††thanks: Corresponding author††thanks: Corresponding author

# Consistency of a kind of general noncanonical warm inflation

Xiao-Min Zhang1<EMAIL_ADDRESS>Ang Fu2<EMAIL_ADDRESS>Kai Li1
Qian Liu1 Peng-Cheng Chu1 Hong-Yang Ma1 Jian-Yang Zhu 3 1School of Science,
Qingdao University of Technology, Qingdao 266033, China
2School of Mathematical Sciences, University of Science and Technology of
China, Hefei 230026, China
3Department of Physics, Beijing Normal University, Beijing 100875, China

###### Abstract

The framework of a kind of noncanonical warm inflation is introduced, and the
dynamical equations of this scenario are presented. We propose the slow roll
approximations and give some redefining slow roll parameters in this scenario
which remain dimensionless. Performing systemic stability analysis, we
calculate the slow roll conditions to guarantee that slow roll approximations
hold. The slow roll conditions suggest slow roll inflation in general
noncanonical warm inflationary scenario can still exist, and in addition, the
slow roll approximations are more easily to be satisfied. Then, a concrete
Dirac-Born-Infeld warm inflationary model is studied.

###### pacs:

98.80.Cq

##  I Introduction

Inflation, as a necessary supplement to the standard model of cosmology, can
solve the problems of horizon, flatness and monopole. Till mow, there are two
candidates for inflation: standard inflation, or sometimes called cold
inflation, and warm inflation. Cold inflation was first proposed in 1981
Guth1981 ; Linde1982 ; Albrecht1982 , which suppose the inflaton is isolated
during inflation, so the Universe is supercool and need a separate reheating
phase to warm the Universe. While in warm inflationary Universe, the inflaton
is not isolated and instead, it has interactions with other subdominated
fields BereraFang ; Berera2000 ; Berera2006 . The Universe is hot during the
whole inflationary phase, so it has an elegant exit and can go smoothly to the
radiation-dominated big-bang phase. Cold inflation can produce seeds to give
rise to the large scale structure and the observed little anisotropy of
cosmological microwave background (CMB) WMAP ; PLANCK  through vacuum
fluctuation Bassett2006 ; LiddleLyth ; Weinberg ; Dodelson , while the density
fluctuations naturally generate from thermal fluctuations in warm inflation.
Warm inflation can cure “$\eta$-problem” etaproblem  and the overlarge
amplitude of the inflaton suffered in cold inflation Berera2006 ;
BereraIanRamos . What’s more, warm inflation has more relaxed slow roll
conditions compared to cold inflation Ian2008 ; Campo2010 ; ZhangZhu ;
Zhang2014 . Warm inflation can broaden the scope of inflation, which means
some models that were ruled out in cold inflation such as the quartic chaotic
potential model, able to again be in a very good agreement with the Planck
results in warm inflationary scenario Sam2014 . The swampland conjecture
swamp1 ; swamp2 ; swamp3 ; swamp4 ; swamp5  proposed recently which may cause
conflicts in slow roll cold inflation, can be consistent with warm
inflationary models swampwarm1 ; swampwarm2 . Some related works reach the
conclusion that very small amounts of dissipation can result in warm
inflation, so the warm inflation can describe the very early Universe more
realizably.

Warm inflation usually takes advantages of canonical field with the Lagrangian
density $\mathcal{L}=X-V$, where
$X=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$ is the kinetic
term and $V$ is the potential term, as the inflaton that drives the inflation.
Most works about warm inflation concentrate on the model building, the
dissipative microphysical mechanism and the cosmological perturbation research
of canonical warm inflation BereraIanRamos ; Lisa2004 ; Mar2007 ; Mar2013 ;
naturalwarm1 ; naturalwarm2 ; Chris2009 ; Taylor2000 ; ZhangZhu . In recent
years, kinds of special noncanonical warm inflationary models have emerged,
such as warm Dirac-Born-Infeld (DBI) inflation Cai2011 , warm tachyon
inflation Herrera2006 ; TachyonZhang ; Xiao2011  and warm k-inflation Peng2018
. As the case in cold inflation, noncanonical inflation also deserve research
in warm inflationary picture. We concentrate on the extension of warm
inflationary theory for years. At first, we construct the noncanonical warm
inflation with a relatively simple Lagrangian density having separable form of
kinetic and potential terms Zhang2014 ; Zhang2015 . Then we extend warm
inflation to the more general noncanonical field with a Lagrangian density
having coupling form of kinetic and potential terms Zhang2018 ; Zhang2019 . In
noncanonical warm inflation, small sound speed and strong dissipation effect
can both influence the perturbation observable quantities such as power
spectrum, spectral index and tensor-to-scalar ratio etc. Zhang2014 ; Zhang2018
. Thanks to the thermal effect and the noncanonical effect, the energy scale
when horizon crossing is depressed, which is good news to the assumption that
the universe inflation can be described well by effective field theory
Zhang2014 ; Zhang2018 . And due to the two effects, the magnitude of
primordial gravitational wave is weaker than canonical warm inflation and cold
inflation. The non-Gaussianity generated in noncanonical warm inflation is
studied and found that large noncanonical effect (in other words, a small
sound speed) and strong thermal dissipation effect can enhance the magnitude
of non-Gaussianity Zhang2019 . The noncanonical warm inflationary models can
safely lie on the allowed region of observations over a wide parameter range.
In original warm inflation, it is found that the slow roll approximations was
more easily to be satisfied Ian2008 ; Campo2010 . The construction of general
noncanonical warm inflation broad the scope of inflation theory. However, the
consistency issue in general warm inflation is still blank and deserve
complete analysis, which is the main task of this paper.

The paper is organized as follows: In Sec. II, we introduce general
noncanonical warm inflationary scenario, and give frame equations containing
slow roll approximations and redefining slow roll parameters. We make a fully
linear stability analysis to obtain slow roll conditions that guarantee the
slow roll approximations are valid in Sec. III. A concrete examples in the
general warm inflationary scenario is studied in Sec. IV. Finally, we
summarize our results in Sec. V.

##  II The framework of general noncanonical warm inflation

In warm inflationary picture, the Universe is a multi-component system, thus
the total matter action can be given as:

$S=\int
d^{4}x\sqrt{-g}\left[\mathcal{L}(X^{\prime},\varphi)+\mathcal{L}_{R}+\mathcal{L}_{int}\right],$
(1)

where
$X^{\prime}=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi$.
The Lagrangian density of the noncanonical field is
$\mathcal{L}_{noncan}=\mathcal{L}(X^{\prime},\varphi)$ in above equation,
which can be an arbitrary function of the inflaton field $\varphi$ and the
kinetic term $X^{\prime}$. $\mathcal{L}_{R}$ is the Lagrangian density of the
radiation fields and $\mathcal{L}_{int}$ denotes the interaction term between
inflaton and other subdominated fields in warm inflationary epoch. There are
two conditions that a workable noncanonical Lagrangian density should meet:
$\mathcal{L}_{X^{\prime}}\geq 0$ and $\mathcal{L}_{X^{\prime}X^{\prime}}\geq
0$ Franche2010 ; Bean2008 , which means we have $\mathcal{L}_{X^{\prime}}>1$
in a normalized field representation Zhang2014 . By convention, the subscripts
$\varphi$ and $X^{\prime}$ denote a derivative.

The motion equation of the inflaton can be obtained by varying above action:

$\frac{\partial(\mathcal{L}(X^{\prime},\varphi)+\mathcal{L}_{int})}{\partial\varphi}-\left(\frac{1}{\sqrt{-g}}\right)\partial_{\mu}\left(\sqrt{-g}\frac{\partial\mathcal{L}(X^{\prime},\varphi)}{\partial(\partial_{\mu}\varphi)}\right)=0.$
(2)

In the flat Friedmann-Robertson-Walker (FRW) Universe, the field is
homogeneous, i.e. $\varphi=\varphi(t)$, so the motion equation of the inflaton
field is reduced to:

$\displaystyle\left[\left(\frac{\partial\mathcal{L}(X^{\prime},\varphi)}{\partial
X^{\prime}}+2X^{\prime}\left(\frac{\partial^{2}\mathcal{L}(X^{\prime},\varphi)}{\partial
X^{\prime 2}}\right)\right)\right]\ddot{\varphi}$
$\displaystyle+\left[3H\left(\frac{\partial\mathcal{L}(X^{\prime},\varphi)}{\partial
X^{\prime}}\right)+\dot{\varphi}\left(\frac{\partial^{2}\mathcal{L}(X^{\prime},\varphi)}{\partial
X^{\prime}\partial\varphi}\right)\right]\dot{\varphi}$
$\displaystyle-\frac{\partial(\mathcal{L}(X^{\prime},\varphi)+\mathcal{L}_{int})}{\partial\varphi}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(3)

where $H$ is the Hubble parameter.

The energy density of the inflaton is:
$\rho(\varphi,X^{\prime})=2X^{\prime}\frac{\partial\mathcal{L}}{\partial
X^{\prime}}-\mathcal{L}$, and the pressure is given by:
$p(\varphi,X^{\prime})=\mathcal{L}$. The sound speed is
$c_{s}^{2}=\frac{\partial p/\partial X^{\prime}}{\partial\rho/\partial
X^{\prime}}=\left(1+2X^{\prime}\frac{\mathcal{L}_{X^{\prime}X^{\prime}}}{\mathcal{L}_{X^{\prime}}}\right)^{-1}$,
describing the traveling speed of scalar perturbations. The Friedmann equation
is

$H^{2}=\frac{\rho}{3M_{p}^{2}}.$ (4)

Considering warm inflationary assumptions, the motion equation of general
noncanonical warm inflation can be given by Zhang2018 :

$\mathcal{L}_{X^{\prime}}c_{s}^{-2}\ddot{\varphi}+(3H\mathcal{L}_{X^{\prime}}+\tilde{\Gamma})\dot{\varphi}+\mathcal{L}_{X^{\prime}\varphi}\dot{\varphi}^{2}+V_{eff,\varphi}(\varphi,T)=0,$
(5)

where $\tilde{\Gamma}$ is the dissipative coefficient in warm inflation and
$V_{eff}$ is the thermal effective potential. In what follows, we will write
$V_{eff}$ as $V$ for simplicity. Because of the existing kinetic potential
coupling term of the general noncanonical Lagrangian density, the above motion
equation has an annoying coupling term
$\mathcal{L}_{X^{\prime}\varphi}\dot{\varphi}^{2}$. This brings difficulties
in solving the kind of more general warm inflation, especially giving slow
roll approximations and calculating perturbations. Fortunately, this problem
usually can be solved by making a field redefinition $\phi=f(\varphi)$ to
eliminate the coupling term Zhang2018 . The solutions can be worked out in
many cases using analytical or sometimes numerical method. Then the Lagrangian
density becomes $\mathcal{L}(X,\phi)$, where $X=\frac{1}{2}\dot{\phi}^{2}$ in
the new field representation. Under the field redefinition, an appropriate
field representation $\phi=f(\varphi)$ is selected out to make the coupling
term disappeared through

$\mathcal{L}_{X\phi}=\frac{1}{f^{4}_{\varphi}}[f_{\varphi}\mathcal{L}_{X^{\prime}\varphi}-2f_{\varphi\varphi}\mathcal{L}_{X}^{\prime}-2f_{\varphi\varphi}\mathcal{L}_{X^{\prime}X^{\prime}}X^{\prime}]=0,$
(6)

where $f_{\varphi}$ and $f_{\varphi\varphi}$ denote the first- and second-
order derivative of the function $f(\varphi)$ respectively. In the redefined
uncoupling $\phi$ representation, the slow roll approximations in general
noncanonical warm inflation can be given out clearly, and we can get its slow
roll conditions by making stability analysis.

