# Time Shifts to Reduce the Size of Reservoir Computers

Thomas L. Carroll<EMAIL_ADDRESS>US Naval Research Lab,
Washington, DC 20375  Joseph D. Hart<EMAIL_ADDRESS>US Naval
Research Lab, Washington, DC 20375

###### Abstract

A reservoir computer is a type of dynamical system arranged to do computation.
Typically, a reservoir computer is constructed by connecting a large number of
nonlinear nodes in a network that includes recurrent connections. In order to
achieve accurate results, the reservoir usually contains hundreds to thousands
of nodes. This high dimensionality makes it difficult to analyze the reservoir
computer using tools from dynamical systems theory. Additionally, the need to
create and connect large numbers of nonlinear nodes makes it difficult to
design and build analog reservoir computers that can be faster and consume
less power than digital reservoir computers. We demonstrate here that a
reservoir computer may be divided into two parts; a small set of nonlinear
nodes (the reservoir), and a separate set of time-shifted reservoir output
signals. The time-shifted output signals serve to increase the rank and memory
of the reservoir computer, and the set of nonlinear nodes may create an
embedding of the input dynamical system. We use this time-shifting technique
to obtain excellent performance from an opto-electronic delay-based reservoir
computer with only a small number of virtual nodes. Because only a few
nonlinear nodes are required, construction of a reservoir computer becomes
much easier, and delay-based reservoir computers can operate at much higher
speeds.

##  I Introduction

Reservoir computers were designed as recurrent neural networks that were easy
to train jaeger2001 ; natschlaeger2002 . As in a recurrent neural network, a
reservoir is typically built by connecting a large number of nonlinear nodes
in a recurrent network, called the reservoir. Unlike a recurrent neural
network, the connections between nodes in the reservoir are not trained;
instead, a linear combination of the time series signals output by the nodes
is used to fit a training signal. Because the training can be done by ridge
regression, it is much faster and more stable than the training required for
traditional recurrent neural networks. Additionally, the fact that the
connections between nodes are fixed has led to the development of analog
hardware reservoir computers, bringing the possibility of fast computation in
a device with low size, weight and power.

Reservoir computers that are all or part analog include photonic systems
appeltant2011 ; larger2012 ; van_der_sande2017 ; hart2019 ; chembo2019 ;
argyris2019 , analog electronic circuits schurmann2004 , mechanical systems
dion2018  and field programmable gate arrays canaday2018 . Many other examples
are included in the review paper tanaka2019 , which describes hardware
implementations of reservoir computers that are very fast, and yet consume
little power, while being small and light. Reservoir computers have been shown
to be useful for solving a number of problems, including reconstruction and
prediction of chaotic attractors lu2018 ; zimmerman2018 ; antonik2018 ; lu2017
; jaeger2004 , recognizing speech larger2017 , handwriting or other images
jalavand2018 , network inference banerjee2019 ; banerjee2021 , or controlling
robotic systems lukosevicius2012  . Reservoir computers have also been used to
better understand the function of neurons in the brain stoop2013 . Lymburn et
al. lymburn2019  studied the relation between generalized synchronization and
reconstruction accuracy, while Herteux and Räth examined how the symmetry of
the activation function affects reservoir computer performance herteux .

The usual theory on why reservoir computers work is that the reservoir maps an
input signal into a high dimensional space. This high dimensional system is
then projected down to a lower dimensional system of interest by a linear
projection. Hart et al. hart2020  showed that there is a positive probability
that a reservoir computer can be an embedding of the driving system, and
therefore it can predict the future of the driving system within an arbitrary
tolerance, while Grigoryeva et al. grigoryeva2021  show conditions under which
a reservoir computer can be in strong generalized synchronization with the
driving system.

The requirement to map an input signal into a high dimensional space suggests
that the reservoir must contain a large number of nodes, making analog
reservoir computers more difficult to build. Time delay-based reservoir
computers appeltant2011 ; larger2012 ; van_der_sande2017 ; hart2019 ;
chembo2019  mitigate this by time-multiplexing a single analog nonlinearity,
but this comes at the cost of a speed reduction. Recent work proposes that the
recurrent network can be eliminated entirely and replaced with nonlinear
combinations of the past input states gauthier2021 . We suggest that there is
no particular reason that a high dimensional space is necessary to form an
embedding of the input system; a smooth dynamical system of $d$ dimensions
requires at most $2d$ dimensions for an embedding. It is necessary for the
reservoir to be in generalized synchronization with the driving system, but it
is not clear what else is required.

We demonstrate here that a reservoir computer may be divided into two parts: a
small recurrent network (“the reservoir”), which we presume creates the
embedding, and a larger matrix consisting of the reservoir output signals
transformed in a way that increases their rank and memory. We accomplish this
rank-increasing transformation using a modified version of the method
developed by Del Frate et al. delfrate2021 , in which values of the reservoir
nodes from previous time steps are used to create a time-shifted reservoir
matrix.

First, we show via simulations of three different types of reservoir computer
that the rank of the covariance matrix of the reservoir output matrix can help
predict the accuracy of a reservoir computer. Then, we show, again through
simulations, that the time-shifted reservoir matrix generally has a
significantly improved covariance rank, memory capacity, and prediction
accuracy compared with the standard reservoir matrix. We confirm that these
results hold in the presence of noise by performing experimental measurements
on an analog opto-electronic delay-base reservoir computer. Our results show
that high accuracy can be obtained with very small reservoirs, which has
important implications for the speed, size, and ease of construction of analog
reservoir computers.

##  II Reservoir Computers

A general form for a reservoir computer can be

$\mathbf{R}\left(n+1\right)=\chi\left(\mathbf{AR}\left(n\right)+\mathbf{W}s\left(n\right)\right)$
(1)

where $\chi()$ is a nonlinear function, ${\bf R}$ is the vector of the states
of the reservoir nodes, ${\bf A}$ is the adjacency matrix, $s(n)$ is the input
signal and ${\bf W}$ is a vector of input signal weights. The individual
components of ${\bf R}(n)$ are $r_{i}(n)$, where $i$ is the index of a
particular node.

In the training stage, the reservoir computer is driven with the input signal
$s(n)$ to produce the reservoir computer output signals $r_{i}(n)$. In all the
examples in this paper, the input signal is normalized to have a mean of zero
and a standatd deviation of 1. The reservoir output matrix $\Omega_{1}$ is
constructed from the reservoir signals as

$\Omega_{1}=\left[{\begin{array}[]{cccccc}{{r_{1}}\left(1\right)}&\cdots&{{r_{M}}\left(1\right)}\\\
{{r_{1}}\left(2\right)}&{}\hfil&{{r_{M}}\left(2\right)}\\\
\vdots&{}\hfil&\vdots\\\
{{r_{1}}\left(N\right)}&\cdots&{{r_{M}}\left(N\right)}\end{array}}\right]$ (2)

where the reservoir has $M$ nodes and the time series have $N$ points. The fit
to the training signal is

${h(n)}={\Omega_{1}}{{\bf C}}$ (3)

where ${\bf C}$ is a vector of training coefficients. The coefficients of
${\bf C}$ are obtained by minimizing the error between $h(n)$ and a training
signal $f(n)$, typically using ridge regression tikhonov  to avoid
overfitting.

In the testing stage, a new input signal $\tilde{s}\left(n\right)$ is
generated from the same dynamical system that generated $s(n)$, but with
different initial conditions. The corresponding test signal is
$\tilde{f}\left(n\right)$. The input signal $\tilde{s}\left(n\right)$ drives
the same reservoir to produce the output signals
${\tilde{r}_{i}}\left(n\right)$, which are arranged in a matrix
$\tilde{\Omega}_{1}$. The testing error is

${\Delta_{tx}}={{\mathrm{std}\left({\tilde{f}\left(n\right)-\tilde{\Omega}_{1}{\bf{C}}}\right)}\mathord{\left/\right.\kern-1.2pt}{\mathrm{std}\left({\tilde{f}\left(n\right)}\right)}}$
(4)

where the coefficient vector ${\bf C}$ was found in the training stage.

##  III Covariance Rank, Memory Capacity, and Testing Error

Equations (3 \- 4) show that the columns of $\Omega_{1}$ are used as a basis
to fit the training and test signals. Principle component analysis joliffe2011
states that the eigenvectors of the covariance matrix of $\Omega$,
$\Theta=\Omega_{1}^{T}\Omega_{1}$, form an uncorrelated basis set. The rank of
the covariance matrix tells us the number of uncorrelated vectors. Therefore,
we will use the rank of the covariance matrix of $\Omega_{1}$,

$\Gamma={\rm{rank}}\left({\Omega_{1}^{T}\Omega_{1}}\right)$ (5)

to characterize the reservoir matrix $\Omega_{1}$. We calculate the rank using
the MATLAB rank() function, which returns the number of singular values above
a certain threshold. The threshold is
$\gamma_{r}={D_{\max}}\delta\left({{\sigma_{\max}}}\right)$, where $D_{max}$
is the largest dimension of $\Omega_{1}$ and $\delta(\sigma_{max})$ is the
difference between the largest singular value of $\Omega_{1}$ and the next
largest double precision number. It was shown in carroll2019  that larger
values of the covariance rank corresponded to smaller values of the testing
and training error.

Memory capacity, as defined in jaeger2002 , is considered to be an important
quantity in reservoir computers. Memory capacity is a measure of how well the
reservoir can reproduce previous values of the input signal.

The memory capacity as a function of delay is

${\rm{M}}{{\rm{C}}_{k}}=\frac{{\sum\limits_{n=1}^{N}{\left[{s\left({n-k}\right)-\left\langle
s\right\rangle}\right]\left[{{h_{k}}\left(n\right)-\left\langle
h_{k}\right\rangle}\right]}}}{{\sum\limits_{n=1}^{N}{\left[{s\left({n-k}\right)-\left\langle
s\right\rangle}\right]\sum\limits_{n=1}^{N}{\left[{{h_{k}}\left(n\right)-\left\langle
h_{k}\right\rangle}\right]}}}}$ (6)

where the $\left\langle\right\rangle$ operator indicates the mean. The signal
$h_{k}(n)$ is the fit of the reservoir signals $r_{i}(n)$ to the delayed input
signal $s(n-k)$. The memory capacity is

${\rm{MC}}=\sum\limits_{k=1}^{\infty}{{\rm{M}}{{\rm{C}}_{k}}}$ (7)

Input signals such as the Lorenz $x$ signal contain correlations in time,
which will cause errors in the memory calculation, so in Eq. (6), $s(n)$ is a
random signal uniformly distributed between -1 and +1. There are some
drawbacks to defining memory in this way; the reservoir is nonlinear, so its
response will be different for different input signals, and the definition
involves a fit to a signal. Table 1 below shows a situation where
supplementing the reservoir signals with a memoryless nonlinearity appears to
increase the memory capacity, even though the nonlinearity has no memory.
Nevertheless, this memory definition is the standard definition used in the
field of reservoir computing.

It is generally believed that the training and testing errors for a reservoir
computer are limited at least in part by the number of nodes; more nodes yield
the possibility of lower error. Here we show that it is not the number of
nodes that are of primary importance but the covariance rank and possibly the
memory capacity of the reservoir computer. More nodes can be better because
the covariance rank is limited by the number of nodes. Memory capacity can be
affected by the number of nodes, although our data suggests that covariance
rank has a larger effect. It has been demonstrated that too much memory
capacity can degrade the performance of a reservoir computer carroll2020b ;
carroll2022 .

###  III.1 Types of Reservoir Computers

In order to demonstrate the generality of the impacts of reservoir size,
covariance rank and memory capacity on reservoir computing accuracy, we used
three different reservoir nonlinearities with two different input signals.

####  III.1.1 Tanh reservoir computer

A reservoir computer based on a hyperbolic tangent is used because it is
common in the literature inubushi2017 . The tanh reservoir computer is
described by

${\bf{R}}\left({n+1}\right)=g\tanh\left({{\bf{AR}}+\varepsilon
s\left(n\right)}\right).$ (8)

For each realization the parameters $g$, $\varepsilon$ and the spectral radius
$\rho$ of the adjacency matrix ${\bf A}$ are each chosen from uniform random
distributions between 0 and 1. Half of the entries in the adjacency matrix
${\bf A}$ are set to numbers drawn from a Gaussian random distribution, while
the others are zero. The reservoir computer parameters, the adjacency matrix
and the input vector are different for each realization of the reservoir
computer.

