lambda}}\int\frac{dzd^{2}\mbox{\boldmath$k$}}{z(1-z)16\pi^{3}}\,\psi_{\lambda\bar{\lambda}}^{(\lambda_{A})}(z,\mbox{\boldmath$k$})\\\
\times\frac{\delta^{i}_{j}}{\sqrt{N_{c}}}|{c_{i}(z,\mbox{\boldmath$k$}+z\mbox{\boldmath$P$})\bar{c}^{j}(1-z,-\mbox{\boldmath$k$}+(1-z)\mbox{\boldmath$P$})}\rangle$
(8)

are then obtained by discretizing the Hamiltonian and evaluating it on a
finite basis. In our calculations, we have used the LFWFs from numerical
results of Li  _et al._ (2017) published in Ref. Li (2017).

Numerical results–In Fig. 2 we show our results for the dimensionless
transition from factor $f_{\rm LT}(Q^{2})\equiv F_{\rm LT}(Q^{2},0)/Q$ over a
broad range of $Q^{2}$. We see considerable dependence on the potential model
at $Q^{2}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}15\,\rm{GeV}^{2}$. Here the BLFQ
approach stands out from the nonrelativistic potential approach. It exhibits a
much weaker dependence on $Q^{2}$. We trace that partially to the broader
distribution in LF momentum fraction $z$ exhibited by the BLFQ-WF. The blue
band in Fig. 2 was obtained in a similar manner as in Li  _et al._ (2016). For
selected $Q^{2}$ values, we calculated $f_{LT}(Q^{2})$ for different values of
the number of basis functions $N_{\rm basis}$. Then $f_{LT}(Q^{2})$ was
obtained by extrapolating $f_{LT}(Q^{2},N_{\rm basis})$ for $1/N_{\rm
basis}\to 0$ by fitting polynomials of different order to points for different
$N_{\rm basis}$. The error band reflects the spread due to the order of the
extrapolating polynomial. The precision of reduced width
${\tilde{\Gamma}}_{\gamma\gamma}$ is better than 2 %.

Now, let us remind the reader of the definition of the so-called reduced
$\gamma\gamma$ decay width of $\chi_{c1}$ given in the limit of the vanishing
projectile photon virtuality Aihara  _et al._ (1988):

$\displaystyle\tilde{\Gamma}_{\gamma\gamma}$ $\displaystyle=$
$\displaystyle\lim_{Q^{2}\to 0}\frac{M_{\chi}^{2}}{Q^{2}}\Gamma^{\rm
LT}_{\gamma^{*}\gamma^{*}}(Q^{2},0,M_{\chi}^{2})$ (9) $\displaystyle=$
$\displaystyle\frac{\pi\alpha^{2}_{\rm em}M_{\chi}}{3}\,f^{2}_{\rm LT}(0)\,.$

Hence, a measurement of the reduced width provides direct access to the
dimensionless transition form factor in the limit $Q^{2}\to 0$. We summarize
our results for $f_{\rm LT}(0)$ and the reduced width in Table 1. We observe a
considerable spread between the results for different potential models. Here
the Cornell potential gives the smallest value of the reduced width, roughly a
factor of five smaller than the value obtained for the harmonic oscillator
potential. Notice that the quark mass substantially influences the
normalization of the form factor $f_{\rm LT}(0)$. The BLFQ-WF gives the
largest result among the considered approaches. However, it is almost a factor
of six smaller than the result obtained by Li et al. Li  _et al._ (2022) using
the same wave functions. These authors, though, do not use the plus-component
of the current and derive a different representation of the transition form
factor.

Table 1: The reduced width of the $\chi_{c1}(2P)$ state for several models of the charmonium wave functions with specific $c$-quark mass. $c\bar{c}$ potential |  $m_{c}$ (GeV) | $f_{\rm LT}(0)$ |  $\tilde{\Gamma}_{\gamma\gamma}$ (keV)
---|---|---|---
harmonic oscillator | 1.4 | 0.041 | 0.36
power-law | 1.334 | 0.033 | 0.24
Buchmüller-Tye | 1.48 | 0.029 | 0.18
logarithmic | 1.5 | 0.025 | 0.14
Cornell | 1.84 | 0.018 | 0.07
BLFQ Li (2017) | 1.6 | 0.044 | 0.42

The Belle collaboration has reported the first evidence for the production of
$\chi_{c1}(3872)$ in single-tag $e^{+}e^{-}$ collisions Teramoto  _et al._
(2021). From three measured events, they provided a range for its reduced
width, $0.02\,\rm{keV}<\tilde{\Gamma}_{\gamma\gamma}<0.5\,\rm{keV}$. This
result has recently been updated by Achasov et al. Achasov  _et al._ (2022)
using a corrected value for the branching ratio ${\rm
Br}(\chi_{c1}(3872)\to\pi^{+}\pi^{-}J/\psi)$ Workman  _et al._ (2022) and
reads

$\displaystyle
0.024\,\rm{keV}<\tilde{\Gamma}_{\gamma\gamma}(\chi_{c1}(3872))<0.615\,\rm{keV}$
(10)

Using nonrelativistic quark model relations, Achasov et al. Achasov  _et al._
(2022) provided the following estimate

$\displaystyle\tilde{\Gamma}_{\gamma\gamma}(\chi_{c1}(3872))\approx
0.35\,\rm{keV}\,\div 0.93\,\rm{keV}\,.$ (11)

Even with the large dependence on the $c\bar{c}$ potential, all our results,
including the BLFQ approach, lie well within the experimentally allowed range.
Therefore, $\gamma\gamma$ data do not exclude the $c\bar{c}$ option, although
there is certainly some room for a contribution from an additional meson-meson
component. Regarding the molecular scenario, no estimates for the reduced
width in the molecular scenario are available.

Summary and outlook–In this work, we concentrated on the lowest $c\bar{c}$
Fock state consistent with the quantum numbers of the axial vector
$\chi_{c1}(3872)$. There are many indications that meson-meson components in
the wave function may also be necessary to understand the decay properties of
$\chi_{c1}(3872)$. The transition amplitude of Eq. (1) can be understood as a
transition dipole moment

$\langle{\chi_{c1}(\lambda_{A})}|J_{+}(0)|{\gamma^{*}_{L}(Q^{2})}\rangle\\\
\propto
i\mbox{\boldmath$q$}_{2}\cdot\langle{\chi_{c1}(\lambda_{A})}|\mbox{\boldmath$r$}|{\gamma^{*}_{L}(Q^{2})}\rangle\,.$
(12)

Evidently, the electric dipoles which enter this transition are controlled by
the photon LFWF to have sizes ${r\sim
1/\epsilon\sim(m_{c}^{2}+Q^{2}/4)}^{-1/2}\lesssim 0.15\,\rm{fm}$. This is much
smaller than the expected size of a molecular component, whose wave function
extends up to distances of around $10\,\rm{fm}$, so no significant overlap
with our $c\bar{c}$ Fock state is expected. Direct calculations for the
molecular scenario are not available yet. Here, the neutral $D,\bar{D}^{*}$
mesons interact through their (transition) magnetic moments. One expects a
much faster falloff at large $Q^{2}$ for these large-size objects. Future
precise measurements in single tagged $e^{+}e^{-}$ collisions can therefore
offer valuable new insight into the structure of this enigmatic hadron and
confirm or rule out an essential role of the $c\bar{c}$ component. One may
also explore the Primakoff–production in the field of a heavy nucleus in high-
energy electron-nucleus collisions at the future electron-ion collider.
Complementary information can be obtained also for timelike
$B_{c}\to\chi_{c1}(3872)$ form factors, see Ref. Colangelo  _et al._ (2022).

Acknowledgements– This work was partially supported by the Polish National
Science Center grant UMO-2018/31/B/ST2/03537 and by the Center for Innovation
and Transfer of Natural Sciences and Engineering Knowledge in Rzeszów. R.P. is
supported in part by the Swedish Research Council grant, contract number
2016-05996, as well as by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No 668679).

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# Quantum-to-classical transition via quantum cellular automata

Pedro C.S. Costa<EMAIL_ADDRESS>Department of Physics and Astronomy,
Macquarie University, Sydney, NSW 2109, Australia

###### Abstract

A quantum cellular automaton (QCA) is an abstract model consisting of an array
of finite-dimensional quantum systems that evolves in discrete time by local
unitary operations. Here we propose a simple coarse-graining map, where the
spatial structure of the QCA is merged into effective ones. Starting with a
QCA that simulates the Dirac equation, we apply this coarse-graining map
recursively until we get its effective dynamics in the semiclassical limit,
which can be described by a classical cellular automaton. We show that the
emergent-effective result of the former microscopic discrete model converges
to the diffusion equation and to a classical transport equation under a
specific initial condition. Therefore, QCA is a good model to validate the
quantum-to-classical transition.

##  I Introduction

In a realistic scenario, quantum systems are never completely isolated from
the environment and its quantum effects naturally start to become suppressed.
Such a phenomenon is known as the decoherence process, and it is the main
ingredient used to explore the quantum-to-classical transition
schlosshauer2014quantum . Completely positive trace-preserving (CPTP) maps are
powerful tools to study such transition. These maps, which can be seen as a
unitary interaction among three systems and further discard two of them
wolf2012quantum , are therefore not reversible and their application implies a
loss of information about the system. Similarly with the classical picture,
where coarse-graining (CG) procedure via the projective operators
PhysicsPhysiqueFizika.2.263 ; ehrenfest1990conceptual  helps us to extract
effective dynamics that have lower computational cost, the CPTP maps also
provide a data compressor for the quantum system. Thus, these maps when
applied in quantum systems can also be seen as a coarse-graining procedure.

Rather than verify the necessary conditions to extract the quantum channels at
each coarse-grained level CG_QM ; duarte2020investigating , here we
concentrate on just one specific quantum dynamics, one map and one CG level.
The Dirac equation is the quantum dynamics chosen, and the coarse-graining
level is the one where the coherence disappears. Given that the quantum model
of computation, known as quantum cellular automata (proposed in QCAviaQws ) is
able to simulate a large class of quantum walks, in particular the one that
simulates the Dirac equation, we use it to study the semiclassical limit of
the Dirac equation Dirac . Our procedure respects the same constraints between
different CG levels for the time evolution shown in CG_QM ; Oleg , which is
represented in figure (3). However, our formalism takes advantage of the space
structure provided by the CA, likewise done in CG_CA ; costaPCA . That is, in
order to analyze the emergent dynamics in the semi-classical limit, we assume
that we cannot resolve neighboring cells and then some information has to be
discarded.

Although in this work is the indistinguishability between the states the
source of decoherence to study the quantum-to-classical transition dynamics,
the procedure here, in analogy to chen2019quantifying , can also be helpful to
track the nonclassicality that emerges due to other sources of decoherence in
the new quantum technologies.

The paper is organized as follows: In section II we defined the quantum model
of computation used to simulate the Dirac equation in the continuous limit. In
section III we introduce the coarse-graining map used to study the decoherence
process for the massless Dirac equation. In section IV the CG map is applied
and our main results are established. Finally, in section V, we conclude with
a discussion of these results and we point out possible directions to go for
next.