In uncoupling $\phi$ representation, the motion equation of the noncanonical
inflaton can be expressed as:

$\mathcal{L}_{X}c_{s}^{-2}\ddot{\phi}+(3H\mathcal{L}_{X}+\Gamma)\dot{\phi}+V_{eff,\phi}(\phi,T)=0.$
(7)

$\Gamma$ in above equation is the new dissipative coefficient in the $\phi$
representation and it has the relation $\tilde{\Gamma}=f_{\varphi}^{2}\Gamma$
with $\tilde{\Gamma}$ Zhang2018 . With the benefit of field redefinition, the
researches on cosmological perturbations in general noncanonical warm
inflation can be easy to handle to some extent Zhang2018 ; Zhang2019 . Also,
we will perform systemic stability analysis using the easy-to-use $\phi$
representation in the following.

The entropy production equation is given by

$T\dot{s}+3HTs=\Gamma\dot{\phi}^{2},$ (8)

where $s$ is the entropy density in warm inflation.

The noncanonical inflaton and the dissipative coefficient may not have the
“right” conventional mass dimension in general noncanonical warm inflation,
different to canonical warm inflation. Given this situation, we redefine the
dimensionless dissipation strength parameter $r$ in new general noncanonical
warm inflation as Zhang2018 :

$r=\frac{\Gamma}{3H\mathcal{L}_{X}}.$ (9)

$r\gg 1$ suggests that warm inflation is in strong regime, while $r\ll 1$ is
corresponding to weak regime in warm inflation.

We propose the probable slow roll approximations of general noncanonical warm
inflation by dropping the highest derivative terms in the motion equations
under the $\phi$ representation. Thus the motion equation of noncanonical
inflaton and entropy production equation can be reduced to:

$3H\mathcal{L}_{X}(1+r)\dot{\phi}+V_{\phi}=0,$ (10)
$3HTs=\Gamma\dot{\phi}^{2}.$ (11)

The validity of these slow roll approximations depends on the concrete slow
roll conditions in general noncanonical warm inflation, which we will give out
in next section. Slow roll conditions of inflation are usually characterized
by slow roll parameters. The slow roll parameters in conventional warm
inflation are usually defined as:

$\tilde{\epsilon}=\frac{M_{p}^{2}}{2}\left(\frac{V_{\phi}}{V}\right)^{2},\tilde{\eta}=M_{p}^{2}\frac{V_{\phi\phi}}{V},\tilde{\beta}=M_{p}^{2}\frac{V_{\phi}\Gamma_{\phi}}{V\Gamma}.$
(12)

Typically, above slow roll parameters are no longer dimensionless in general
kind noncanonical warm inflation, different from canonical cold and warm
inflation. In general noncanonical warm inflation, we define some new slow
roll parameters that remain dimensionless for convenience:

$\epsilon=\frac{M_{p}^{2}}{2\mathcal{L}_{X}}\left(\frac{V_{\phi}}{V}\right)^{2},~{}~{}~{}\eta=\frac{M_{p}^{2}}{\mathcal{L}_{X}}\frac{V_{\phi\phi}}{V},~{}~{}~{}\beta=\frac{M_{p}^{2}}{\mathcal{L}_{X}}\frac{V_{\phi}\Gamma_{\phi}}{V\Gamma}.$
(13)

Now the “new” slow roll parameters are all dimensionless, as the “old” slow
roll parameters in canonical warm inflation. A good feature for the “new” slow
roll parameters is their invariance under field transformations.

Describing temperature influence in warm inflation, two additional slow roll
parameters are given by:

$b=\frac{TV_{\phi T}}{V_{\phi}},$ (14)

and

$c=\frac{T\Gamma_{T}}{\Gamma}.$ (15)

Then the number of e-folds in general noncanonical warm inflation can be
obtained:

$N=\int
Hdt=\int\frac{H}{\dot{\phi}}d\phi\simeq-\frac{1}{M_{p}^{2}}\int_{\phi_{\ast}}^{\phi_{end}}\frac{V\mathcal{L}_{X}(1+r)}{V_{\phi}}d\phi,$
(16)

where $M_{p}^{2}=\frac{1}{8\pi G}$ and the subscript $\ast$ denoting the time
of Hubble horizon crossing.

##  III Stability analysis

In order to perform a systematic stability analysis conveniently, we define
$u=\dot{\phi}$ and then the Eqs. (7) and (8) can be rewritten as:

$\dot{u}=-c_{s}^{2}\left[3H(1+r)u+\frac{V_{\phi}(\phi,T)}{\mathcal{L}_{X}}\right],$
(17) $\dot{s}=-3Hs+\frac{\Gamma u^{2}}{T}.$ (18)

Inflation is generally associated and dealt with slow roll approximations, and
it is the same in general warm inflationary picture. We have proposed its slow
roll approximations in Sec. II. The slow roll approximations indicate that the
energy density is dominated by potential, the production of radiation is
quasi-static, and the evolution of inflaton field is quite slow.

For convenience, $u_{0}$, $\phi_{0}$ and $s_{0}$ is used to express slow roll
solutions satisfying these slow roll equations:

$3H\mathcal{L}_{X}(1+r)u_{0}+V_{\phi}(\phi,T)=0,$ (19)
$3H_{0}T_{0}s_{0}=\Gamma u_{0}^{2}.$ (20)

The variables $u$, $\phi$ and $s$ denote the exact solutions satisfying the
exact inflationary equations (4), (17), (18). We expand the exact solutions
around the slow roll solutions as: $u=u_{0}+\delta u,$
$\phi=\phi_{0}+\delta\phi$ and $s=s_{0}+\delta s$, where the perturbation
terms $\delta u,$ $\delta\phi$, and $\delta s$ are much smaller than the
background ones $u_{0}$, $\phi_{0}$ and $s_{0}$.

In order to find out conditions to guarantee that slow roll solutions can
really act as formal attractor solutions for general warm inflationary
dynamical system, we perform stability analysis around the slow roll
solutions. Using the new variable $u$, we can rewrite $X=\frac{1}{2}u^{2}$,
then $\delta X=u\delta u$, and $\delta\mathcal{L}_{X}=\mathcal{L}_{XX}u\delta
u$. With the help of the thermal relation $s=-V_{T}$, we get $\delta
s=-V_{TT}\delta T-V_{\phi T}\delta\phi$. The relation

$2H\delta H=\frac{1}{3M_{P}^{2}}[\mathcal{L}_{X}c_{s}^{-2}u_{0}\delta
u+(V_{\phi}+\frac{1}{3}V_{\phi}b)\delta\phi+\frac{4}{3}T_{0}\delta s],$ (21)

can be obtained by varying the Friedmann equation. Then considering the
definition of new slow roll parameters, we can work out the variations of
$V_{\phi}$, $\Gamma$ and $r$ etc in terms of the fundamental perturbation
variables:

$\frac{\delta\Gamma}{\Gamma_{0}}=\left(\frac{1}{M_{p}^{2}}\frac{V_{0}}{V_{\phi}}\mathcal{L}_{X}\beta+\frac{c}{3s_{0}T_{0}}V_{\phi}b\right)\delta\phi+\frac{c}{3s_{0}}\delta
s,$ (22) $\frac{\delta
V_{\phi}}{V_{\phi}}=\left(\frac{V_{\phi\phi}}{V_{\phi}}+\frac{V_{\phi}}{3s_{0}T_{0}}b^{2}\right)\delta\phi+\frac{b}{3s_{0}}\delta
s,$ (23) $\displaystyle\frac{\delta r}{r_{0}}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{M_{p}^{2}}\frac{V_{0}}{V_{\phi}}\mathcal{L}_{X}\beta+\frac{bcV_{\phi}}{3s_{0}T_{0}}-\frac{1}{2V}(V_{\phi}+\frac{1}{3}V_{\phi}b)\right]\delta\phi$
(24)
$\displaystyle-\left(\frac{1}{2V}\mathcal{L}_{X}c_{s}^{-2}+\frac{\mathcal{L}_{XX}}{\mathcal{L}_{X}}\right)u_{0}\delta
u+\left(\frac{c}{3s_{0}}-\frac{2T_{0}}{3V}\right)\delta s.$

Above equations will be used repeatedly thereinafter.