Because the tanh reservoir does not contain any even nonlinearities, the set
of reservoir computer output signals $r_{i}(n)$ were supplemented with their
squares, $r_{i}^{2}(n)$, as is commonly done for this type of nonlinearity.

####  III.1.2 polyODE reservoir computer

A second reservoir computer is based on a polynomial ordinary differential
equation, and will be called the polyODE reservoir computer. This reservoir
computer is described by

$\begin{split}&\frac{{d{r_{i}}\left(t\right)}}{{dt}}\\\
&=\alpha\left[{{p_{1}}{r_{i}}\left(t\right)+{p_{2}}r_{i}^{2}\left(t\right)+{p_{3}}r_{i}^{3}\left(t\right)+\sum\limits_{j=1}^{M}{{A_{ij}}{r_{j}}\left(t\right)}+{W_{i}}s\left(t\right)}\right].\end{split}$
(9)

The adjacency matrix was chosen in the same way as for the tanh reservoir
computer. The entries in the input vector ${\bf W}$ were drawn from a uniform
random distribution between -1 and 1. For each realization the parameter
$p_{1}$ was chosen from a uniform random distribution between -5 and 0,
$p_{2}$ was between 0 and 1, $p_{3}$ was between -1 and 1, the adjacency
matrix spectral radius $\rho$ was between 0 and 1 and the time constant
$\alpha$ was between 0 and 5. The equations were integrated with a 4th order
Runge-Kutta with a time step of 1.

####  III.1.3 Opto-electronic delay-based reservoir computer

The third reservoir computer is a simulation of the opto-electronic delay-
based reservoir computer of Ref. larger2012 . In this reservoir computer
multiple virtual nodes were created by time-multiplexing a driven opto-
electronic oscillator. The virtual nodes were coupled because the delay loop
contained a low pass filter, so each virtual node was coupled to the preceding
nodes hart2019 .

The opto-electronic reservoir computer can be modeled by

${T_{L}}\dot{\nu}(t)=-\nu\left(t\right)+\beta\sin^{2}{\left({\nu\left({t-{\tau_{D}}}\right)+\phi+\rho
W\left(t\right)s_{in}\left(t\right)}\right)}.$ (10)

As with the other reservoir computers, the time step for this model was fixed
at $t_{S}=1$. The time per node was $\theta=50$, which was chosen to give a
large number of time steps per node. The total delay time was
$\tau_{D}=M_{1}\times\theta$, where $M_{1}$ was the number of virtual nodes.
The input mask $W(t)$ is a piecewise constant function of time defined as
$W(t)=W_{k}$ where $k=(t{\rm mod}\tau_{D})/\theta$, where the elements of the
length $M_{1}$ vector $W_{k}$ were randomly set to either 1 or -1. Note that
$W$ is periodic in $\tau_{D}$.

Equation 10 was integrated with a 4th order Runge-Kutta routine. The 4th order
routine evaluates the function at the beginning and end of the integration
interval. For the time $t$ and $t+1$, the input signals were

${s_{in}}\left(t\right)=s\left({\left\lfloor{\frac{t}{{{\tau_{D}}}}}\right\rfloor}\right)\quad{s_{in}}\left({t+1}\right)=s\left({\left\lfloor{\frac{{t+1}}{{{\tau_{D}}}}}\right\rfloor}\right)$
(11)

where $\left\lfloor\right\rfloor$ is the floor operator.

After numerical integration, the time series $\nu(t)$ was reshaped into
reservoir computer form as

$\Omega_{1}=\left[\begin{array}[]{cccc}\nu\left(\theta\right)&\nu\left(2\theta\right)&\ldots&\nu\left(M\theta\right)\\\
\nu\left(\theta+\tau_{D}\right)&&&\vdots\\\ \vdots&&&\vdots\\\
\nu\left(\theta+N\tau_{D}\right)&\ldots&\ldots&\nu\left(M\theta+N\tau_{D}\right)\end{array}\right]$
(12)

For each realization, the parameter values were drawn from uniform random
distributions. For $T_{L}$ the limits were 0 to 300, for $\rho$ and $\beta$
the limits were 0 to 1, and for $\phi$ the limits were 0 to $\pi$.

###  III.2 Reservoir observer task

We qualify these reservoir computers by their ability to perform the “observer
task” of performing model-free inference of unmeasured variables lu2017 . To
demonstrate generality, we test the reservoir computers’ ability to perform
the observer task on both the Lorenz system and the Rössler system; that is,
we ask the reservoir computer to infer the $z$ variable given only the $x$
variable for a given system.

The input signals for the simulations were taken from the Lorenz or Rössler
chaotic systems. The Lorenz system is described by lorenz1963

$\begin{array}[]{l}\frac{{dx}}{{dt}}=T_{l}\left({p_{1}}\left({y-x}\right)\right)\\\
\frac{{dy}}{{dt}}=T_{l}\left(x\left({{p_{2}}-z}\right)-y\right)\\\
\frac{{dy}}{{dt}}=T_{l}\left(xy-{p_{3}}z\right)\end{array}$ (13)

with $p_{1}=10$, $p_{2}=28$, $p_{3}=8/3$ and $T_{l}=0.1$. The Lorenz equations
were numerically integrated with a time step of 1. The input to the reservoir
computer was $x$, and the reservoir computer was trained to output $z$.

The Rössler system is described by rossler1976

$\begin{array}[]{*{20}{l}}{\frac{{dx}}{{dt}}=T_{r}\left(-y-{p_{1}}z\right)}\\\
{\frac{{dy}}{{dt}}=T_{r}\left(x+{p_{2}}y\right)}\\\
{\frac{{dz}}{{dt}}=T_{r}\left({p_{3}}+z\left({x-{p_{4}}}\right)\right)}\end{array}$
(14)

The Rössler equations were numerically integrated with a time step of 1,
$T_{r}=0.65$ and parameters $p_{1}=1$, $p_{2}=0.2$, $p_{3}=0.2$, $p_{4}=5.7$.
As with the Lorenz system, the input to the reservoir computer was $x$, and
the reservoir computer was trained to output $z$.

####  III.2.1 Results

Figure 1 shows the mean testing error $\Delta_{tx}$ as a function of
covariance rank $\Gamma$ for six combinations of input signal and reservoir
computer.

Figure 1:  Mean testing error $\left<\Delta_{tx}\right>$ as a function of
covariance rank $\Gamma$ for Lorenz or Rössler $x$ signals driving either a
tanh, a polynomial ODE or a opto-electronic reservoir computer. The reservoir
computers were trained on the $z$ signals from the corresponding input
systems. Reservoir computers containing 50 or 100 nodes were used. The tanh
data has twice the rank because the set of reservoir computer signals
$r_{i}(t)$ were supplemented with their squares $r_{i}^{2}(t)$.

In Fig. 1 the mean testing error depends on the covariance rank $\Gamma$, not
on the number of nodes in the reservoir computer. The tanh data has twice the
rank because the set of reservoir computer signals $r_{i}(t)$ were
supplemented with their squares $r_{i}^{2}(t)$. Using more nodes allows the
covariance rank to be larger, since it is limited by the size of the
reservoir. For a particular rank there is a spread in testing error because
there are other factors that affect the error, such as the reservoir Lyapunov
exponents carroll2020b  or the memory carroll2022 , but the main effect of the
number of nodes is to increase the possible rank. The point of Fig. 1 is that
the curve of testing error versus rank does not depend on the number of nodes
in the reservoir computer, but rather on the covariance rank.

Figure 1 does show that for the highest ranks in the laser system the testing
error actually increases. The laser nonlinearity is bounded, so for some
parameter combinations it is possible the laser equation has one or more
positive Lyapunov exponents. Positive Lyapunov exponents would lead to complex
chaotic signals that had high rank, but they would also cause the breakdown of
generalized synchronization, so the testing error would be large.

It is generally accepted that the larger memory capacity for a reservoir
computer leads to lower testing errors, although carroll2022  pointed out that
the memory capacity must be matched to the problem at hand, so memory capacity
in some cases can be too large. We may ask if larger reservoir computers
increase the memory capacity; Fig. 2 and Table 1 show that the memory capacity
increases as the reservoir computer becomes larger, but the increase scales
slower than linear with the number of nodes.

Figure 2:  Mean testing error $\left<\Delta_{tx}\right>$ as a function of
memory capacity MC for Lorenz or Rössler $x$ signals driving either a tanh, a
polynomial ODE or a opto-electronic reservoir computer. The reservoir
computers were trained on the $z$ signals from the corresponding input
systems. Reservoir computers containing 50 or 100 nodes were used.

Table 1 shows a shortcoming in the standard definition of memory capacity in
Eqs. (6-7). Table 1 shows the measured memory capacity for the tanh reservoir
computers with 50 or 100 nodes. The training signal may be fit using only the
reservoir signals $r_{i}(t)$ or supplementing these signals with
$r_{i}^{2}(t)$. Table 1 shows that the increase in calculated memory capacity
when supplementing the reservoir signals with their squares is comparable to
the increase in memory capacity when doubling the number of nodes. The
operation of squaring the reservoir signals is a memoryless nonlinearity,
however, so it should not change the actual memory of the reservoir computer.
Adding the squared signals does improve the reservoir computer fit to the
training signal, causing an apparent increase in memory capacity when there is
no actual increase.

Table 1:  Mean memory capacity when a uniformly distributed random signal drove a tanh reservoir computer, where either $r_{i}(t)$ or $r_{i}(t)$ and $r_{i}^{2}(t)$ are used to fit the training signal 50 nodes $r_{i}(t)$ | 50 nodes $r_{i}(t)$, $r_{i}^{2}(t)$ | 100 nodes $r_{i}(t)$ | 100 nodes $r_{i}(t)$, $r_{i}^{2}(t)$
---|---|---|---
6.2 | 7.0 | 7.5 | 9.2

Figure 2 does show that, as with the covariance rank, the curve of testing
error versus memory capacity mostly follows the same curve independent of the
number of nodes. The curves do diverge in the tanh and opto-electronic
reservoirs when the testing error actually increases with memory capacity.
Memory capacity is not as simple as covariance rank because effects that
increase memory capacity, such as increased Lyapunov exponents, can also
increase testing error carroll2020b .

Figures 1 and 2 suggest that in order to reduce the testing error, we would
like to increase the covariance rank and find the optimum memory capacity. The
following section (IV) shows that increasing the rank or memory does not
necessarily require increasing the size of the reservoir; we find that using a
matrix of time-delayed signals from the reservoir can give us the necessary
rank or memory. We find that the reservoir itself requires only a small number
of nodes. Creating and connecting nonlinear nodes in analog hardware can be
difficult, so using small reservoir computers should make them much easier to
build.

##  IV Time-Shifted Reservoir Computers

Section III showed that increased covariance rank leads to better reservoir
computer performance, regardless of the number of nodes. In a sense, this is
what was done when augmenting the tanh reservoir with the $r_{i}^{2}$ values.
One way to optimize reservoir computer performance therefore is to maximize
the rank for a given number of nodes. One way to do this is to oversample the
reservoir; this has been shown to improve performance for small oversampling
factors such that the number of “effective nodes” is increased by a factor of
2-3 takano2018 ; harkhoe2020 ; sunada2021 . Jaurigue et al. jaurigue2021
showed that augmenting the reservoir matrix with a delayed version of the
input signal can also improve the quality of time series prediction.

Another way to increase the covariance rank is based on the time shifting
method of Del Frate et al. delfrate2021 ; this technique can result in a very
large number of “effective nodes”. In that paper, the authors applied random
time shifts to the set of reservoir node states and showed that this approach
could reduce the testing error. Their goal was to improve the reservoir
computer performance without optimizing all parameters. A reduced version of
this method using a single time shift has been shown to improve the
performance of a photonic analog reservoir computer harkhoe2020 .

We have simplified the method of delfrate2021  by using an ordered set of time
shifts instead of random shifts. In this case our goal was to see if we could
achieve a small testing error with only a small number of nodes, so we
simulated a reservoir computer with $M_{1}$ nodes and used the output signals
to create a matrix of $M_{2}$ signals, where $M_{2}\geq M_{1}$.

