The general solution to differential equations (99) for $\theta_{t}^{[1|m]}$
is analytically well-known, and for example using the method of ‘variation of
parameters’, can be written as follows,

$\hskip
28.45274pt\theta_{t}^{[1|m]}=\;e^{-\lambda^{I}t}\;\big{(}\;c^{[1|m]}\;+\int
dt\;e^{\lambda^{I}t}\;J_{t}^{[1|m]}\;\big{)}\;\;\;;\;\;\;\forall\;m\in\mathbb{N}.$
(101)

The coefficients $c^{[1|m]}$ are determined by imposing the following boundary
conditions,

$\;\;\;\;\;\theta_{t_{0}+a}^{[0]}\;=\;\theta_{t_{0}+a}^{[1]}\;\;\Longrightarrow\;\;\theta_{t_{0}+a}^{[1|m>1]}=0\;\;\;;\;\;\;h_{t_{0}+a}^{[0]}\;=\;\theta_{t_{0}+a}^{[1|1]}.$
(102)

The above patters goes on, almost identically, to the subsequent length-$a$
time-steps. That is, one obtains the same expressions for all fixed-step
$\theta$-orbit state-histories $\theta_{t}^{[1|n>1]}$. The only change that we
need to implement at each time-step is to re-set the initial quantum history,
being the state-history in the previous step, and re-compute the source
functions $J_{t}^{[n|m]}$ accordingly. To be clear, let us now consider the
$\theta$-orbit state-history in the second interval $[t_{0}+2a,t_{0}+3a]$,
$\theta_{t}^{[2]}$. The corresponding initial quantum history is similarly
marginalized by the $\epsilon$ parameter, but also receives the higher order
terms in $\epsilon$ in the following form,

$\begin{split}\theta_{t}^{[1]}\;\equiv\;\epsilon\;h_{t}^{[1]}\;\;\;;\;\;\;h_{t}^{[1]}\;=\;\theta_{t}^{[1|1]}\;+\;\sum_{m=1}^{\infty}\epsilon^{m}\;\theta_{t}^{[1|m+1]}.\end{split}$
(103)

Being so, the solutions $\theta_{t}^{[2|m]}$ will be found to be of the same
form as (104), with the only change being in the expressions of the source
functions $J_{t}^{[2|m]}$, in comparison with their counterparts (100), given
all the higher order terms in $\epsilon$ which enter into (103). In
conclusion, the fixed-step $\theta$-orbit state-histories are all in the
following form,

$\hskip
85.35826pt\theta_{t}^{[n|m]}=\;e^{-\lambda^{I}t}\;\big{(}\;c^{[n|m]}\;+\int
dt\;e^{\lambda^{I}t}\;J_{t}^{[n|m]}\;\big{)}\;\;\;;\;\;\;\forall\;(n,m)\in\mathbb{N}^{2}.$
(104)

The boundary conditions which fix the coefficients $c^{[n|m]}$, for every
$n>1$, are as follows,

$\;\;\;\;\;\theta_{t_{0}+na}^{[n-1]}\;=\;\theta_{t_{0}+na}^{[n]}\;\;\Longrightarrow\;\;\theta_{t_{0}+na}^{[n-1|m\geq
1]}=\theta_{t_{0}+na}^{[n|m\geq 1]}.$ (105)

####  5.2.3 Determining The Thresholds Along The $\theta-$Orbit

One intriguing characteristic of all Purely-QMM-UEs is that they are developed
only if their corresponding largest QMDs are larger than (or at least equal
to) some thresholds whose finite values depend on the defining couplings of
the QMM-H: $a\geq a^{\star}=a^{\star}(\vec{\lambda})$. This general aspect was
established in Subsection 3.3 and we will furthermore verify it in all
numerical solutions to the QMM Schrödinger equations presented in Section 6.
Now, one asks: what functions of the Hamiltonian couplings $\vec{\lambda}$ are
the thresholds $a^{\star}$? The fact is, given the mathematical significant
complexity of generic delay-differential QMM-Schrödinger equations even for
dimensionally-very-small quantum systems, the above dependencies are
generically too complicated to be analytically computable by the available
methods. But, there are rare cases where we can analytically determine
$a^{\star}(\vec{\lambda})$. The simplest example of these rare cases is
presented in this subsection, mainly in order to shed analytical and
quantitative light on the ‘Robust Non-Markovianity’ of Purely-QMM-UEs.

We determine in what follows the threshold $a^{\star}$ of the $(2,2)$ QMM-UE
of the one-qubit closed system along the $\theta$-orbit. In Subsection 5.2.1,
the profile-invariant solution to the $\phi$-orbit equation (85) was obtained.
The solution, given by (87,88), corresponds to the one-qubit unitary state-
histories with constant $\phi$ angles and linear-in-time $\theta$ angles.
According to (88), the slopes $\alpha$ of the linear-in-time $\theta$ angles
must be a real root of the algebraic equation,

$a\;\alpha+\arcsin\big{(}\frac{\alpha}{\;\lambda^{I}}\big{)}\;=\;0.$ (106)

Now, we analyze the _full_ Taylor-series expansion of the LHS of equation
(106) as a function of $\alpha$, around $\alpha=0$ as its trivial root. Notice
that, only the convergence of the Taylor series concerns us in this analysis
(rather than its perturbative utility), therefore, there is not any
small-$\alpha$ restriction involved in here. Equation (85) implies that every
$\theta$-orbit state-history must validate the dynamical inequality:
$|\dot{\theta}_{t}|<|\lambda^{I}|$. Remembering the convergence of the Taylor-
series expansion of $\arcsin(x)$ around any point on the interval $|x|\leq 1$,
we deduce that the Taylor series of the LHS of equation (106) is convergent
for any profile-invariant unitary state-history (87,88), resulting at the
following strict inequality,

$\begin{split}&a_{\,|_{\;\text{profile-invariant
solution}\;\eqref{ts5}}}\,=\,\frac{1}{-\lambda^{I}}\;\big{\\{}\,1+\sum_{n=1}^{\infty}\;\dfrac{(2n!)}{(2^{n}\;n!)^{2}\;(2n+1)}\;(\frac{\alpha}{\lambda^{I}})^{2n}\;\big{\\}}\;\;\;\;\text{
s.t.}\\\ &\;a_{\,|_{\;\text{profile-invariant
solution}\;\eqref{ts5}}}>\;\dfrac{1}{|\lambda^{I}|}\;\equiv\;a^{\star}_{\;|_{\;\text{profile-
invariant solution}\;(\ref{ts5},\ref{eq:alphaII})}}.\end{split}$ (107)

Result (107) is indeed satisfied by every nontrivial solution of (88) and
quantifies the Robust Non-Markovianity of the non-constant profile-invariant
solutions on the $\theta$-orbit. We remind that for positive values of
$\lambda^{I}$, the $\theta$-orbit (85) admits only trivial constant solutions,
such that the negativity of $\lambda^{I}$ is assumed in resulting the
inequality (107). Before going on, it is worth knowing quantitatively how the
Robust Non-Markovianity feature, manifested by the strict inequality (107), is
satisfied by the ‘highest-rate’ profile-invariant solution:
$|\dot{\theta}_{t}|=|\alpha|=-\lambda^{I}$. Either by looking into the
corresponding principal value of the RHS of the equation in the relation
(107), or directly from (88), the measure of the Robust Non-Markovianity of
the highest-rate solution is obtained to be:
$a=\frac{\pi}{2}\frac{1}{|\lambda^{I}|}>\frac{1}{|\lambda^{I}|}$. As proved,
result (87) places a strict lower bound on the largest QMDs of all the non-
constant profile-invariant solutions on the $\theta$-orbit.

Moreover, one sees that the validity of the above inequality extends much
beyond the solutions (87,88) for which we laid out the given proof. There are
two independent ways in which a much wider coverage of the lower bound (107)
can be confirmed and also specified. First, as proved in Subsection 5.2.1, the
profile-invariant solution is the late time asymptotics of all solutions on
the $\theta$-orbit. As such, with the largest QMD $a$ being a constant, its
late-time-asymptotics lower bound is necessarily imposed on the entire-time
solution. Second, any smooth and differentiable solution along the
$\theta$-orbit must become reducible in sufficiently time windows around every
generic point on $\mathbb{R}_{t}$ to the profile-invariant solution (87)
corresponding to a real root of (88). Given this requirement, the largest QMD
$a$ of every nontrivial differentiable solution of the $\theta$-orbit
dynamical equation (85) must admit $\frac{1}{|\lambda^{I}|}$ as a strict lower
bound. We conclude that, _a quantification of the Robust Non-Markovianity of
all $\theta$-orbit unitary state-histories of the one-qubit Purely-QMM-UE
under $H_{{\rm QMM}}^{(2|2)}(t)$_ is stated by the following lower bound,

$a^{*}_{\,|_{\;\text{all}\;\theta\text{-orbit
solutions}}}\;\geq\;a^{*}_{\,|_{\;\text{profile-invariant
solution}\;(\ref{ts5},\ref{eq:alphaII})}}\,=\;\dfrac{1}{|\lambda^{I}|}.$ (108)