##  II A PUQCA that simulates the (1+1) Dirac equation

In one spatial and one temporal dimension, the Dirac equation of a free
spin-$1/2$ particle of mass $m$ may be written as follows:

$\left({\rm i}\frac{\hbar}{c}\partial_{t}+{\rm
i}\hbar\sigma_{z}\partial_{x}-mc\sigma_{x}\right)\psi=0,$ (1)

where $c$ is the velocity of light, $\hbar$ is Planck’s constant divided by
$2\pi$, and $\sigma_{j}$, with $j\in\\{x,z\\}$, are the usual Pauli matrices.
We want to construct a partitioned unitary quantum cellular automata (PUQCA)
whose dynamics, in the continuous limit, is the same as the Dirac equation
(1).

A PUQCA is a discrete model of quantum computation, which is most easily
understood in its graph perspective. We start by discretizing the space
coordinate, and thus constructing a 1D linear graph–see figure 1$a$. Each
vertex in this graph represents a point in space, and we will assume that the
distance between two vertices is $\Delta x$. In this way, the vertices are
located at positions $x=k\,\Delta x$, with $k\in\mathbb{Z}$.

Figure 1:  $a$ Graph representation of the space coordinates. Labeling the
vertex positions by $k$ their positions in space can be calculated, i.e.,
$x=k\Delta x$. $)$ The vertices are related to the PUQCA cells, with two
subcells each.

In order to describe the dynamics of the particle, an excitation moving in the
space, each vertex is split into two subcells and in each subcell, we place a
qubit–see figure 1$b$. More concretely, to the vertex at position $x$ we
associate a Hilbert space
$(\mathcal{H})_{x}=(\mathcal{H}_{0})_{x}\otimes(\mathcal{H}_{1})_{x}$ with
${\cal H}_{0}\simeq{\cal H}_{1}\simeq\mathbb{C}^{2}$, for the left and right
subcells, respectively. The state $\ket{0}$ will be associated with no
particles, while the state $\ket{1}$ will be associated with the presence of a
particle/excitation.

In this model, the evolution is also discrete. Defining the time steps size as
$\Delta t$, the values for time are $t=m\,\Delta t$, with $m\in\mathbb{N}$.
For the cell structure defined, the transition function $\mathcal{E}$, i.e.,
the operator that updates the PUQCA state
$\mathcal{E}\ket{\Psi(t)}=\ket{\Psi(t+\Delta t)}$, is composed by two parts.
First, each vertex is individually and simultaneously updated by the
application of a unitary
$(W_{0})_{x}:(\mathcal{H}_{0})_{x}\otimes(\mathcal{H}_{1})_{x}\rightarrow(\mathcal{H}_{0})_{x}\otimes(\mathcal{H}_{1})_{x}$.
Then, to allow for the particle propagation a second unitary is applied, but
now acting on the graph edges, i.e., $(W_{1})_{x}^{x+\Delta
x}:(\mathcal{H}_{1})_{x}\otimes(\mathcal{H}_{0})_{x+\Delta
x}\rightarrow(\mathcal{H}_{1})_{x}\otimes(\mathcal{H}_{0})_{x+\Delta x}$. The
transition function is then

$\mathcal{E}=\bigotimes_{k\in\mathbb{Z}}(W_{1})_{k\Delta x}^{(k+1)\Delta
x}\bigotimes_{k^{\prime}\in\mathbb{Z}}(W_{0})_{k^{\prime}\Delta x}.$ (2)

Following the construction of a quantum walk that simulates the 1+1 Dirac
equation PhysRevA.97.042131 , and the translation between quantum walks and
the PUQCA QCAviaQws , we choose the unitary transformations as follows. For
$W_{0}$ we allow, a rotation in the single-excitation subspace for each cell:

$\displaystyle W_{0}$ $\displaystyle=|00\rangle\langle 00|-{\rm
i}\sin\theta|01\rangle\langle 01|+\cos\theta|01\rangle\langle 10|$
$\displaystyle\quad+\cos\theta|10\rangle\langle 01|-{\rm
i}\sin\theta|10\rangle\langle 10|+|11\rangle\langle 11|.$ (3)

The restriction to the single excitation subspace guarantees that the number
of particles is fixed to one. To interact the cells, we employ a simple swap
gate between the subcells connected by an edge. That means, we take:

$\displaystyle W_{1}=|00\rangle\langle 00|+|01\rangle\langle
10|+|10\rangle\langle 01|+|11\rangle\langle 11|.$ (4)

Given the above construction, we can now see the evolution on the one-particle
sector of this QCA. To do that we write the system state at time $t$,
$\ket{\Psi(t)}$ as:

$\displaystyle\sum_{k\in\mathbb{Z}}\Bigg{[}\psi^{0}(k\Delta
x,t)|\ldots(00)_{(k-1)\Delta x}(10)_{k\Delta x}(00)_{(k+1)\Delta
x}\cdots\rangle$ $\displaystyle\quad+\psi^{1}(k\Delta
x,t)|\cdots(00)_{(k-1)\Delta x}(01)_{k\Delta x}(00)_{(k+1)\Delta
x}\cdots\rangle\Bigg{]}.$ (5)

Applying the evolution $\epsilon$ to this state the following recurrence
relation for the amplitudes can be derived:

$\displaystyle\psi^{0}\left(x,t+\Delta t\right)$
$\displaystyle=\cos\theta\psi^{0}\left(x-\Delta x,t\right)-{\rm
i}\sin\theta\psi^{1}\left(x-\Delta x,t\right)$
$\displaystyle\psi^{1}\left(x,t+\Delta t\right)$
$\displaystyle=\cos\theta\psi^{1}\left(x+\Delta x,t\right)-{\rm
i}\sin\theta\psi^{0}(x+\Delta x,t).$ (6)

Associating $\psi_{0}$ and $\psi_{1}$ with the spinor components of $\psi$,
the last step is just to take the continuous limit. Making
$\theta=mc^{2}\Delta t/\hbar$, and taking the limit $\Delta x\rightarrow 0$
and $\Delta t\rightarrow 0$, such that $\Delta x/\Delta t\rightarrow c$, in
first order we get the massive Dirac equation in (1+1)-dimensional spacetime
(1). We call such quantum automaton as the Dirac PUQCA.

##  III Coarse-graining a 1D PUQCA

Now that we have established the quantum cellular automaton that simulates the
1+1 Dirac equation, we use the automaton spatial structure to define a coarse-
graining map. The physical idea is that an observer with low resolution will
not be able to discriminate between different automaton cells.

To incorporate this idea in our approach, we employ completely positive and
trace preserving (CPTP) maps, a.k.a. quantum channels, to throw away
inaccessible information in a general way. This approach has been developed
and used in other scenarios pedrinho ; Oleg ; mermin1980 ; isadora2020 ; CG_QM
; cris2019 ; alicki2009 ; poulin2005 ; caslavLG ; Wang2013 ; Jeong2014 ;
silva2020macro . In general, a coarse-graining channel will be defined as a
quantum channel which is a trace preserving and completely positive map that
reduces the description of the system:
$\Lambda:\mathcal{L}(\mathcal{H}_{D})\rightarrow\mathcal{L}(\mathcal{H}_{d})$
with $D>d$, where $\mathcal{L}(\mathcal{H})$ is the set of all linear
operators acting on $\mathcal{H}$.

Here, we will apply a coarse-graining channel in each cell of the automaton,
reducing their two subcells into a single subcell. Locally the coarse-graining
map acts as follows:

$\Lambda_{k\Delta x}:\mathcal{L}\left(\left(\mathbb{C}_{0}^{2}\right)_{k\Delta
x}\otimes\left(\mathbb{C}_{1}^{2}\right)_{k\Delta
x}\right)\rightarrow\mathcal{L}\left(\left(\mathbb{C}_{k\mod
2}^{2}\right)_{\lfloor{\frac{k}{2}}\rfloor\Delta x}\right).$ (7)

The application of the coarse-graining map in the whole automaton is then
given by:

$\Lambda_{CG}=\bigotimes_{k\in\mathbb{Z}}\Lambda_{k\Delta x}.$ (8)

Notice that the application of the coarse-graining map preserves the automaton
local structure. It is also important to realize that after “compressing” the
two subcells at a position $x$ into an effective subcell, this subcell is
merged with a neighboring effective subcell to form an effective cell. The
distance between two effective cells is still $\Delta x$. By this procedure,
we establish a new level of description of the automaton, with a new 1-d
lattice.

To finalize our procedure to obtain an effective description, we must define
the action of $\Lambda_{x}$. To model the lack of resolution, we will employ a
coarse-graining map that cannot distinguish if an excitation is in the left or
right subcell of a point $x$. The action of $\Lambda_{x}$, which was first
suggested in CG_QM  in the context of atoms in a optical lattice, is as
follows:

$\Lambda_{x}\left(\left|01\rangle\\!\langle 01\right|\right)=\left|1\rangle\\!\langle 1\right|$ |  |  | $\Lambda_{x}\left(\left|01\rangle\\!\langle 00\right|\right)=\left|1\rangle\\!\langle 0\right|/\sqrt{3}$
---|---|---|---
$\Lambda_{x}\left(\left|10\rangle\\!\langle 10\right|\right)=\left|1\rangle\\!\langle 1\right|$ |  |  | $\Lambda_{x}\left(\left|10\rangle\\!\langle 00\right|\right)=\left|1\rangle\\!\langle 0\right|/\sqrt{3}$
$\Lambda_{x}\left(\left|00\rangle\\!\langle 00\right|\right)=\left|0\rangle\\!\langle 0\right|$ |  |  | $\Lambda_{x}\left(\left|10\rangle\\!\langle 01\right|\right)=0$
(9)

The first column contains the diagonal elements and expresses the inability to
distinguish whether the excitation is on the left or the right subcell. If no
excitation is present, then effectively no excitation is observed. It is
clearly observed that this map is not equivalent to a partial trace. The
second column contains the off-diagonal terms. As the the base vectors in the
excited subspace cannot be discriminated, then there can be no coherence among
them. The only possible coherence terms are in between the no excitation
subspace and the single-excitation subspace. The factor $1/\sqrt{3}$
guarantees the complete positivity of $\Lambda_{x}$. As quantum channels
preserve the Hermiticity, the action on the Hermitian conjugated terms can be
apprehended from (9). Further discussion on this map and its action on two-
particle subspace can be found in CG_QM ; pedrinho .

Given the map $\Lambda_{x}$, we are now ready to find the coarse-grained 1D
PUQCA. As the coarse-graining channel maps the 1-d lattice to another similar
1-d lattice, we can apply this map $L$ times on the underlying automation to
obtain its $L$th level of description. In the one-particle sector, we can
write the automaton state at level $L$ of description as

$\varrho_{L}=\sum_{k,k^{\prime}\in\mathbb{Z}}\sum_{a,a^{\prime}=0}^{1}(\varrho_{L})^{k\Delta
x,a}_{k^{\prime}\Delta x,a^{\prime}}\ket{(k,a)}\bra{(k^{\prime},a^{\prime})},$
(10)

where $\ket{(k,a)}$ represents the state with an excitation at position
$k\Delta x$ and subcell $a$. The state at level $L+1$, i.e., after we apply
the coarse-graining map $\Lambda_{CG}$ in the whole automaton, is given by:

$\displaystyle\varrho_{L+1}=\sum_{k,k^{\prime}\in\mathbb{Z}}\sum_{a,a=0}^{1}\left(\delta_{k,k^{\prime}}\delta_{a,a^{\prime}}+\frac{\left(1-\delta_{k,k^{\prime}}\right)}{3}\right)$
$\displaystyle\quad\times(\varrho_{L})^{k\Delta x,a}_{k^{\prime}\Delta
x,a^{\prime}}\left|\left(\left\lfloor{\frac{k}{2}}\right\rfloor,k\;\text{mod}\;2\right)\right\rangle\left\langle\left(\left\lfloor{\frac{k^{\prime}}{2}}\right\rfloor,k^{\prime}\;\text{mod}\;2\right)\right|.$
(11)

The derivation of this expression is done in Appendix VI.1.