We decided to perform these simulations for the opto-electronic reservoir
computer of Eq. (10) because a reduction in $M_{1}$ results in a proportional
increase in the computation speed of delay-based reservoir computers. Further,
we are able to confirm these simulations with a laboratory experiment (see
section V). The parameters for these simulations were $T_{L}=200$,
$\beta=0.5$, $\rho=1$, $\theta=50$ and $\phi=\pi/4$.

The matrix of signals from the reservoir computer was $\Omega_{1}$ as in Eq.
(12). The time-shifted reservoir was

$\Omega_{2}\left(i,j\right)=\nu\left(k\theta+\left(i-1\right)\tau_{D}-\tau_{j}\right)$
(15)

where $k={\rm mod}(j,M_{1})$, $i=1,2,\ldots,N$ and $j=1,2,\ldots,M_{2}$

The maximum time shift value was $\tau_{max}$, while the individual time
shifts were $\tau_{j}=j\times\tau_{max}/M_{2}$. When the time shift $\tau_{j}$
was not an integer multiple of $\theta$, linear interpolation was performed
between the two nearest $\theta$-sampled values of the opto-electronic
reservoir computer output signal $\nu$ to get the shifted values. In
delfrate2021  the delay where the autocorrelation of the input signal was
reduced to half its peak value was used to set $\tau_{max}$; we found that
because the Rössler system had a very slowly decaying autocorrelation, that
method gave a value of $\tau_{max}$ that was too large. Instead we simulated
the testing error as $\tau_{max}$ was varied with $M_{1}=M_{2}=100$ nodes and
picked a value that gave a small testing error.

###  IV.1 Vary Number of Nodes

To find out how many actual nodes are needed for a reservoir computer, the
laser delay reservoir computer was simulated for a variety of reservoir sizes
$M_{1}=2,3,\ldots M_{2}$ and with a fixed size $M_{2}$ for the time-shifted
matrix. The first task was the Lorenz observer task as described in Sec.
III.2, in which case the maximum delay value $\tau_{max}$ was 10$\tau_{D}$,
which was equivalent to 10 Lorenz time steps. The parameters for the Lorenz
system were the same as in Eq. (13). Figure 3 shows the training error,
covariance rank and memory capacity based on either $\Omega_{1}$ or
$\Omega_{2}$ as $M_{1}$ varies. The elements of the input vector ${\bf W}$
were randomly assigned to -1 or 1. Because the training and testing error can
depend on the input vector, 20 different realizations were performed, each
with a different input mask, and all reported results are the mean values over
these 20 realizations.

Figure 3:  The top plot is the mean testing error for the opto-electronic
reservoir computer when the input signal is the Lorenz $x$ signal and the
training signal is the $z$ signal. The trace labeled $\Omega_{1}$ is the error
for the original reservoir computer with $M_{1}$ nodes. The traces labeled as
$\Omega_{2}$ are for the time-shifted matrix with different fixed values of
$M_{2}$. The middle plot shows the covariance rank $\Gamma$ while the bottom
plot shows the memory capacity MC.

In Fig. 3, the covariance rank for the reservoir output matrix $\Omega_{1}$
increases with the number of nodes, but the increase slows as the number of
nodes increases. Except for the case where the time-shifted reservoir matrix
contains $M_{2}=50$ nodes, the covariance rank of the shifted reservoir matrix
is always higher than the covariance rank of the unshifted matrix
$\Omega_{1}$. The rank of the time-shifted reservoir saturates when $M_{1}$ is
approximately equal to $M_{2}/10$. The time-shifted reservoir matrices also
have a larger memory capacity than the unshifted reservoir matrix. Simply
adding a delayed input signal with a delay of $\tau_{max}$ can increase the
memory capacity, but Fig. 3 shows that adding more nodes also increases the
memory capacity. Whether this is an actual increase of memory or an artifact
of the better fit to the training signal used in the definition of memory
capacity is not possible to say.

Figure 3 also shows that the testing error for the time-shifted reservoir
computer is always lower than the error for the unshifted version. This agrees
with delfrate2021 , where the time-shifted reservoir with random time shifts
was used to decrease the testing error in a large (N=100) reservoir computer.

What Fig. 3 shows is that the reservoir itself only needs a few nodes to give
a small training error. The set of signals in the matrix $\Omega$ still needs
a large covariance rank, but this rank may be accomplished by creating a
matrix with time shifts. There appear to be two phenomena at work here; the
networked nonlinear nodes in the reservoir produce enough information to re-
create an approximation of the Lorenz $z$ signal, possibly by creating an
embedding of the Lorenz system, while the time-shifted reservoir increases the
covariance rank to improve the approximation.

The time-shifting technique was also tested on a simulated opto-electronic
reservoir computer driven by the Rössler $x$ signal and trained on the Rössler
$z$ signal. In this case the maximum time shift $\tau_{max}$ was 20$\tau_{D}$.
The parameters for the Rössler system were the same as those used in Eq. (14).

Figure 4:  The top plot is the mean training error for the opto-electronic
reservoir computer when the input signal is the Rössler $x$ signal and the
training signal is the $z$ signal. The trace labeled $\Omega_{1}$ is the error
for the original reservoir computer with $M_{1}$ nodes. The traces labeled as
$\Omega_{2}$ are for the time-shifted matrix with different fixed values of
$M_{2}$. The bottom plot shows the covariance rank $\Gamma$.

Figure 4 shows that the covariance rank for the time-shifted reservoir matrix
saturates when $M_{1}\approx M_{2}/20$. For the Lorenz system, this number was
$M_{2}/10$. As with the Lorenz system, adding time-shifted signals to the
opto-electronic reservoir computer driven by a Rössler signal decreases the
testing error.

##  V Experiment

We performed an experiment to confirm that the simulations on the opto-
electronic reservoir computer from Section IV hold in the real world, where
noise can degrade reservoir computer performance soriano2013 ; semenova2019 .
The experiment was similar to that described in larger2012 ; hart2019 ; the
details are given in the supplementary material . Based on the model Eq. (10),
the parameters were the same as in the simulations: $\beta=0.5$, $\rho=1$,
$\phi=\pi/4$ and $T_{L}=4\theta$. The experiment was run with
$M_{1}=\tau_{D}/\theta$ set to 5, 10, 50, 100, 150 or 200 virtual nodes. The
number of time-shifted signals used in the time-shifted matrix $\Omega_{2}$
was fixed at $M_{2}=200$. As in the simulations, the elements of the input
mask ${\bf W}$ were randomly set to either $\pm 1$. For each value of $M_{1}$
the experiment was repeated with 20 different input masks, and the testing
errors and ranks shown are the mean error and rank for all 20 experiments. The
input signal $s_{in}$ was set to the Lorenz or Rössler $x$ variable for
$8000\times\tau_{D}$, then set to zero for $100\times\tau_{D}$ to allow the
experiment to reset, and then again set to the $x$ variable for
$4000\times\tau_{D}$. The opto-electronic reservoir response signal $\nu(t)$
was sampled at intervals of $\theta$ and arranged in a reservoir matrix
($\Omega_{1}$) or time-shifted matrix ($\Omega_{2}$) as in Eqs. (12) and (15).
After removing an initial transient of $200\times\tau_{D}$ reservoir time
steps the next $7800\times\tau_{D}$ time steps were used to train the
experiment. Because the 8000 point initial signal was followed by 100 points
set to zero, the 3800 time series points from 8300 to the end were used for
training the experiment. The number of signals in the time-shifted matrix was
$M_{2}=200$. The mean testing errors for the experiment are plotted in Fig. 5.

Figure 5:  Mean testing errors $\left<\Delta_{tx}\right>$ as a function of the
number of nodes $M_{1}$ in the non-time-shifted reservoir computer. The red
symbols are for the reservoir computer without time shifts, while the blue
symbols are for fits using the time-shifted matrix $\Omega_{2}$ with 200
nodes. These plots follow the same trends as the simulations in Figs. 3 and 4.

As in the simulations, using a matrix of time-shifted reservoir states
$\Omega_{2}$ significantly decreases the testing error for both the Lorenz and
Rössler observer tasks. The testing error for the time-shifted matrix bottoms
out for $M_{1}=5$ for these experiments, meaning that one may obtain small
testing errors for a small reservoir size. In delay-based reservoir computers
like our opto-electronic one, the computation rate is inversely proportional
to the number of nodes. By using our time-shifting technique with $M_{1}=5$
reservoir nodes and $M_{2}=200$, we are able to increase the computation rate
by a factor of 40 while decreasing the training error by more than a factor of
2 compared to traditional reservoir computing with $\Omega_{1}=200.$

##  VI Conclusions

In this work we showed that excellent reservoir computing performance can be
obtained by taking the signals from a small reservoir and creating a matrix of
time-shifted signals. We demonstrated this technique in simulations of three
different types of reservoir computer and in an opto-electronic analog
reservoir computer experiment. We ascribed this improvement in testing error
to an increase in the covariance rank and memory capacity of the time-shifted
reservoir matrix compared to the reservoir matrix without time shifts.
Depending on the relative sizes of the reservoir computer and the time-shifted
matrix, the improvement in testing error was in the range of 10 to 20. An
immediate benefit of this technique is that the same or better testing error
can be accomplished with a much smaller reservoir computer, an important
consideration if creating and coupling a large number of nodes is difficult or
expensive. In addition, choosing an optimal reservoir network should be much
easier when only a few nodes are required.

This work also illuminates the components that are necessary to create a
reservoir computer. The network of nonlinear nodes creates nonlinearity and
memory, while the time-shifted matrix increases the rank and memory. It is
known from work on nonlinear series approximations boyd1985 ; orcioni2014 ;
gauthier2021  that memory in the form of delays may be combined with
memoryless nonlinearities to approximate signals, so it is not surprising in
hindsight that the nonlinearity and memory may be distributed differently.

##  VII Data Availability

The data that support the findings of this study are available on request from
the corresponding author. The data are not publicly available because they
have not been approved for public release.

##  VIII Acknowledgements

This work was supported by the Naval Research Laboratory’s Basic Research
Program and the Department of Defense Applied Research for Advancement of S&T
Priorities Neuropipe program..

##  IX Supplemental Materials

The experimental reservoir computer (RC) is based on an opto-electronic
oscillator with time-delayed feedback, similar to the ones used in Refs.
paquot2012optoelectronic ; larger2012 ; antonik2017brain ;
dai2021classification . A traditional opto-electronic oscillator consists of
an opto-electronic nonlinearity, an optical delay line, and photoreceiver, the
output of which is fed back to the electrical input of the opto-electronic
nonlinearity chembo2019optoelectronic . Typically, the nonlinearity is an
intensity modulator and the photoreceiver also acts as an analog lowpass or
bandpass filter. Properly optimized opto-electronic reservoir computers have
been shown to be capable of extremely high data processing rates larger2017 .
An extensive review of opto-electronic reservoir computers can be found in
Ref. chembo2020machine .

In this work, we use a hybrid analog-digital opto-electronic oscillator
similar to those used in Refs. antonik2017brain ; murphy2010complex ;
hart2019laminar . The main difference from the traditional opto-electronic
oscillators is that the delay line and/or the filtering are implemented in
digital electronics. This setup provides greater flexibility and control over
the delay and filter parameters, and has even allowed for the implementation
of arbitrary networks of coupled maps to study basic network science hart2017
and reservoir computing hart2019 ; soriano2019optoelectronic ; however, it
does cap the maximum speed of the device to be somewhat slower than fully
analog opto-electronic oscillators.

A schematic of the opto-electronic oscillator used here is shown in Fig. 6. A
fiber-coupled CW laser emits light of constant intensity at 1550 nm. That
light passes through a fiber-coupled intensity modulator and converted to an
electrical signal by a photoreceiver (Terahertz Technologies TIA-525). The
electrical signal is read into a field-programmable gate array (FPGA) by an
analog to digital converter (ADC). An digital electronic delay and digital
lowpass filtering are implemented on the FPGA. Additionally, the input (drive)
signal is added electronically in the FPGA before it is output via a digital
to analog converter (DAC). The ADC (AD9254) and DAC (TI DAC 5672) used here
have 14-bits of resolution. The DAC output is amplified and used to drive the
RF-port of the intensity modulator.