####  5.2.4 The $\theta$-orbit Purely-QMM-UEs In Their Infancy

We have shown that the asymptotic behavior of all $\theta$-orbit state-
histories, developed in the late-times regime $\frac{t}{a}\gg 1$, is the
profile-invariant solution (87,88). Naturally, one now examines the period
when the Purely-QMM-UEs are in their ‘infancy’, $\frac{t}{a}\ll 1$, and asks
what analytic solutions the $\theta$-orbit state-histories approximate in
these very early times. We have shown that under the $(N,L)$ purely-QMM-Hs,
there always are finite thresholds ${a}^{\star}$, such that non-constant
wavefunctions are developed only if $a\geq{a}^{\star}$. Hence, an interesting
subset of infant unitary state-histories are constructible by taking
$a^{\star}$ thresholds which are sufficiently large and lead to the infancy
periods $\frac{t}{a}\leq\frac{t}{a^{\star}}\ll 1$ which are sufficiently long.
For $\theta$-orbit state-histories which obey (108), the sufficient largeness
of $a^{\star}$ is secured by taking sufficiently small values of the
interaction coupling $|\lambda^{I}|$. Moreover, we remember that the unitary
evolution under any $H^{(N,L)}_{{\rm QMM}}(t)$ is to be measured from the
outset of having the quantum-memory-pool filled completely: $t=t_{*}$. All in
all, if one introduces the dimensionless time variable:
$\tau\;\equiv\;|\lambda^{I}|\;(t-{t}_{\star})$, rewrites the $\theta$-orbit
equation (85) as a delay-differential equation in terms of $\tau$, and obtains
polynomial solutions $\hat{\theta}(\tau)$ to it as an order-by-order
perturbation in terms of $\tau\ll 1$, an interesting family of $\theta$-orbit
infant unitary state-histories are constructed. Finally, because (for constant
$a$,) ${t}_{\star}=t_{0}+a$, and that with no loss of generality one can
choose $t_{0}=-a$, we set $\tau\;=\;|\lambda^{I}|\;t$ in what follows.

We now implement the recipe prescribed above. First, the QMD measured in the
new time variable $\tau$ which parametrizes the post-$t^{\star}$ evolution in
$\frac{1}{|\lambda^{I}|}$ units, reads as:
$\hat{a}\equiv|\lambda^{I}|\;a\geq\frac{a}{a^{\star}}\geq 1$. Redefining Bloch
variable: $\hat{\theta}(\tau)\equiv\theta(-\lambda^{I}\;t)$, (85) takes the
form,

$\frac{d\hat{\theta}(\tau)}{d\tau}\;\equiv\;\dot{\hat{\theta}}(\tau)\;=\;-\sin\big{(}\;\hat{\theta}(-\hat{a}+\tau)-\hat{\theta}(\tau)\;\big{)}.$
(109)

We confine to the regime $\tau\ll 1$ which given (108) guarantees the
evolution infancy: $\frac{t}{a^{\star}}\leq\tau\ll 1$. Next, for solving the
above QMM delay-differential equation, we expand the time-shifted
$\hat{\theta}(-a+\tau)$, around the moment $\tau_{0}=-a$, as a Taylor series
for $\tau\ll 1$,

$\hat{\theta}(-\hat{a}\;+\;\tau)=\hat{\theta}(-\hat{a})\;+\;\hat{\mathcal{R}}_{0}^{(-\hat{a})}(\tau)\;=\;\hat{\theta}_{-\hat{a}}\;+\;\dot{\hat{\theta}}_{-\hat{a}}\;\tau\;+\;\dfrac{1}{2}\;\ddot{\hat{\theta}}_{-\hat{a}}\;\tau^{2}\;+\;\dfrac{1}{3!}\;\dddot{\hat{\theta}}_{-\hat{a}}\;\tau^{3}\;+\;\cdots\;$
(110)

Before working out the perturbative solutions as described, we ensure the
convergence of the Taylor series (110), by obtaining an upper bound on the
remainder $\hat{\mathcal{R}}_{0}^{(-\hat{a})}(\tau)$. As equation (109)
implies, during the entire QMM-UE $\tau\geq\tau_{*}=0$, one has:
$|\dot{\hat{\theta}}(\tau)|\leq 1$. Hence, using the Remainder Estimation
Theorem, we conclude that at every moment $\tau\in[0,\infty)$, the zeroth-
order absolute value of the remainder function
$\hat{\mathcal{R}}_{0}^{(-\hat{a})}(\tau)$ in $(\ref{def:exp:thetahat})$
respects the upper bound,

$|\hat{\mathcal{R}}_{0}^{(-\hat{a})}(\tau)|\;\leq\;\frac{\tau^{1}}{1!}\cdot{\rm
Max}\big{(}\;|\dot{\hat{\theta}}(\tau^{\prime}-a)|\;\big{)}_{\;\big{|}_{\tau^{\prime}\in\;[0,\tau]}}\;=\;\tau\;\cdot\;1\;=\;\tau.$
(111)

Hence, imposing $\tau\ll 1$ which secures the QMM-UE infancy, we obtain:
$\hat{\mathcal{R}}_{0}^{(-\hat{a})}(\tau)\ll\hat{1}$. Therefore, the
convergence of the Taylor series (110) is guaranteed.

Now, everything is set for us to combine (109) and (110), and solve the
following first-order differential equation for $\hat{\theta}(\tau)$,

$\begin{split}&\dot{\hat{\theta}}(\tau)\;=\;-\sin\big{(}\;\;\hat{\mathcal{R}}_{0}^{(-\hat{a})}(\tau)\;-\;\hat{\theta}(\tau)\;\;+\;\hat{\theta}_{-\hat{a}}\;\big{)},\\\
&\hat{\mathcal{R}}_{0}^{(-\hat{a})}(\tau)\;=\;\dot{\hat{\theta}}_{-\hat{a}}\;\tau\;+\;\dfrac{1}{2}\;\ddot{\hat{\theta}}_{-\hat{a}}\;\tau^{2}\;+\;\dfrac{1}{3!}\;\dddot{\hat{\theta}}_{-\hat{a}}\;\tau^{3}\;+\;\cdots\;\end{split}$
(112)

It is natural to impose polynomial-in-time profiles on the solutions
${\hat{\theta}}(\tau)$,

${\hat{\theta}}(\tau)\;=\;{\hat{\theta}}_{0}\;+\;{\hat{\theta}}_{1}\;\tau\;+\;{\hat{\theta}}_{2}\;\tau^{2}\;+\;{\hat{\theta}}_{3}\;\tau^{3}\;+\;\cdots$
(113)

Applying (113), we solve (112) in perturbation over $\tau$, order by order.
One notices that in the perturbative fixation of the solutions (113), the
$n$-th order derivatives of
$\hat{\theta}(\tau^{\prime})_{|_{\tau^{\prime}=-a}}$, namely
$({\hat{\theta}}_{-\hat{a}},\dot{\hat{\theta}}_{-\hat{a}},\ddot{\hat{\theta}}_{-\hat{a}},\dddot{\hat{\theta}}_{-\hat{a}},\cdots)$,
play the role of the ‘boundary-conditions’ constants, in terms of which the
polynomial coefficients $(\theta_{0},\theta_{1},\theta_{2},\theta_{3},\cdots)$
are to be found as the equation variables. Assuming the perfect
differentiability of the entire solution, these boundary constants
$\frac{d^{n}\hat{\theta}(\tau^{\prime})}{d\tau^{\prime\;n}}_{|_{\tau^{\prime}=-a}}$
must be read from the pre-$t_{*}$ solution developed by the Kicker
Hamiltonian. It is straightforward to obtain perturbative polynomial solutions
of differential equation (112) to arbitrary orders in the expansion (113). For
example, one obtains the first three variables
$(\hat{\theta_{1}},\hat{\theta_{2}},\hat{\theta_{3}})$ in terms of (the
arbitrary constant) $\hat{\theta}_{0}$ and
$({\hat{\theta}}_{-\hat{a}},\dot{\hat{\theta}}_{-\hat{a}},\ddot{\hat{\theta}}_{-\hat{a}})$
as follows,

$\begin{split}&\hat{\theta}_{1}=\sin(\hat{\theta}_{0}-\hat{\theta}_{-\hat{a}}),\\\
&\hat{\theta}_{2}=\frac{1}{2}\;\cos(\hat{\theta}_{0}-\hat{\theta}_{-\hat{a}})\;\cdot\;[\;\sin(\hat{\theta}_{0}-\hat{\theta}_{-\hat{a}})-\dot{\hat{\theta}}_{-\hat{a}}\;],\\\
&\hat{\theta}_{3}=\frac{1}{12}\;\big{\\{}\;\sin[\;3(\hat{\theta}_{0}-\hat{\theta}_{-\hat{a}})\;]-\sin(\hat{\theta}_{0}-\hat{\theta}_{-\hat{a}})\;+\\\
&\;\;\;\;\;+\;[\;1-3\;\cos[2(\hat{\theta}_{0}-\hat{\theta}_{-\hat{a}})\;]-2\;\sin(\hat{\theta}_{0}-\hat{\theta}_{-\hat{a}})\;\dot{\hat{\theta}}_{-\hat{a}}\;]\;\dot{\hat{\theta}}_{-\hat{a}}-2\;\cos(\hat{\theta}_{0}-\hat{\theta}_{-\hat{a}})\;\ddot{\hat{\theta}}_{-\hat{a}}\;\big{\\}}.\end{split}$
(114)

###  5.3 Purely-QMM Unitary State-Histories On The ‘$\phi$-Orbit’

The counterpart of the $\theta$-orbit is also a special orbit of the one-qubit
Purely-QMM-UE. This second special orbit, the ‘$\phi$-orbit’, is such that the
$\theta$ angle is kept constant, but one lets the $\phi$ angle evolve purely
based on its two interacting memories:
$\dot{\phi}_{t}=f(\phi_{t-a},\phi_{t})$. Although the defining constraint of
the $\phi$-orbit is the exact counterpart to that of the $\theta$-orbit, it
calls for an independent analysis in regard with its solutions. Indeed, the
ways in which the two Bloch angles appear in the dynamical equations of the
one-qubit closed system, in this case equations (4.3), are distinctive enough
to make the moduli spaces of the purely-QMM unitary state-histories developed
along these counterpart special orbits nonidentical. This analysis is
undertaken carefully in what follows.