Taking the variation of the Eqs. (17) and (18), we can obtain:

$\left(\begin{array}[]{c}\delta\dot{\phi}\\\ \delta\dot{u}\\\
\delta\dot{s}\end{array}\right)=E\cdot\left(\begin{array}[]{c}\delta\phi\\\
\delta u\\\ \delta s\end{array}\right)-F.$ (25)

The $E$ is a $3\times 3$ matrix which can be expressed as:

$E=\left(\begin{array}[]{ccc}0&1&0\\\ A&\lambda_{1}&B\\\
C&D&\lambda_{2}\end{array}\right).$ (26)

The matrix elements of $E$ can be calculated out:

$\displaystyle
A=3H_{0}^{2}c_{s}^{2}\left[\frac{1}{1+r}\epsilon-\eta+\frac{r}{1+r}\beta-\frac{(1+r)^{2}}{r}b^{2}\right.$
$\displaystyle\left.+(1+r)bc+\frac{1}{3}\frac{1}{1+r}b\epsilon\right],~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(27)
$B=\frac{H_{0}T_{0}}{\mathcal{L}_{X}u_{0}}c_{s}^{2}\left[-c-\frac{4}{3}\frac{\epsilon}{(1+r)^{2}}-\frac{1+r}{r}b\right],$
(28) $\displaystyle
C=\frac{3H_{0}^{2}\mathcal{L}_{X}u_{0}}{T_{0}}\left[\frac{r}{1+r}\epsilon-\frac{r}{1+r}\beta\right.$
$\displaystyle\left.+(1+r)(1-c)b+\frac{1}{3}\frac{r}{1+r}b\epsilon\right],$
(29)
$D=\frac{H_{0}\mathcal{L}_{X}u_{0}}{T_{0}}\left[6r-\frac{rc_{s}^{-2}}{(1+r)^{2}}\epsilon\right],$
(30)
$\lambda_{1}=-3H_{0}\left[1-(2+r)c_{s}^{2}-\frac{2}{3}\mathcal{S}-\frac{1+c_{s}^{2}+rc_{s}^{2}}{3(1+r)^{2}}\epsilon\right],$
(31) $\lambda_{2}=H_{0}(c-4)-\frac{4}{3}H_{0}\frac{r\epsilon}{(1+r)^{2}}.$
(32)

The column matrix $F$ acts as a small “forcing term”, which can be expressed
as:

$F=\left(\begin{array}[]{c}0\\\ \dot{u}_{0}\\\ \dot{s}_{0}\end{array}\right).$
(33)

The parameter $\mathcal{S}$ in Eq. (31) is a characteristic quantity in
noncanonical inflation expressed as $\mathcal{S}=\frac{\dot{c_{s}}}{Hc_{s}}$,
describing the variation rate of the noncanonical effect. The slow roll
solutions can behave as an attractor for general warm inflationary dynamic
system only when the eigenvalues of the matrix $E$ are negative or possibly
positive but small quantities of order $\mathcal{O}(\frac{\epsilon}{1+r})$
(which means we can have slow growth), and the the “forcing term” $F$ is small
enough, i.e. $|\frac{\dot{u}_{0}}{H_{0}u_{0}}|$ ,
$|\frac{\dot{s}_{0}}{H_{0}s_{0}}|\ll 1$. Now we analyze the “forcing term” $F$
first. Taking the time derivative of the slow roll equations (19) and (20),
and after some calculations, we get

$\dot{u}_{0}=\frac{BC-A\lambda_{2}}{\lambda_{1}\lambda_{2}-BD}u_{0},$ (34)
$\dot{s}_{0}=\frac{AD-C\lambda_{1}}{\lambda_{1}\lambda_{2}-BD}u_{0}.$ (35)

With the help of the matrix elements of $E$, we can finally calculate out

$\displaystyle\frac{\dot{u}_{0}}{H_{0}u_{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\left[\frac{4-c-rc}{1+r}\epsilon+\frac{4r}{1+r}\beta+(c-4)\eta\right.$
(36) $\displaystyle+$ $\displaystyle\left.3(1+r)bc\right],$
$\displaystyle\frac{\dot{s}_{0}}{H_{0}s_{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}\left[\frac{6-\mathcal{G}}{1+r}\epsilon+\frac{6+\mathcal{G}}{1+r}\beta-6\eta\right.$
(37) $\displaystyle+$
$\displaystyle\left.6(1+r)bc+\frac{1+r}{r}(c-1)\mathcal{G}b\right].$

The quantity $\Delta$ in above two equations is
$\Delta\simeq\mathcal{G}(c-4)+2rc$, where
$\mathcal{G}=c_{s}^{-2}-(2+r)-\frac{2}{3}\mathcal{S}c_{s}^{-2}$.

In slow roll inflationary epoch, the Hubble parameter should also be slowly
varying, i.e.
$\frac{\dot{H}_{0}}{H_{0}^{2}}\simeq-\frac{1}{\mathcal{L}_{X}(1+r)}\tilde{\epsilon}=-\frac{1}{1+r}\epsilon\ll
1$. Then the sufficient conditions characterized by new defined slow roll
parameters to satisfy all above requirements can be obtained:

$\epsilon\ll 1+r,~{}~{}\beta\ll 1+r,~{}~{}\eta\ll r,~{}~{}b\ll\frac{1}{1+r};$
(38)

when $c_{s}^{2}$ is not far less than unity; and while $c^{2}_{s}\ll 1$, we
have

$\epsilon\ll c_{s}^{-2}+r,~{}~{}\beta\ll c_{s}^{-2}+r,~{}~{}\eta\ll
c_{s}^{-2}+r,~{}~{}b\ll\frac{c_{s}^{-2}+r}{r}.$ (39)

Thus the validity of slow roll approximations can be measured by above slow
roll conditions. The conditions are dimensionless and invariant under field
transformations. If we use the traditional slow roll parameters, above
conditions can be expressed as:

$\tilde{\epsilon}\ll\mathcal{L}_{X}(1+r),~{}~{}\tilde{\beta}\ll\mathcal{L}_{X}(1+r),~{}~{}\tilde{\eta}\ll\mathcal{L}_{X}r,~{}~{}b\ll\frac{1}{1+r};$
(40)

when $c_{s}^{2}$ is not far less than unity; and when $c^{2}_{s}\ll 1$, we
have

$\displaystyle\tilde{\epsilon}\ll\mathcal{L}_{X}(c_{s}^{-2}+r),~{}~{}~{}~{}\tilde{\beta}\ll\mathcal{L}_{X}(c_{s}^{-2}+r),$
$\displaystyle\tilde{\eta}\ll\mathcal{L}_{X}(c_{s}^{-2}+r),~{}~{}~{}~{}~{}~{}b\ll\frac{c_{s}^{-2}+r}{r}.$
(41)

From above slow roll conditions, it is founded that the slow roll
approximations we proposed in general noncanonical warm inflation is proper,
and the validity of slow roll approximations can be easily guaranteed by the
more relaxed slow roll conditions. Since the “new” slow roll parameters are
invariant under field transformations, we can discuss the range of slow roll
conditions in normalized field representation (the normalized field can always
be get by making $\psi=\int\sqrt{\mathcal{L}_{X}}d\phi$). Then the parameter
$\mathcal{L}_{X}$ is larger than one as discussed in Sec. II, so the slow roll
requirements on inflaton potential are really much more relaxed. The slow roll
conditions in general noncanonical warm inflation are much broader than
canonical warm inflation, let alone cold inflation, resulting from the strong
noncanonical effect and thermal dissipation effect. Thus many kinds potentials
that are excluded by cold inflation or even canonical warm inflation, can be
consistent with noncanonical warm inflation. In “coupling” noncanonical warm
inflation (i.e. the noncanonical field with a general Lagrangian density
having coupling form of kinetic and potential terms), the slow roll
approximations can be also given out, as in “uncoupling” noncanonical warm
inflation (i.e. the noncanonical field with a somewhat easy Lagrangian density
having separate form of kinetic and potential terms). The slow roll conditions
suggest the noncanonical effect in general noncanonical warm inflation has
more evident impact on the validity of slow roll inflation. The slow roll
condition for parameter $b$ indicates thermal correction to the inflaton
potential should be small, as in canonical warm inflation Ian2008 ; Campo2010
.

In the following, we analyze the matrix $E$ to give additional slow roll
conditions. Using slow roll conditions we have obtained above, we reach the
conclusion that

$\displaystyle\det(\lambda I-E)$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\lambda&-1&0\\\
-A&\lambda-\lambda_{1}&-B\\\ -C&-D&\lambda-\lambda_{2}\end{array}\right|$
$\displaystyle=$
$\displaystyle\lambda(\lambda-\lambda_{1})(\lambda-\lambda_{2})-BD\lambda-A(\lambda-\lambda_{2})-BC$
$\displaystyle=$ $\displaystyle 0$ (46)

exists a very small eigenvalue
$\lambda\simeq\frac{BC-A\lambda_{2}}{\lambda_{1}\lambda_{2}-BD-A}\ll\lambda_{1},\lambda_{2}$.
The other two eigenvalues are the solutions of the following equation:

$\lambda^{2}-(\lambda_{1}+\lambda_{2})\lambda+\lambda_{1}\lambda_{2}-BD=0.$
(47)

In order that the slow roll solutions can be an attractor for general warm
inflationary dynamic system, the two eigenvalues should be both negative,
which implies that $\lambda_{1}+\lambda_{2}<0$ and
$\lambda_{1}\lambda_{2}-BD>0$. Finally we get

$|c|<4.$ (48)

According to the slow roll conditions we have got above, the relation
$\frac{\rho_{r}}{V}=\frac{r\epsilon}{2(1+r)^{2}}\ll 1$ is hold, which
indicates radiation energy density is subdominated during the slow roll
inflationary epoch. The conclusion is consistent with the slow roll
requirement that the inflation is potential dominated. Compared to cold
inflation and canonical warm inflation, the slow roll approximations are
easily to be guaranteed under the condition of noncanonical warm inflation.

##  IV  A concrete example of general warm inflation

Now we begin with a kinetic and potential coupling noncanonical warm
inflationary model with the Lagrangian density:

$\mathcal{L}(X^{\prime},\varphi)=\Lambda^{4}\left[1-\sqrt{1-\Lambda^{-4}\varphi^{2}X^{\prime}}\right]-V(\varphi).$
(49)

Obviously, it has a “coupling” form with $\mathcal{L}_{X^{\prime}\varphi}\neq
0$. In order to eliminate the coupling term, we define a new field
$\phi=\frac{1}{2\sqrt{2}}\varphi^{2}$, then using the $\phi$ representation,
the Lagrangian density becomes:

$\mathcal{L}(X,\phi)=\Lambda^{4}\left[1-\sqrt{1-2\Lambda^{-4}X}\right]-V(\phi).$
(50)

Then the new Lagrangian density has the DBI form with a constant warp factor
$f=\Lambda^{-4}$ as in Franche2010 , which is decoupled. For convenience, we
shall consider the strong regime of warm inflation with $r\gg 1$. The
dissipation coefficient is assumed to have the form
$\Gamma=C_{\phi}\frac{T^{3}}{\phi^{2}}$ as indicated in Mar2007 . The
quantities associate with the strength of noncanonical effect is
$\mathcal{L}_{X}=c_{s}^{-1}=\frac{1}{\sqrt{1-2fX}}>1$.