The parameter $\beta$ represents the round-trip gain and is controlled by the
amplifier gain on the DAC output. A variable voltage power supply drives the
DC-port of the intensity modulator and is used to control the bias phase
$\phi$.

The digital filter used here is a single-pole lowpass infinite impulse
response (IIR) filter. The filter sampling rate $F_{clk}=2$ MHz and the filter
time constant $T_{L}$=40 $\mu$s so that the digital filter is an excellent
approximation of an analog single-pole lowpass filter. The node time $\theta$
(i.e., the time over which the input is held constant) is chosen to be 10
$\mu$s so that $\tau_{L}/\theta=4$.

We choose to operate our opto-electronic oscillator at such a slow rate
(relative to the state of the art) in order to reduce the cost and complexity
of the electronics required. This is sufficient for our proof of principle
experiments that show that adding ordered time shifts to a reservoir computer
output can dramatically improve performance of a photonic reservoir computer,
even in the presence of noise and slight parameter drifts inherent in any
experimental setup. This same time shifting technique can be applied directly
to higher rate photonic reservoir computers to either reduce the number of
nodes or increase the accuracy.

Figure 6: Schematic of the opto-electronic reservoir computer used here. The
intensity modulator implements an opto-electronic sinusoidal nonlinearity. The
operations in the shaded green box occur digitally inside the FPGA. The FPGA
clock rate $F_{clk}=2$ MHz, which is sufficiently fast relative to $\theta=10$
$\mu$s that the discretization in time is negligible.

## References

  * (1) H. Jaeger, German National Research Center for Information Technology GMD Technical Report 148 (1), 34 (2001)http://publica.fraunhofer.de/documents/B-73135.html.
  * (2) T. Natschlaeger, W. Maass and H. Markram, Special Issue on Foundations of Information Processing of TELEMATIK 8 (1), 39-43 (2002).
  * (3) L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso and I. Fischer, Nature Communications 2, 468 (2011)10.1038/ncomms1476 https://www.nature.com/articles/ncomms1476
  * (4) L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso and I. Fischer, Opt. Express 20 (3), 3241-3249 (2012)10.1364/oe.20.003241.
  * (5) G. Van der Sande, D. Brunner and M. C. Soriano, Nanophotonics 6 (3), 561-576 (2017)10.1515/nanoph-2016-0132.
  * (6) J. D. Hart, L. Larger, T. E. Murphy and R. Roy, Phil. Trans. R. Soc. 377, 20180123 (2019)http://doi.org/10.1098/rsta.2018.0123.
  * (7) Y. K. Chembo, Chaos 30 (1), 013111 (2019).
  * (8) A. Argyris, J. Bueno and I. Fischer, IEEE Access 7, 37017-37025 (2019)10.1109/ACCESS.2019.2905422.
  * (9) F. Schurmann, K. Meier and J. Schemmel, presented at the Advances in Neural Information Processing Systems 17, Vancouver, British Columbia, Canada, 2004
  * (10) G. Dion, S. Mejaouri and J. Sylvestre, Journal of Applied Physics 124 (15), 152132 (2018)10.1063/1.5038038.
  * (11) D. Canaday, A. Griffith and D. J. Gauthier, Chaos 28 (12), 123119 (2018)10.1063/1.5048199.
  * (12) G. Tanaka, T. Yamane, J. B. Héroux, R. Nakane, N. Kanazawa, S. Takeda, H. Numata, D. Nakano and A. Hirose, Neural Networks 115, 100-123 (2019)https://doi.org/10.1016/j.neunet.2019.03.005.
  * (13) Z. Lu, B. R. Hunt and E. Ott, Chaos: An Interdisciplinary Journal of Nonlinear Science 28 (6), 061104 (2018)10.1063/1.5039508.
  * (14) R. S. Zimmermann and U. Parlitz, Chaos: An Interdisciplinary Journal of Nonlinear Science 28 (4), 043118 (2018)10.1063/1.5022276.
  * (15) P. Antonik, M. Gulina, J. Pauwels and S. Massar, Phys. Rev. E 98 (1), 012215 (2018)10.1103/PhysRevE.98.012215.
  * (16) Z. Lu, J. Pathak, B. Hunt, M. Girvan, R. Brockett and E. Ott, Chaos: An Interdisciplinary Journal of Nonlinear Science 27 (4), 041102 (2017)10.1063/1.4979665.
  * (17) H. Jaeger and H. Haas, Science 304 (5667), 78-80 (2004)10.1126/science.1091277.
  * (18) L. Larger, A. Baylon-Fuentes, R. Martinenghi, V. S. Udaltsov, Y. K. Chembo and M. Jacquot, Physical Review X 7 (1) (2017)ARTN 011015 10.1103/PhysRevX.7.011015.
  * (19) A. Jalalvand, K. Demuynck, W. De Neve and J.-P. Martens, Neurocomputing 277, 237-248 (2018)https://doi.org/10.1016/j.neucom.2016.11.100.
  * (20) A. Banerjee, J.D. Hart, R. Roy, and E. Ott, Phys. Rev. X 11 (3), 031014 (2021).
  * (21) A.Banerjee, J. Pathak, R.Roy, J. Restrepo, E. Ott, Chaos 30 (12) 121104 (2019).
  * (22) Lukoševičius, Mantas, H. Jaeger and B. Schrauwen, KI - Künstliche Intelligenz 26 (4), 365-371 (2012)10.1007/s13218-012-0204-5.
  * (23) R. Stoop, V. Saase, C. Wagner, B. Stoop and R. Stoop, Phys. Rev. Lett. 110 (10), 108105 (2013)10.1103/PhysRevLett.110.108105.
  * (24) T. Lymburn, A. Khor, T. Stemler, D. C. Corrêa, M. Small, and T. Jüngling, Chaos: An Interdisciplinary Journal of Nonlinear Science 29 (2), 023118 (2019)10.1063/1.5079686.
  * (25) J. Herteux and C. Räth, Chaos: An Interdisciplinary Journal of Nonlinear Science 30 (12), 123142 (2020)10.1063/5.0028993.
  * (26) A. Hart, J. Hook and J. Dawes, Neural Networks 128, 234-247 (2020)https://doi.org/10.1016/j.neunet.2020.05.013.
  * (27) L. Grigoryeva, A. Hart and J.-P. Ortega, Phys. Rev. E 103 (6), 062204 (2021)10.1103/PhysRevE.103.062204.
  * (28) D. J. Gauthier, E. Bollt, A. Griffith and W. A. S. Barbosa, Nature Communications 12 (1), 5564 (2021)10.1038/s41467-021-25801-2.
  * (29) E. D. Frate, A. Shirin and F. Sorrentino, Chaos: An Interdisciplinary Journal of Nonlinear Science 31 (12), 121103 (2021)10.1063/5.0068941.
  * (30) I. T. Jolliffe, Principal component analysis. (Springer, New York; London, 2011)
  * (31) T. L. Carroll and L. M. Pecora, Chaos 29 (8), 083130 (2019)10.1063/1.5097686.
  * (32) H. Jaeger, GMD-Forschungszentrum Informationstechnik (Technical report GMD-Forschungszentrum Informationstechnik) (2002)
  * (33) T. L. Carroll, Chaos 30 (12), 121109 (2020)10.1063/5.0038163.
  * (34) T. L. Carroll, Chaos: An Interdisciplinary Journal of Nonlinear Science 32 (2), 023123 (2022)10.1063/5.0078151.
  * (35) M. Inubushi and K. Yoshimura, Scientific Reports 7 (1), 10199 (2017)10.1038/s41598-017-10257-6.
  * (36) E. N. Lorenz, Journal of Atmospheric Science 20 (2), 130-141 (1963)10.1175/1520-0469(1963)020¡0130:DNF¿2.0.CO;2.
  * (37) O. E. Rössler, Phys. Lett. A 57 (5), 397-398 (1976)https://doi.org/10.1016/0375-9601(76)90101-8.
  * (38) L. Jaurigue, E. Robertson, J. Wolters and K. Lüdge, Entropy 23 (12), 1560 (2021)
  * (39) K. Harkhoe, G. Verschaffelt, A. Katumba, P. Bienstman and G. Van der Sande, Opt. Express 28 (3), 3086-3096 (2020)10.1364/Oe.382556.
  * (40) K. Takano, C. Sugano, M. Inubushi, K. Yoshimura, S. Sunada, K. Kanno and A. Uchida, Opt. Express 26 (22), 29424-29439 (2018)10.1364/Oe.26.029424.
  * (41) Q. Q. Zeng, Z. M. Wu, D. Z. Yue, X. S. Tan, J. Y. Tao and G. Q. Xia, Appl Optics 59 (23), 6932-6938 (2020)10.1364/Ao.394999.
  * (42) S. Sunada and A. Uchida, Optica 8 (11), 1388-1396 (2021)10.1364/Optica.434918.
  * (43) S. Boyd and L. Chua, IEEE Trans. CAS 32 (11), 1150-1161 (1985)10.1109/TCS.1985.1085649.
  * (44) S. Orcioni, Nonlinear Dynamics 78 (4), 2861-2869 (2014)10.1007/s11071-014-1631-7.
  * (45) A. N. Tikhonov, V. I. Arsenin, and V. Y. Arsenin, Solutions of ill-posed problems, Vol. 14 (Winston, Washington, DC, 1977).
  * (46) M. C. Soriano, S. Ortín, D. Brunner, L. Larger, C. Mirasso, I. Fischer, and L. Pesquera, Opt. Express, 21 (1) 12-20 (2013).
  * (47) N. Semenova, X. Porte, L. Andreoli, M. Jacquot, L. Larger, and D. Brunner, Chaos 29(10) 103128 (2019).
  * (48) Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman and S. Massar, Scientific Reports 2 (1), 287 (2012)10.1038/srep00287.
  * (49) M. C. Soriano, P. Massuti-Ballester, J. Yelo, ús and I. Fischer, presented at the International Conference on Artificial Neural Networks, 2019
  * (50) H. Dai and Y. K. Chembo, IEEE Journal of Quantum Electronics 57 (3), 1-8 (2021)10.1109/JQE.2021.3074132.
  * (51) D. Brunner, B. Penkovsky, B. A. Marquez, M. Jacquot, I. Fischer and L. Larger, Journal of Applied Physics 124 (15), 152004 (2018)10.1063/1.5042342.
  * (52) P. Antonik, M. Haelterman and S. Massar, Physical Review Applied 7 (5), 054014 (2017)10.1103/PhysRevApplied.7.054014.
  * (53) Y. K. Chembo, D. Brunner, M. Jacquot and L. Larger, Rev. Mod. Phys. 91 (3), 035006 (2019)10.1103/RevModPhys.91.035006.
  * (54) J. D. Hart, R. Roy, D. Müller-Bender, A. Otto and G. Radons, Phys. Rev. Lett. 123 (15), 154101 (2019)10.1103/PhysRevLett.123.154101.
  * (55) J. D. Hart, D. C. Schmadel, T. E. Murphy and R. Roy, Chaos 27 (12), 7 (2017)10.1063/1.5016047.
  * (56) S. Yanchuk and G. Giacomelli, Journal of Physics A: Mathematical and Theoretical 50 (10), 103001 (2017)10.1088/1751-8121/50/10/103001.
  * (57) L. Larger, B. Penkovsky and Y. Maistrenko, Nature Communications 6 (1), 7752 (2015)10.1038/ncomms8752.
  * (58) T. E. Murphy, A. B. Cohen, B. Ravoori, K. R. Schmitt, A. V. Setty, F. Sorrentino, C. R. Williams, E. Ott and R. Roy, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368 (1911), 343-366 (2010)
  * (59) Y. K. Chembo, Chaos: An Interdisciplinary Journal of Nonlinear Science 30 (1), 013111 (2020)10.1063/1.5120788.