Along the $\phi$-orbit which is constrained by $\theta_{t}=\theta_{t_{0}}$,
the functions $\mathcal{F}_{t-a,t}$ and $\mathcal{G}_{t-a,t}$ defined in (4.2)
are given by,

$\mathcal{F}_{t-a,t}=\;\sin(2\theta_{t_{0}})\;\sin^{2}(\frac{\phi_{t}-\phi_{t-a}}{2})\;\;\;;\;\;\;\mathcal{G}_{t-a,t}=\;-\sin(\theta_{t_{0}})\;\sin(\phi_{t}-\phi_{t-a}).$
(115)

Moreover, imposing the constancy of the $\theta_{t}$ angle on the general
expressions (56,57), the QM-TPF and its observable norm for a $\phi$-orbit
unitary state-history are found as,

$m_{t-a\;t}\;=\;\sin^{2}(\frac{\theta_{t_{0}}}{2})\;\big{(}\;e^{i(\phi_{t}-\phi_{t-a})}\;-1\big{)}\;+1\;\;\;;\;\;\;w_{t-a\;t}^{2}\;=\;-\sin^{2}(\theta_{t_{0}})\;\sin^{2}\big{(}\frac{\phi_{t}-\phi_{t-a}}{2}\big{)}\;+\;1.$
(116)

Now, we return to the one-qubit $(2,2)$-Purely-QMM delay-differential
equations (4.3), apply to them the $\phi$-orbit traits $\dot{\theta}_{t}=0$,
and (115), and work out all the implications. The outcomes depend on the
couplings $\lambda^{I}$ and $\hat{\mu}_{t-a}$ and the Bloch-angle constant
$\theta_{t_{0}}$. Let us present the final results as classified in the
following five distinct categories.

. _Generic $(\hat{\mu}_{t-a},\lambda^{I},\theta_{t_{0}})$ Triplet_
This is the generic category with the triplet $(\lambda^{I}$,
$\hat{\mu}_{t-a},\theta_{t_{0}})$ being only constrained by:
$\lambda^{I}\hat{\mu}_{t-a}\sin(\theta_{t_{0}})\neq 0$. Modulo the
arbitrariness in setting $\phi_{t_{0}}$, there is nothing but one unitary
state-history realizable in this category along the $\phi$-orbit, given by the
solution,

$\theta_{t}=\;\theta_{t_{0}}\;\;\;;\;\;\;\phi_{t}=\;\beta+\;\big{[}\;\frac{2\;\cos(\theta_{t_{0}})\;\big{(}\;(\lambda^{I})^{2}+\hat{\mu}_{t-a}^{2}\;\big{)}\;\hat{\mu}_{t-a}}{(\lambda^{I})^{2}\;\cos^{2}(\theta_{t_{0}})\;+\;\hat{\mu}_{t-a}^{2}}\;\big{]}\;t.$
(117)

. $\hat{\mu}_{t-a}$ and $\lambda^{I}$ _Non-Vanishing, Fine-Tuned_
$\theta_{t_{0}}:$ $\sin(\theta_{t_{0}})=0$
This class is very special. The $\theta$-orbit dynamical equations are solved
automatically, regardless of what function $\phi_{t}$ is. That is, one has a
whole arbitrary family of unitary state-histories realizable on the
$\phi$-orbit,

$\Big{(}\;\frac{\theta_{t}}{\pi}\;,\;\phi_{t}\;\Big{)}\;=\;\big{(}k\in\\{0,1\\}\;,\;f_{t}\;\big{)}.$
(118)

. $\lambda^{I}=0\;$, _Non-Vanishing_ $\hat{\mu}_{t-a}$, _Fine-Tuned_
$\theta_{t_{0}}$
If $\hat{\mu}_{t-a}$ is the only non-vanishing coupling of the $(2,2)$-Purely-
QMM-H, modulo one’s freedom in setting $\phi_{t_{0}}$, only a discrete family
of non-constant unitary state-histories is realizable along the $\phi$-orbit,
upon fine-tuning $\theta_{t_{0}}$. The solutions, being labeled by
$k\in\mathbb{Z}$, are given by,

$\theta_{t}=\;\arccos\;[\;\frac{(2\;k+1)\;\pi}{2\;a\;\hat{\mu}_{t-a}}\;]\;\;\;,\;\;\;\phi_{t}=\;\beta\;+\;\frac{(2\;k+1)\;\pi}{a}\;\;t.$
(119)

Notice that although (119) is from (25(b)), this is not a sub-class
. _Only One Non-Vanishing Coupling, Generic_ $\theta_{t_{0}}$
When either $\hat{\mu}_{t-a}$ or $\lambda^{I}$ is the only non-vanishing
coupling, and $\theta_{t_{0}}$ is not fine-tuned to the values specified in
the previous or the next category, the only $\phi$-orbit solutions are
constant-in-times state-histories,

$(\;\theta_{t}\;,\;\phi_{t}\;)\;=\;(\;\theta_{t_{0}}\;,\;\phi_{t_{0}}\;).$
(120)

. $\hat{\mu}_{t-a}=0\;$, _Non-Vanishing_ $\lambda^{I}$, _Fine-Tuned_
$\theta_{t_{0}}$
Finally, when the only coupling which is turned on in the $(2,2)$-Purely-QMM-H
given in (66) is $\lambda^{I}$, and furthermore one implements the fine-
tuning: $\cos(\theta_{t_{0}})=0$, the unitary state-histories along the
$\phi$-orbit are realized by solutions to the equation,

$\dot{\phi}_{t}\;=\;-\lambda^{I}\;\sin(\phi_{t}-\phi_{t-a}).$ (121)

Because the above delay-differential equation is mathematically identical to
(85), it is immediate to draw three conclusions as follows.

First, the only profile-invariant $\phi$-orbit unitary state-history in this
category is given by the following solution,

$\big{(}\;\frac{\theta_{t}}{\pi}\;,\;\phi_{t}\;\big{)}\;=\;\big{(}\;\frac{k\in\\{1,3\\}}{2}\;,\;\beta\;+\;\alpha\;t\;\big{)}\;\;\;;\;\;\;\alpha\;+\;\lambda^{I}\;\sin(a\;\alpha)=\;0\;\;\;;\;\;\;w^{2}_{t-a\;t}=\;\cos^{2}(\frac{a\;\alpha}{2}).$
(122)

Second, the profile-invariant solution (122) is the dynamical attractor of all
solutions to the $\phi$-orbit dynamical equation (121). This already secures
that the profile-invariant solution (122) is stable against the perturbations
which are confined to the $\phi$-orbit. Equation (121) develops a large family
of generic profile-varying unitary state-histories, some of which (modulo its
different angular identifications) are given by the numerical solutions
presented in Subsection 6 for its $\theta$-orbit counterpart.

Third, an analysis similar to what was done in Subsection 5.2.1 demonstrates
that the profile-invariant solution (121) is however unstable against the more
general family of perturbations with non-constant initial-history of
$(\delta\theta)_{t}$ is also turned on.

####  5.3.1 The $\phi$-Orbit Thresholds

Now, we move on to quantify analytically the Robust Non-Markovianity of the
unitary state-histories along the $\phi$-orbit. The procedure is very
straightforward and similar to its $\phi$-orbit counterparts. All it takes is
to explicitly reconstruct only one section of the entire analysis we had to do
in Subsection 5.3 in order to completely classify the $\phi$-orbit solutions.
As such, in what follows, it suffices to this relevant part of the
$\phi$-orbit impositions on the one-qubit dynamical system (4.3). Imposing the
constancy of the $\theta_{t}$ angle on the $\dot{\theta}_{t}$ equation in the
system (4.3) states that $\mathcal{F}_{t-a,t}$ and $\mathcal{G}_{t-a,t}$ must
be proportional:
$\mathcal{F}_{t-a,t}=\frac{\hat{\mu}_{t-a}}{\lambda^{I}}\;\mathcal{G}_{t-a,t}$.
Using this and the $\mathcal{G}_{t-a,t}$ in (115), the $\dot{\phi}_{t}$
equation in (4.3) yields,

$\;\;\;\;\;\;\;\;\;\;\;\dot{\phi_{t}}\;=-\;\big{(}\;\lambda^{I}+\,\frac{\hat{\mu}_{t-a}^{2}}{\lambda^{I}}\;\big{)}\;\sin\big{(}\phi_{t}-\phi_{t-a}\big{)}.$
(123)

Because the dynamical equation (123) is mathematically identical to equation
(85), doing the same analysis as above, the Robust Non-Markovinaity of all the
$\phi$-orbit unitary state-histories is immediately quantified likewise,

$a^{*}_{\,|_{\;\text{all}\;\phi\text{-orbit
solutions}}}\;\geq\;\;\frac{|\lambda^{I}|}{(\lambda^{I})^{2}\;+\;\hat{\mu}_{t-a}^{2}}.$
(124)

###  5.4 Hybrid Unitary Evolutions: General-Orbit Near-Markovian Solutions

We now look for one-qubit unitary state-histories made in the general context
of (1), when the dynamics of the qubit is generated by a linear combination of
both QMM and conventional Hamiltonians, namely ‘Hybrid Hamiltonians’. Let us
recall that by the term ‘conventional Hamiltonians’, we refer collectively to
all possible (local-in-time or nonlocal-in-time) Hamiltonians made only from
the building-block operators which, in their definitions, are entirely
independent of the system’s state-history, as the discipline commonly used in
formulating quantum systems. There are plenty of intriguing novel questions to
address about the observables of closed quantum systems whose unitary
evolutions are generated by the joining of conventional Hamiltonians and QMM-
Hs. Here, we address only one such questions: the simplest one. We work out
analytically the Near-Markovian limit of the hybrid-UE of the most elementary
quantum system, namely the one-qubit closed system we have been considering
from Section 4.