There are two kinds of DBI inflationary models: named the ultraviolet (UV)
model and the infrared (IR) model IRDBI ; IRDBI1 . The UV DBI model is already
at odds with observations, so we consider only the IR DBI model here. The IR
DBI potential $V(\phi)$ in Eq. (50) can be expressed as
$V(\phi)=V_{0}-\frac{1}{2}\alpha H^{2}\phi^{2}$ IRDBI ; IRDBI1 . Then the slow
roll conditions are given in the $\phi$ representation:

$\displaystyle\epsilon=\frac{\tilde{\epsilon}}{\mathcal{L}_{X}}=\frac{\alpha^{2}\phi^{2}}{18M_{p}^{2}\mathcal{L}_{X}}\ll
r;$
$\displaystyle\eta=\frac{\tilde{\eta}}{\mathcal{L}_{X}}=-\frac{\alpha}{3\mathcal{L}_{X}}\ll
r;$
$\displaystyle\beta=\frac{\tilde{\beta}}{\mathcal{L}_{X}}=\frac{2\alpha}{3\mathcal{L}_{X}}\ll
r.$ (51)

Just meeting the conditions that the parameter $\alpha$ is not too large and
the magnitude of inflaton is sub-Planckian, the slow roll conditions can be
easily hold. We have proposed a necessary criterion for a successful warm
inflationary model that the Hubble slow roll (HSR) parameter $\epsilon_{H}$
must be an increasing function of time during the inflationary phase, thus
$\epsilon_{H}$ can increase to 1 to end the inflation naturally and the
Universe can turn smoothly into the radiation dominated phase TachyonZhang . A
crucial requirement for a warm inflationary model is having an increasing HSR
parameter $\epsilon_{H}$. In this IR DBI warm inflationary model, the Hubble
slow roll parameter
$\epsilon_{H}=-\frac{\dot{H}_{0}}{H_{0}^{2}}\simeq\frac{\epsilon}{r}=\frac{\alpha^{2}\phi^{2}}{18M_{p}^{2}\mathcal{L}_{X}r}\propto\phi^{4}$.
Along with the proceeding of inflation, the inflaton field is increasing, so
we can have an increasing $\epsilon_{H}$, which is also consistent with the
criterion. The conclusion can be reached that the IR DBI warm inflation model
we assumed here is an practicable model.

##  V Conclusions

We review kinds of inflationary theories, and introduce the theory of general
noncanonical warm inflation. General noncanonical warm inflation generalize
the scope the inflation. We give the basic equations of this scenario, and
review that an inconvenient “coupling” noncanonical model in $\varphi$
representation can turn to an easy-to-use “uncoupling” model in $\phi$
representation through field transformation $\phi=f(\varphi)$. We introduce
the slow roll approximations and new slow roll parameters in the new scenario.
The new slow roll parameters are all dimensionless as in canonical inflation,
which have different form with the traditional definition of slow roll
parameters. The main task of this paper is performing a systemic stability
analysis to give out that under which conditions the slow roll solutions can
really act as an attractor for the inflationary system. Then we get the slow
roll conditions assuring the validity of slow roll approximations of the
general warm inflationary picture which expressed by several slow roll
parameters. The sound speed $c_{s}$ , measuring the strength of noncanonical
effect, is invariant under field transformations. The slow roll conditions we
got imply that slow roll inflation can still be tenable in general
noncanonical warm inflation. The noncanonical effect resulting from kinetic
and potential “coupling” will not destroy the definition of slow roll. What’s
more, the slow roll conditions are more relaxed compared to canonical warm
inflation, let alone cold inflation, which is due to the enhanced Hubble
damped term and thermal damped term in motion equation. The differences of
conditions on $b$ and $c$ in canonical and noncanonical warm inflation are
proved to be not huge, for they only describe the temperature dependence. Then
we study the IR DBI warm inflationary model as a concrete example in general
noncanonical warm inflation. We decouple the inflationary model by using an
easy field transformation, calculate its slow roll conditions and a necessary
warm inflationary criterion. And then we reach the conclusion that the choice
of potential and dissipation coefficient etc. are suitable. This model can be
a successful and self-consistent noncanonical warm inflationary model.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grant No. 11605100, No. 11704214, and No. 11975132).

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# Entanglement filter with Rydberg atoms

Gen-Sheng Ye1∗, Biao Xu1∗, Yue Chang2,3∗, Shuai Shi1, Tao Shi4,5, Lin Li1†
1MOE Key Laboratory of Fundamental Physical Quantities Measurement,
Hubei Key Laboratory of Gravitation and Quantum Physics, PGMF,
Institute for Quantum Science and Engineering, School of Physics,
Huazhong University of Science and Technology, Wuhan 430074, China  2Beijing
Automation Control Equipment Institute, Beijing 100074, China  3Quantum
Technology R$\&$D Center of China Aerospace Science and Industry Corporation,
Beijing 10019, China  4Institute of Theoretical Physics, Chinese Academy of
Sciences, P.O. Box 2735, Beijing 100190, China  5CAS Center for Excellence in
Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijing 100049, China  $\ast$These authors contributed equally.  $\dagger$ To
whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
###### Abstract

Devices capable of deterministically manipulating the photonic entanglement
are of paramount importance, since photons are the ideal messengers for
quantum information. Here, we report a Rydberg-atom-based entanglement filter
that preserves the desired photonic entangled state and deterministically
blocks the transmission of the unwanted ones. Photonic entanglement with near-
unity fidelity can be extracted from an input state with an arbitrarily low
initial fidelity. The protocol is inherently robust, and succeeds both in the
Rydberg blockade regime and in the interaction-induced dissipation regime.
Such an entanglement filter opens new routes toward scalable photonic quantum
information processing with multiple ensembles of Rydberg atoms.

  *

Advancing the efficient quantum control of photonic entanglement is at the
heart of quantum science (1–4). As one of the key elements in quantum
photonics, an entanglement filter (EF) transmits the entanglement of the
desired quantum states, while blocking the transmission of unwanted photonic
components (5). It has a plethora of potential applications, including
photonic entanglement generation (6), all-optical quantum information
processing (7,8) and entanglement distillations (9,10). However, its
scalability and applicability have been limited by the fact that all photonic
entanglement filter protocols to date are based on linear-optical approaches
(5,11–13), which remove unwanted photonic states only in a probabilistic way.
The probabilistic nature and the requirement of ancillary quantum resources
lead to poor scalability and overwhelming resource consumption. Moreover, the
output entanglement fidelity in the linear-optical approach is ultimately
limited by the finite interference visibility between photons.

Therefore, scalable quantum photonic applications (1–3,14) will largely
benefit from an ideal entanglement filter that deterministically removes
undesired states, unconditionally achieves a high entanglement fidelity, and
demands no extra photonic resources. Unfortunately, the realization of such a
superior entanglement-filter protocol remains elusive hitherto, due to the
lack of a strong and controllable photon-photon interaction in linear-optical
approaches. In recent years, cold Rydberg atoms have been employed to
facilitate the interactions between photons and to achieve intrinsically
deterministic quantum photonic operations, such as single-photon generation
(15–17) and manipulation (18–22), atom-photon entanglement preparation
(23,24), and photon-photon gate (25).

Here, we propose and demonstrate an above-envisioned superior entanglement
filter by exploiting cold Rydberg atomic ensembles to mediate the interaction
between photons. The qubits are encoded in the horizontal (H) and vertical (V)
polarizations of photons a and b. With a product state of
$(\ket{H}_{a}+\ket{V}_{a})(\ket{H}_{b}+\ket{V}_{b})$ as input, the filter
removes the $\ket{H}_{a}\ket{H}_{b}$ and $\ket{V}_{a}\ket{V}_{b}$ components,
resulting in a maximally entangled Bell state
$\ket{\Psi^{+}}=1/\sqrt{2}\left(\ket{H}_{a}\ket{V}_{b}+\ket{V}_{a}\ket{H}_{b}\right)$.
In contrast to previous probabilistic linear-optical approaches, our protocol
eliminates the unwanted states in a deterministic way, outputs high-fidelity
entangled states, and requires no extra quantum resources.

Figure 1:  Illustration of experimental protocol. (A) The horizontally (H) and
vertically (V) polarized components of the $780\text{\,}\mathrm{n}\mathrm{m}$
photonic qubits $a$ and $b$ are separated into the down (D) and upper (U)
paths by a polarization beam splitter (PBS), and stored as Rydberg excitations
by the $480\text{\,}\mathrm{n}\mathrm{m}$ laser fields
$\Omega_{a}^{\mathrm{U},\mathrm{D}}$ and $\Omega_{b}^{\mathrm{U},\mathrm{D}}$.
The $780\text{\,}\mathrm{n}\mathrm{m}$ and $480\text{\,}\mathrm{n}\mathrm{m}$
fields are counter-propagating and focused on the cold 87Rb atomic ensembles
with waists of $6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ and
$15\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$, respectively. In the read-
out stage, the $480\text{\,}\mathrm{n}\mathrm{m}$ laser fields
$\Omega_{a}^{\mathrm{U},\mathrm{D}}$ and $\Omega_{b}^{\mathrm{U},\mathrm{D}}$
are sequentially applied to convert the Rydberg excitations
$\ket{D}_{a}/\ket{U}_{a}$ and $\ket{D}_{b}/\ket{U}_{b}$ to photonic qubits
$\ket{H}_{a}/\ket{V}_{a}$ and $\ket{H}_{b}/\ket{V}_{b}$. The photons then go
through a polarization-projective measurement setup before being collected by
single-mode fibers and detected by SPCMs. The inserts show the timing sequence
(B) and the relevant 87Rb atomic levels (C): ground state
$\ket{\mathrm{g}}=\ket{5S_{1/2},F=2,m_{F}=2}$, intermediate state
$\ket{\mathrm{e}}=\ket{5P_{3/2},F=3,m_{F}=3}$, and Rydberg states
$\ket{r_{a}}=\ket{n_{a}D_{5/2},J=5/2,m_{j}=5/2}$ and
$\ket{r_{b}}=\ket{n_{b}D_{5/2},J=5/2,m_{j}=5/2}$.