# String amplitudes and mutual information in confining backgrounds:
the partonic behavior

Mahdis Ghodrati<EMAIL_ADDRESS>Asia Pacific Center for Theoretical
Physics, Pohang University of Science and Technology, Pohang 37673, Korea

###### Abstract

We present the connections between the behaviors of string scattering
amplitudes and mutual information, in several holographic confining
backgrounds, demonstrating further the ability of these quantum information
measures in capturing the QCD phenomena. We show analogies between the
logarithmic branch cut behavior of the string scattering amplitude in $4d$,
$\mathcal{A}_{4}$, at low and medium Mandelstam variable $s$ observed in
Bianchi  _et al._ (2022), which is due to the dependence of the string tension
on the holographic coordinate, and the branch cut behavior observed in mutual
information and critical distance $D_{c}$ observed at low-cut-off variable
$u_{KK}$ observed in our previous work Ghodrati (2022a). It can also be seen
that in both cases, as $s$ or $u_{KK}$ increases, the peaks in the branch cuts
fade away in the form of $\text{Re}[\mathcal{A}_{4}]\propto s^{-1}$. This
observation can further establish the ER$=$EPR conjecture and the general
interdependence between the scattering amplitudes and entanglement entropy.

##  I Introduction

In the SLAC laboratory, it has been observed that at high energy and fixed
angle, the hadronic scattering amplitudes fall off based on power low behavior
in the Mandelstam variable $s$ (the center of mass energy) and not based on
exponential behavior in $s$. This observation is incompatible with the results
coming from string theory. On the other hand, using holography, an interesting
connection between entanglement entropy (EE) and the string scattering
amplitude has been noticed in Seki and Sin (2014); Hubeny and Semenoff (2014);
Semenoff and Zarembo (2011); Hubeny and Semenoff (2014, 2015). According to
these connections, Seki and Sin (2014); Hubeny and Semenoff (2014); Semenoff
and Zarembo (2011); Hubeny and Semenoff (2014, 2015); Polchinski and Strassler
(2003); Andreev and Siegel (2005); Hatta  _et al._ (2008); Pire  _et al._
(2008); Ballon Bayona  _et al._ (2008); Brower  _et al._ (2000); Brünner  _et
al._ (2015), we can now show that the richer mixed correlation measures such
as negativity and mutual information (MI) can further establish connections
between the behaviors of scattering amplitudes and the pattern of quantum
entanglement. Specifically, we noticed that the shapes of the singularities in
the behavior of scattering amplitudes, observed in Bianchi  _et al._ (2022),
that form logarithmic branch cuts (rather than poles), and are more pronounced
in small $s$, and fade away in the larger $s$, are indeed correct and
physical, and can be explained by the connections with entanglement entropy
and mutual information, as we have observed numerically the same structure in
our previous works on MI in confining geometries Ghodrati (2022a, b).

In Bianchi  _et al._ (2022), using holographic confining backgrounds, the
partonic behavior of scattering processes at large and medium ranges of
Mandelstam variable $s$ have been explored, where the branch cut singularities
at smaller values of $s$ have been observed, whereas the peaks decline at
large values $s$. This work was based on the previous works of Polchinski and
Strassler Polchinski and Strassler (2002), whose main result was writing the
$4d$ amplitude in terms of the $10d$ scattering amplitude as

$\displaystyle\mathcal{A}_{4}(s,t,u)\propto\int
dr\sqrt{-g}\times\mathcal{A}_{10}[\tilde{s}(r),\tilde{t}(r),\tilde{u}(r)],$
(1)

where here $\mathcal{A}_{10}$ is the $10d$ string amplitude, $r$ is the
holographic radial coordinate, and $\Psi(r)$ is the wave function of the
scattered state. In the holographic QCD models with a non-trivial dilaton
field such as Maldacena-Nunez Maldacena and Nunez (2001) or even Witten-QCD
Witten (1997); Seiberg and Witten (1994), where the dilaton field depends on
the radial coordinate, the above relation should instead be written as

$\displaystyle\mathcal{A}_{4}(s,t,u)=\int_{u_{KK}}^{\infty}du\sqrt{-g}e^{-2\phi}\Big{(}\prod_{i=1}^{4}\psi_{i}(u)\Big{)}\mathcal{A}_{10}[\tilde{s}(r),\tilde{t}(r),\tilde{u}(r)].$
(2)

Then, there, the amplitudes has been expanded around its poles and it has been
shown that the singularities, instead of being the actual poles, would be in
the form of branch points or branch cuts with finite imaginary parts (which is
the sign of bound state creation) and therefore, the authors of Bianchi  _et
al._ (2022), suggested that the prescription of Polchinski and Strassler
Polchinski and Strassler (2002) would fail in this limit. Then, in these
scenarios, one might think that other tools such as new holographic mixed
quantum information and correlation measures, such as mutual information,
negativity or entanglement of purification could be new tools to probe these
scenarios and give new information.

In order to reinforce such implementations, we propose a strong connection
between the behavior of entanglement entropy and quantum correlation measures
among two mixed subsystems at various energy and scales of the setup, and the
behaviors of string scattering amplitudes in the holographic QCD backgrounds,
where the quantized behavior and the creation of bound states could be
observed in both cases. This connection is specifically found using these
three works Ghodrati (2022a); Bianchi  _et al._ (2022); Hubeny and Semenoff
(2014), and based on the numerical studies performed in confining geometries.

The main point is that the spectrum of QCD, unlike theories such as
$\mathcal{N}=4$ SYM, is discrete, which is the result of the IR hard wall and
the boundary condition that it imposes. This discrete spectrum then can be
caught by the quantum information correlation measures. Also, measures similar
to the case of amplitude, as found in Bianchi  _et al._ (2022), could catch
the transition from soft to hard scattering, or the “bending” trajectories in
confining models. Moreover, entanglement entropy and mutual information can
detect the effects of asymptotic freedom where the gauge couplings and the
binding energies become smaller. At lower energies they could also detect the
“quantized” behavior of such binding energies.

It is worths to notice that most of the closed strings reside near the wall
which causes the wave function to be peaked there at the wall, instead of
vanishing there. This fact then could show an effect on the entanglement
patterns and mixed correlations saddles, specially, this would be more
pronounced at lower energies.

Therefore, due to the connections between the partonic nature of hadron
scattering and holography Bianchi  _et al._ (2022); Kang and Nastase (2005);
Pire  _et al._ (2008), one would expect that the entanglement patterns and
these scattering structures would have strong connections among them as well,
which we can now show here.

##  II Critical distance in confining backgrounds

First, turning to the EE side, in Kol  _et al._ (2014); Brandhuber  _et al._
(1998), it has been shown that for a general confining background with the
geometry

$\displaystyle
ds^{2}=\alpha(u)[\beta(u)du^{2}+dx^{\mu}dx_{\mu}]+g_{ij}d\theta^{i}d\theta^{j},$
$\displaystyle u_{KK}<u<\infty,\ \ x^{\mu}(\mu=0,1,...,d),\ \
\theta^{i}(i=d+2,...9),$ (3)

the length of a strip and its entanglement entropy, in terms of the minimum of
the holographic radial coordinate, $u_{0}$, can be written as

$\displaystyle
L(u_{0})=2\int_{u_{0}}^{\infty}du\sqrt{\frac{\beta(u)}{\frac{H(u)}{H(u_{0})}-1}},$
$\displaystyle
S(u_{0})=\frac{V_{0}}{2G_{N}}\int_{u_{0}}^{\infty}du\sqrt{\frac{\beta(u)H(u)}{1-\frac{H(u_{0})}{H(u)}}},$
(4)

where $H(u)=e^{-4\phi}V^{2}_{\text{int}}\alpha(u)^{d}$ and
$V_{\text{int}}=\int d\vec{\theta}\sqrt{\text{det}[g_{ij}]}$. Then, for two
strips of width $L$ and distance $D$ between them, the mutual information
would be

$\displaystyle I(D,L)=2S(L)-S(D)-S(2L+D),$ (5)

and the critical distance $D_{c}$ is where $I(D_{c},L)=0$. This quantity,
$D_{c}$, has been implemented in our previous works Ghodrati (2022c, b, 2021,
a); Ghodrati  _et al._ (2019) to probe the quantum correlation properties of
various geometries.

In Ghodrati (2022a), we noticed that, in the background of confining
geometries, the critical distance between two strips, which is related to
mutual information and in fact is a measure of mixed correlations, shows
singularities at smaller values of $u_{KK}$ and statistically smooths out in
the bigger values of $u_{KK}$. This behavior is especially pronounced in the
Witten-QCD geometry, figure 19 of Ghodrati (2022a), which is shown again in
figure 1 here. The same singularity jumps can also be seen in the background
of Klebanov-Tseytlin geometry Klebanov and Tseytlin (2000); Klebanov and
Strassler (2000), shown in figure 2 here.

Figure 1: The behavior of $D_{c}$ which comes from the mutual information
between two mixed strips, versus $u_{KK}$, in the Witten-QCD background. This
figure can simply be compared with the figure 6 of Bianchi  _et al._ (2022),
for the “real parts of the amplitude”, as one can see the behavior of branch
cuts and then the decay at large $s$ are indeed very similar to the behavior
of MI and $D_{c}$ in this model. This is the main observation of our work
here.

Similar to the results of Bianchi  _et al._ (2022), it could be seen that the
peaks of $D_{c}$ which correspond to the peaks of mutual information and mixed
correlation measures are stronger and more noticeable in the medium (or rather
smaller) regions of $u_{KK}$ which corresponds to small values of Mandelstam
variable $s$. For bigger $s$ (and bigger $t$ where the angle is fixed),
corresponding to bigger values of $u_{KK}$, the peaks would fade away. This is
due to the effects of asymptotic freedom where the gauge couplings and the
binding energies of the bound states are small. However, from figures 1 and 2,
it could be seen again that there is no peak at larger $u_{KK}$, which is due
to the fact that the binding energy, length and mass of the strings become
smaller in that regimes too.

Figure 2: The relationship between the critical distance $D_{c}$ among two
strips, versus $u_{KK}$, in the background of Klebanov-Tseytlin geometry.
Figure 3: The relationship between the critical distance $D_{c}$ among the two
strips versus $u_{KK}$ in the background of Sakai-Sugimoto geometry. Again,
one can see that the peaks become less pronounced as $u_{KK}$ increases and
they fade away at large values of $s$. The decay at large $u_{KK}$ is also a
power law.

The connection between critical distance $D_{c}$ and $u_{KK}$ in the
background of Sakai-Sugimoto Sakai and Sugimoto (2005) is shown in figure 3.
As this figure has been created from the real part of entanglement entropy
$S(u_{0})$ versus the real part of the size of the strips $L(u_{0})$, one
would expect that, similarly in the “real” part of the amplitude, a power low
decay behavior should be observed at bigger values of $s$ or $u$, as indeed
one can see such behavior from figure 3. Note that, the behavior of the
amplitude is in fact $\text{Re}[\mathcal{A}_{4}]\sim s^{-1}$ and
$\text{Im}[\mathcal{A}_{4}]\sim s^{2-\Delta/2}$, where $\Delta$ is the sum of
the scaling dimensions of the operators that contribute as
$\prod_{i=1}\psi_{i}(r)\sim r^{-\Delta}$, ($\Delta$ can be replaced by the
twist in QCD models).

##  III String scattering amplitude in confining geometries

Now turning to the string amplitude side, the $10d$ amplitude in Bianchi  _et
al._ (2022) has been found to be

$\displaystyle\mathcal{A}_{10}=4^{4}\sum_{n=0}^{\infty}(-1)^{n}\frac{(n+1)^{2}}{(n!)^{2}}\frac{(1+c_{+}^{4}+c_{-}^{4})}{\alpha^{\prime}s/4-(n+1)}\times$
$\displaystyle\frac{\Gamma\left((n+1)c_{+}\right)\Gamma\left((n+1)c_{-}\right)}{\Gamma\left(1-(n+1)c_{+}\right)\Gamma\left(1-(n+1)c_{-}\right)}\equiv\sum_{n=0}^{\infty}\frac{\mathcal{R}(\theta)}{\alpha^{\prime}s/4-(n+1)}.$
(6)

Higher $\alpha^{\prime}s$ corresponds to higher energies and bigger $u_{KK}$.