To be concrete, let us focus on the Hybrid-QMM-UE of the one-qubit closed
system in which the $(2,2)$-Purely-QMM-H is deformed by the conventional
Hamiltonian of a constant magnetic filed. Because the roles of the independent
couplings $\mu_{t-a}$ and $\lambda^{R}$ in the QMM-Schrödinger equations (4.3)
are additively combined in a single coupling $\hat{\mu}_{t-a}$, we can safely
restrict the monomial interactions of the two quantum memories to the second-
order couplings $(\lambda^{R},\lambda^{I})$. The hybrid Hamiltonian of the
qubit is given by,

$H^{\text{Hybrid}}_{{\text{QMM}}}(t)\;\equiv\;H^{(2,2)}_{{\text{QMM}}}(t)\;+H_{\text{C}}\;=\;\lambda^{R}\;\\{\rho_{t-a},\rho_{t}\\}\;+\;i\;\lambda^{I}\;[\rho_{t-a},\rho_{t}]\;+\;\frac{1}{2}\;\vec{B}.\vec{\sigma}.$
(125)

As we detailed in Subsection 4.3, and using the $(2,2)$-QMM magnetic field
$\vec{{\mathcal{B}}}^{(2,2)}_{\text{QMM}}(t)$ as given in (68), the QMM-
Schrödinger equation for the hybrid Hamiltonian (125) is,

$\dot{\vec{r}}_{t}\;=\big{[}\;\vec{{\mathcal{B}}}^{(2,2)}_{\text{QMM}}(t)\;+\;\vec{B}\;\big{]}\;\times\;\vec{r}_{t}.$
(126)

Let us assume, with no loss of generality, that the memory-free magnetic field
is coupled to only one of the Pauli matrices which we choose to be
$\sigma^{3}$, $\vec{B}=B_{z}\;\hat{z}$. Recalling the general-orbit equations
(4.3) expressed in terms of the functions (4.2), the one-qubit delay-
differential dynamical system for the Bloch angles $(\theta_{t},\phi_{t})$ is
given by,

$\displaystyle\dot{\theta}_{t}$
$\displaystyle=\lambda^{R}\;\mathcal{G}_{t-a,t}\;-\;\lambda^{I}\;\mathcal{F}_{t-a,t}$
$\displaystyle\sin\theta_{t}\;\dot{\phi}_{t}.$
$\displaystyle=\lambda^{R}\;\mathcal{F}_{t-a,t}\;+\;\lambda^{I}\;\mathcal{G}_{t-a,t}+B_{z}\;\sin\theta_{t}.$
(127)

As explained, we only focus on the near-Markovian limit of the above Hybrid-
QMM system which corresponds to arbitrarily-small values of the largest QMD.
In this limit, we solve the system (5.4) order by order in terms of the
parameter $a\ll 1$.

Being so, we introduce an infinite doublet-series of unknown angular functions
$(\theta^{(n)}_{t},\phi^{(n)}_{t})$, as the to-be-obtained solutions at the
Near-Markovian perturbation order: $n\leq N\to\infty$, such that the full
solutions to the dynamical system (5.4) are so constructed,

$\begin{split}&\theta_{t}\;\equiv\;\sum_{n=0}^{N\to\;\infty}\;a^{n}\;\theta^{(n)}_{t}\;\;\;;\;\;\;\phi_{t}\;\equiv\;\sum_{n=0}^{N\to\;\infty}\;a^{n}\;\phi^{(n)}_{t}.\end{split}$
(128)

Because (5.4) is a system of delay-differential equations involving functions
$(\mathcal{F}_{t-a,t},\mathcal{G}_{t-a,t})$, the Near-Markovian expansions of
the time-shifted functions $(\theta_{t-a},\phi_{t-a})$, being deduced from the
above expansions, also come into play in every chosen order of perturbation.
Applying the perturbative series of $(\dot{\theta}_{t},\dot{\phi}_{t})$ and
$(\mathcal{F}_{t-a,t},\mathcal{G}_{t-a,t})$ to the LHS and the RHS of
equations (5.4) respectively, we can analytically solve the system to
arbitrarily-high order $N$ in the Near-Markovian perturbation (LABEL:nme),
with the solutions $(\theta^{(n\leq N)}_{t},\phi^{(n\leq N)}_{t})$ being
polynomials of maximum order $n$ in time. For concreteness, we have obtained
the above Near-Markovian solution all the way to the order $N=6$. However, it
suffices to detail the Near-Markovian solution to second order in the
perturbation (LABEL:nme), as obtained according to the above procedure. The
resulted $\mathcal{F}_{t-a,t}$ and $\mathcal{G}_{t-a,t}$ are given by,

$\begin{split}&\mathcal{F}_{t-a,t}^{\;(\text{up to
O}[a^{3}])}\;=\;a\;\dot{\theta}^{(0)}_{t}\;+\;a^{2}\;\big{[}\;\dot{\theta}^{(1)}_{t}+\frac{1}{4}\;\sin(2\theta^{(0)}_{t})\;(\dot{\phi}^{(0)}_{t})^{2}-\;\frac{1}{2}\;\ddot{\theta}^{(0)}_{t}\;\big{]},\\\
&\mathcal{G}_{t-a,t}^{\;(\text{up to
O}[a^{3}])}\;=-a\;\sin(\theta^{(0)}_{t})\;\dot{\phi}^{(0)}_{t}\;+\\\ &\;\hskip
62.04112pt+\;a^{2}\;\big{[}\;\cos(\theta^{(0)}_{t})\;(\dot{\theta}^{(0)}_{t}\;-\;\theta^{(1)}_{t})\;\;\dot{\phi}^{(0)}_{t}+\frac{1}{2}\;\sin(\theta^{(0)}_{t})\;\big{(}\;\ddot{\phi}^{(0)}_{t}-2\;\dot{\phi}^{(1)}_{t}\big{)}\;\big{]}.\end{split}$
(129)

Given (129), to second order in the perturbation, the delay-differential
system (5.4) is turned to the following chain of ordinary differential
equations for the six unknown contributing functions $\theta_{t}^{(0,1,2)}$
and $\phi_{t}^{(0,1,2)}$,

$\begin{split}&\dot{\theta}^{(0)}_{t}=0\;\;\;;\;\;\;B_{z}-\dot{\phi}^{(0)}_{t}=0,\\\
&\dot{\theta}^{(1)}_{t}+\lambda^{I}\dot{\theta}^{(0)}_{t}+\lambda^{R}\sin(\theta^{(0)}_{t})\dot{\phi}^{(0)}_{t}=0\;\;\;;\;\;\;\sin(\theta^{(0)}_{t})\;\dot{\phi}^{(1)}_{t}+\lambda^{I}\sin(\theta^{(0)}_{t})\;\dot{\phi}^{(0)}_{t}-\lambda^{R}\;\dot{\theta}^{(0)}_{t}=0,\\\
&\dot{\theta}^{(2)}_{t}+\lambda^{I}\;\big{[}\;\dot{\theta}^{(1)}_{t}+\frac{1}{4}\sin(2\theta^{(0)}_{t})(\dot{\phi}^{(0)}_{t})^{2}-\frac{1}{2}\ddot{\theta}^{(0)}_{t}\;\big{]}\;+\;\\\
&\hskip
16.92941pt+\lambda^{R}\;\big{[}\;\cos(\theta^{(0)}_{t})\theta^{(1)}_{t}\dot{\phi}^{(0)}_{t}-\cos(\theta^{(0)}_{t})\dot{\theta}^{(0)}_{t}\dot{\phi}^{(0)}_{t}+\sin(\theta^{(0)}_{t})\dot{\phi}^{(1)}_{t}-\frac{1}{2}\sin(\theta^{(0)}_{t})\ddot{\phi}^{(0)}_{t}\;\big{]}=0,\\\
&\sin(\theta^{(0)}_{t})\;\dot{\phi}^{(2)}_{t}+\lambda^{I}\;\big{[}\;\sin(\theta^{(0)}_{t})\dot{\phi}^{(1)}_{t}-\cos(\theta^{(0)}_{t})\dot{\theta}^{(0)}_{t}\dot{\phi}^{(0)}_{t}-\frac{1}{2}\sin(\theta^{(0)}_{t})\ddot{\phi}^{(0)}_{t}\;\big{]}+\\\
&\hskip
50.83212pt\;\;\;\;+\lambda^{R}\;\big{[}\;\cot(\theta^{(0)}_{t})\theta^{(1)}_{t}\dot{\theta}^{(0)}_{t}-\dot{\theta}^{(1)}_{t}-\frac{1}{4}\sin(2\theta^{(0)}_{t})(\dot{\phi}^{(0)}_{t})^{2}+\frac{1}{2}\ddot{\theta}^{(0)}_{t}\;\big{]}=0.\end{split}$
(130)

The general solutions to equations (130), modulo angular identifications and
upon ignoring the redundant constant parameters in the higher order
contributions, are given by the following polynomial functions of the time
variable,