From the above expression, we can derive, by recursion, the density matrix
coefficients for level $L$ in terms of the density matrix coefficients of
level 0:

$\left(\varrho_{L}\right)_{x,b}^{x,b}=\sum_{s=0}^{2^{L-1}-1}\sum_{a=0}^{1}\left(\varrho_{0}\right)_{2^{L}x+2^{L-1}b\Delta
x+s\Delta x,a}^{2^{L}x+2^{L-1}b\Delta x+s\Delta x,a},$ (12)

and

$\left(\varrho_{L}\right)_{x^{\prime},b^{\prime}}^{x,b}=\frac{1}{3^{L}}\sum_{s,s^{\prime}=0}^{2^{L-1}-1}\sum_{a,a^{\prime}=0}^{1}\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x+s^{\prime}\Delta x,a^{\prime}}^{2^{L}x+2^{L-1}b\Delta x+s\Delta x,a},$ (13)

for $x\neq x^{\prime}$ and $b,\;b^{\prime}\in\\{0,1\\}$. From the recursive
relation above for the off-diagonal elements of $\varrho_{L}$, we show in
Appendix VI.2 that
$\left|\left(\varrho_{L}\right)_{x^{\prime},b^{\prime}}^{x,b}\right|\leq(2\sqrt{2}/3)^{L}$,
i.e. the coherence elements converge exponentially fast to zero with the
coarse-graining level. To illustrate this exponential decay, consider the case
where at $L=0$ we start the automaton in the state
$\ket{\Psi(0)}=(\ket{(0,0)}+\ket{(0,1)})/\sqrt{2}$ and we evolve it with the
Dirac PUQCA, setting $\theta=\pi/4$ in equation (II), for 200 time steps.
Figure 2 shows how the sum of all absolute values of the off-diagonal terms
decays with level of coarse-graining.

Figure 2:  This plot shows the semilog sum of the off-diagonal elements in
different levels of CG for a fixed time. Starting with a QCA state at $t=0$ in
$L=0$, the sum of the absolute values of the coherence elements is taken from
$L=0$ to $L=6$. An exponential decay behavior can be noticed.
Figure 3:  The coarse-graining procedure for an arbitrary level of $L$ and
time step $h\Delta t$. While at the level $L=0$ the time evolution operator,
$\mathcal{E}$, is given by a unitary map, at the level $L$ the time evolution
is given by a quantum channel $\tilde{\mathcal{E}}$ derived from
$\mathcal{E}$.

##  IV Coarse-grained Dirac PUQCA

In the previous section, we computed the coarse-grained density operator for
an arbitrary level $L$ and we wrote the recursive equations over different
$L$. In this section, we will compute the coarse-grained QCA. Before the
coarse-graining map, the QCA is governed by equation (II) and it is well known
that such evolution models a free evolution of Dirac particles, as we recalled
in the first section. Now, what can we say about the dynamics at a generic
level $L$? We certainly know that the QCA will lose coherence at each
application of the coarse-graining map. For sake of simplicity, we start
restricting ourselves to $\theta=0$, the massless case. This means, from
equation (II), that $\psi^{0}$ and $\psi^{1}$ are fully decoupled:
left(right)-moving modes at level $L=0$ maps to left(right)-moving modes at
level $L\geq 1$. Thus, let us focus on the population terms. Before we coarse-
grain the density operator at time $t$, we add a new information about the
initial condition. Here we assume that we start with a superposed state in the
center of the lattice of size $2^{L}+1$, that is for $l=0,1,\ldots,L-1$

$\displaystyle\left(\varrho_{0}(0)\right)^{2^{L-1-l},0}_{2^{L-1-l},0}$
$\displaystyle=\alpha^{(0)}_{l},$ (14)
$\displaystyle\left(\varrho_{0}(0)\right)^{2^{L-1+l},1}_{2^{L-1+l},1}$
$\displaystyle=\alpha^{(1)}_{l},$

where $\alpha^{(1)}_{l},\alpha^{(0)}_{l}\in\mathbb{R}$ and
$\sum_{l=0}^{L-1}(\alpha^{(0)}_{l}+\alpha^{(1)}_{l})=1$. Now, for a given time
$t\ll 2^{L}$ from the massless dynamics, equation (II), since any
superposition is created anymore when $\theta=0$, all amplitudes that started
on the left(right) of the center remain in the subcells 0(1) that is

$\displaystyle\left(\varrho_{L}(t)\right)^{x,0}_{x,0}$ $\displaystyle=$
$\displaystyle\sum_{s=0}^{2^{L-1}-1}\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x,0}_{2^{L}x+s\Delta x,0}$ (15)
$\displaystyle\left(\varrho_{L}(t)\right)^{x,1}_{x,1}$ $\displaystyle=$
$\displaystyle\sum_{s=2^{L-1}}^{2^{L}-1}\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x,1}_{2^{L}x+s\Delta x,1}.$ (16)

Now, let us coarse-graining the density operator $\varrho$ at time $t+\Delta
t$ and then use equation (II) for $\theta=0$:

$\displaystyle\varrho^{(0)}_{L}(t+\Delta t,x)$ $\displaystyle=$
$\displaystyle\sum_{s=0}^{2^{L-1}-1}\left(\varrho_{0}(t+\Delta
t)\right)^{2^{L}x+s\Delta x,0}_{2^{L}x+s\Delta x,0}$ $\displaystyle=$
$\displaystyle\sum_{s=0}^{2^{L-1}-1}\left(\varrho_{0}(t)\right)^{2^{L}x-\Delta
x+s\Delta x,0}_{2^{L}x-\Delta x+s\Delta x,0},$
$\displaystyle\varrho^{(1)}_{L}(t+\Delta t,x)$ $\displaystyle=$
$\displaystyle\sum_{s=0}^{2^{L-1}-1}\left(\varrho_{0}(t+\Delta
t)\right)^{2^{L}x+s\Delta x,1}_{2^{L}x+s\Delta x,1}$ $\displaystyle=$
$\displaystyle\sum_{s=2^{L-1}}^{2^{L}-1}\left(\varrho_{0}(t)\right)^{2^{L}x+\Delta
x+s\Delta x,1}_{2^{L}x+\Delta x+s\Delta x,1},$

where we adopted the notation $\varrho^{(i)}_{L}(t+\Delta
t,x)=\left(\varrho_{L}(t)\right)^{x,i}_{x,i}$ for $i=0,1$ to describe the the
density matrix elements in the level $L$. The challenge here is to see whether
in the right-hand side of equations (IV) and (IV), there is a density matrix
in the level $L$ at time $t$ which is derived from the a density matrix of
level 0 at time $t$. It turns out that the diagonal elements of the density
matrix of level 0 after we coarse-grained $L$ times are
$\varrho^{(i)}_{L}(t,x-(-1^{i})\Delta x/2^{L})$, and therefore,

$\displaystyle\varrho^{(0)}_{L}(t+\Delta t,x)$ $\displaystyle=$
$\displaystyle\varrho^{(0)}_{L}(t,x-\Delta x/2^{L})$ (19)
$\displaystyle\varrho^{(1)}_{L}(t+\Delta t,x)$ $\displaystyle=$
$\displaystyle\varrho^{(1)}_{L}(t,x+\Delta x/2^{L}).$

Now, Taylor expands the above equation around $x$ and $t$ up to a first order,
and the following continuous differential equations can be established

$\displaystyle\partial_{t}\varrho^{(0)}_{L}(t,x)$ $\displaystyle=$
$\displaystyle-\frac{c}{2^{L}}\partial_{x}\varrho^{(0)}_{L}(t,x)$ (20)
$\displaystyle\partial_{t}\varrho^{(1)}_{L}(t,x)$ $\displaystyle=$
$\displaystyle\frac{c}{2^{L}}\partial_{x}\varrho^{(1)}_{L}(t,x).$

where $c=\Delta x/\Delta t$. From the results above, it is clear that the
factor $1/2^{L}$ starts to create a stationary state as we increase the level
$L$, see figures (4). In these plots, we can see the contrast between the
light cones after the CG map generated from a massless particle which started
in a superposition state in $x=0$ at $t=0$. In other words, by doing only CG
in space, trivial dynamics start to appear fast. Inspired by the results given
in costaPCA  more interesting cases can be derived when the coarse-graining in
time is also done. In this procedure, the transition function, $\mathcal{E}$,
can be applied $h$ times before each CG application. The number of
$\mathcal{E}$ applications satisfies the following upper bound $h\leq s$,
where $s$ is the amount of grouped subcells where the local maps $\Lambda_{x}$
are applied. In our case, from equation (7), $s=2$.

(a)
(b)

Figure 4:  Light cone propagation before (a) and after (b) two space coarse-
grained transformation. The particle propagation becomes slower at each CG
transformation, where the speed of the particle is rescaled by the factor
$1/2^{L}$.

In figure (3) we elucidated the procedure with an arbitrary value for $h$.
Notice that in the new level the update state is achieved by only one
application of $\tilde{\mathcal{E}}$, i.e., the quantum channel emerged from
the CG process. However, in the level $L$ we rescaled the time from $t+h\Delta
t$ to $t+\Delta t$ when $h\neq 1$. As explained in costaPCA , we do not accept
$h>s$ since we want to preserve the local structure in the emergent dynamics
which comes from the light cone existent in the QCA evolution.

Let us go back to our previous problem, given by equation (20), where the
localization starts to appear. Combining the temporal with the spatial CG,
this no physical situation generated by the particle localization can be
avoided. In fact, we can keep with the same light cone structure if before
each CG operation we compensate it by applying the transition function twice.
Thus, since we are moving from the level 0 straight to the level $L$, which is
equivalent of we group $2^{L}$ cells, the velocity can be kept constant if the
compensation by $h=2^{L}$ is done before $\Lambda_{CG}^{L}$. In other words,
we have $t+2^{L}\Delta t$ in the level 0 equivalent to $t+\Delta t$ in the
level $L$,

$\displaystyle\left(\varrho_{L}(t+\Delta t)\right)^{x,0}_{x,0}$
$\displaystyle=$
$\displaystyle\sum_{s=0}^{2^{L-1}-1}\left(\varrho_{0}(t+2^{L}\Delta
t)\right)^{2^{L}x+s\Delta x,0}_{2^{L}x+s\Delta x,0}$
$\displaystyle\left(\varrho_{L}(t+\Delta t)\right)^{x,1}_{x,1}$
$\displaystyle=$
$\displaystyle\sum_{s=2^{L-1}}^{2^{L}-1}\left(\varrho_{0}(t+2^{L}\Delta
t)\right)^{2^{L}x+s\Delta x,1}_{2^{L}x+s\Delta x,1}.$

Therefore, $x\pm\Delta x/2^{L}$, goes to $x\pm\Delta x$, when we have
$t+2^{L}\Delta t$ in the level 0 instead $t+\Delta t$, and consequently the
factor $1/2^{L}$ in equation (20) is removed, that is

$\displaystyle\partial_{t}\varrho^{(0)}_{L}(t,x)$ $\displaystyle=$
$\displaystyle-c\partial_{x}\varrho^{(0)}_{L}(t,x)$ (21)
$\displaystyle\partial_{t}\varrho^{(1)}_{L}(t,x)$ $\displaystyle=$
$\displaystyle c\partial_{x}\varrho^{(1)}_{L}(t,x).$

Before we move to the case where $m\neq 0$, here we can already point out that
the emergent dynamics derived in the classical limit can be easily described
by the partitioned cellular automata costaPCA ; Wol00 ; Margolus  with two
subcells, which is equivalent to two bits at each cell in the classical case.
The transition function is composed of two permutation operators. Both
permutations are the SWAP operator, analogous to the PUQCA used to describe
the massless case, that is, the first one acting locally inside all vertices,
moving the excitation from the subcell 0 (1) to the he subcell 1 (0), and the
second interacting the vertices, sending the excitation from the vertex $x$,
in the subcell 1 (0) to $x+1$ $(x-1)$ in the subcell 0 (1).