As illustrated in Fig. 1, our experiment employs two Rydberg ensembles to
achieve polarization-selective interaction between photonic qubits. The
working principle is to convert the unwanted photonic components into double
Rydberg excitations in the same atomic ensemble and to achieve the
deterministic removal of these states with either the Rydberg blockade (26) or
the interaction induced two-body dissipation (27). Using the photon storage
based on the Rydberg electromagnetically induced transparency (EIT), the
$480\text{\,}\mathrm{nm}$ control laser field
$\Omega_{a}^{\mathrm{U}}(\Omega_{a}^{\mathrm{D}})$ coherently transfers the
$780\text{\,}\mathrm{nm}$ photon in the state $\ket{H}_{a}$ ($\ket{V}_{a}$)
into a collective atomic excitation in the down (upper) ensemble:

$\displaystyle\ket{D}_{a}=\sum^{N}_{j=1}{\ket{\mathrm{g^{D}}}_{1}\ldots\ket{\mathrm{{r_{a}}^{D}}}_{j}\ldots\ket{\mathrm{g^{D}}}_{N}}/\sqrt{N}$
(1)
$\displaystyle\ket{U}_{a}=\sum^{N}_{j=1}{\ket{\mathrm{g^{U}}}_{1}\ldots\ket{\mathrm{{r_{a}}^{U}}}_{j}\ldots\ket{\mathrm{g^{U}}}_{N}}/\sqrt{N},$

where $\ket{\mathrm{g}}$ is the atomic ground state, $\ket{\mathrm{r}_{a}}$ is
a Rydberg state, and U/D denotes the upper/down ensemble. To induce the
interaction between photons a and b, an adjacent Rydberg level
$\ket{\mathrm{r}_{b}}$ is employed to transfer the qubit state $\ket{H}_{b}$
($\ket{V}_{b}$) to the excitation $\ket{D}_{b}$ ($\ket{U}_{b}$) in the down
(upper) ensemble.

To implement the entanglement filter in the blockade regime, high-lying
Rydberg states with principal quantum numbers $n_{a}=76$ and $n_{b}=77$ are
employed. As a result, the storage of unwanted components
$\ket{H}_{a}\ket{H}_{b}$ and $\ket{V}_{a}\ket{V}_{b}$ into double Rydberg
excitations in the same ensemble, i.e., $\ket{D}_{a}\ket{D}_{b}$ or
$\ket{U}_{a}\ket{U}_{b}$, is strongly suppressed by the blockade effect. On
the other hand, two ensembles are separated by
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$, well beyond the Rydberg blockade
radius, such that the desired entangled state $\ket{\Psi^{+}}$ can be stored
as $1/\sqrt{2}\left(\ket{D}_{a}\ket{U}_{b}+\ket{U}_{a}\ket{D}_{b}\right)$ of
the atoms. After the storage process, the Rydberg excitations are converted
back to the photonic qubits in the state $\ket{\Psi^{+}}$ through collective
emission by sequentially applying $480\text{\,}\mathrm{nm}$ read-out light
fields.

Figure 2:  Blockade-based entanglement filter. (A and B)  Normalized two-
photon populations of the input (A) and output (B) state, obtained from photon
correlation measurements. (C and D)  The real part of the reconstructed
density matrix of the input (C) and the output (D) state. (E) The measured
input- (green diamonds) and output- (red circles) state fidelities as
functions of the input-state amplitude ratio $\alpha/\beta$. Solid line:
expected input fidelity at different $\alpha/\beta$. Error bars: $1\sigma$
standard deviation from photoelectric counting events.  Figure 3:  Interaction
induced two-body decoherence. (A and B) Illustration of two-body decoherence
induced by distance-dependent interaction. (C and D) The suppression ratio
$\chi=\frac{P_{HH}+P_{VV}}{P_{HV}+P_{VH}}$ as a function of evolution time $t$
for $n_{a}=47,n_{b}=48$ (C) and $n_{a}=55,n_{b}=56$ (D). Error bars: $1\sigma$
standard deviation from photoelectric counting events. Solid curves:
simulations based on the two-body decoherence mechanism.

The polarization-selective photon blockade effect is shown in Fig. 2, A and B.
Two weak coherent laser pulses with average photon number $\braket{n}\sim 0.1$
are used to approximate single photons a and b. After passing through the
entanglement filter, photons are detected in the $\ket{H}/\ket{V}$ basis by
single-photon counting modules (SPCMs) . Figure 2A displays the measured two-
photon populations of the input state, which are distributed equally ($\sim
0.25$) over all bases. The populations of the output state are shown in Fig.
2B, where the $\ket{H}_{a}\ket{H}_{b}$ and $\ket{V}_{a}\ket{V}_{b}$ components
are strongly suppressed. The measured suppression ratio
$\chi=\frac{P_{HH}+P_{VV}}{P_{HV}+P_{VH}}\sim$4(1)\text{\,}\times\mathrm{1}\mathrm{0}^{-3}$$
demonstrates the strong blockade effect in the entanglement filter.

To characterize the entanglement filter in the non-classical regime, quantum
state tomography for the input and output states are performed. The density
matrix $\rho_{\mathrm{in}}$ of the input state is reconstructed and shown in
Fig. 2C, which has a fidelity of $F_{\mathrm{in}}=49.3(7)\%$ overlapping with
the target state $\ket{\Psi^{+}}$. The entanglement filter removes the
undesired components and improves the fidelity up to
$F_{\mathrm{out}}=98.8(5)\%$ (Fig. 2D). The reduction of $F_{\mathrm{out}}$
from unity is mainly caused by the $0.61(6)\%$ infidelity from background
detection events of SPCMs, and the background-corrected fidelity is improved
to $F_{\mathrm{cor}}=99.4(5)\%$. The multi-photon components from the weak
coherent light and the crosstalk between different polarization bases also
cause infidelities of $\leq 0.58\%$ and $0.25(23)\%$, respectively. The above-
mentioned sources of infidelities are not intrinsic to our protocol and can be
further suppressed with technical efforts. The fundamental limitation on the
fidelity comes from the imperfect suppression of double Rydberg excitations,
which gives an upper bound of $F_{out}\sim 99.98(6)\%$ (see supplementary
materials).

An ideal entanglement filter should post no prerequisite on the fidelity of
the input state and extract the desired entangled state from an arbitrarily
noisy input. To demonstrate this essential capability,
($\alpha\ket{H}_{a}+\beta\ket{V}_{a}$)($\alpha\ket{H}_{b}+\beta\ket{V}_{b}$)
is employed as the input state. By varying the amplitude ratio $\alpha/\beta$,
the input fidelity $F_{in}$ can be tuned in the classical regime between 0 and
0.5 (green diamonds in Fig. 2E). When the entanglement filter is applied, the
fidelities of the corresponding output states are improved to a near-unity
level (red circles in Fig. 2E). In the case of $\alpha/\beta=0.185$, our
entanglement filter improves the input state fidelity $F_{in}=0.070(12)$ by
more than one order of magnitude. The measured state fidelity
$F_{out}=0.955(22)$ is limited by the decreased signal-to-background ratio for
small $\alpha/\beta$, and the background-corrected fidelity
$F_{cor}=0.992(25)$ is not affected by the low input state fidelity.

Developing novel Rydberg quantum photonic protocols with low principal quantum
number $n$ holds the promise to alleviate some of the decoherence and loss
mechanisms associated with high $n$ states, such as long-lived Rydberg
contaminants, energy level drifts induced by background electric fields, and
density-dependent dephasing. To this end, we demonstrate an entanglement
filter in the absence of blockade effect using lower-lying Rydberg states
$n_{a}=47$ and $n_{b}=48$, whose van der Waals interaction coefficient $C_{6}$
is nearly two orders of magnitudes weaker than that of the states $n_{a}=76$
and $n_{b}=77$. Therefore, the $\ket{H}_{a}\ket{H}_{b}$ and
$\ket{V}_{a}\ket{V}_{b}$ components can be stored as double Rydberg
excitations in the same ensemble.

The distance-dependent van der Waals interaction strength
$V_{jj^{\prime}}=\frac{C_{6}}{R_{jj^{\prime}}^{6}}$ varies strongly for
Rydberg atom pairs with different separations $R_{jj^{\prime}}$, and leads to
the accumulation of random phases and a fast two-body decoherence during the
quantum evolution (27):

$\ket{U}_{a}\ket{U}_{b}\propto\sum^{N}_{j,j^{\prime}\neq
j}{e^{i\frac{V_{jj^{\prime}}}{\hbar}t}\ket{\mathrm{g^{U}}}_{1}\ldots\ket{\mathrm{{r_{a}}^{U}}}_{j}\ldots\ket{\mathrm{{r_{b}}^{U}}}_{j^{\prime}}\ldots\ket{\mathrm{g^{U}}}_{N}.}$
(2)

If the read-out is performed immediately after storage, i.e., without two-body
decoherence, the spatial mode of the retrieved photons will be highly
directional, as a result of the mode-matched collective emission (Fig. 3A).
However, the accumulation of interaction-induced random phases deteriorates
the collective coherence of the excitations and leads to the spontaneous
photon emission into random directions during the read-out (Fig. 3B). In the
experiment, only photons in the mode-matched direction are collected and
therefore the retrieved $\ket{H}_{a}\ket{H}_{b}$ and $\ket{V}_{a}\ket{V}_{b}$
components are suppressed due to the dissipation. The entangled state
$\ket{\Psi^{+}}$ is stored in the decoherence-free-subspace
$1/\sqrt{2}\left(\ket{D}_{a}\ket{U}_{b}+\ket{U}_{a}\ket{D}_{b}\right)$, hence
immune to the above dissipation. Figure 3C displays the measured suppression
ratio $\chi$ as a function of the dissipative evolution time $t$. The ratio
$\chi$ decreases to $0.440(45)$ with just a short evolution time of
$0.2\text{\,}\mu\mathrm{s}$ and is further suppressed to $0.037(17)$ after
$4\text{\,}\mu\mathrm{s}$.

Figure 4:  Entanglement filter via dissipative quantum evolution. (A) The
measured state fidelity as a function of dissipative evolution time $t$ with
$n_{a}=47,n_{b}=48$. (B) The reconstructed density matrices for input state
and output states with different evolution times. (C) Same as (A), but for
$n_{a}=55,n_{b}=56$. Error bars: $1\sigma$ standard deviation from
photoelectric counting events. Solid curves: simulations based on the two-body
decoherence model.

Similar to $\chi$, the measured output-state fidelity depends on the quantum
evolution time $t$ in the dissipative regime. The observed temporal dynamics
of output-state fidelity is shown in Fig. 4A, which agrees well with the
result simulated by the two-body decoherence model (27). The undesired
components in the density matrix $\rho$ are gradually suppressed with a longer
dissipative evolution time (Fig. 4B). After
$4\text{\,}\mathrm{\SIUnitSymbolMicro s}$ of evolution, the entanglement
filter improves the state fidelity from $F_{in}=0.495(7)$ to $F=0.951(17)$.