$\begin{split}&\theta^{(0)}_{t}=\;c_{1}\;\;\;;\;\;\;\phi^{(0)}_{t}=\;B_{z}\;t+c_{2},\\\
&\theta^{(1)}_{t}=-\sin(c_{1})\;\lambda^{R}\;B_{z}\;t\;\;\;;\;\;\;\phi^{(1)}_{t}=\;-\;\lambda^{I}\;B_{z}\;t,\\\
&\theta^{(2)}_{t}\;=\;\frac{1}{4}\;\sin(2\;c_{1})\;(\lambda^{R})^{2}\;B_{z}^{2}\;t^{2}\;+\;\frac{1}{4}\;\big{[}\;8\;\sin(c_{1})\;\lambda^{I}\;\lambda^{R}-\sin(2\;c_{1})\;\lambda^{I}\;B_{z}\;\big{]}\;B_{z}\;t,\\\
&\phi^{(2)}_{t}=\frac{1}{2}\;\big{[}\;B_{z}\;\lambda_{R}\;\cos(c_{1})\;+\;2\;\big{(}(\lambda^{I})^{2}-(\lambda^{R})^{2}\big{)}\;\big{]}\;B_{z}\;t.\end{split}$
(131)

We remark that the polynomial structures of the corresponding higher order
solutions, which we have obtained up to $\text{O}[a^{7}]$, remain as observed
in (131). For example, $\phi^{(6)}_{t}$ and $\theta^{(6)}_{t}$ are
respectively found to be likewise-simple polynomials of orders five and six.
Moreover, we note that the complete perturbative solution (LABEL:nme), being
truncated to arbitrary order $N$ in the perturbation, is uniquely known by
determining the constant doublet $(c_{1},c_{2})$ appearing in (131). This
doublet is in turn fixed by choosing the initial value of
$(\theta_{t},\phi_{t})$ at $t=t_{0}+a$, namely at the outset of the hybrid-
QMM-UE.

Let us end this subsection with highlighting the most important message we
learn from the above investigation which comes in contrast with the
corresponding aspect of purely-QMM-UE analyzed in Subsection 3.3. The lesson
is that the hybrid-QMM-UE breaks free from the lower bound of Robust Non-
Markovianity: its Near-Markovian limit is dynamically non-trivial. In other
worlds, the threshold of the largest-QMD $a^{\star}$ vanishes identically for
generic hybrid-QMM-UE, such that no matter how infinitesimal the finite value
of $a$ is set to be, solutions to the hybrid-QMM-Schrödinger equations realize
unitary state-histories (2) with non-constant $\rho(t^{\prime})$. On the other
hand, as we observe, upon setting the $|\vec{B}|$ in (125) zero which takes us
back to purely-QMM-UE, the solutions (131) and all their high-order
counterparts become constants-in-time.

##  6 Mapping The Phase Diagrams of One-Qubit Pure States Evolving Under
Purely-QMM-Hs: Numerical Investigations

###  6.1 Generalities About The Numerical Analyses And The Results

The unitary state-histories which correspond to QMM-UEs are realized by
solutions to QMM Schrödinger equations (27) or (in their alternative
equivalent form) (24). Mathematically, they are highly-nonlinear delay-
differential equations with complicated forms. Thus, exact analytical
solutions to these equations, or even ‘good’ approximate analytic solutions to
them which survive sufficiently-long times in units of the largest QMD, are
rarely attainable using the known methods. This is a mathematically-known fact
which we have concretely examined even in some simplest examples of the most
elementary quantum setting: the closed system of only one qubit which, driven
by its interacting quantum memories, evolves unitarily.

Therefore, we resort in the present section to (a couple of) numerical
approaches which enable us to obtain generic unitarily-evolving wavefunction
solutions to the one-qubit QMM Schrödinger equations developed by two or three
interacting quantum memories. We numerically construct these generic solutions
in a sufficiently-wide range of all the control parameters which
quantitatively configure the setting: the QMM-H couplings, the one or the two
QMM distance(s), and magnetic fields of the Kicker Hamiltonians. Carrying out
such a broad numerical analysis, we succeed in mapping the complete phase
diagram of the one-qubit unitary pure-state-histories generated by the least
order polynomial Hamiltonians made by two interlinked quantum states chosen as
$(\rho_{t-a},\rho_{t})$. The specific cases, analyzed in their increasing
complexities, are given as follows: the $(1,1)$, $(2,2)$ and finally $(2,3)$
QMM-H. Moreover, we implement one of these numerical methods in exploring the
example of a one-qubit closed system whose unitary evolution is generated by
the simplest interactions of a triplet of its quantum memories: the $(3,3)$
QMM-UE. Mapping the phase diagram of this example is much more involved,
however we are able to classify’ the qualitatively-more-significant phases of
the $(3,3)$ one-qubit QMM-UE. In each one of the above cases, we choose one of
the quantum memories to be the present moment state of the system, and also
take the QMDs to be constants. As such, we numerically solve equations (64),
(4.3), (LABEL:SE23POLAR) and (82).

The first numerical method which we employ relies on Mathematica’s developed
routine for numerically solving systems of coupled differential and delay-
differential equations: NDSolve. The second approach is however an independent
numerical solver. Here we remark briefly on the implementation and the
specifics of both the approaches.

Let us begin with explaining the algorithms of the Mathematica codes we have
used. The first remark is that the NDSolve routine has both a ‘standard’ and a
‘generalized’ version, and that, it is the _standard version_ which we utilize
throughout Section 6, except than Subsection 6.3.2 where, given its purpose,
it suffices to use the alternative. The standard version construct numerical
solutions to differential equations, using the given values of the unknown
functions at an initial time, as the boundary condition. Similarly, the
generalized version constructs numerical solutions to delay-differential
equations, defined with a constant delay time, upon implementing the method of
steps and using the given initial histories of the unknown functions as input
of the procedure. As remarked above, the numerical code we have developed to
construct the QMM-UEs of the one-qubit closed system uses the standard NDSolve
routine, in a way that the method of steps is automatically, but only
implicitly, realized by running it.

The code considers a single qubit whose density operator at $t_{0}=0$ is the
arbitrarily chosen pure state identified with initial Bloch coordinates:
$\ket{\rho_{0}}\longleftrightarrow(\theta_{0},\phi_{0})$, and then numerically
the unitary time evolution of the state under the two-part Hamiltonian (10).
Unitary evolution preserves the purity of states. Therefore, the time-
dependent state of the qubit remains to be uniquely coordinated by the two
Bloch angles which, modulo their defining angular identifications and based on
the two-part Hamiltonian (10), must be both continuous functions of time:
$(\theta_{t},\phi_{t})\longleftrightarrow\ket{\rho_{t}}$. In accordance with
the total Hamiltonian (10), the code divides the numerical construction of the
time evolution of the qubit state into two successive distinctive parts. In
the first part, which begins at $t_{0}=0$ and continues for a time duration of
length $\Delta^{(t)}_{1}=a$, the quantum memory pool is filled. In this part,
the qubit-state time evolution is Markovian and is generated by a ‘Kicker
Hamiltonian’ which we set to be that of constant magnetic field:
$\vec{B}=B_{y}\;\hat{y}$. Immediately after having the quantum-memory pool
filled, the unitary time evolution of the qubit state becomes Non-Markovian,
generated by the $(N,L)$-QMM-H under consideration. This is the second part
which begins from $t=a$ and goes on ‘forever’, numerically meaning up to a
sufficiently-long time $t_{\text{end}}\equiv K\;a$, with $K\gg 1$ being a few
thousands in most examples.

This straightforward code gives us enough flexibility to range over
sufficiently-wide spectra of all the control parameters of the QMM-UE of the
qubit: the initial Bloch angles, the Kicker magnetic field $B_{y}$, the QMDs,
and all the couplings of the QMM-H. As such, we have been able to efficiently
explore the novel phases of the one-qubit $(N,L)$-Purely-QMM-UEs with $N\leq
3$ and $L\leq 3$. In fact, in the examples with $N<3$, we have done the
complete classification of the phases of one-qubit QMM-UEs.

Moreover, we have developed an independent numerical solver, in order to cross
check the reliability of the above code, and also have a more direct control
of the numerical solutions. It consists of a modified Runge-Kutta-4 algorithm
with discrete fixed time step (see e.g. [66]). To incorporate the QMM
character of the evolution, the subroutines implementing a single Runge-Kutta
step were modified so as to use the information of the dependent variables not
only at the present time step, but also at an older time at fixed delay. We
have tested both of these approaches in a wide range of configurations,
especially with regard to the QMDs, and all the defining couplings of the QMM-
Hs. We have found in all cases that, within acceptable numerical
uncertainties, the two codes give same results, leading to the same phase
diagrams of the one-qubit QMM-UEs.

Now, we begin to work out numerical constructions of the state-histories
(LABEL:hps) which are realized by the unitarily-evolving wavefunctions of a
qubit whose quantum memories interact and compose each one of the four
specific choices of the $(N,L)$-Purely-QMM-Hs (14) detailed in Section 4.