The starting point for the massive case is the recurrence relation for the
probabilities. From equation (II) when $\sin\theta\approx\theta$, in terms of
the density matrix elements, we have

$\displaystyle\varrho(t+\Delta t)_{x,0}^{x,0}$
$\displaystyle\approx\varrho(t)_{x-\Delta x,0}^{x-\Delta x,0}$ (22)
$\displaystyle\quad+i\frac{mc^{2}\Delta t}{\hbar}\left(\varrho(t)_{x-\Delta
x,1}^{x-\Delta x,0}-\varrho(t)_{x-\Delta x,0}^{x-\Delta x,1}\right)$
$\displaystyle\varrho(t+\Delta t)_{x,1}^{x,1}$
$\displaystyle\approx\varrho(t)_{x+\Delta x,1}^{x+\Delta x,1}$
$\displaystyle\quad+i\frac{mc^{2}\Delta t}{\hbar}\left(\varrho(t)_{x+\Delta
x,0}^{x+\Delta x,1}-\varrho(t)_{x+\Delta x,1}^{x+\Delta x,0}\right),$

Notice that in the pair of equations above the second-order terms in time,
i.e., $\Delta t^{2}$, are not included. Now, we will not assume anything about
the initial condition, and thus, in the level $L$ we have

$\displaystyle\left(\varrho_{L}(t)\right)^{x,0}_{x,0}$
$\displaystyle=\sum_{s=0}^{2^{L-1}-1}\left[\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x,0}_{2^{L}x+s\Delta x,0}+\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x,1}_{2^{L}x+s\Delta x,1}\right]$
$\displaystyle=\sum_{s=0}^{2^{L-1}-1}\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x}_{2^{L}x+s\Delta x},$ $\displaystyle\left(\varrho_{L}(t)\right)^{x,1}_{x,1}$
$\displaystyle=\sum_{s=2^{L-1}}^{2^{L}-1}\left[\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x,0}_{2^{L}x+s\Delta x,0}+\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x,1}_{2^{L}x+s\Delta x,1}\right],$
$\displaystyle=\sum_{s=2^{L-1}}^{2^{L}-1}\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x}_{2^{L}x+s\Delta x},$

wherein the dependence of the internal state of the particle was eliminated in
the right part of the equality above. This elimination can be done since the
probability of finding a particle in the cell $x$ is given by
$\varrho(t)^{x,0}_{x,0}+\varrho(t)^{x,1}_{x,1}$. If we also add both equations
above, we get

$\left(\varrho_{L}(t)\right)^{x}_{x}=\sum_{s=0}^{2^{L}-1}\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x}_{2^{L}x+s\Delta x}.$ (23)

Likewise the massless case, our next step is to look up our density matrix at
time $t+\Delta t$ and then making use of equations (22). Thus, the update
state at level $L$ writing in terms of the level 0 is

$\displaystyle\left(\varrho_{L}(t+\Delta t)\right)^{x,0}_{x,0}$
$\displaystyle\approx\sum_{s=0}^{2^{L-1}-1}\left[\left(\varrho_{0}(t)\right)^{2^{L}x-\Delta
x+s\Delta x,0}_{2^{L}x-\Delta x+s\Delta
x,0}+\left(\varrho_{0}(t)\right)^{2^{L}x+\Delta x+s\Delta x,1}_{2^{L}x+\Delta
x+s\Delta x,1}\right.$ $\displaystyle\quad+i\frac{mc^{2}\Delta
t}{\hbar}\left(\left(\varrho_{0}(t)\right)^{2^{L}x-\Delta x+s\Delta
x,0}_{2^{L}x-\Delta x+s\Delta x,1}-\left(\varrho_{0}(t)\right)^{2^{L}x-\Delta
x+s\Delta x,1}_{2^{L}x-\Delta x+s\Delta x,0}\right.$
$\displaystyle\quad+\left.\left.\left(\varrho_{0}(t)\right)^{2^{L}x+\Delta
x+s\Delta x,1}_{2^{L}x+\Delta x+s\Delta
x,0}-\left(\varrho_{0}(t)\right)^{2^{L}x+\Delta x+s\Delta x,0}_{2^{L}x+\Delta
x+s\Delta x,1}\right)\right],$ (24) $\displaystyle\left(\varrho_{L}(t+\Delta
t)\right)^{x,1}_{x,1}$
$\displaystyle\approx\sum_{s=2^{L-1}}^{2^{L}-1}\left[\left(\varrho_{0}(t)\right)^{2^{L}x-\Delta
x+s\Delta x,0}_{2^{L}x-\Delta x+s\Delta
x,0}+\left(\varrho_{0}(t)\right)^{2^{L}x+\Delta x+s\Delta x,1}_{2^{L}x+\Delta
x+s\Delta x,1}\right.$ $\displaystyle\quad+i\frac{mc^{2}\Delta
t}{\hbar}\left(\left(\varrho_{0}(t)\right)^{2^{L}x-\Delta x+s\Delta
x,0}_{2^{L}x-\Delta x+s\Delta x,1}-\left(\varrho_{0}(t)\right)^{2^{L}x-\Delta
x+s\Delta x,1}_{2^{L}x-\Delta x+s\Delta x,0}\right.$
$\displaystyle\quad+\left.\left.\left(\varrho_{0}(t)\right)^{2^{L}x+\Delta
x+s\Delta x,1}_{2^{L}x+\Delta x+s\Delta
x,0}-\left(\varrho_{0}(t)\right)^{2^{L}x+\Delta x+s\Delta x,0}_{2^{L}x+\Delta
x+s\Delta x,1}\right)\right].$ (25)

In this stage, we can already do a Taylor expansion in both sides. On the left
side, we expand until the first order in time. However, on the right side
there is a subtlety that should be noticed. First, the expansion is done
around the point $2^{L}x+p$, where $p=\pm s\Delta x$. Second, while for the
terms without the mass the expansion is done until the first order, that is

$\displaystyle\left(\varrho_{0}(t)\right)^{2^{L}x-\Delta x+s\Delta
x,0}_{2^{L}x-\Delta x+s\Delta x,0}+\left(\varrho_{0}(t)\right)^{2^{L}x+\Delta
x+s\Delta x,1}_{2^{L}x+\Delta x+s\Delta x,1}$ (26)
$\displaystyle\quad\approx\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x}_{2^{L}x+s\Delta x}-\frac{\Delta
x}{2^{L}}\partial_{x}\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x,0}_{2^{L}x+s\Delta x,0}$ $\displaystyle\quad+\frac{\Delta
x}{2^{L}}\partial_{x}\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x,1}_{2^{L}x+s\Delta x,1},$

the ones where the mass appears we only consider the zero order in the
expansion. This difference in the order considered during the Taylor expansion
appears because the ones which includes the mass are already first order terms
in $\Delta t$. However, it turns out that the result of adding all zero orders
terms eliminates the mass in the emergent dynamics, since

$\left(\varrho_{0}(t)\right)^{2^{L}x-\Delta x+s\Delta x,0}_{2^{L}x-\Delta
x+s\Delta x,1}-\left(\varrho_{0}(t)\right)^{2^{L}x+\Delta x+s\Delta
x,0}_{2^{L}x+\Delta x+s\Delta x,1}\approx O\left(\frac{\Delta
x}{2^{L}}\right).$ (27)

Thus, the following dynamics is derived for the massive case

$\displaystyle\partial_{t}\left(\varrho_{L}(t)\right)^{x}_{x}$
$\displaystyle=\sum_{s=0}^{2^{L}-1}\frac{c}{2^{L}}\partial_{x}\left(\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x,1}_{2^{L}x+s\Delta x,1}-\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x,0}_{2^{L}x+s\Delta x,0}\right).$ (28)

Therefore, all the interference terms that would appear as a consequence of
the mass are not detectable in the level $L$. Moreover, since in equation (IV)
the dynamics of the level $L$ is level 0 dependent, there is not a model in
the level $L$ which is derived from the level 0 when the mass of the particle
is included.

However, in the level $L$ the internal degree of the particle is not perceived
anymore, and then, since the states 0 and 1 belong to the same cell but
located in different points, as shown in figure (1b), we can say that the
displacement between them is $\Delta x$ as well. Therefore, by setting

$\left(\varrho_{0}(t)\right)^{x,1}_{x,1}=\left(\varrho_{0}(t)\right)^{x}_{x},$
(29)

the following equality holds

$\left(\varrho_{0}(t)\right)^{x,0}_{x,0}=\left(\varrho_{0}(t)\right)^{x-\Delta
x}_{x-\Delta x}.$ (30)

Using this equivalence into equation (IV), and considering that we have
$\Delta x^{2}/\Delta t=\lambda$, we get

$\displaystyle\partial_{t}\varrho_{L}(t,x)$
$\displaystyle=\sum_{s=0}^{2^{L}-1}\frac{\Delta x^{2}}{2^{L}\Delta
t}\partial_{x}^{2}\left(\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x}_{2^{L}x+s\Delta x}\right)$ (31) $\displaystyle=\frac{\Delta
x^{2}}{2^{L}\Delta
t}\partial_{x}^{2}\left(\sum_{s=0}^{2^{L}-1}\left(\varrho_{0}(t)\right)^{2^{L}x+s\Delta
x}_{2^{L}x+s\Delta x}\right)$
$\displaystyle=\frac{\lambda}{2^{L}}\partial_{x}^{2}\varrho_{L}(t,x).$

which is the diffusion equation.

In figure (5) we contrast the variance between the level 0, that simulates the
Dirac equation, and the emerged one. In the level 0, we can see the character
linear of the variance, as expected from the quantum walks dynamics. However,
in the level 3 we can notice a nonlinear behavior, similar to the dispersion
relation provided by the random walk. Such result shown in figure (5) supports
the continuous limit derived in equation (31).

Figure 5:  Variance against time from the QCA at level 0 and the emergent
dynamics derived at level 3. In both cases, the same value $\theta=0.2$ is
used.