In terms of $s$ and the angle $\theta$, this amplitude can be written as

$\displaystyle\mathcal{A}_{10}=\frac{1}{32}\alpha^{4}s^{4}(\cos(2\theta)+7)^{2}\frac{\Gamma\left(-\frac{1}{4}(s\alpha)\right)}{\Gamma\left(\frac{s\alpha}{4}+1\right)}\times$
(7)
$\displaystyle\frac{\Gamma\left(-\frac{1}{8}s\alpha(\cos(\theta)-1)\right)}{\Gamma\left(\frac{1}{8}(-s\alpha+s\text{$\alpha$p}\cos(\theta)+8)\right)}\times\frac{\Gamma\left(\frac{1}{8}s\alpha(\cos(\theta)+1)\right)}{\Gamma\left(1-\frac{1}{8}s\alpha(\cos(\theta)+1)\right)}.$
(8)

Its behavior versus the Mandelstam variable $s$ in confining holographic
backgrounds such as Witten-QCD has been shown in Bianchi  _et al._ (2022). The
specific confining backgrounds considered there include the hard wall model,
soft wall model and Witten’s QCD model, showing the same singularity patterns
as the EE and MI as shown in figure 4 here. Then, using the relations 1 and 2,
$\mathcal{A}_{10}$ can be integrated and the scattering amplitude in $4d$ can
be derived. The results for different values of $\theta$ are shown in figure
5, as for our main figure, which should be compared with those of $D_{c}$ for
our various confining models.

The reason for the peculiar behavior in generating singularities in the form
of branch cuts at low energies, is the dependence of the string tension on the
holographic radial coordinate. In their work, one could see that in $10d$ or
$11d$ geometries, the amplitude drops faster compared to lower $d$ geometries,
which is similar to the smoothing out of the singularities in the mixed-
correlation measures in higher dimensions.

Figure 4: The relationship between $A_{10}$ versus $\alpha^{\prime}s$ for
$\cos\theta=0.8$. Again, the fading away of the branch cut singularities at
large Mandelstam variable $s$ (center of mass energy) is similar to the case
of $D_{c}$ coming from mutual information in confining backgrounds. Figure 5:
The relationship between the real part of $[\mathcal{A}_{4}]$ versus
$\alpha^{\prime}s$ using the result of Bianchi  _et al._ (2022) for different
values of $\theta$, and in the hard wall confining model. The main behaviors
are similar to the case of Witten-QCD. The logarithmic branch cut
singularities at lower $s$, and the damping at larger $s$, as
$[\mathcal{A}_{4}]\sim s^{-1}$, are similar to the general behaviors of the
mutual information (MI) and critical distance $D_{c}$, observed in the
confining backgrounds.

The connections between $s$ and $\theta$ have also been further investigated
in figure 6. Note that, there, $0<\theta<9\pi$, where 10 branch cuts can be
observed. In general, with the upper bound of $n\pi$, $n+1$ branch cuts would
be present. Also, it can be seen that for bigger $s$, the branch cuts become
dispersed, which is also the case for mixed correlation quantum information
measures such as mutual information and $D_{c}$.

Figure 6: The $3d$ plot of the $10d$ scattering amplitude, $A_{10}$, versus
$s$ and $\theta$, with different angles We set $\alpha^{\prime}=1$ here.

One could see that in the simple case of fixed $t$, $\Delta S_{E}\propto(\log
s)^{2}$, as shown in figure 7. For the mixed correlation measures and in the
presence of a hard wall which creates the “four” main saddles Dong  _et al._
(2022); Ghodrati (2022c), the behaviors becomes more interesting and rich as
shown in figures 1 and 2. So as the parameter $D_{c}$ which comes from the
mutual information is related to the sum of several entanglement entropies,
therefore, figure 7 is comparable to the case of the confining and mixed setup
of figures 1 and 2.

Figure 7: The relationship between $\Delta S_{EE}$ and $s$ in pure $AdS$.

It is worthwhile to mention here that in Bianchi  _et al._ (2022), the fixed
angle limit approximation where $s\to\infty$ while $s/t$ is fixed, has also
been studied, where the behavior of the real part of the $A_{10}$ is shown in
figure 8. Note that, this limit is not physical as there is in fact imaginary
contributions and therefore this case is not related to the quantum mixed
correlation measure which we have seen in the full physical $A_{10}$ case
without this approximation.

The case of the Regge limit, where $s\to\infty$ while $t$ is fixed, gives
figures 9, where again the full phase diagram cannot be observed in this setup
and taking the full scenario is needed. This case then can only detect two
main saddles.

Figure 8: The relationship between $A_{10}$ versus $s$ and $\theta$ for
$\alpha=1$ in the fixed angle approximated limit.

Figure 9: The relationship between $A_{10}$ versus $s$ and $\theta$ for
$\alpha=1$ in the Regge limit.

One important point worths to emphasize again here is that the scattering
process will always be dominated by the interior regions which is close to the
IR wall at $u_{KK}$, or $u_{\Lambda}$, and therefore this parameter should
indeed be the main factor in determining the saddle. The mixed quantum
correlations would also be more dominant close to that wall leading to the
result that it can probe the phase transitions rather effectively.

Another point is that in Hubeny and Semenoff (2014), the disc amplitude of the
scattering of two heavy quark-anti-quark pairs which are accelerating in the
opposite direction, has been studied by employing the perturbation theory, and
then using that the entanglement entropy of the quark and anti-quark has been
found approximately as $S=\sqrt{\lambda}$, where $\lambda$ is the ’t Hooft
coupling, indicating again the connections between these quantities.

Also, in reference Seki and Sin (2014), it has been shown that as any
interaction between particles would change the entanglement entropy (EE),
therefore the change of EE should be related to the scattering amplitude. This
is also the case for all other mixed correlation measures such as mutual
information and negativity. So the scattering amplitude should also be able to
track the four saddles of mixed correlations Dong  _et al._ (2022); Ghodrati
(2022c), as we have noticed. The entanglement entropy can also be written in
terms of a Wilson loop $\langle W\rangle$ as
$S_{E}=(1-c\lambda\partial_{\lambda})\log\langle W\rangle$, and thus at large
’Hooft couplings $\lambda$, the change of EE at leading order in $\lambda$ is
related to the gluon scattering amplitude as

$\displaystyle\Delta
S_{E}\sim\frac{(1-\frac{1}{2}c)\sqrt{\lambda}}{8\pi}\left(\log\frac{s}{t}\right)^{2}.$
(9)

Accordingly, the same relations should be present between the mixed
correlation measures such as MI, negativity, entanglement and complexity of
purification, and the string scattering amplitudes, as we have traced out
here.

Another interesting observation is to note the links between the change of EE,
or other correlation measures, and the Bremsstralung of radiative corrections
as $\lambda\partial_{\lambda}\log\langle W\rangle$ in the case of accelerating
quark-antiquarks pair, which is proportional to the Bremsstralung function. So
the spikes that we noticed during the phase transitions in the plot of $D_{c}$
could be considered as the K-lines in Bremsstralung radiation.

Bear in mind that our observation for the connections is only for closed
strings (glueballs). In the case of open strings, which describe mesons or
baryons, there would be two kinds of entanglement entropy, which makes finding
the connections between mutual information and scattering amplitude more
difficult. The EE there includes shares from the entanglement of strings
endpoints in the gluons (or flavor branes) and the entanglement between the
gluons themselves, making the framework more complicated.

##  IV Conclusion

We showed connections between the logarithmic branch cut singularities, at low
Mandelstam variable $s$, of the open string scattering amplitude and the
singular behaviors observed in the mutual information and critical distance
$D_{c}$, in holographic confining backgrounds. These singularities actually
come from the dependence of the string tension on the holographic radial
coordinate and are a sign of the creation of bound states at low energies.

The power low decay of both quantities matches also at larger energies, $s$ or
$u_{KK}$, where the real part of both follows a $s^{-1}$ curve. This
observation can further establish the interconnections announced previously
between the patterns of entanglement entropy and the scattering amplitude, and
also can further strengthen the ER$=$EPR conjecture. In addition, it proposes
that the string scattering amplitude can serve in detecting the phase
transitions.

Hence, one could expect that in confining backgrounds, there would be other
connections between the phase diagrams of mixed quantum correlation measures
such as mutual information or critical distance $D_{c}$, negativity,
entanglement and complexity of purification, and the open string scattering
amplitude $\mathcal{A}$, where specifying the details for each case is the aim
of our future works.

## acknowledgments

We would like to thank Dorin Weissman for sharing his codes and useful
discussions. I also thank Stephen Angus and Jesus Cruz Rojas for reading the
manuscrip and their comments. This work has been supported by an appointment
to the JRG Program at the APCTP through the Science and Technology Promotion
Fund and Lottery Fund of the Korean Government. It has also been supported by
the Korean Local Governments - Gyeongsangbuk-do Province and Pohang City - and
by the National Research Foundation of Korea (NRF) funded by the Korean
government (MSIT) (grant numbers 2021R1F1A1048531 and 2021R1A2C1010834).

## References

  * Bianchi  _et al._ (2022) M. Bianchi, M. Firrotta, J. Sonnenschein, and D. Weissman, JHEP 05, 058 (2022), arXiv:2111.12106 [hep-th] .
  * Ghodrati (2022a) M. Ghodrati, Eur. Phys. J. C 82, 531 (2022a), arXiv:2110.12970 [hep-th] .
  * Seki and Sin (2014) S. Seki and S.-J. Sin, Phys. Lett. B 735, 272 (2014), arXiv:1404.0794 [hep-th] .
  * Hubeny and Semenoff (2014) V. E. Hubeny and G. W. Semenoff, (2014), arXiv:1410.1172 [hep-th] .
  * Semenoff and Zarembo (2011) G. W. Semenoff and K. Zarembo, Phys. Rev. Lett. 107, 171601 (2011), arXiv:1109.2920 [hep-th] .
  * Hubeny and Semenoff (2015) V. E. Hubeny and G. W. Semenoff, JHEP 10, 071 (2015), arXiv:1410.1171 [hep-th] .
  * Polchinski and Strassler (2003) J. Polchinski and M. J. Strassler, JHEP 05, 012 (2003), arXiv:hep-th/0209211 .
  * Andreev and Siegel (2005) O. Andreev and W. Siegel, Phys. Rev. D 71, 086001 (2005), arXiv:hep-th/0410131 .
  * Hatta  _et al._ (2008) Y. Hatta, E. Iancu, and A. H. Mueller, JHEP 01, 026 (2008), arXiv:0710.2148 [hep-th] .
  * Pire  _et al._ (2008) B. Pire, C. Roiesnel, L. Szymanowski, and S. Wallon, Phys. Lett. B 670, 84 (2008), arXiv:0805.4346 [hep-ph] .
  * Ballon Bayona  _et al._ (2008) C. A. Ballon Bayona, H. Boschi-Filho, and N. R. F. Braga, JHEP 03, 064 (2008), arXiv:0711.0221 [hep-th] .
  * Brower  _et al._ (2000) R. C. Brower, S. D. Mathur, and C.-I. Tan, Nucl. Phys. B 587, 249 (2000), arXiv:hep-th/0003115 .
  * Brünner  _et al._ (2015) F. Brünner, D. Parganlija, and A. Rebhan, Phys. Rev. D 91, 106002 (2015), [Erratum: Phys.Rev.D 93, 109903 (2016)], arXiv:1501.07906 [hep-ph] .
  * Ghodrati (2022b) M. Ghodrati, J. Korean Phys. Soc. 81, 77 (2022b), arXiv:2205.03565 [hep-th] .
  * Polchinski and Strassler (2002) J. Polchinski and M. J. Strassler, Phys. Rev. Lett. 88, 031601 (2002), arXiv:hep-th/0109174 .
  * Maldacena and Nunez (2001) J. M. Maldacena and C. Nunez, Int. J. Mod. Phys. A 16, 822 (2001), arXiv:hep-th/0007018 .
  * Witten (1997) E. Witten, Nucl. Phys. B 507, 658 (1997), arXiv:hep-th/9706109 .
  * Seiberg and Witten (1994) N. Seiberg and E. Witten, Nucl. Phys. B 431, 484 (1994), arXiv:hep-th/9408099 .
  * Kang and Nastase (2005) K. Kang and H. Nastase, Phys. Rev. D 72, 106003 (2005), arXiv:hep-th/0410173 .
  * Kol  _et al._ (2014) U. Kol, C. Nunez, D. Schofield, J. Sonnenschein, and M. Warschawski, JHEP 06, 005 (2014), arXiv:1403.2721 [hep-th] .
  * Brandhuber  _et al._ (1998) A. Brandhuber, N. Itzhaki, J. Sonnenschein, and S. Yankielowicz, JHEP 06, 001 (1998), arXiv:hep-th/9803263 .
  * Ghodrati (2022c) M. Ghodrati, (2022c), arXiv:2209.04548 [hep-th] .
  * Ghodrati (2021) M. Ghodrati, Phys. Rev. D 104, 046004 (2021), arXiv:2012.04386 [hep-th] .
  * Ghodrati  _et al._ (2019) M. Ghodrati, X.-M. Kuang, B. Wang, C.-Y. Zhang, and Y.-T. Zhou, JHEP 09, 009 (2019), arXiv:1902.02475 [hep-th] .
  * Klebanov and Tseytlin (2000) I. R. Klebanov and A. A. Tseytlin, Nucl. Phys. B 578, 123 (2000), arXiv:hep-th/0002159 .
  * Klebanov and Strassler (2000) I. R. Klebanov and M. J. Strassler, JHEP 08, 052 (2000), arXiv:hep-th/0007191 .
  * Sakai and Sugimoto (2005) T. Sakai and S. Sugimoto, Prog. Theor. Phys. 113, 843 (2005), arXiv:hep-th/0412141 .
  * Dong  _et al._ (2022) X. Dong, S. McBride, and W. W. Weng, JHEP 06, 094 (2022), arXiv:2110.11947 [hep-th] .