###  6.2  The Phase Diagram of One-Qubit Wavefunctions Evolving Under The
(1,1) Purely-QMM Hamiltonians

Let us begin with solving numerically the simplest setting, that is, the
unitarily-evolving wavefunctions of the one-qubit closed system generated by
the QMM-H of the type $(1,1)$. As we see in (15), the Hamiltonian at the
present moment consists of the pure density operator of the qubit at a
uniquely-specified moment in the past: that which is of a temporal distance
$\Delta_{t}^{(t)}=a$ from the present moment. Moreover, because of the assumed
purity of the unitary state-history (LABEL:hps), the Hamiltonian admits only
one control parameter, being the real coupling $\mu$ introduced in (15).
Although, in $(1,1)$ theory, one has a QMM-H which simply is a linear monomial
of the system’s quantum memory, the theory is actually nonlinear, interacting
and also non-trivial dynamically. One really has a physically-interacting
theory because the QMM-H, although being linear in quantum memory, imposes
observable interaction-type correlations between the two wavefunctions which
participate in the dynamics. We can understand this fact most easily from the
corresponding nonlocal-in-time and nonlinear QMM Schrödinger equation (62), or
equivalently (63), according to which $\ket{\psi_{t-a}}$ and $\ket{\psi_{t}}$
are made to interact due to the QM-TPF and its observable norm (56) as
explicitly given in (57) and (56). Therefore, the phase diagram of the unitary
state-histories which the $(1,1)$ theory develops is non-trivial and deserves
its independent numerical exploration, as we present here in what follows.

We now present a careful classification of the phases by scanning over the
profiles of the one-qubit-wavefunction dynamics, and the norm of its QMM-TPF,
when $(\mu,a)$, the only coupling of the $(1,1)$ QMM-H and the QMD, are chosen
freely in $\mathbb{R}\times\mathbb{R}^{+}$. For sufficiently-many choices of
the real coupling, the QMD, and all the other control parameters, the system
(64,4.2) is solved numerically for the Bloch angles $(\theta_{t},\phi_{t})$,
applying the methods of Subsection 6.1. Briefing the procedure, the
arbitrarily-chosen values of
$(\mu,a;\theta_{0},\phi_{0};B^{y}_{\text{Kicker}})$, and the numerically-
filled quantum-memory pool of the two Bloch angles in the duration $t\in[0,a]$
as developed by the Kicker Hamiltonian, are fed to this nonlinear nonlocal-in-
time infinite dimensional dynamical system as the input data. Henceforth, we
construct the input-resulted numerical solution to (64), beginning from the
outset of the QMM-UE, that is $t=a$, up to a sufficiently-long time t =
$t_{\text{completion}}$.

The complete solutions and the phase diagram resulting from the above
procedure can be demonstrated optimally in three plots as identified now.
Obviously, the outputs of solving the system (64,4.2) are two dynamical plots:
one for $\theta_{t}$ and one for $\phi_{t}$. But, the phases of the purely-
QMM-UEs of the one-qubit closed system can be shown and understood in a more
clear way, if (instead) two alternative plots are presented, together with a
supplementary third one. As for the alternative independent plots, we choose
two quantities which straightly represent the one-qubit wavefunctions (48), so
as to have the advantage of being more directly related to the physical
observables. One-qubit wavefunctions $\ket{\Psi_{t}}$ have three real valued
independent constituents in their vector parametrization (48): the up
component, $\Psi_{t}^{\text{up}}\equiv\cos(\frac{\theta_{t}}{2})$, the
imaginary part of the down component:
$\text{Im}[\Psi_{t}^{\text{down}}]\equiv\sin(\frac{\theta_{t}}{2})\sin(\phi_{t})$,
and its counterpart, the real part of the down component:
$\text{Re}[\Psi_{t}^{\text{down}}]\equiv\sin(\frac{\theta_{t}}{2})\cos(\phi_{t})$.
Because by unitarity, these three quantities satisfy:
$|\braket{\Psi_{t}}{\Psi_{t}}|^{2}=1$, there are two independent real valued
constituents among them, with the third one being deducible. We have proved,
and also specifically checked in Subsections 3.2 and 2, that QMM-Hs generate
unitary time evolutions. We choose the doublet
$(\text{Re}[\Psi_{t}^{\text{down}}],\Psi_{t}^{\text{up}})$ as the two
alternative independent quantities which represent the QMM-UE of the one-qubit
closed system. Moreover, the dynamics of the norm of the QM-TPF is the natural
observable which serves as the discerning signature of the phases of the QMM-
UE. Therefore, although being deducible from the other two quantities, it is
crucial to present the plot of the QM-TPF observable for every solution. On
the whole, an optimal representation of the distinct phases of the
wavefunction solutions to _all_ one-qubit QMM Schrödinger equations marked
with delay spectrum $(\Delta^{(t)}_{2},\Delta^{(t)}_{1})=(0,a)$ is showing the
numerical results in the figures each of which containing these three plots:
one for $w_{t-a\;a}^{2}$, one for $\text{Re}[\Psi_{t}^{\text{down}}]$ and one
for $\Psi_{t}^{\text{up}}$. Indeed, from now on, for every numerical solution
which represents one distinct phase of a one-qubit QMM-UE, we present the
plots of these wavefunction quantities.

Now, we report and show the results of our scanning of the phase diagram of
$(1,1)$ QMM-UE of the one-qubit closed system. For utter clarity, let us
emphasize what we mean by ‘the phase diagram’ of the QMM one-qubit
wavefunctions for each one choice of $(N,L)$: _the classification of the
defining dynamical profiles of the ‘post-early-times’ solutions to the
associated QMM Schrödinger equations_. In the case at hand, the QMM
Schrödinger equations correspond to the delay-differential equations (64,4.2).

We begin with a specially remarkable phase of the one-qubit (1,1) Purely-QMM-
UE whose representative is the solution shown in Fig.1. The dynamical profiles
of the triple $(\text{Re}[\Psi_{t}^{\text{down}}],$
$\Psi_{t}^{\text{up}},w_{t-a\;a}^{2})$ that uniquely identify this phase are
shown in the four, 3 + 1 = 4, plots of Fig.1 which, to enhance visibility,
correspond to two sample temporal sub-zones, $[100a,138a]$ and $[5a,1250a]$,
of the entire numerical solution: $t\in[0,2000a]$. As one sees, _in this
phase, the post-early-times wavefunction of the one-qubit closed system
undergoes everlasting ‘regular’ oscillations which are large and synchronized
in such a way that the one-qubit QM-TP observable sticks effectively to a
constant value_. All numerical solutions confirm that, in this phase, the
dynamics of $w_{t-a\;t}$ is suppressed down to extremely narrow strips which
are made by tiny fluctuations of almost-constant width around constant values,
while the two independent components of the one-qubit wavefunction experience
synchronized, large, regular oscillations. To distinguish it from the
forthcoming distinct classes of all the solutions to the dynamical system
(64,4.2), this phase of unitary state-histories is to be called the phase ‘A’
of the one-qubit (1,1) QMM-UE.

Figure 1: Numerical solution of the QMM-UE-System (64,4.2) with parameters:
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\mu\leftrightarrow\hat{\mu}_{t-a}|a;t_{\text{completion}}$)=($7\;;1.001,0.089\;|\;1.85\;|\;1.73;\;2000a$).
‘Phase A’ of the one-qubit $(1,1)$ QMM-UE $\cong$ ‘Phase Two’ of the one-qubit
(2,2) QMM-UE.

The numerical solution presented in Fig.2 shows another remarkable phase of
the one-qubit (1,1) Purely-QMM-UE: the phase ‘B’. As the representative
solution of Fig.2 shows in a sample time zone of the entire numerical solution
specified in the caption, the one-qubit wavefunction and its QM-TPF observable
form _large oscillations whose dynamical profiles are notably structured_.
Specially, the hallmark of the phase B is the ‘pronounced’ structure in the
oscillations of the QM-TPF: the oscillations are _formed in structural units
of ‘bump modules’ which are marked with robustly-ordered well-gapped local
extrema_. In particular, in the third plot of Fig.2, one sees that
$w_{t-a\;a}^{2}$ picks up four ordered extrema over each bump module: the
global maximum $\sim 0.79$, one local maximum $\sim 0.71$, one local minimum
$\sim 0.4$ and the global minimum $\sim 0.17$. To observe how robustly these
defining structures are preserved across different time scales of the
corresponding QMM-UE, Fig.3 presents the two plots for $w_{t-a\;a}^{2}$ and
$\Psi_{t}^{\text{up}}$ of the representative solution shown in Fig.2, but in
the time zone shifted by $+1500a$.

(a)

(b)

(c)

Figure 2: Numerical solution of the QMM-UE-System (64,4.2) with parameters:
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\mu\leftrightarrow\hat{\mu}_{t-a}|a;t_{\text{completion}}$)=($7\;;1.001,0.089\;|\;2.96\;|\;1.73;\;2000a$).
‘Phase B’ of the one-qubit $(1,1)$ QMM-UE $\cong$ ‘Phase Four’ of the one-
qubit (2,2) QMM-UE.

(a)

(b)

Figure 3: Numerical solution of the QMM-UE-System (64,4.2) with parameters:
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\mu\leftrightarrow\hat{\mu}_{t-a}|a;t_{\text{completion}}$)=($7\;;1.001,0.089\;|\;2.96\;|\;1.73;\;2000a$).
‘Phase B’ of the one-qubit $(1,1)$ QMM-UE $\cong$ Phase Four’ of the one-qubit
(2,2) QMM-UE.

Now, we come to the phase ‘C’ as represented by the specific solution shown in
the three plots for $\text{Re}[\Psi_{t}^{\text{down}}],\Psi_{t}^{\text{up}}$
and $w_{t-a\;a}^{2}$ in Fig.4. As in the previous cases, it suffices to show
the plots in a sample zone $t\in[165\;a,178\;a]$ of the total time span of the
numerical solution, specified in the caption together with other control
parameters, merely to enhance the fine structure visibility of the solution
profile. As we see in the three plots of the Fig.4, specifically that for
$w_{t-a\;a}^{2}$, the first phase of the one-qubit $(1,1)$ Purely-QMM-UE is
characterized by _irregular oscillations_ of the wavefunction and its QM-TPF
observable. Indeed, all the numerical simulations performed in sufficiently
wide temporal spans in this phase region show that, _these unstructured
oscillations of this phase go on likewise everlastingly_.