##  V Conclusion

In this work, we studied emergent dynamics starting from the Dirac like
Hamiltonian, i.e., $H_{D}=\sigma_{x}\otimes p_{x}+m\sigma_{z}\otimes I$, where
the dynamics is driven by $\partial_{t}\varrho=\left[H,\varrho\right]$. We
used the partitioned unitary quantum cellular automata QCAviaQws  as the
computational model to derive $\exp(iH_{D})$. Inspired by costaPCA , we
provided an analogous coarse-graining scheme to explore emergent dynamics, but
now for its quantum version, namely the PUQCA.

We presented a numerical instance and analytical evidence that the coarse-
graining map employed here generates decoherence in our quantum system. As a
consequence of multiple CG map applications, a semiclassical system can be
achieved. Furthermore, we explicitly show that under a particular initial
condition for the massless case, the classical transport given by
$\partial_{t}\varrho_{L}+v\sigma_{z}\partial_{x}\varrho_{L}=0$ is the emergent
dynamics by computing the continuous limit. In this emergent system, the
velocity $v$ can either go exponentially faster to zero, i.e., $v\sim
1/2^{L}$, or remain the same if the temporal CG is also included. On the other
hand, when the mass is included and no assumptions about the initial
conditions are done, the diffusion equation is recovered. It is well known
that the diffusion equation is achieved by computing the continuous limit of
the random walk. Therefore, the CG procedure developed here provided the right
tools to see how the random walk can be derived from the quantum walk.

Differently from costaPCA  that shows a general procedure to study emergent
dynamics, here just one map and one type unitary operators are used. However,
our results are pointing out that this procedure can be generalized for
different complete positive trace-preserving maps as well as for different
dynamics, including the cases where we have interacting quantum walks twoWalk1
; twoWalk2 ; twoWalk3 , and we intend to address these possibilities in future
publications. Another question that immediately raises from our procedure is
whether we can derive emergent local quantum channels $\tilde{\mathcal{E}}$
arising from QCA, following the procedure shown in CG_QM ;
duarte2020investigating . If such quantum channels can be achieved, we will
provide a powerful tool to generate transition functions such that noisy
quantum cellular automata, as described in NoiseQCA , naturally appear during
the decoherence process.

Therefore, this formalism might be an excellent candidate to study open
quantum systems and also to shed some light on more fundamental questions in
the quantum theory, e.g., a tool to explore quantum-to-classical transition
via quantum cellular automata. Finally, another possible direction to go is
contrasting the emergent dynamics derived here against alternative formulation
to study different coarse levels like the use of the wavelets basis wavelet .

## Acknowledgements

I am really grateful to Fernando de Melo who introduced the subject of coarse-
graining and proposed the topic of emergent dynamics via quantum cellular
automata during my PhD I also would like to thank Giuseppe Di Molfetta for
helpful discussion during this project, Yuval R. Sanders for careful reading
of the manuscript, and Gavin Brennen for helpful comments. This work was
partially supported by ARC Discovery Projects.

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##  VI Appendices

###  VI.1 Effective state at level $L+1$ from the state at level $L$

In the following, we apply the coarse-graining local map to the operator (10),
getting a general expression for the coarse-grained density operator in level
$L+1$. To get this general form, note that we can distinguish three different
cases: (i) $x=x^{\prime}$ and $a=a^{\prime}$, (ii) $a\neq a^{\prime}$, for all
for $x=x$, and (iii) $x\neq x$ . In case (i), the state at level $L+1$ reads:

$\displaystyle\Lambda_{CG}\left(\left|k,a\rangle\\!\langle k,a\right|\right)$
$\displaystyle=\left|\cdots\left(0\right)_{k-1}\left(1\right)_{k}\left(0\right)_{k+1}\cdots\right\rangle\left\langle\cdots\left(0\right)_{k-1}\left(1\right)_{k}\left(0\right)_{k+1}\cdots\right|.$

Note that the three conditions for $x$ shown above imply the same for $k$. The
number of subcells for each position is reduced to one, and we have to space
group subcells at different spatial positions, to build a new cell hosting two
qubits. Consequently, the choice of the space grouping of the subcells, at a
higher level, might be determinant and may depend on the spatial position at
level $L$. In fact, applying the map at even position $x$ leads to:

$\left|\cdots\left(1\right)_{\frac{k}{2}}\cdots\right\rangle\left\langle\cdots\left(1\right)_{\frac{k}{2}}\cdots\right|=\left|\frac{k}{2},0\right\rangle\left\langle\frac{k}{2},0\right|.$

where the one particle excitation takes now this place at $a=0$, the left
subcell. While if the local map is applied at odd position $x$, the single-
particle excitation will be placed in the right subcell $a=1$:

$\left|\cdots\left(1\right)_{\frac{k-1}{2}}\cdots\right\rangle\left\langle\cdots\left(1\right)_{\frac{k-1}{2}}\cdots\right|=\left|\frac{k-1}{2},1\right\rangle\left\langle\frac{k-1}{2},1\right|.$

Notice once again that, without lack of generality, we space-group from the
left to the right.

Both the above cases can be recast as follows:

$\Lambda_{CG}\left(\left|k,a\rangle\\!\langle
k,a\right|\right)=\left|\frac{k}{2},k\;\text{mod}\;2\rangle\\!\langle\frac{k}{2},k\;\text{mod}\;2\right|$
(32)

Case (ii), i.e., $a\neq a^{\prime}$ and $x=x^{\prime}$ leads directly to

$\Lambda_{CG}\left(\left|k,a\right\rangle\left\langle
k,a^{\prime}\right|\right)=0,$ (33)

because $\Lambda_{x}\left(\left|01\right\rangle\left\langle
10\right|\right)=0$.

Finally, for $x\neq x^{\prime}$, we follow the same procedure, and we apply
the local map for each spatial position $x$:

$\displaystyle\Lambda_{CG}\left(\left|k,a\right\rangle\left\langle
k^{\prime},a^{\prime}\right|\right)=$
$\displaystyle\left|\ldots\left(1\right)_{k},\ldots\left(0\right)_{x_{L}^{\prime}},\ldots\right\rangle\left\langle\ldots,\left(0\right)_{k},\ldots\left(1\right)_{k^{\prime}},\ldots\right|,$

and then, from equation (9), we see that the CPTP map covering the coherence
terms leads us to

$\displaystyle\Lambda_{CG}\left(\left|k,a\right\rangle\left\langle
k^{\prime},a^{\prime}\right|\right)$
$\displaystyle=\frac{1}{3}\left|\left\lfloor\frac{k}{2}\right\rfloor,k\;\text{mod}\;2\right\rangle\left\langle\left\lfloor\frac{k^{\prime}}{2}\right\rfloor,k^{\prime}\;\text{mod}\;2\right|.$
(34)

Finally, recast together equations (32), (33) and (34), we get the final
expression shown in equation (11), which is the coarse grained density
operator at level $L+1$.

###  VI.2 Decoherence by coarse-graining

######  Lemma 1.

Let $p\in[0,1]^{D}$, with $D\geq 2$, be a probability distribution, i.e.,
$\sum_{i=1}^{D}p_{i}=1$. Then the inequality

$\sum_{i=1}^{D}\sqrt{p_{i}}\leq\sqrt{D}$ (35)

holds, and it is saturated when $p_{i}=1/D$ for all $i\in[D]$.

###### Proof.

The proof goes by the way of induction. For the base case, $D=2$, we define
the function $f_{2}:[0,1]\rightarrow\mathbb{R}$ as

$f_{2}(p_{1})=\sqrt{p_{1}}+\sqrt{1-p_{1}};$

where we already employed the normalization property. Setting
$df_{2}/dp_{1}=0$ is simple to obtain that the maximum is obtained for
$p_{1}=1/2$. Therefore, $\sqrt{p_{1}}+\sqrt{1-p_{1}}\leq\sqrt{2}$.

For the inductive step, we assume that
$\sum_{i=0}^{D-1}\sqrt{p_{i}}\leq\sqrt{D-1}$. For $D$ we have:

$\displaystyle\sum_{i=0}^{D}\sqrt{p_{i}}$
$\displaystyle=\sum_{i=0}^{D-1}\sqrt{p_{i}}+\sqrt{p_{D}}$
$\displaystyle=\sqrt{\sum_{j=0}^{D-1}p_{j}}\left(\sum_{i=0}^{D-1}\sqrt{\frac{p_{i}}{\sum_{j=0}^{D-1}p_{j}}}\right)+\sqrt{p_{D}}.$
(36)

Noting that defining $p^{\prime}_{i}=\sqrt{p_{i}/\sum_{j=0}^{D-1}p_{j}}$, then
$p^{\prime}$ is a probability distribution in $[0,1]^{D-1}$, for which the
induction hypothesis can be applied, and that $\sum_{j=0}^{D-1}p_{j}=1-p_{D}$
we have:

$\sum_{i=0}^{D}\sqrt{p_{i}}\leq\sqrt{1-p_{D}}\sqrt{D-1}+\sqrt{p_{D}}.$ (37)

Finally, the function $f_{D}(p_{D})=\sqrt{1-p_{D}}\sqrt{D-1}+\sqrt{p_{D}}$ has
its maximum value, $\sqrt{D}$, at $p_{D}=1/D$, what finishes the proof. ∎

Before we proof that the off-diagonal elements of the density matrix go
exponentially faster to zero, the simplest case of the upper bound value from
the level 0 to the level 1 is explicitly computed.

From equation (13), for $b=b^{\prime}=0$, we see that when we move from the
level $0$ to $1$ we have,

$\displaystyle\left(\varrho_{1}\right)^{x,0}_{x,0}=$
$\displaystyle\frac{1}{3}\left[\left(\varrho_{0}\right)^{2x,0}_{2x^{\prime},0}+\left(\varrho_{0}\right)^{2x,1}_{2x^{\prime},1}\right.$
(38)
$\displaystyle\quad+\left.\left(\varrho_{0}\right)^{2x,1}_{2x^{\prime},0}+\left(\varrho_{0}\right)^{2x,0}_{2x^{\prime},1}\right].$

Now, from the triangle inequality, i.e.,
$\left|a+b\right|\leq\left|a\right|+\left|b\right|$, we can write

$\displaystyle\left|\left(\varrho_{1}\right)^{x,0}_{x^{\prime},0}\right|\leq$
$\displaystyle\frac{1}{3}\left[\left|\left(\varrho_{0}\right)^{2x,0}_{2x^{\prime},0}\right|+\left|\left(\varrho_{0}\right)^{2x,1}_{2x^{\prime},1}\right|\right.$
(39)
$\displaystyle\quad+\left.\left|\left(\varrho_{0}\right)^{2x,1}_{2x^{\prime},0}\right|+\left|\left(\varrho_{0}\right)^{2x,0}_{2x^{\prime},1}\right|\right].$

Using the following inequality for the density matrix

$\left|\varrho^{r,a}_{r^{\prime},a^{\prime}}\right|^{2}\leq\varrho_{r,a}^{r,a}\varrho_{r^{\prime},a^{\prime}}^{r^{\prime},a^{\prime}},$
(40)

equation (39) can be written as

$\displaystyle\left|\left(\varrho_{1}\right)^{x,0}_{x^{\prime},0}\right|\leq$
$\displaystyle\frac{1}{3}\left[\sqrt{\left(\varrho_{0}\right)^{2x,0}_{2x,0}\left(\varrho_{0}\right)^{2x^{\prime},0}_{2x^{\prime},0}}+\sqrt{\left(\varrho_{0}\right)^{2x,1}_{2x,1}\left(\varrho_{0}\right)^{2x^{\prime},1}_{2x^{\prime},1}}\right.$
$\displaystyle\quad+\left.\sqrt{\left(\varrho_{0}\right)^{2x,1}_{2x,1}\left(\varrho_{0}\right)^{2x^{\prime},0}_{2x^{\prime},0}}+\sqrt{\left(\varrho_{0}\right)^{2x,0}_{2x,0}\left(\varrho_{0}\right)^{2x^{\prime},1}_{2x^{\prime},1}}\right].$