#  Period-doubling Continued Fractions are Algebraic in Characteristic $2$

Yining Hu  School of Mathematics and Statistics, Huazhong University of
Science and Technology, Wuhan, PR China<EMAIL_ADDRESS>and  Alain
Lasjaunias  Mathematical Institute Bordeaux University, Bordeaux, France.
<EMAIL_ADDRESS>
###### Abstract.

Considering an arbitrary pair of distinct and non constant polynomials, $a$
and $b$ in $\mathbb{F}_{2}[t]$, we build a continued fraction in
$\mathbb{F}_{2}((1/t))$ whose partial quotients are only equal to $a$ or $b$.
In a previous work of the first author and Han (to appear in Acta
Arithmetica), the authors considered two cases where the sequence of partial
quotients represents in each case a famous and basic $2$-automatic sequence,
both defined in a similar way by morphisms. They could prove the algebraicity
of the corresponding continued fractions for several pairs $(a,b)$ in the
first case (the Prouhet-Thue-Morse sequence) and gave the proof for a
particular pair for the second case (the period-doubling sequence). Recently
Bugeaud and Han (arXiv:2203.02213) proved the algebraicity for an arbitrary
pair in the first case. Here we give a short proof for an arbitrary pair in
the second case.

###### Key words and phrases:

algebraicity, automatic sequence, continued fraction, period-doubling sequence

###### 2010 Mathematics Subject Classification:

11B85, 11J70, 11B50, 11Y65, 05A15, 11T55

##  1\. Introduction

Let $\mathbb{F}_{2}((1/t))$ be the field of power series in $1/t$, where t is
a formal indeterminate, over the finite field $\mathbb{F}_{2}$. A non-zero
element is $\alpha=t^{n}+\sum_{i<n}a_{i}\cdot t^{i}$ where $n$ belongs to
$\mathbb{Z}$ and $a_{i}=0$ or $1$. An absolute value on
$\mathbb{F}_{2}((1/t))$ is defined by $|0|=0$ and $|\alpha|=|t|^{n}$ where
$|t|>1$ is a fixed given real number .

So $\mathbb{F}_{2}((1/t))$ is the completion of $\mathbb{F}_{2}(t)$ for this
absolute value. Every irrational (resp. rational) element in
$\mathbb{F}_{2}((1/t))$ can be expanded in an infinite (resp. finite)
continued fraction: $\alpha=[a_{0};a_{1},...,a_{n},...]$ where the $a_{i}$ are
in $\mathbb{F}_{2}[t]$ and $\deg(a_{i})>0$ for $i>0$. For a basic introduction
on fields of power series and continued fractions, the reader may consult [6].

The origin of the question discussed here is due to G.-N. Han and the first
author. In [4], the authors considered two basic sequences, formed by an
infinite word with two letters $a$ and $b$, belonging to the family of
$2$-automatic sequences (see [1] p.173 and p.176) . Both sequences are
obtained in a similar way, as fixed point of a morphism. For the first one,
the $(a;b)$-Prouhet-Thue-Morse sequence, denoted by $\mathbf{t}$, the morphism
$\tau$ is defined by $\tau(a)=ab$ and $\tau(b)=ba$, and we have

$\mathbf{t}=\tau^{\infty}(a)=abbabaabbaab\ldots$

Note that this famous sequence was considered very long ago and has been the
starting point of various studies. For the second one , the $(a;b)$-period-
doubling sequence, denoted by $\mathbf{p}$, the morphism $\sigma$ is defined
by $\sigma(a)=ab$ and $\sigma(b)=aa$, and we have

$\mathbf{p}=\sigma^{\infty}(a)=abaaabababaaabaa\ldots$

In both cases, we consider a pair $(a,b)$ of distinct and non-constant
polynomials in $\mathbb{F}_{2}[t]$, and we can associate with each sequence an
infinite continued fraction in $\mathbb{F}_{2}((1/t))$,
$\operatorname{CF}(\mathbf{t})$ and $\operatorname{CF}(\mathbf{p})$, where the
sequence of partial quotients is derived from the sequences $\mathbf{t}$ and
$\mathbf{p}$ :

$\alpha_{\mathbf{t}}=\operatorname{CF}(\mathbf{t})=[a;b,b,a,b,a,a,\ldots]$

and

$\alpha_{\mathbf{p}}=\operatorname{CF}(\mathbf{p})=[a;b,a,a,a,b,a,\ldots].$

It was proved in [4] that, for pairs $(a,b)$ with $\deg a+\deg b\leq 7$,
$\alpha_{\mathbf{t}}$ was a root of a polynomial of degree $4$, with five
coefficients in $\mathbb{F}_{2}[t]$ depending on $(a,b)$. In a recent work,
extending the case of $\operatorname{CF}(\mathbf{t})$, Bugeaud and Han [3]
obtained the same result for all pairs $(a,b)$ and they gave the explicit
formulas for the $5$ coefficients as polynomials in $a$ and $b$.

Also in [4], it was proved that for $(a,b)=(t^{3},t^{2}+t+1)$ we have

$(t^{5}+t^{3}+t^{2})\cdot\alpha_{\mathbf{p}}^{4}+(t^{8}+t^{6}+t^{5}+t^{3})\cdot\alpha_{\mathbf{p}}^{3}+(t^{5}+t^{4}+t^{3})\cdot\alpha_{\mathbf{p}}^{2}+1=0.$

In this note, we give a short proof of the general case for the sequence
$\mathbf{p}$ with the following theorem, thus confirming Conjecture 1.5 from
[4].

######  Theorem 1.1.

Let $a,b$ be two distinct non constant elements in $\mathbb{F}_{2}[t]$. Let
$\alpha_{\mathbf{p}}=[\sigma^{\infty}(a)]\in\mathbb{F}_{2}((1/t))$. Define
$P(x)\in\mathbb{F}_{2}(t)[x]$ to be

(1.1) $P(x)=Ax^{4}+Bx^{3}+Cx^{2}+1$

with

$A=ab+b^{2}+1,\quad B=ab(a+b),\quad C=ab.$

Then $P(\alpha_{\mathbf{p}})=0$.

######  Remark 1.

An elementary proof shows that $P(x)$ has no solution in $\mathbb{F}_{2}(t)$.
Consequently, since $\alpha_{\mathbf{p}}$ is not quadratic, $P(x)$ is
irreducible and $\alpha_{\mathbf{p}}$ has degree $4$ over $\mathbb{F}_{2}(t)$.

######  Remark 2.

The theorem remains true when we replace $\mathbb{F}_{2}$ by any other field
$K$ of characteristic $2$.

######  Remark 3.

In fact we have proven that the continued fraction $\alpha_{\mathbf{p}}$ as a
series in the ring $\mathbb{F}_{2}((1/a,1/b))$ is algebraic over
$\mathbb{F}_{2}(a,b)$.

In Section 2 we recall notation and formulas for continued fractions. In
Section 3 we give the proof of Theorem 1.1 and in a last section we make some
comments about the link with Riccati differential equations.

##  2\. Notation and basic formulas for continued fractions

We use the same notation as in [6], which we recall in this section.

Let $W=w_{1},w_{2},\ldots,w_{n}$ be a sequence of variables over a ring
$\mathbb{A}$. We set $|W|=n$ for the length of the word $W$. We define the
following operators for the word $W$.

$\displaystyle W^{\prime}$ $\displaystyle=w_{2},w_{3},\ldots,w_{n}\mbox{ or
}W^{\prime}=\emptyset\mbox{ if }|W|=1.$ $\displaystyle W^{\prime\prime}$
$\displaystyle=w_{1},w_{2},\ldots,w_{n-1}\mbox{ or
}W^{\prime\prime}=\emptyset\mbox{ if }|W|=1.$ $\displaystyle W^{*}$
$\displaystyle=w_{n},w_{n-1},\ldots,w_{1}.$

We consider the finite continued fraction associated to $W$ to be

(2.1)
$[W]=[w_{1},\ldots,w_{n}]=w_{1}+\cfrac{1}{w_{2}+\cfrac{1}{w_{3}+\cfrac{1}{\ddots+\cfrac{1}{w_{n}}}}}$

The continued fraction $[W]$ is a quotient of multivariate polynomials in the
variables $w_{1},w_{2},\ldots,w_{n}$. These polynomials are called continuants
built on $W$. They are defined inductively as follows:

Set $\langle\emptyset\rangle=1$. If the sequence $W$ has only one element,
then we have $\langle W\rangle=W$. Hence, with the above notation, the
continuants can be computed, recursively on the length $|W|$, by the following
formula

(2.2) $\langle W\rangle=w_{1}\langle
W^{\prime}\rangle+\langle(W^{\prime})^{\prime}\rangle\mbox{ for }|W|\geq 2.$

Thus, with this notation, for any finite word $W$, the finite continued
fraction $[W]$ satisfies

(2.3) $[W]=\frac{\langle W\rangle}{\langle W^{\prime}\rangle}.$

It is easy to prove by induction that $\langle W^{*}\rangle=\langle W\rangle$.
For any finite sequences $A$ and $B$ of variables over $\mathbb{A}$, defining
$A,B$ as the concatenation of the sequences, by induction on $|A|$, we also
have the following generalization of (2.2)

(2.4) $\langle A,B\rangle=\langle A\rangle\langle B\rangle+\langle
A^{\prime\prime}\rangle\langle B^{\prime}\rangle.$

Using induction of $|W|$, we have the following classical identity

(2.5) $\langle W\rangle\langle(W^{\prime})^{\prime\prime}\rangle-\langle
W^{\prime}\rangle\langle W^{\prime\prime}\rangle=(-1)^{|W|}\mbox{ for }|W|\geq
2.$

##  3\. Proof of Theorem 1.1

For $n\geq 0$, set

$W_{n}=(\sigma^{n}(a))^{\prime\prime}.$

We have $W_{0}=\emptyset$, $W_{1}=a$, $W_{2}=aba$, etc., and
$|W_{n}|=2^{n}-1$.

We will prove that $P([W_{n}])$ converges to $0$ . For this, we need to
following two lemmas.