(a)

(b)

(c)

Figure 4: Numerical solution of the QMM-UE-System (64,4.2) with parameters:
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\mu\leftrightarrow\hat{\mu}_{t-a}|a;t_{\text{completion}}$)=($7;1.001,0.089\;|\;5.124\;|\;1.73;\;2000a$).
‘Phase C’ of the One-qubit $(1,1)$ QMM-UE $\cong$ ‘Phase Three’ of the one-
qubit (2,2) QMM-UE.

Indeed, the fact that the $(1,1)$ QMM-UEs form a special sub-family of their
$(2,2)$ counterparts is a clear statement, which is true more generally: given
definition (14), the phase diagram of any system under $(N,L)$-QMM-UE must be
consistently locatable inside the phase-diagram of that system under
$(N+1,L)$-QMM-UE. The other way around, one sensibly regards the phases of the
$(N,L)$ system as _‘the building blocks’_ of the ‘compositionally-reducible
subsets’ of the phases of the higher-$(N,L)$ systems. Henceforth, for clarity
in both identifying and understanding the phases of all one-qubit $(N\leq
3,L\leq 3)$ QMM-UEs, we employ _the five phases of $(2,2)$ one-qubit closed
system_, specified in the next subsection, as a convenient ‘reference system’.
Having it as a useful reference system at hand, we now summarize the statement
of one-qubit (1,1) QMM-UE.

One indeed validates, by constructing sufficiently-many numerical solutions to
(64,4.2) upon varying $(\mu,a)$ and other control parameters in sufficiently-
wide ranges, that the phase diagram of the (1,1) QMM-UE of the one-qubit
closed system consists of the above _three phases: A, B and C_. Moreover,
insensitive to the sign of the real coupling $\mu$, these three phases take
place in the alphabetic order: $A\to B\to C$, if one increases
$a|\mu|\in\mathbb{R}^{+}$ monotonically. Finally, the phases A, B and C are
equivalent with the three (to be identified) compositionally-reducible phases
of the one-qubit (2,2) QMMUE, _the phases two, four and three_ , respectively.
Accordingly, the representative solutions shown in Figs.1,2,4, correspond
respectively to the phases two, four and three of the one-qubit (2,2)-QMMUE.

As observed above, the particular QMM-UEs generated by (60) develop three
classes of one-qubit state-histories with _significant profile variability_.
Certainly, the one-qubit $(1,1)$ system is _the most elementary quantum system
featuring the simplest QMM-UE_. In the landscape of all possibilities, it is
the most minimalistic setting: A two-state degree of freedom is evolving
unitarily as generated by momentary Hamiltonians which are essentially nothing
but one of its quantum memories in its constant-distant past. To us, the fact
that the most elementary implementation of QMM-UE leads to the above phase
diagram, consisting of three phases with such level of distinctiveness in
their defining qualities, is remarkable.

###  6.3 The Phase Diagram of One-Qubit Wavefunctions Evolving Under The (2,2)
Purely-QMM Hamiltonians

####  6.3.1 The General-Orbit Phase Diagram

Now we turn to Unitary Non-Markovian dynamical system (4.2,4.3) and construct
its numerical solutions, ranging over the spectra of its control parameters,
to explore and make a complete map of its phase diagram. The novel dynamical
system (4.2,4.3) is the nonlocal-in-time nonlinear Schrödinger equation for
the (2,2)-Purely-QMM-UE which is given in (32,33), applied to the closed
system of one single qubit to become (69), and re-written as the Non-Markovian
unitary time evolution of the one-qubit Bloch polar coordinates:
$(\theta_{t},\phi_{t})$. To conclude, the delay-differential equations (69)
describe how pure states of a qubit unitarily evolve under the first-and-
second degree monomials made of the chosen pair of its quantum memories:
$\rho_{t-a}$ and $\rho_{t}$.

To construct solutions to the delay-differential equations (4.2,4.3), we
follow the procedures detailed in Subsection 6.1. Moreover, we check and
indeed confirm that, a) there always is a finite threshold
$a^{\star}(\hat{\mu}_{t-a},\lambda^{I})$ as the strict lower bound of the
largest QMD of any dynamical solution, b) if $\lambda^{I}$ is the only QMM-H
coupling, the numerical threshold and the analytical determination (108) are
consistent. To characterize all one-qubit phases optimally, the numerical
solutions are shown in three (‘two + one’) plots:
$(\text{Re}[\Psi_{t}^{\text{down}}],\Psi_{t}^{\text{up}})$, supplemented with
QMM signature of the corresponding phase of the system, being
$w_{t-a\;a}^{2}$. Ranging over sufficiently-wide spectra of all the control
parameters:
$a,\theta_{0},\phi_{0},B^{y}_{\text{Kicker}},\lambda^{I},\hat{\mu}_{t-a},t_{\text{completion}}$,
the complete map of the phase diagram made by solutions to (4.2,4.3) is
presented in what follows.

One firstly sees that, upon the following mapping of the associated couplings:
$\hat{\mu}_{t-a}\leftrightarrow\mu$, the $(2,2)$ one-qubit QMM-UE becomes
mathematically equivalent with one-qubit $(1,1)$ system: QMM-Hs (60,66), and
their QMM Schrödinger equations (64,4.3) show this fact. Accordingly, by one’s
turning on a finite value of $|\lambda^{I}|$ which relative to
$|\hat{\mu}_{t-a}|$ is sufficiently small, no sudden change is to be seen in
the wavefunction behavior. Hence, imposing
$\frac{|\lambda^{I}|}{|\hat{\mu}_{t-a}|}\ll 1$, every one-qubit (2,2) phase
must fall into one of the (1,1) phases: A, B, C. This fact can be also checked
by numerical investigations when $|\hat{\mu}_{t-a}|$ becomes, typically by
something like one order of magnitude, dominant: by turning on sufficiently
small values of $|\lambda^{I}|$, the corresponding phase of the solution stays
invariant, but it receives sub-structures which gradually become richer. Fig.5
shows an example of this fact: phase 4 $\sim B$ same but the effect of
$|\lambda^{I}|$ multiplying the structured pattern extrema: in the left, only
one minimum between every, but on the right, one global minimum and 8 local
extrema between every subsequent pair of global maxima.

As explained above, in the regions of the coupling space corresponding to
$\frac{|\lambda^{I}|}{|\mu_{t-a}+\lambda^{R}|}\ll 1$, the $(2,2)$ system
reproduces the three phases of the $(1,1)$ system as prescribed in the
previous subsection. Therefore, we focus on describing the behavioral phases
of the $(2,2)$ one-qubit QMM-UE in the ‘non-redundant’ regions of the
$(\mu,\lambda^{R},\lambda^{I})$-coupling space in which the effective
$\mu$-coupling, $|\hat{\mu}_{t-a}|=|\mu_{t-a}+\lambda^{R}|$, is either
sufficiently small or its strength is comparable with that of $|\lambda^{I}|$
but does not overwhelm it too much. We now briefly describe the _‘five’_
behavioral phases which are observed, carefully examined and validated upon
doing many simulations in our exhaustive numerical analysis.

(a)

(b)

Figure 5: QM-TPFs of the numerical solutions of (64,4.2) with
$(\lambda^{I},\mu)=(0,3.202)$ in the left, and (4.2,4.3) with
$(\lambda^{I},\hat{\mu}_{t-a})=(-0.35,3.202)$ in the right, with the other
parameters being given by
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|a;t_{\text{completion}}$)=($2.75;1.001,0.089|1.73;2600\;a$).

_PHASE 1: The Fixed-Point-Attractor Dynamics_
In phase $1$ of one-qubit $(2,2)$-Purely-QMM-UEs, all the solutions develop
_fixed point attractors_ at sufficiently late times. A representative solution
of this phase is presented in Fig. 6. As one observes in the
$(\text{Re}[\Psi_{t}^{\text{down}}],\Psi_{t}^{\text{up}})$ plots, the one-
qubit wavefunction becomes monotonically static. Moreover, the signature of
the attractor phase is seen in the third plot of figure 6: the fundamental QM-
TPF observable, that is $w_{t-a\;a}$, evolves monotonically to its fixed-point
attractor whose unique value can be nothing but $1$. As our numerical
investigations confirm, irrespective of the initial control parameters
$(B^{y}_{\text{Kicker}};\theta_{0},\phi_{0})$ and for arbitrarily values of
the largest QMD which do satisfy $a\geq a^{\star}$, this attractor phase is
formed in the following regime of the QM-couplings:
$\lambda^{I}\in\mathbb{R}^{+}$, while $|\hat{\mu}_{t-a}|$ is either zero or
sufficiently smaller than $\lambda^{I}$.

Moreover, we highlight that in this phase, the (2,2)-QMM-UE can drive the one-
qubit system to its fixed-point attractor within durations as short as several
tens or several hundreds of length-$a$ time interval, as simulations show. For
example, in the specific solution shown in Fig. 6, we see that the one-qubit
wavefunction has already settled down to its attractor in about $35\;a$ after
$t_{*}=a$, namely the outset of QMM-UE.