After a simple algebraic manipulation in the equation above, we have

$\displaystyle\left|\left(\varrho_{1}\right)^{x,0}_{x^{\prime},0}\right|\leq$
$\displaystyle\frac{1}{3}\sqrt{\left(\varrho_{0}\right)^{2x,0}_{2x,0}}\left(\sqrt{\left(\varrho_{0}\right)^{2x^{\prime},0}_{2x^{\prime},0}}+\sqrt{\left(\varrho_{0}\right)^{2x^{\prime},1}_{2x^{\prime},1}}\right)$
$\displaystyle\quad+\frac{1}{3}\sqrt{\left(\varrho_{0}\right)^{2x,1}_{2x,1}}\left(\sqrt{\left(\varrho_{0}\right)^{2x^{\prime},0}_{2x^{\prime},0}}+\sqrt{\left(\varrho_{0}\right)^{2x^{\prime},1}_{2x^{\prime},1}}\right).$

Noticing that

$\sqrt{\left(\varrho_{0}\right)^{2x,a}_{2x,a}}\left(\sqrt{\left(\varrho_{0}\right)^{2x^{\prime},0}_{2x^{\prime},0}}+\sqrt{\left(\varrho_{0}\right)^{2x^{\prime},1}_{2x^{\prime},1}}\right)\leq\sqrt{\left(\varrho_{0}\right)^{2x^{\prime},0}_{2x^{\prime},0}}+\sqrt{\left(\varrho_{0}\right)^{2x^{\prime},1}_{2x^{\prime},1}},$
(41)

since $\left(\varrho_{0}\right)^{2x,a}_{2x,a}\leq 1$ for $a=0,1$ we have that

$\left|\left(\varrho_{1}\right)^{x,0}_{x^{\prime},0}\right|\leq\frac{2}{3}\left(\sqrt{\left(\varrho_{0}\right)^{2x^{\prime},0}_{2x^{\prime},0}}+\sqrt{\left(\varrho_{0}\right)^{2x^{\prime},1}_{2x^{\prime},1}}\right).$
(42)

Now, from lemma 1

$\displaystyle\sqrt{\left(\varrho_{0}\right)^{2x^{\prime},0}_{2x^{\prime},0}}+\sqrt{\left(\varrho_{0}\right)^{2x^{\prime},1}_{2x^{\prime},1}}\leq\sqrt{2},$

therefore,

$\left|\left(\varrho_{1}\right)^{x,0}_{x^{\prime},0}\right|\leq\frac{2\sqrt{2}}{3}<1.$
(43)

######  Theorem 2.

Let
$\left(\varrho_{L}\right)\in\mathbb{C}^{n/2^{L}}\times\mathbb{C}^{n/2^{L}}$ be
the density matrix in the coarse level $L$, with $n\gg 2^{L}$, where their
off-diagonal elements in terms of the level 0 are given by equation (13).
Then, the absolute value of the coherence terms of $\left(\varrho_{L}\right)$
is upper bounded by

$\left(\frac{2\sqrt{2}}{3}\right)^{L},$ (44)

for $x\neq x^{\prime}$ and $b,b^{\prime}=0,1$.

###### Proof.

From equation (13), the following inequality can be written as

$\displaystyle\left|\left(\varrho_{L}\right)_{x^{\prime},b^{\prime}}^{x,b}\right|\leq\frac{1}{3^{L}}\sum_{s,s^{\prime}=0}^{2^{L-1}-1}\left(\left|\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x+s^{\prime}\Delta x,0}^{2^{L}x+2^{L-1}b\Delta x+s\Delta x,0}\right|\right.$
$\displaystyle\quad+\left|\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x+s^{\prime}\Delta x,0}^{2^{L}x+2^{L-1}b\Delta x+s\Delta
x,1}\right|\left|\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x+s^{\prime}\Delta x,1}^{2^{L}x+2^{L-1}b\Delta x+s\Delta x,0}\right|$
$\displaystyle\quad+\left.\left|\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x+s^{\prime}\Delta x,1}^{2^{L}x+2^{L-1}b\Delta x+s\Delta x,1}\right|\right).$
(45)

Now, using the inequality given by equation (40)

$\displaystyle\left|\left(\varrho_{L}\right)_{x^{\prime},b^{\prime}}^{x,b}\right|\leq\frac{1}{3^{L}}\sum_{s=0}^{2^{L-1}-1}$
$\displaystyle\quad\times\left[\sqrt{\left(\varrho_{0}\right)_{2^{L}x+2^{L-1}b\Delta
x+s\Delta x,0}^{2^{L}x+2^{L-1}b\Delta x+s\Delta
x,0}}\sum_{s^{\prime}=0}^{2^{L-1}-1}\left(\sqrt{\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s^{\prime}\Delta x,0}^{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s\Delta x^{\prime},0}}\right.\right.$
$\displaystyle\quad+\left.\sqrt{\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s^{\prime}\Delta x,1}^{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s\Delta x^{\prime},1}}\right)$
$\displaystyle\quad+\sqrt{\left(\varrho_{0}\right)_{2^{L}x+2^{L-1}b\Delta
x+s\Delta x,1}^{2^{L}x+2^{L-1}b\Delta x+s\Delta
x,1}}\sum_{s^{\prime}=0}^{2^{L-1}-1}\left(\sqrt{\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s^{\prime}\Delta x,0}^{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s\Delta x^{\prime},0}}\right.$
$\displaystyle\quad+\left.\left.\sqrt{\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s^{\prime}\Delta x,1}^{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s\Delta x^{\prime},1}}\right)\right].$

Now we can apply the inequality used in equation (41) to get

$\displaystyle\left|\left(\varrho_{L}\right)_{x^{\prime},b^{\prime}}^{x,b}\right|$
$\displaystyle\leq\frac{2^{L-1}}{3^{L}}\left[\sum_{s^{\prime}=0}^{2^{L-1}-1}\left(\sqrt{\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s^{\prime}\Delta x,0}^{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s\Delta x^{\prime},0}}\right.\right.$
$\displaystyle\quad+\left.\sqrt{\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s^{\prime}\Delta x,1}^{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s\Delta x^{\prime},1}}\right)$
$\displaystyle\quad+\sum_{s^{\prime}=0}^{2^{L-1}-1}\left(\sqrt{\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s^{\prime}\Delta x,0}^{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s\Delta x^{\prime},0}}\right.$
$\displaystyle\quad+\left.\left.\sqrt{\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s^{\prime}\Delta x,1}^{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s\Delta x^{\prime},1}}\right)\right],$

or

$\displaystyle\left|\left(\varrho_{L}\right)_{x^{\prime},b^{\prime}}^{x,b}\right|$
$\displaystyle\leq\frac{2^{L}}{3^{L}}\sum_{s^{\prime}=0}^{2^{L-1}-1}\left(\sqrt{\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s^{\prime}\Delta x,0}^{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s\Delta x^{\prime},0}}\right.$
$\displaystyle\quad+\left.\sqrt{\left(\varrho_{0}\right)_{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s^{\prime}\Delta x,1}^{2^{L}x^{\prime}+2^{L-1}b^{\prime}\Delta
x^{\prime}+s\Delta x^{\prime},1}}\right).$ (46)

The proof can be conclude by using the lemma 1 in the inequality above for
$D=2^{L}$ ∎


# Covering shrinking polynomials by quasi progressions

Norbert Hegyvári  Norbert Hegyvári, ELTE TTK, Eötvös University, Institute of
Mathematics, Pázmány st. 1/c, Budapest, Hungary and associated member of
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Science, H-1364
Budapest, P.O.Box 127<EMAIL_ADDRESS>
###### Abstract.

Erdős introduced the quantity $S=T\sum^{T}_{i=1}|X_{i}|$, where
$X_{1},\dots,X_{T}$ are arithmetic progressions that cover the squares up to
$N$. He conjectured that $S$ is close to $N$, i.e. the square numbers cannot
be covered ”economically” by arithmetic progressions. Sárközy confirmed this
conjecture and proved that $S\geq cN/\log^{2}N$. In this paper we extend this
to shrinking polynomials and so-called $\\{X_{i}\\}$ quasi progressions.

AMS 2010 Primary 11B75, 11B25 Secondary 11B83,

Keywords: Covering problems, quasi progressions,

##  1\. Introduction

A long-standing and challenging problem in combinatorial number theory is to
give an upper bound on the number of squares in any arithmetic progression. In
relation to this problem, Erdős posed the following question, which he
formulated as follows: Is it true that square numbers cannot be ”economically”
covered by arithmetic progressions? More precisely let
$X_{i}=\\{m_{i}j+r_{i}\\}_{j=1}^{k_{i}}\subseteq\\{1,2\dots,,N\\}$,
$i=1,2,\dots,T$ be a system of arithmetic progressions such that
$\bigcup_{i=1}^{T}\\{m_{i}j+r_{i}\\}_{j=1}^{k_{i}}\supseteq\mathcal{Q}_{N}$,
where $\mathcal{Q}_{N}$ is the set of squares up to $N$, i.e.
$\mathcal{Q}_{N}:=\\{1^{2},2^{2},3^{2},\dots,\lfloor\sqrt{N}\rfloor^{2}\\}$.
Let $F(\mathcal{Q}_{N}):=\min_{\\{X_{i}\\},T}T\sum_{i=1}^{T}k_{i}$. Is it true
that $F(\mathcal{Q}_{N})>N^{1-\varepsilon}$, ($\varepsilon>0)$?

Maybe this conjecture was motivated by the following two examples. When we
cover $\mathcal{Q}_{N}$ by arithmetic progression with length two (i.e. the
squares are covered by piecewise), then $T\sim\sqrt{N}$, so
$T\sum_{i=1}^{T}k_{i}\sim 2N$. The other example, when $\mathcal{Q}_{N}$ is
covered by the interval $\\{1,2,\dots,N\\}$, then $T=1$, so
$T\sum_{i=1}^{T}k_{i}=N$.

In $[8]$ this conjecture was proved in a sharper form:

######  Theorem 1.1 (Sárközy).

There exists an $N_{0}$ such that for all $N>N_{0}$ and

$F(\mathcal{Q}_{N})>\frac{1}{700}\frac{N}{\log^{2}N}.$

The purpose of this note is to investigate what happens if we ask quasi-
progression to efficiently cover shrinking polynomials (see Section 3). Quasi
progression properties have been studied by several authors (see [2],[3] and
[4]).

###  1.1. Erdős conjecture under some arithmetic constrain

We may ask, if we restrict ourselves to some additively behaving $m_{i}$s
modulus, what happens?

The following result shows that for ”good covering” the moduli must be highly
composite.