The dissipative entanglement filter protocol is very robust to the principal
quantum number $n$. For higher Rydberg states $n\sim 55$, the larger
interaction variation leads to a quicker dissipation of the unwanted photonic
components (Fig. 3D) and consequently a faster improvement of the state
fidelity with the evolution time $t$. In Fig. 4C, an entanglement fidelity of
$F=0.978(15)$ is achieved with $t=$2.8\text{\,}\mu\mathrm{s}$$. In principle,
the dissipative scheme can also be realized with very low $n$, by harnessing
the resonant dipole-dipole interactions. Our simulation shows that, efficient
entanglement filtering can be achieved even with
$\ket{r_{a}}=\ket{19D_{5/2},m_{j}=3/2}$ and
$\ket{r_{b}}=\ket{20P_{3/2},m_{j}=3/2}$ (see supplementary materials).
Combined with the blockade-based scheme at high $n$, the working range of our
entanglement filter spans over a large spectrum of Rydberg states.

Although the transmission of the entanglement filter is determined by the
efficiency of photon storage-and-retrieval processes, our protocol is
intrinsically deterministic. Currently, the measured storage efficiency
$\eta_{s}=0.24$ and read-out efficiency $\eta_{r}=0.36$ are mainly limited by
the finite optical depth ($\sim 3.5$) and can be further improved. By
incorporating the atomic ensemble into an optical resonator, $\eta_{s}$ and
$\eta_{r}$ can be increased to near unity and the high entanglement
transmission could be achieved. Moreover, the target state of our entanglement
filter is not limited to $\ket{\Psi^{+}}$. In principle, our entanglement
filter can extract any of the four Bell states from an arbitrary input states.
For example, we also demonstrate the entanglement filtering for
$\ket{\Psi^{-}}$ from an input state
$\ket{H}_{a}\ket{V}_{b}=1/\sqrt{2}(\ket{\Psi^{+}}+\ket{\Psi^{-}})$, with an
output state fidelity of $0.989(10)$ (see supplementary materials).

In summary, we report the realization of a novel entanglement filter enabled
by Rydberg atoms. In contrast to previous demonstrations, our scheme features
superior scalability, since it works without extra photonic resources, removes
undesired states deterministically, and features the high entanglement
fidelity.

Our entanglement filter opens new avenues for a number of novel quantum
photonic applications and studies. First of all, by scaling up to an array of
Rydberg ensembles, the efficient generation and manipulation of multi-photon
entanglements, such as Dicke states and GHZ states, can be realized. Moreover,
our protocol is not based on the photon-photon interference, therefore, the
qubits do not need to be indistinguishable. This unique feature allows an
effective quantum control between photons with different temporal-spatial
profiles and even with different colors, as long as they can be coupled to
appropriate atomic transitions. Last but not least, our entanglement filter
also succeeds in the dissipative regime by exploiting the interaction-induced
two-body decoherence. The extension of such a scheme to an array of
interacting Rydberg excitations could enable the dissipative preparation of
long-range correlated states, such as Wigner crystal (28), and the exploration
of novel many-body quantum dynamics (29,30) with interaction disorders that
can be tuned by orders of magnitude.

## References

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## Acknowledgments

The authors thank Daiqin Su and Kuan Zhang for valuable discussions and
Chenhao Du, Feng-Yuan Kuang and Yafen Cai for experimental assistance. This
work was supported by the National Key Research and Development Program of
China under Grants No. 2021YFA1402003, the National Natural Science Foundation
of China (Grant No. U21A6006, No. 12004127, and No. 12104173), and the
Fundamental Research Funds for the Central Universities, HUST. Y.C. is
supported by the National Natural Science Foundation of China (Grant No.
U2141237). T.S. is supported by National Key Research and Development Program
of China (Grant No. 2017YFA0718304), and the National Natural Science
Foundation of China (Grants No. 11974363, No. 12135018, and No. 12047503)

## Author contributions

All authors contributed substantially to this work.

## Competing interests

The authors declare no competing interests.

## Data and materials availability

Data that support the plots within this paper are available from the
corresponding authors upon reasonable request.

See pages - of Supplementary


In this section we define some standard functions that are either mentioned in
the Table 2 or used somewhere in the paper. We also state some of their
properties that we need for our proofs. We start by defining some basic
Boolean functions.

######  Definition B.1.

Define $\mathsf{PARITY}:\\{0,1\\}^{n}\to\\{0,1\\}$ to be the
$\mathsf{PARITY}(x_{1},\dots,x_{n})=\sum x_{i}\text{mod}\leavevmode\nobreak\
2$. We use the notation $\oplus$ to denote $\mathsf{PARITY}$.

######  Definition B.2.

Define $\mathsf{AND}:\\{0,1\\}^{n}\to\\{0,1\\}$ to be the
$\mathsf{AND}(x_{1},\dots,x_{n})=0$ if and only if there exists an $i\in[n]$
such that $x_{i}=0$. We use the notation $\wedge$ to denote $\mathsf{AND}$.

######  Definition B.3.

Define $\mathsf{OR}:\\{0,1\\}^{n}\to\\{0,1\\}$ to be the
$\mathsf{OR}(x_{1},\dots,x_{n})=1$ if and only if there exists an $i\in[n]$
such that $x_{i}=1$. We use the notation $\vee$ to denote $\mathsf{OR}$.

######  Definition B.4.

Define $\mathsf{MAJORITY}:\\{0,1\\}^{n}\to\\{0,1\\}$ as
$\mathsf{MAJORITY}(x)=1$ if and only if $|x|>\frac{n}{2}$.

We need the following definition of composing iteratively with itself.

######  Definition B.5 (Iterative composition of a function).

Let $f:\\{0,1\\}^{n}\to\\{0,1\\}$ a Boolean function. For $d\in\mathbb{N}$ we
define the function $f^{d}:\\{0,1\\}^{n^{d}}\to\\{0,1\\}$ as follows: if $d=1$
then $f^{d}=f$, otherwise

$\displaystyle
f^{d}(x_{1},\dots,x_{n^{d}})=f\left(f^{d-1}(x_{1},\dots,x_{n^{d-1}}),\dots,f^{d-1}(x_{n^{d}-n+1},\dots,x_{n^{d}})\right).$

######  Definition B.6.

For $d\in\mathbb{N}$ define $\mathsf{NAND}$-tree of depth $d$ as
$\mathsf{NAND}^{d}$ where $\mathsf{NAND}:\\{0,1\\}^{2}\to\\{0,1\\}$ is defined
as: $\mathsf{NAND}(x_{1},x_{2})=0$ if and only if $x_{1}\neq x_{2}$. We use
the notation $\widetilde{\wedge}$-tree to denote $\mathsf{NAND}$-tree.

The now define a function that gives a quadratic separation between
sensitivity and block sensitivity.

######  Definition B.7 (Rubinstein’s function ([Rub95])).

Let $g:\\{0,1\\}^{k}\to\\{0,1\\}$ be such that $g(x)=1$ iff $x$ contains two
consecutive ones and the rest of the bits are $0$. The Rubinstein’s function,
denoted by $\mathsf{RUB}:\\{0,1\\}^{k^{2}}\to\\{0,1\\}$ is defined to be
$\mathsf{RUB}=\mathsf{OR}_{k}\circ g$.

######  Theorem B.8.

[Rub95] For the Rubinstein’s function in Definition B.7
$\mathsf{s}(\mathsf{RUB})=k$ and $\mathsf{bs}(\mathsf{RUB})=k^{2}/2$. Thus
$\mathsf{RUB}$ witnesses a quadratic gap between sensitivity and block
sensitivity.

[NS94] first introduced a function whose $\mathrm{deg}$ is significantly
smaller than $\mathsf{s}$ or $\mathsf{bs}$. This appears in the footnote in
[NW94] that E. Kushilevitz also introduced a similar function with $6$
variables which gives slightly better gap between $\mathsf{s}$ and
$\mathrm{deg}$. Later Ambainis computed $\mathsf{Q_{E}}$ of that function and
gave a separation between $\mathsf{Q_{E}}$ and $\mathsf{D}$ [Amb16]. This
function is fully sensitive at all zero input, consequently this gives a
separation between $\mathsf{Q_{E}}$ and $\mathsf{s}$.

######  Definition B.9 ([NS94]).

Define $\mathsf{NW}$ as follows:

$\displaystyle\mathsf{NW}(x_{1},x_{2},x_{3})=\begin{cases}1\text{ iff
}x_{i}\neq x_{j}\text{ for some }i,j\in\\{1,2,3\\}\\\ 0\text{
otherwise}.\end{cases}$

Now define the $d$-th iteration $\mathsf{NW}^{d}$ on
$(x_{1},x_{2},\dots,x_{3^{d}})$ as Definition B.5 where $d\in\mathbb{N}$.

######  Definition B.10 (Kushilevitz’s function).

Define $\mathsf{K}$ as follows:

$\displaystyle\mathsf{K}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})=\Sigma
x_{i}+\Sigma
x_{i}y_{i}+x_{1}x_{3}x_{4}+x_{1}x_{4}x_{5}+x_{1}x_{2}x_{5}+x_{2}x_{3}x_{4}+x_{2}x_{3}x_{5}+x_{1}x_{2}x_{6}$
$\displaystyle+x_{1}x_{3}x_{6}+x_{2}x_{4}x_{6}+x_{3}x_{5}x_{6}+x_{4}x_{5}x_{6}.$

Now define the $d$-th iteration $\mathsf{K}^{d}$ on
$(x_{1},x_{2},\dots,x_{6^{d}})$ as Definition B.5 where $d\in\mathbb{N}$.

Next we will describe two example that was introduced in [GSS16] and gives the
separation between $\mathsf{C}$ vs. $\mathsf{RC}$ and $\mathsf{RC}$ vs.
$\mathsf{bs}$ respectively . They also have introduces some new complexity
measures for iterative version of a function and how to use them to get the
critical measure between two complexity measures. For more details we refer to
[GSS16].

######  Definition B.11 ([GSS16]).

Let $n$ be an even perfect square, let $k=2\sqrt{n}$ and $d=\sqrt{n}$. Divide
the $n$ indices of the input into $n/k$ disjoint blocks. Define
$\mathsf{GSS}_{1}:\\{0,1\\}^{n}\to\\{0,1\\}$ as follows:
$\mathsf{GSS}_{1}(x)=1$ if and only if $|x|\geq d$ and all the $1$’s in $x$
are in a single group. Define $\mathsf{GSS}_{1}^{d}$ with
$\mathsf{GSS}_{1}^{1}=f$.