######  Lemma 3.1.

For all $n\geq 0$,

(3.1) $W_{n+1}=W_{n},\varepsilon_{n},W_{n}$

where $\varepsilon_{n}=a$ if $n$ is even and $\varepsilon_{n}=b$ if $n$ is
odd. In consequence, for $n\geq 0$, we have

$W_{n}=W_{n}^{*}.$

###### Proof.

We prove by induction. Suppose identity (3.1) holds for $n$, then

$\displaystyle W_{n+2}$ $\displaystyle=\sigma(\sigma^{n+1}(a))^{\prime\prime}$
$\displaystyle=\sigma(W_{n+1})a$
$\displaystyle=\sigma(W_{n}),\sigma(\varepsilon_{n}),\sigma(W_{n})a$
$\displaystyle=W_{n+1},\varepsilon_{n+1},W_{n+1}.$

In the last equality, we use the fact that
$\sigma(\varepsilon_{n})=a\varepsilon_{n+1}$. ∎

For $n\geq 1$, define $u_{n}=\langle W_{n}\rangle$ and $v_{n}=\langle
W_{n}^{\prime}\rangle$, so that

$[W_{n}]=\frac{u_{n}}{v_{n}}.$

######  Lemma 3.2.

For all $n\geq 1$, we have

$\displaystyle u_{n+1}$ $\displaystyle=\varepsilon_{n}u_{n}^{2},$
$\displaystyle v_{n+1}$ $\displaystyle=\varepsilon_{n}u_{n}v_{n}+1.$

###### Proof.

We use Lemma 3.1, identity (2.4), and the fact that we are in characteristic
$2$.

$\displaystyle u_{n+1}$ $\displaystyle=\langle W_{n+1}\rangle$
$\displaystyle=\langle W_{n},\varepsilon_{n},W_{n}\rangle$
$\displaystyle=\langle W_{n}\rangle\langle\varepsilon_{n},W_{n}\rangle+\langle
W_{n}^{\prime\prime}\rangle\langle W_{n}\rangle$ $\displaystyle=\langle
W_{n}\rangle(\varepsilon_{n}\langle W_{n}\rangle+\langle
W_{n}^{\prime}\rangle)+\langle W_{n}^{\prime\prime}\rangle\langle
W_{n}\rangle$ $\displaystyle=\langle W_{n}\rangle(\varepsilon_{n}\langle
W_{n}\rangle+\langle W_{n}^{\prime}\rangle)+\langle
W_{n}^{\prime}\rangle\langle W_{n}\rangle$ $\displaystyle=\langle
W_{n}\rangle\varepsilon_{n}\langle W_{n}\rangle$
$\displaystyle=\varepsilon_{n}u_{n}^{2}.$ $\displaystyle v_{n+1}$
$\displaystyle=\langle W_{n+1}^{\prime}\rangle$ $\displaystyle=\langle
W_{n}^{\prime},\varepsilon_{n},W_{n}\rangle$ $\displaystyle=\langle
W_{n}^{\prime}\rangle\langle\varepsilon_{n},W_{n}\rangle+\langle(W_{n}^{\prime})^{\prime\prime}\rangle\langle
W_{n}\rangle$ $\displaystyle=\langle
W_{n}^{\prime}\rangle(\varepsilon_{n}\langle W_{n}\rangle+\langle
W_{n}^{\prime}\rangle)+\langle(W_{n}^{\prime})^{\prime\prime}\rangle\langle
W_{n}\rangle$ $\displaystyle=\varepsilon_{n}u_{n}v_{n}+\langle
W_{n}^{\prime}\rangle\langle
W_{n}^{\prime}\rangle+\langle(W_{n}^{\prime})^{\prime\prime}\rangle\langle
W_{n}\rangle$ $\displaystyle=\varepsilon_{n}u_{n}v_{n}+1.$

In the last step, we also use (2.5), and the symmetry of $W_{n}$. ∎

###### Proof of Theorem 1.1.

For $n\geq 1$, set

$X_{n}=Au_{n}^{4}+Bu_{n}^{3}v_{n}+Cu_{n}^{2}v_{n}^{2}+v_{n}^{4},$

so that $P(u_{n}/v_{n})=X_{n}/v_{n}^{4}$.

Using Lemma 3.2 and noticing that
$\varepsilon_{n}^{2}+ab=\varepsilon_{n}(a+b)$ , we obtain

(3.2)
$X_{n+1}+1=\varepsilon^{4}_{n}u_{n}^{4}(X_{n}+1)+\varepsilon_{n}^{3}u_{n}^{4}(a+b)(1+abu_{n}^{2}).$

We observe taht

$X_{1}+1=a^{3}(a+b)=\varepsilon_{0}u_{1}^{2}(a+b),$

this allows us, using (3.2) and the fact that
$\varepsilon_{n}\varepsilon_{n-1}=ab$, to get by induction that for all $n\geq
1$

(3.3) $X_{n}+1=\varepsilon_{n-1}u_{n}^{2}(a+b).$

Therefore

$P\left(\frac{u_{n}}{v_{n}}\right)=\frac{\varepsilon_{n-1}u_{n}^{2}(a+b)+1}{v_{n}^{4}}$

This is a quotient of polynomials in $\mathbb{F}_{2}[t]$, and it can be easily
seen from Lemma 3.2 that the degree in $t$ of the denominator minus that of
the numerator tends to plus infinity as $n$ goes to infinity. And therefore
the sequence converges to $0$ in $\mathbb{F}_{2}((1/t))$. ∎

##  4\. Link with Riccati differential equations

In both cases, the Prouhet-Thue-Morse and the period-doubling continued
fraction, a particular choice of the pair $(a,b)$ in $\mathbb{F}_{2}[t]^{2}$,
brings us back to an investigation undertaken 45 years ago. Indeed, in these
two cases, taking $(a,b)=(t,t+1)$, all the partial quotients of these
continued fractions are $t$ or $t+1$. In 1977, Baum and Sweet [2] considered
all the infinite continued fractions in $\mathbb{F}_{2}((1/t))$ whose partial
quotients have all degree one.

They considered the subset $D$ of irrational continued fractions $\alpha$ in
$\mathbb{F}_{2}((1/t))$ such that

$\alpha=[0;a_{1},a_{2},\ldots,a_{n},\ldots]$

where $\deg(a_{i})=1$ (i.e., $a_{i}=t$ or $t+1$) for $i\geq 1$. Let $P$ the
subset of $\mathbb{F}_{2}((1/t))$ containing all $\alpha$ such that
$|\alpha|<1$. Hence $P$ contains $D$. We have :

######  Theorem 4.1.

[Baum-Sweet, 1977] An element $\alpha\in P$ is in $D$ if and only if $\alpha$
satisfies

$\alpha^{2}+t\alpha+1=(1+t)\beta^{2}$

for some $\beta\in P$.

In order to obtain another formulation of Theorem 4.1, we will now use formal
differentiation in $\mathbb{F}_{2}((1/t))$. Continued fractions and formal
differentiation in power series fields are related, these are the main tools
in Diophantine approximation (see [5], p.221-225). We recall that in
characteristic $2$, which is the case considered here, an element has
derivative zero if and only if this element is a square. Moreover the
derivative of an element is a square (hence the second derivative is zero) .
We have the following theorem.

######  Theorem 4.2.

An element $\alpha\in P$ is in $D$ if and only if $\alpha$ satisfies

$(\alpha\cdot t(t+1))^{\prime}=\alpha^{2}+1.$

###### Proof.

If

$\alpha^{2}+t\alpha+1=(1+t)\beta^{2}$

then by derivation we get

$(t\alpha)^{\prime}=\beta^{2}.$

Consequently, we have

$\alpha^{2}+1=t\alpha+(1+t)(t\alpha)^{\prime}=(\alpha\cdot t(t+1))^{\prime}.$

Conversely, if

$(\alpha\cdot t(t+1))^{\prime}=\alpha^{2}+1.$

then

$\alpha^{2}+t\alpha+1=(\alpha\cdot t(t+1))^{\prime}+t\alpha=(t+1)(\alpha\cdot
t)^{\prime}=(t+1)\beta^{2},$

for some $\beta\in P$ since $(t\alpha)^{\prime}$ is a square in $P$. ∎

Thus the Riccati differential equation

(R0) $(x\cdot t(t+1))^{\prime}=x^{2}+1$

is characteristic of infinite continued fraction expansions in
$\mathbb{F}_{2}((1/t))$, having all partial quotients of degree one. Indeed,
there exists a direct proof, without Theorem 4.1, showing that a solution in
$\mathbb{F}_{2}((1/t))$ of (R0) is irrational and has all partial quotients of
degree one (see [5], p. 225).

Now we turn to elements in $\mathbb{F}_{2}((1/t))$ which are algebraic over
$\mathbb{F}_{2}(t)$ of degree $d\geq 1$. If $\alpha$ is such an element, we
have (see [5], p. 221)

(4.1) $\alpha^{\prime}=a_{0}+a_{1}\alpha+\cdots+a_{d-1}\alpha^{d-1}$

where the coefficients $a_{i}$ belong to $\mathbb{F}_{2}(t)$.

Concerning the algebraic continued fraction
$\alpha_{\mathbf{t}}=\operatorname{CF}(\mathbf{t})\in\mathbb{F}_{2}((1/t))$
associated with the Prouhet-Thue-Morse sequence, we have observed that it
satisfies the Riccati differential equation (see [3], Proposition 2.4):

(R) $(ab(a+b)x)^{\prime}=(ab)^{\prime}(1+x^{2}).$

In fact, the continued fraction $\operatorname{CF}(\mathbf{p})$ (also
algebraic of degree $4$), associated with the period-doubling sequence
satisfies the same Riccati differential equation.

######  Proposition 4.3.

The continued fraction $\alpha_{\mathbf{p}}=\operatorname{CF}(\mathbf{p})$
satisfies the Riccati differential equation (R)

###### Proof.

We know from Theorem 1.1 that

(4.2)
$A\alpha_{\mathbf{p}}^{4}+B\alpha_{\mathbf{p}}^{3}+C\alpha_{\mathbf{p}}^{2}+1=0$

with

$A=ab+b^{2}+1,\quad B=ab(a+b),\quad C=ab.$

Hence, by straightforword derivation of (4.2), we get

$B\alpha_{\mathbf{p}}^{\prime}+B^{\prime}\alpha_{\mathbf{p}}=A^{\prime}\alpha_{\mathbf{p}}^{2}+C^{\prime}.$

Therefore, observing that $A^{\prime}=C^{\prime}=(ab)^{\prime}$, we obtain

$(ab(a+b)\alpha_{\mathbf{p}})^{\prime}=(B\alpha_{\mathbf{p}})^{\prime}=A^{\prime}\alpha_{\mathbf{p}}^{2}+C^{\prime}=(ab)^{\prime}(1+\alpha_{\mathbf{p}}^{2}).\qed$

Note that for the pair $(a,b)$=$(t,t+1)$, equation (R) reduces to (R0), the
one stated in Theorem 4.2. This was the way, taking in consideration
differential aspects for $\operatorname{CF}(\mathbf{t})$ and
$\operatorname{CF}(\mathbf{p})$, knowing the equation (R0) in the basic case,
and inspired by the particular case published in [4], that allowed us to guess
the coefficients for the algebraic equation satisfied by
$\operatorname{CF}(\mathbf{p})$.

We have observed that the two particularly simple sequences, the Prouhet-Thue-
Morse sequence and the period-doubling sequence, are close in many different
ways, and not only in their generation process. This leads to the intuition
that other $2$-automatic sequences, defined on an alphabet of two letters,
could have a representation as an algebraic continued fraction in
$\mathbb{F}_{2}((1/t))$ (not necessarily of degree $4$), as it happens for the
two famous ones considered up to now.

## References

  * [1] Jean-Paul Allouche and Jeffrey Shallit.  Automatic sequences. Theory, Applications, Generalizations.  Cambridge University Press, Cambridge, 2003.
  * [2] Leonard E. Baum and Melvin M. Sweet.  Badly approximable power series in characteristic $2$.  Ann. of Math. (2), 105(3):573–580, 1977.
  * [3] Yann Bugeaud and Guo-Niu Han.  The Thue-Morse continued fractions in characteristic $2$ are algebraic.  arXiv preprint arXiv:2203.02213, 2022.
  * [4] Yining Hu and Guo-Niu Han.  On the algebraicity of Thue-Morse and period-doubling continued fractions.  Acta Arithmetica, 2022 (to appear).
  * [5] Alain Lasjaunias.  A survey of diophantine approximation in fields of power series.  Monatshefte für Mathematik, 130(3):211–229, 2000.
  * [6] Alain Lasjaunias.  Continued fractions.  arXiv preprint arXiv:1711.11276, 2017.