######  Theorem 1.2.

Assume that $K:=\max\\{\tau(m_{i}):i=1,2,\dots T\\}$ and
$\bigcup_{i=1}^{T}\\{m_{i}j+r_{i}\\}_{j=1}^{k_{i}}\supseteq Q_{N}$ where
$\tau(x)$ is the divisor function. Then
$T\sum_{i=1}^{T}k_{i}\geq\frac{N}{4K\log N}$.

###### Proof.

In [5] I proved the following lemma:

######  Lemma 1.3.

For every $i\in[T]$

$|\mathcal{Q}_{N}\cap\\{r_{i}+jm_{i}\\}_{j=1}^{k_{i}}|\leq
2\sqrt{K}\sqrt{k_{i}\log k_{i}}.$

###### Proof of the lemma.

Write shortly $m=m_{i};\ r=r_{i};\ k=k_{i}$.

Let $J:=\\{j_{1}<j_{2}<\dots<j_{t}\\}$ be the sequence of indices for which
$r+i_{j}m\in\mathcal{Q}$, $r=1,2,\dots,t$. Let $J_{1}\subseteq J$ be the set
of indices for which $i_{j_{s+1}}-i_{j_{s}}>\sqrt{\frac{k}{K\log k}}$.

Clearly $|J_{1}|\leq k/\sqrt{\frac{k}{K\log k}}=\sqrt{K}\sqrt{k\log k}$.

Now let $J_{2}:=J\setminus J_{1}$. For these indices we have
$i_{j_{s+1}}-i_{j_{s}}\leq\sqrt{\frac{k}{K\log k}}$ and so

(1)
$i_{j_{s+1}}m-i_{j_{s}}m=(r+i_{j_{s+1}}m)-(r+i_{j_{s}}m)=x^{2}-y^{2}=(x-y)(x+y)$

for some $x,y$. Fix the couple $i_{j_{s+1}},i_{j_{s}}$. Denote by $M$ the
number of pairs $(x,y)$ for which (1) holds. Then
$M\leq\tau(i_{j_{s+1}}m-i_{j_{s}}m)$. Thus we have

$|J_{2}|\leq\sum_{x\leq\sqrt{\frac{k}{K\log
k}}}\tau(mx)\leq\tau(m)\sum_{x\leq\sqrt{\frac{k}{K\log k}}}\tau(x)\leq
K\sum_{x\leq\sqrt{\frac{k}{K\log k}}}\tau(x)$

using the fact that $\tau(mx)\leq\tau(m)\tau(x)$. It is well-known that

$\sum_{x\leq\sqrt{\frac{k}{K\log k}}}\tau(x)\leq\sqrt{\frac{k}{K\log
k}}\log\Big{(}\sqrt{\frac{k}{K\log k}}\Big{)}\leq\sqrt{\frac{k\log k}{K}}.$

Finally

$|J_{1}|+|J_{2}|=|J|\leq\sqrt{K}\sqrt{k\log k}+\sqrt{K}\sqrt{k\log k}.$

∎

Suppose now that some system of arithmetic progressions covers the squares up
to $N$, i.e. $\bigcup_{i=1}^{T}\\{m_{i}j+r_{i}\\}_{j=1}^{k_{i}}\supseteq
Q_{N}$. Thus
$\sqrt{N}\leq\sum_{i=1}^{T}|\mathcal{Q}_{N}\cap\\{r_{i}+jm_{i}\\}_{j=1}^{k_{i}}|$.
Now by Lemma 1.3 and the Cauchy inequality we get

$N\leq(\sum_{i=1}^{T}|\mathcal{Q}_{N}\cap\\{r_{i}+jm_{i}\\}_{j=1}^{k_{i}}|)^{2}\leq(\sum_{i=1}^{T}2\sqrt{K}\sqrt{k_{i}\log
k_{i}})^{2}\leq$ $\leq(\sum_{i=1}^{T}2\sqrt{K}\sqrt{k_{i}\log N})^{2}\leq
4KT\sum_{i=1}^{T}k_{i}\log N$

which implies that $T\sum_{i=1}^{T}k_{i}\geq\frac{N}{4K\log N}$ as we wanted.

∎

##  2\. Covering by Quasi Progressions

In [3] Brown, Erdős and Freedman introduced the generalization of arithmetic
progressions. For $k\geq 1$ let $X=\\{x_{1}<x_{2}<\dots<x_{k}\\}$. The
sequence $X$ is said to be a k-term quasi (or combinatorial) progression of
order $d$ (briefly CP-$d$) if the diameter of the set of $\\{x_{i+1}-x_{i}:\
1\leq i\leq k-1\\}$ is bounded by $d$.

In this section we prove that $\mathcal{Q}_{N}$ can be covered with quasi-
progressions far more efficiently than with arithmetic progressions.

We will consider the case of $d=2$, especially when
$x_{i+1}-x_{i}\in\\{D,D+1\\}$, as this is the most similar to arithmetic
progression. In this case we write that $X$ is CP-$\\{D,D+1\\}$

######  Definition.

Let $X_{1},X_{2},\dots,X_{T}\subseteq\\{1,2\dots,,N\\}$ be a system of
sequences, where for each $1\leq i\leq T$, $X_{i}$ is
CP-$\\{D_{i},D_{i}+1\\}$. Assume that
$\bigcup_{i=1}^{T}X_{i}\supseteq\mathcal{Q}_{N}$. Let
$G(\mathcal{Q}_{N}):=\min_{\\{D_{i}\\},T}T\sum_{i=1}^{T}|X_{i}|$.

######  Theorem 2.1.

We have

$G(\mathcal{Q}_{N})<CN^{3/4}\log_{2}N,$

where $C=\frac{\sqrt[4]{3}}{2}\frac{2^{3/4}}{2^{3/4}-1}$, ($\log_{2}N$ is the
logarithm in base $2$).

###### Proof.

Let $T=\lfloor\log_{2}(N)\rfloor$ and for every $1\leq i\leq T$ let
$I_{i}:=[N/2^{i},N/2^{i-1}]$.

We cover first $I_{1}$ (and since each $I_{i}$ and $X_{i}$ the treatment will
be similar to $I_{1}$ and $X_{i}$ respectively, we can briefly give a bound
for $G(\mathcal{Q}_{N})$).

Let $n^{2}_{1}<n^{2}_{2}<\dots<n^{2}_{k}$ be the squares in $I_{1}$ and let
$D_{1}=\lfloor\sqrt{n_{1}}\rfloor$.

We have that
$\frac{n^{2}_{2}-n^{2}_{1}}{D_{1}}\geq\frac{2n_{1}+1}{\sqrt{n_{1}}}\geq
2\sqrt{n_{1}}\geq\sqrt{n_{1}}+1\geq D_{1}+1$ if $n_{1}\geq 1$. Note that
$\Delta:=n^{2}_{2}-n^{2}_{1}-\lfloor\frac{n^{2}_{2}-n^{2}_{1}}{D_{1}}\rfloor
D_{1}<D_{1}$ and $\lfloor\frac{n^{2}_{2}-n^{2}_{1}}{D_{1}}\rfloor>D_{1}$.

Now we are going to define the elements of $X_{1}$ in the interval
$[n^{2}_{1},n^{2}_{2}]$. Let $x_{1}=n^{2}_{1}$ and from the differences
$x_{i+1}-x_{i}$ ($i=1,2,\dots\lfloor\frac{n^{2}_{2}-n^{2}_{1}}{D_{1}}\rfloor$)
let $\Delta$ be the number of $D_{1}+1$ and
$\lfloor\frac{n^{2}_{2}-n^{2}_{1}}{D_{1}}\rfloor-\Delta$ be the number of
$D_{1}$. From the definition of $X_{1}$ we obtain that
$\lfloor\frac{n^{2}_{2}-n^{2}_{1}}{D_{1}}\rfloor th$ element is just
$n_{2}^{2}$.

Now consider the interval $[n^{2}_{j},n^{2}_{j+1}]$, $j\geq 2$. Since
$\lfloor\frac{n^{2}_{j+1}-n^{2}_{j}}{D_{1}}\rfloor>\lfloor\frac{n^{2}_{2}-n^{2}_{1}}{D_{1}}\rfloor$
we can repeat the previous process; i.e. let
$n^{2}_{j+1}-n^{2}_{j}-\lfloor\frac{n^{2}_{j+1}-n^{2}_{j}}{D_{1}}\rfloor
D_{1}$ be the number of the consecutive differences $D_{1}+1$, and let the
rest be $D_{1}$.

Now we calculate the cardinality of the quasi-progression $X_{1}$. $N/2\leq
n^{2}_{1}$, so
$D_{1}=\lfloor\sqrt{n_{1}}\rfloor>\sqrt{n_{1}}-1\geq\sqrt[4]{N/2}-1>\sqrt[4]{N/3}$.
Furthermore $X_{1}\subseteq[N/2,N]$, and $x_{i+1}-x_{i}\geq D_{1}$ for all
$i$, hence
$|X_{1}|\leq\frac{N/2}{D_{1}}<\frac{N/2}{\sqrt[4]{N/3}}=\frac{\sqrt[4]{3}}{2}N^{3/4}$.

The calculation for every $X_{i}$, $i=2,\dots,T$ is the same so we have

$T\sum_{i=1}^{T}|X_{i}|<\frac{\sqrt[4]{3}}{2}\sum_{i=1}^{T}\Big{(}\frac{N}{2^{i}}\Big{)}^{3/4}\log_{2}N<\frac{\sqrt[4]{3}}{2}N^{3/4}\sum_{i=1}^{\infty}\frac{1}{(2^{3/4})^{i}}\log_{2}N=CN^{3/4}\log_{2}N,$

where $C=\frac{\sqrt[4]{3}}{2}\frac{2^{3/4}}{2^{3/4}-1}$. ∎

######  Remark.

The reason we can cover the squares ”well” is that we use a lot of $D_{i}+1$
in addition to $D_{i}$. The above proof works even if we require that the
number of ($D_{i}+1$)s in each quasi-progression is at most $N^{\varepsilon}$
(in the sense that the squares are ”well” covered, i.e.
$G(\mathcal{Q}_{N})\leq N^{1-c(\varepsilon)}$, where $0<c(\varepsilon)<1$).

In the next section we extend the result of Sárközy to the Erdős problem for a
wide class of polynomials.

##  3\. covering shrinking polynomials by quasi progressions

A sub-sequence of prime numbers
$P^{\prime}=\\{p_{1}<p_{2}<\dots<p_{s}<\dots\\}$ is said to be $\eta-$dense if
there exists an $\eta>0$ for which $|P^{\prime}\cap[1,x]|/\pi(x)>\eta$ holds
for every large $x$.