######  Definition B.12 ([GSS16]).

Define $\mathsf{GSS}_{2}:\\{0,1\\}^{n}\to\\{0,1\\}$ where $n$ is of the form
$\binom{t}{2}$ for some integer $t$. Identify the input bits of
$\mathsf{GSS}_{2}$ with the edges of the complete graph $K_{t}$. An input
$x\in\\{0,1\\}^{n}$ induces a subgraph of $K_{t}$ consisting of edges assigned
$1$ by $x$. The function $\mathsf{GSS}_{2}(x)$ is defined to be $1$ iff the
subgraph induces by $x$ has a star graph.

######  Definition B.13.

For $\Sigma=[n^{k}]$ the function
$\mathsf{k\mbox{-}sum}:\Sigma^{n}\to\\{0,1\\}$ is defined as follows: on input
$x_{1},x_{2},\dots,x_{n}\in\Sigma$, if there exists $k$ element
$x_{i_{1}},\dots,x_{i_{k}}$, $i_{1},\dots,i_{k}\in[n]$, that sums to
$0\leavevmode\nobreak\ (\text{mod}\leavevmode\nobreak\ |\Sigma|)$ then output
$1$, otherwise output $0$.

######  Theorem B.14 ([ABK16, BS13]).

For the function $\mathsf{k\mbox{-}sum}:\Sigma^{n}\to\\{0,1\\}$, if
$|\Sigma|\geq 2\binom{n}{n}$ then

$\displaystyle\mathsf{Q}(\mathsf{k\mbox{-}sum})=\Omega(n^{k/(k+1)}/\sqrt{k}).$

Finally, we that the pointer functions and its variants introduced in
[ABB+17]. They are used for demonstrate separation between several complexity
complexity measures like _deterministic query complexity_ , _Randomized query
complexity_ , _Quantum query complexity_ etc. These functions was originally
motivated from [GPW18] function. The functions that we construct for many of
our theorems is a composition function whose outer function is these pointer
function, or a slight variant of these. In the next section we present the
formal definition of pointer functions.

###  B.1 Pointer function and some of its variant

For the sake of completeness first we will describe the three variants of the
function introduced in [ABB+17] that achieves separation between several
complexity complexity measures like _Deterministic query complexity_ ,
_Randomized query complexity_ , _Quantum query complexity_ etc. These
functions was originally motivated from [GPW18] function. The three variants
of [ABB+17] function have some special kind of non-Boolean domain, which we
call _Pointer matrix_. Our function is a special "encoding" of that non-
Boolean domain such that the resulting function becomes transitive and
achieves the separation between complexity measures that matches the known
separation between the general functions. We will start by defining different
types of domain we need for the three variants of [ABB+17] function in
Definition B.15.

######  Definition B.15 (Pointer matrix over $\Sigma$).

For $m,n\in\mathbb{N}$, let $M$ be a $(m\times n)$ matrix with $m$ rows and
$n$ columns. We refer to each of the $m\times n$ entries of $M$ as _cells_.
Each cell of the matrix is from a alphabet set $\Sigma$ where
$\Sigma=\\{0,1\\}\times\widetilde{P}\times\widetilde{P}\times\widetilde{P}$
and $\widetilde{P}=\\{(i,j)|i\in[m],j\in[n]\\}\cup\\{\perp\\}$. We call
$\widetilde{P}$ as set of pointers where, pointers of the form
$\\{(i,j)|i\in[m],j\in[n]\\}$ pointing to the cell $(i,j)$ and $\perp$ is the
null pointer. Hence,each entry $x_{(i,j)}$ of the matrix $M$ is a $4$-tuple
from $\Sigma$. The elements of the $4$-tuple we refer as _value_ , _left
pointer_ , _right pointer_ and _back pointer_ respectively and denote by
$\text{Value}(x_{(i,j)})$, $\text{LPointer}(x_{(i,j)})$,
$\text{RPointer}(x_{(i,j)})$ and $\text{BPointer}(x_{(i,j)})$ respectively
where $\text{Value}\in\\{0,1\\}$,
$\text{LPointer},\text{RPointer},\text{BPointer}\in\widetilde{P}$. We call
this type of matrix as _pointer matrix_ and denote by $\Sigma^{n\times n}$.

A special case of the pointer-matrix, which we call _$\mathsf{Type}_{1}$
pointer matrix over $\Sigma$_, is when for each cell of $M$,
$\text{BPointer}\in\\{[n]\cup\perp\\}$ that is backpointers are pointing to
the columns of the matrix.

Also, in general when,
$\text{BPointer}\in\\{(i,j)|i\in[m],j\in[n]\\}\cup\\{\perp\\}$, we call it a
_$\mathsf{Type}_{2}$ pointer matrix over $\Sigma$_.

Now we will define some additional properties of the domain that we need to
define [ABB+17] function.

######  Definition B.16 (Pointer matrix with marked column).

Let $M$ be an $m\times n$ _pointer-matrix_ over $\Sigma$. A column $j\in[n]$
of $M$ is defined to be a _marked column_ if there exists exactly one cell
$(i,j)$, $i\in[m]$, in that column with entry $x_{(i,j)}$ such that
$x_{(i,j)}\neq(1,\perp,\perp,\perp)$ and every other cell in that column is of
the form $(1,\perp,\perp,\perp)$. The cell $(i,j)$ is defined to be the
_special element_ of the _marked column_ $j$.

Let $n$ be a power of $2$. Let $T$ be a rooted, directed and balanced binary
tree with $n$-leaves and $(n-1)$ internal vertices. we will use the following
notations that will be used in defining some functions formally.

######  Notation B.17.

Let $n$ be a power of $2$. Let $T$ be a rooted, directed and balanced binary
tree with $n$-leaves and $(n-1)$ internal vertices. Labels the edges of $T$ as
follows: the outgoing edges from each node are labeled by either $left$ or
$right$. The leaves of the tree are labeled by the elements of $[n]$ from left
to right, with each label used exactly once. For each leaf $j\in[n]$ of the
tree, the path from the $root$ to the leaf $j$ defines a sequence of $left$
and $right$ of length $O(\log n)$, which we denote by $T(j)$.

When $n$ is not a power of $2$, choose the largest $k\in\mathbb{N}$ such that
$2^{k}\leq n$, consider a complete balanced tree with $2^{k}$ leaves and add a
pair of child node to to each $n-2^{k}$ leaves starting from left. Define
$T(j)$ as before.

More details about _partial assignment_ and _certificates_ can be found in
Definition A.7. Now we are ready to describe the _Variant 1_ of [ABB+17]
function.

######  Definition B.18 (Variant 1 [ABB+17]).

Let $\Sigma^{m\times n}$ be a $\mathsf{Type}_{1}$ _pointer matrix_ where
BPointer is a pointer of the form $\\{j|j\in[n]\\}$ that points to other
column and LPointer, RPointer are as usual points to other cell. Define
$\mathsf{A1}_{(m,n)}:\Sigma^{m\times n}\to\\{0,1\\}$ on a $\mathsf{Type}_{1}$
_pointer matrix_ such that for all $x=(x_{i,j})\in\Sigma^{m\times n}$, the
function $\mathsf{A1}_{(m,n)}(x_{i,j})$ evaluates to $1$ if and only if it has
a _$1$ \- cell certificate_ of the following form:

  1. 1.

there exists exactly one _marked column_ $j^{\star}$ in $M$,

  2. 2.

There is a special cell in the the _marked column_ and there is a balanced
binary tree $T$ rooted at the special cell

  3. 3.

for each non-marked column $j\in[n]\setminus\\{b\\}$ there exist a _cell_
$l_{j}$ such that $\text{Value}(l_{j})=0$ and $\text{BPointer}(l_{j})=b$ where
$l_{j}$ is the end of the path that starts at the _special element_ and
follows the pointers LPointer and RPointer as specified by the sequence
$T(j)$. $l_{j}$ exists for all $j\in[n]\setminus\\{b\\}$ i.e. no pointer on
the path is $\perp$. We refer $l_{j}$ as the leaves of the tree.

The above function achieves the separation between $\mathsf{D}$ vs.
$\mathsf{R_{0}}$ and $\mathsf{D}$ vs. $\mathsf{Q}$ for $m=2n$. Here we will
restate some of the results from [ABB+17] which we will use to prove the
results for our function:

######  Theorem B.19 ([ABB+17]).

The function $\mathsf{A1}_{(m.n)}$ in Definition B.18 satisfies

$\displaystyle\mathsf{D}$ $\displaystyle=\Omega(n^{2})\text{ for $m=2n$ where
$m,n\in\mathbb{N}$},$ $\displaystyle\mathsf{R_{0}}$
$\displaystyle=\widetilde{O}(m+n)\text{ for any $m,n\in\mathbb{N}$},$
$\displaystyle\mathsf{Q}$
$\displaystyle=\widetilde{O}(\sqrt{m}+\sqrt{n})\text{ for any
$m,n\in\mathbb{N}$}.$

Though [ABB+17] gives the deterministic lower bound for the function
$\mathsf{A1}$ precisely for $2m\times m$ matrices following the same line of
argument it can be proved that $\mathsf{D}(\Omega(n^{2}))$ holds for $n\times
n$ matrices also. For sake of completeness we give a proof for $n\times n$
matrices.

######  Theorem B.20.

$\mathsf{D}(\mathsf{A1}_{(n,n)})=\Omega(n^{2})$.

Adversary Strategy for $\mathsf{A1}_{(n,n)}$: We describe an adversary
strategy that ensures that the value of the function is undetermined after
$\Omega(n^{2})$ queries. Assume that deterministic query algorithm queries a
cell $(i,j)$. Let $k$ be the number of queried cell in the column $j$. If
$k\leq\frac{n}{2}$ adversary will return $(1,\perp,\perp,\perp)$. Otherwise
adversary will return $(0,\perp,\perp,n-k)$.

######  Claim B.21.

The value of the function $\mathsf{A1}_{(n,n)}$ will be undetermined if there
is a column with at most $n/2$ queried cells in the first $\frac{n}{2}$
columns $\\{1,2,\dots,\frac{n}{2}\\}$ and at least $3n$ unqueried cells in
total.

###### Proof.

Adversary can always set the value of function to $0$ if the conditions of the
claim are satisfied.

Adversary can also set the value of the function to $1$: If
$s\in[\frac{n}{2}]$ be the column with at most $\frac{n}{2}$ queried cell,
then all the queried cells of the column are of the form
$(1,\perp,\perp,\perp)$. Assign $(1,\perp,\perp,\perp)$ to the other cell and
leave one cell for the $special$ $element$ $a_{p,s}$(say).

For each non-marked column $j\in[n]\ {s}$ define $l_{j}$ as follows: If column
$j$ has one unqueried cell then assign $(0,\perp,\perp,s)$ to that cell. If
all the cells of the column $j$ were already queried then the column contains
a cell with $(0,\perp,\perp,s)$ by the adversary strategy. So, in either case
we are able to form a _leave_ $l_{j}$ in each of the non-marked column.

Now using the cell of _special element_ $a_{p,s}$ construct a rooted tree of
pointers isomorphic to tree $T$ as defined in Definition B.18 such that the
internal nodes we will use the other unqueried cells and assign pointers such
that $l(j)$’s are the leaves of the tree and the _special element_ $a_{p,s}$
is the $root$ of the tree. Finally assign anything to the other cell. Now the
function will evaluates to $1$.

To carry out this construction we need at most $3n$ number of unqueried cells.
Outside of the $marked$ $column$ total $n-2$ cells for the internal nodes of
the tree, atmost $n-1$ unqueried cell for the $leaves$ and the $all-1$
$unique$ $marked$ $column$ contains total $n$ cell, so total $3n$ unqueried
cell will be sufficient for our purpose.

Now there are total $n$ number of columns and to ensure that each of the
column in $\\{1,2,\dots,\frac{n}{2}\\}$ contains at least $\frac{n}{2}$
queried cell we need at least $\frac{n^{2}}{4}$ number of queries. Since
$n^{2}-3n\geq\frac{n^{2}}{4}$ for all $n\geq 6$. Hence
$D(\mathsf{A1}_{(n,n)})=\Omega(n^{2})$. ∎

Hence Theorem B.20 follows.

Also [GPW18]’s function realises quadratic separation between $\mathsf{D}$ and
$\mathrm{deg}$ and the proof goes via $\mathsf{UC}_{min}$ upper bound. But
$\mathsf{A1}_{(n,n)}$ exhibits the same properties corresponding to
$\mathsf{UC}_{min}$. So, from the following observation it follows that
$\mathsf{A1}_{(n,n)}$ also achieves quadratic separation between $\mathsf{D}$
and $\mathrm{deg}$.

######  Observation B.22.

It is easy to observe that for each positive input $x$ to the function
$\mathsf{A1}_{(n,n)}$, the _marked column_ together with the rooted tree of
pointers with _leaves_ in every other column gives a unique minimal
$1$-certificate of $x$. Thus,
$\mathsf{UC}_{1}(\mathsf{A1}_{(n,n)})=\widetilde{O}(n)$. Now, from the
definition of $\mathsf{UC}_{min}$ it follows that
$\mathsf{UC}_{min}(\mathsf{A1}_{(n,n)})\leq\mathsf{UC}_{1}(\mathsf{A1}_{(n,n)})$.
Hence, $\mathsf{UC}_{min}(\mathsf{A1}_{(n,n)})=\widetilde{O}(n)$.
$\mathsf{UC}_{min}(\mathsf{A1}_{(n,n)})=\widetilde{O}(n)$. From the fact
$\mathsf{UC}_{min}\geq\mathrm{deg}$ it follows that
$\mathrm{deg}(\mathsf{A1}_{(n,n)})=O(n)$.

######  Observation B.23 ([ABB+17]).

For any input $\Sigma^{n\times n}$ to the function $\mathsf{A1}_{(n,n)}$ (in
Definition B.18) if we permute the rows of the matrix using a permutation
$\sigma_{r}$ and permute the columns of the matrix using a permutation
$\sigma_{c}$ and we update the pointers in each of the cells of the matrix
accordingly then the function value does not change.

Now we describe _Variant 2_ of [ABB+17] function where the domain is slightly
different from that of _Variant 1_.

######  Definition B.24 (Variant 2 [ABB+17]).

Let $\Sigma^{m\times n}$ be a $\mathsf{Type}_{2}$ _pointer matrix_ such that
$\text{BPointer}\in\\{(i,j)|i\in[m],j\in[n]\\}$. Note that here the BPointer
points to a cell of $M$, not a column. Let $n$ be even. Define
$\mathsf{A2}_{(m,n)}:\Sigma^{m\times n}\to\\{0,1\\}$ on $\mathsf{Type}_{2}$
_pointer matrix_ such that for all $x=(x_{i,j})\in\Sigma^{m\times n}$, the
function $\mathsf{A2}_{(m,n)}(x_{i,j})$ evaluates to $1$ if and only if it has
a $1$-_cell certificate_ of the following form:

  1. 1.

there exists exactly one _Marked column_ $j^{\star}$ in $M$. Let
$(i^{\star},j^{\star})$ be the _special element_.

  2. 2.

for each non-marked column $j\in[n]\setminus\\{b\\}$ there exist an element
$l_{j}$ such that $\text{Value}(l_{j})=0$ where $l_{j}$ be the end of the path
that starts at the _special element_ and follows the pointers LPointer and
RPointer as specified by the sequence $T(j)$. $l_{j}$ exists in all
$j\in[n]\setminus\\{b\\}$. We call that $l_{j}$ the $Leaves$ of the tree.

  3. 3.

The size of the set
$\\{j\in[n]\setminus\\{b\\}|\text{BPointer}(l_{j})=(i^{\star},j^{\star})\\}$
is exactly $\frac{n}{2}$.

_Variant 2_ achieves the separation between $\mathsf{R_{0}}$ vs. $\mathsf{R}$
and $\mathsf{R_{0}}$ vs $\mathsf{Q_{E}}$. The following results are some of
the properties of _Variant 2_ function:

######  Theorem B.25.

[ABB+17] For any $m,n\in\mathbb{N}$, the function $\mathsf{A2}_{(m,n)}$ in
Definition B.24 satisfies

$\displaystyle\mathsf{R_{0}}$ $\displaystyle=\Omega(mn),$
$\displaystyle\mathsf{R}$ $\displaystyle=\widetilde{O}(m+n)\text{ and}$
$\displaystyle\mathsf{Q_{E}}$ $\displaystyle=\widetilde{O}(m+n).$

Finally we will describe _Variant 3_ of [ABB+17] function, where we have one
extra parameter, the number of _marked column_. So, we will address the
function as $\mathsf{A3}_{(k,m,n})$ where $m,n$ are number of rows and columns
respectively and $k$ is the number of _marked column_.

######  Definition B.26 (Variant 3 [ABB+17]).

Define $\mathsf{A3}_{(k,m,n)}:\Sigma^{m\times n}\to\\{0,1\\}$ on
_$\mathsf{Type}_{2}$ pointer-matrix_ $\Sigma$ such that for all
$x=(x_{i,j})\in\Sigma^{m\times n}$, the function
$\mathsf{A3}_{(k,m,n)}(x_{i,j})$ evaluates to $1$ if and only if it has a
$1$-_cell certificate_ of the following form:

  1. 1.

there exists $k$ _Marked column_ in $M$. Let $b_{1},b_{2},\dots,b_{k}\in[n]$
denotes the _Marked column_ in $M$ and $(a_{k},b_{k})$ be the _special
element_ in each _Marked column_. Let us denote the cell of _special element_
by $a$

  2. 2.

We have $\text{BPointer}(x_{a_{j},b_{j}})=(a_{j+1},b_{j+1})$ for all
$j\in[k-1]$ and $\text{BPointer}(x_{a_{k},b_{k}})=(a_{1},b_{1})$. Also,
$\text{LPointer}(x_{a_{s},b_{s}})=\text{LPointer}(x_{a_{t},b_{t}})$ and
$\text{RPointer}(x_{a_{s},b_{s}})=\text{RPointer}(x_{a_{t},b_{t}})$ for all
$s,t\in[k]$.

  3. 3.

for each non-marked column $j\in[n]\setminus\\{b_{1},b_{2},\dots,b_{k}\\}$
there exist an element $\ell_{j}$ such that $\text{Value}(l_{j})=0$ and
$\text{BPointer}(l_{j})=b$ where $l_{j}$ be the end of the path that starts at
the _special element_ and follows the pointers LPointer and RPointer as
specified by the sequence $T(j)$. We need that $l_{j}$ exists in each
$j\in[n]\setminus\\{b_{1},b_{2},\dots,b_{k}\\}$ i.e. no pointer on the path is
$\perp$. We call that $l_{j}$ the $Leaves$ of the tree.

_Variant 3_ realises the separation between $\mathsf{R_{0}}$ vs. $\mathsf{Q}$,
$\mathsf{R}$ vs. $\mathsf{Q_{E}}$, $\mathsf{R}$ vs. $\mathrm{deg}$ and
$\widetilde{\mathrm{deg}}$ for different $m,n$ and $k$. The followings are
some important result for _variant 3_ from [ABB+17]:

######  Theorem B.27.

[ABB+17] For sufficiently large $m,n\mathbb{N}$ and a natural number $k<n$,
the function $\mathsf{A3}_{(k,m,n)}$ in Definition B.26 satisfies

$\displaystyle\mathsf{R_{0}}$ $\displaystyle=\Omega(mn)\text{ for
$k<\frac{n}{2}$},$ $\displaystyle\mathsf{R_{0}}$
$\displaystyle=\Omega(\frac{mn}{\log n})\text{ for $k=1$},$
$\displaystyle\mathsf{Q}$
$\displaystyle=\widetilde{O}(\sqrt{mn/k}+\sqrt{km}+k+\sqrt{n}),$
$\displaystyle\mathsf{Q_{E}}$
$\displaystyle=\widetilde{O}(m\sqrt{n/k}+km+n)\text{ and}$
$\displaystyle\widetilde{\mathrm{deg}}$
$\displaystyle=\widetilde{O}(\sqrt{m}+\sqrt{n})\text{ for }k=1.$

######  Observation B.28.

It is easy to observe that for each positive input $x$ to the function
$\mathsf{A3}_{(1,n,n)}$, the _marked column_ together with the rooted tree of
pointers with _leaves_ in every other column gives a unique minimal
$1$-certificate of $x$. Thus,
$\mathsf{UC}_{1}(\mathsf{A3}_{(1,n,n)})=\widetilde{O}(n)$. Now, from the
definition of $\mathsf{UC}_{min}$ it follows that
$\mathsf{UC}_{min}(\mathsf{A3}_{(n,n)})=\widetilde{O}(n)$. Also it follows
from the fact $\mathsf{UC}_{min}\geq\mathrm{deg}$ that
$\mathrm{deg}(\mathsf{A3}_{(n,n)})=O(n)$.