######  Definition.

Let $f(x)\in\mathbb{Z}[x]$. We say that $f$ is an $(\eta,\mu)-$shrinking
polynomial if there is an $\eta-$dense sequence of primes
$P^{\prime}=\\{p_{1}<p_{2}<\dots<p_{s}<\dots\\}$ such that for all $i$ large
enough, $|f(\mathbb{F}_{p_{i}})|<\mu p_{i}$ (here
$f(\mathbb{F}_{p_{i}}):=\\{f(x):x\in\mathbb{F}_{p_{i}}\\}$). We say the $f$ is
shrinking if for some $0<\eta,\mu\leq 1$ $f$ is $(\eta,\mu)-$shrinking
polynomial

Clearly all functions $f(x)=x^{n}$; $n\geq 2$ are shrinking, since for
$1<d=g.c.d.(p-1,n)$, $|\\{x^{n}:x\in\\{1,2,\dots,p-1\\}=\frac{p-1}{d}$, and
$\pi(x,1,d)=(1+o(1))\frac{1}{\phi(d)}\frac{x}{\ln x}$, where $\pi(x,1,d)$
denotes the number of primes $\equiv 1\pmod{d}$ which are less than or equal
to $x$. Further example is the polynomial $f(x)=x^{3}+x$. In [6] it was proved
for any prime $p>2$ the number of distinct residues in the form $x^{3}+x$
$\pmod{p}$ is $2p/3+O(\sqrt{p}\log p)$. For a generic polynomial $g(x)$ (i.e.
where the integral coefficients distributed uniformly and independently in
$\mathbb{Z}$) it is known that the number of distinct residues represented by
$g(x)$ is $p(1-1/2+1/3-\dots-(-1)^{d}/d!)+O(\sqrt{p})$, where $d$ is the
degree of $g$ and $p\geq 2$ prime (see [1]). Note that the sequence in the
first brackets tends to $1-1/e$ as $d\to\infty$.

The aim of the present section is to extend Erdős’ problem of covering
shrinking polynomials.

So let $f$ be an $(\eta,\mu)-$shrinking polynomial
$f_{N}=f(\mathbb{N})\cap\\{1,2,\dots,N\\}$ and let
$X_{1},X_{2},\dots,X_{T}\subseteq\\{1,2\dots,,N\\}$ be a system of quasi
progressions, where for each $1\leq i\leq T$, $X_{i}$ is
CP-$\\{D_{i},D_{i}+1\\}$. As we have seen in the previous section we have to
bound the number of $(D_{i}+1)$s. So we assume that for every $1\leq i\leq T$
$(D_{i}+1)$ occurs at most $\log^{A}N$ times for some $A>0$. We will write
that $X_{i}$ is CP-$\\{D_{i},(D_{i}+1)_{\log^{A}N}\\}$.

######  Theorem 3.1.

Let $f$ be an $(\eta,\mu)-$shrinking polynomial with degree $d$. Assume that
$\bigcup_{i=1}^{T}X_{i}\supseteq f_{N}$. Let
$H_{A,d}(f_{N}):=\min_{\cup_{i=1}^{T}X_{i}\supseteq
f_{N}}T\sum_{i=1}^{T}|X_{i}|$. We have

$H_{A,d}(f_{N})\geq(1+o(1))\frac{C^{\prime}N^{2/d}}{\log^{A+2}N}$

where $C^{\prime}=\frac{(1-\mu)^{2}\eta^{2}}{200}$

Note that trivially $T\sum_{i}|X_{i}|\leq CN^{2/d}$; when $f_{N}$ is covered
by singletons.

###### Proof.

Assume that $\eta,\mu$ is fixed and $N>N(\eta,\mu)$ is large enough. Let $X$
be a CP-$\\{D,D+1\\}$ quasi-progression where the number of gaps $D+1$ is at
most $M\leq\log^{A}N$. Let $I=\\{i_{1}<i_{2}<\dots i_{M}\\}$ be the sequence
of subscripts for which $x_{i_{j}+1}-x_{i_{j}}=D+1$. Furthermore let
$I^{\prime}\subseteq I$ be the sub-sequence of indices for which
$i_{j+1}-i_{j}\geq\log^{2}N$. Write
$Z_{i_{j}}=\\{x_{i_{j}+1}<x_{i_{j}+2}<\dots<x_{i_{j+1}-1}\\}$. Note that
$Z_{i_{j}}$ is an arithmetic progression with difference $D$ or $D+1$ (and
$D+1$ can be only if $A\geq 2$). Here we assume that $Z_{i_{j}}$ is non-empty
set. To do this, you need to have $|X|\geq\log^{A+2}N$. Actually we split $X$
into not too short arithmetic progressions with difference $D$ or $D+1$. In
the sequel we assume that the difference is $D$, the argument completely the
same when the difference is $D+1$.

From this point by some modification of the proof of Sárközy, we could extend
his result. For this we need the arithmetic form of the large sieve (see [7]
p. 560):

######  Lemma 3.2.

Let $U\subseteq\\{1,2,\dots,M\\}$. Denote by $U(p,h):=|\\{u:u\in U;\ u\equiv
h\pmod{p}\\}|$ we have

$\sum_{p\leq
W}p\sum_{h=0}^{p-1}\Big{|}U(p,h)-\frac{1}{p}|U|\Big{|}^{2}\leq(M+W^{2})|U|.$

For some fixed $i_{j}$ let $U=\\{t:x_{t}\in Z_{i_{j}}\cap f_{N}\\}$,
$M=s=|Z_{i_{j}}|$, and $W=\sqrt{s}$. We are going to give an upper bound for
$U$, so without loss of generality we can assume that
$U>\sqrt{\frac{2}{(1-\mu)\eta}}$ (i.e. $U$ is larger than a fixed constant).

Now we use the large sieve in the form
$\sum_{p\leq\sqrt{s}}p\sum_{h=0}^{p-1}\Big{|}U(p,h)-\frac{1}{p}|U|\Big{|}^{2}\leq
2s|U|.$

Now we are going to sieve just for the primes for which $f$ is shrinking, i.e.
if $\sqrt{s}>x_{0}$ and then there is an $\eta-$dense sequence of primes
$p_{1}<p_{2}<\dots<p_{t}\leq\sqrt{s}$ such that for all $i$,
$|f(\mathbb{F}_{p_{i}})|<\mu p_{i}$. Write
$\mathcal{I}=\\{p_{1}<p_{2}<\dots<p_{t}\\}$. Here $t\geq\eta\pi(\sqrt{s})$.

Since $f$ is $(\eta,\mu)$-shrinking we conclude that for every $1\leq i\leq t$
the number of resides $h$ modulo $p_{i}$ for which $U(h,p_{i})=0$ is at least
$(1-\mu)p_{i}$. Furthermore we have to leave those $p$ from $\mathcal{I}$ for
which $p|D$. Clearly it is $\omega(D)$ (the number of distinct prime factors
of $D)$ which is at most $(1+o(1))\log N/\log\log N$. Denote the remaining set
by $\mathcal{I}^{\prime}$. We have

$\sum_{p\leq\sqrt{s}}p\sum_{h=0}^{p-1}\Big{|}U(p,h)-\frac{1}{p}|U|\Big{|}^{2}\geq\sum_{p\in\mathcal{I^{\prime}}}p(1-\mu)p\Big{|}\frac{1}{p}|U|\Big{|}^{2}=$
$=(1-\mu)|U|^{2}(\eta\pi(\sqrt{s})-(1+o(1))\log N/\log\log
N)>\frac{(1-\mu)\eta}{5}|U|^{2}\frac{\sqrt{s}}{\log s},$

since we assume that $s=|X_{j}|\geq\log^{2}N$ and
$U>\sqrt{\frac{2}{(1-\mu)\eta}}$. Comparing the left and right hand side of
the sieve inequality we get

$|U|\leq\frac{10}{(1-\mu)\eta}\sqrt{s}\log s.$

Now we are going to estimate the number of values of $f$ in $X$. Write
$X=\cup_{i_{j}\in\mathcal{I^{\prime}}}Z_{i_{j}}\cup(X\setminus\cup_{i_{j}\in\mathcal{I^{\prime}}}Z_{i_{j}})$.
By the definition of $\mathcal{I^{\prime}}$, the number of values of $f$ is at
most
$|X\setminus\cup_{i_{j}\in\mathcal{I^{\prime}}}Z_{i_{j}}|\leq\log^{A}N\cdot\log^{2}N=\log^{A+2}N$.

The number of values of $f$ in the rest of $X$ can be calculated by

$\frac{10}{(1-\mu)\eta}\sum_{i_{j}\in\mathcal{I^{\prime}}}\sqrt{|Z_{i_{j}}|}\log|Z_{i_{j}}|+\log^{A+2}N\leq\frac{20}{(1-\mu)\eta}\sqrt{\log^{A}N\sum_{i_{j}\in\mathcal{I^{\prime}}}|Z_{i_{j}}|}\log
N\leq$ $\leq\frac{20}{(1-\mu)\eta}\sqrt{|X|}\log^{A/2+1}N$

since the function $\sqrt{x}$ is concave function (the estimation comes from
the Jensen inequality).

Now we complete the proof of the theorem. Since the degree of $f$ is $d$ thus
$|f_{N}|\geq(1+o(1))N^{1/d}$. Assume that the union of
$X_{1},X_{2},\dots,X_{T}$ covers $f_{N}$, and for every $1\leq i\leq T$,
$X_{i}$ is CP-$\\{D_{i},(D_{i}+1)_{\log^{A}N}\\}$ quasi progression. Write
$X^{\prime}_{i}=X_{i}\cap f_{N}$. The above sieving estimation can only be
used for ”long” quasi-progressions, so we divide the sum
$\sum_{i}|X^{\prime}_{i}|$ into two parts. We have

$(1+o(1))N^{2/d}\leq\big{(}\sum^{T}_{i=1}|X^{\prime}_{i}|\big{)}^{2}\leq
2\big{(}\sum_{|X_{i}|\leq\log^{A+2}N}|X^{\prime}_{i}|\big{)}^{2}+2\big{(}\sum_{|X_{i}|>\log^{A+2}2N}|X^{\prime}_{i}|\big{)}^{2}$

since $(a+b)^{2}\leq 2a^{2}+2b^{2}$. Let $T_{1}$ be the number of terms in the
first sum and $T_{2}=T-T_{1}$. Then
$(\sum_{|X_{i}|\leq\log^{A+2}N}|X^{\prime}_{i}|)^{2}\leq
T_{1}\log^{A+2}N(\sum_{|X_{i}|\leq\log^{A+2}N}|X_{i}|)$. By the Cauchy
inequality the second sum can be estimated as

$\big{(}\sum_{|X_{i}|>\log^{A+2}N}|X^{\prime}_{i}|\big{)}^{2}\leq
T_{2}\sum_{|X_{i}|>\log^{A+2}N}|X^{\prime}_{i}|^{2}\leq
T_{2}\frac{400}{(1-\mu)^{2}\eta^{2}}\sum_{|X_{i}|>\log^{A+2}N}|X_{i}|\log^{A+2}N.$

Putting everything together we get

$(1+o(1))N^{2/d}\leq
2T_{1}\log^{A+2}N\sum_{|X_{i}|\leq\log^{A+2}N}|X_{i}|+T_{2}\frac{200}{(1-\mu)^{2}\eta^{2}}\sum_{|X_{i}|>\log^{A+2}N}|X_{i}|\log^{A+2}N\leq$
$\leq\frac{200}{(1-\mu)^{2}\eta^{2}}\log^{A+2}N\big{(}T_{1}\sum_{|X_{i}|\leq\log^{A+2}N}|X_{i}|+T_{2}\sum_{|X_{i}|>\log^{A+2}N}|X_{i}|\big{)}$
$=\frac{200}{(1-\mu)^{2}\eta^{2}}\log^{A+2}N\big{(}T\sum^{T}_{i=1}|X_{i}|\big{)}.$

Rearranging the inequality we obtain the statement. ∎

## Acknowledgment

The research is supported by the National Research, Development and Innovation
Office NKFIH Grant No K-129335.

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