Indeed, the fact that the QMM-UE of closed quantum systems can develop phases
in which the global wavefunctions experience fixed-point attractor behavior, a
typical hallmark of open quantum systems, is indeed remarkable. Indeed, going
on with further explorations of the phase diagrams under consideration, we
find that other celebrated features of open quantum system are naturally
realized by (even the simplest versions of) QMM-UEs. Finally, we highlight in
reference to the phase diagram of the $(1,1)$ system, that the phase one,
namely the fixed-point attractor behavior, is new to the enlarged system of
the $(2,2)$ QMM-UE.

(a)

(b)

(c)

Figure 6: Numerical Solution of The QMM-UE system (4.2,4.3) with parameters:
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\lambda^{I},\hat{\mu}_{t-a}|a;t_{\text{completion}}$)=($2.75;1.001,0.089|6.111,0.592|1.73;2600\;a$).
‘Phase 1’ of one-qubit (2,2)-Purely-QMM-UE.

_PHASE 2: Regular Oscillations With (Almost-)Constant Attractors of The QM-
TPF_
The second phase of the one-qubit system under $(2,2)$ QMM-UE is equivalent
with the phase ‘A’ of its $(1,1)$ counterpart: the curious case of an
_‘oscillatory-attractor mixed behavior’_. The system’s QMM-UE suppresses the
sufficiently-late-times oscillations of the QM-TPF down to infinitesimal
fluctuations around initial-data-dependent constant values, while ordering the
asymptotic dynamics of the global wavefunction in the form of large
oscillations which are extremely regular.

Interestingly enough, one finds that the second phase of the $(2,2)$ system
can be easily formed even in the non-compact regions of couplings where the
equivalence with the (1,1) system is broken completely. Specifically,
simulations confirm the formation of the second phase with arbitrary
_negative_ values of (the ‘new’ coupling) $\lambda^{I}$ and with (zero or
finite) $|\hat{\mu}_{t-a}|$ upper bounded as
$\frac{|\hat{\mu}_{t-a}|}{-\lambda^{I}}\in[0,1+\epsilon]$ where $\epsilon$ is
a case-dependent parameter whose value is typically smaller than one.

A representative solution of the second phase, with its double-faced quality
clearly manifested in the sample time windows, is presented in Fig. 7. Indeed,
the solution of Fig. 7 belongs to the non-compact region of the
$(\mu,\lambda^{R},\lambda^{I})$-coupling-space where the one-qubit $(2,2)$
system is far beyond the reach of its $(1,1)$ counterpart.

Figure 7: Numerical Solution of The QMM-UE system (4.2,4.3) with parameters:
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\lambda^{I},\hat{\mu}_{t-a}|a;t_{\text{completion}}$)=($5.75;1.001,0.089|-2.07,1.85|1.73;6000\;a$).
‘Phase 2’ of one-qubit $(2,2)$-Purely-QMM-UE.

_PHASE 3: Unstructured Oscillations_
The third phase of the one-qubit $(2,2)$-Purely-QMM-UE is identical to the
phase ‘C’ of its $(1,1)$ counterpart in which the global wavefunction and the
QM-TPF observable experience everlasting irregular oscillations. Fig. 4
presents _this ‘disordered’ phase_. It is formed only when $\hat{\mu}$ has a
finite value of either sign, sufficiently larger than $\lambda^{I}$.

_PHASE 4: Oscillations which are clearly Structured or even Regular_
This phase is equivalent with the phase ‘B’ of the $(1,1)$ system. However,
simulations show that it can be developed even going far beyond the regime of
small $|\lambda^{I}|$ coupling. Specifically, choosing an arbitrary
$\lambda^{I}\in\mathbb{R^{-}}$, while
$\frac{|\hat{\mu}_{t-a\;t}|}{-\lambda^{I}}$ being neither less than one, nor
too large, results in the formation of the phase four of the $(2,2)$-Purely-
QMM-UE of the one-qubit closed system. The major hallmark, once more is the
post-early-times behavior of QM-TPF: it develops _repetitions of large
‘oscillatory patterns’ which are robustly structured and are featured with
several well-gapped extrema_. The wavefunction oscillations are not disordered
likewise. In Fig. 8 a representative of the phase is shown.

(a)

(b)

(c)

Figure 8: Numerical Solution of The QMM-UE system (4.2,4.3) with parameters:
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\lambda^{I},\hat{\mu}_{t-a}|a;t_{\text{completion}}$)=($7;1.001,0.089|-2,3.36|1.73;2200\;a$).
‘Phase 4’ of one-qubit $(2,2)$-Purely-QMM-UE.

_PHASE 5: BiState Meta-Stable Dynamics_
This one is the most intriguing phase qualitatively. Moreover, among the five
phases, it is the most structurally-ordered one: both in terms of the QM-TPF
profile and the wavefunction oscillations themselves. Like other phases, the
only condition for a generic solution of the dynamical system (4.3) to be in
this phase is one’s choosing the ‘right’ range of the couplings of the QMM-H
(66). As our numerical investigations confirm, the corresponding right range
of the couplings for this phase is this: $\lambda^{I}\in\mathbb{R}^{+}$, while
one sets a comparable strength for its competitor coupling
$\hat{\mu}_{t-a\;t}$, that is, the ratio
$\frac{|\hat{\mu}_{t-a\;t}|}{\lambda^{I}}$ is neither too small, nor too
large. Hence, the (1,1) system cannot form this phase.

One representative solution of the fifth phase is shown in the three plots of
Fig. 9. We highlight the temporal relations between the dynamical patterns of
plots ((a),(b)) for the one-qubit wavefunction oscillations, and that of plot
(c) for the oscillations of the fundamental QM observable: the emergent
patterns are all synchronized robustly. Intriguingly, in the sufficiently-long
duration of the representative solution of Fig. 9,
$\frac{t_{\text{completion}}}{a}=3600$, one clearly sees a _Two-State
Metastable Dynamics_. This two-state meta-sable dynamics, resembling an
‘exhaling-holding-inhaling’ pattern, emerges from the everlasting trails of
fast dense oscillations, and is highly robust even quantitatively. Moreover,
the one-qubit closed system switches the two metastable states by means of
_sharp transitions_. The time scale of the transitions is about one order of
magnitude shorter than the duration of the metastable states. One discerns
discerns the unique hallmarks of the phase five of the one-qubit
$(2,2)$-Purely-OMM-UE, shown by the three plots of Fig. 9. The pure state and
the QM-TPF of the one-qubit closed system form highly-ordered temporally-
correlated oscillatory behavior as such: (i) reach at one of their up-or-down
levels quickly, (ii) settle down and very tinily fluctuate there for
$\text{O}[100a]$, (iii) swiftly transition to the switched-level round in
succession. Remarkably, the QM-TPF transitions between a typically-deep
minimum and its maximum being $1$, while holding to $1$ for $\text{O}[100a]$.

(a)

(b)

(c)

Figure 9: Numerical Solution of The QMM-UE system (4.2,4.3) with parameters:
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\lambda^{I},\hat{\mu}_{t-a}|a;t_{\text{completion}}$)=($7.5;1.001,0.089|5.921,3.786|1.73;3600\;a$).
‘Phase 5’ of one-qubit (2,2)-Purely-QMM-UE.

We _conclude_ that the one-qubit unitary state-histories which solve (4.2,4.3)
yield five behavioral phases of Purely-QMM-Us generated by the second order
polynomial made from two quantum memories, as explained above. In Section 7,
we come back to this surprisingly-rich five-region phase diagram, and its even
richer generalizations in the next two subsections, to present a selection of
interesting lessons.

####  6.3.2 Supplementary Numerical Solutions Along The $\theta$-orbit

The unitary state-history (87,88) features unique and significant
characteristics among the theta-orbit solutions. These properties were
analytically derived, and extensively discussed, in Subsection 5.2.1. First
and foremost, although being the simplest of all nontrivial solutions to the
dynamical system (85), this special unitary state-history has an interesting
and important relation with _all_ the unitary state-histories in the
$\theta$-orbit moduli space, as follows. The unitary state-history (87,88) is
the dynamic-or-curve attractor of all one-qubit unitary state-histories formed
along the $\theta$-orbit. We now present a supplementary numerical study with
double purposes. First, this numerical analysis re-validates, actually in a
most general setting as explained below, the above statement. Second, as
promised in Subsection 5.2.1, it demonstrates ‘the fast tendency’ (in $a$
units) of the $\theta$-orbit unitary state-histories in approaching (87,88).

Fig. 10 presents six solutions $\theta_{t}$ to the dynamical system (85),
numerically constructed based on six qualitatively-distinct functions $f(t)$
taken as their initial histories, $\theta(t\leq a)$. These exemplary functions
are specified in the caption. The left and right plot show the dynamical
behavior of $\theta_{t}$-and-$\dot{\theta}_{t}$ in the two time intervals of
the Purely-QMM-UE written on the vertical axes. For enhanced visibility and
differentiation: (i) these six solutions are coloured: Light Blue (exponential
$f_{t}$), Red (non-special linear $f_{t}$), Green (cubic polynomial $f_{t}$),
Orange (oscillatory $f_{t}$), Strong Blue (complicated $f_{t}$) and Yellow
(special linear $f_{t}$), see the caption; (ii) the curves are shown with
exaggerated thickness. The yellow solution whose linear $f(t)$ realizes (88)
is ‘a’ profile-invariant one.

(a)

(b)

Figure 10: Purely-QMM-UEs solving (85) with $(\lambda,a)=(-2,5)$, assuming the
six initiations $\theta(t\leq a)=f(t)=$ $\frac{1}{5}t^{3}-t^{2}+t+8$ (Green);
$1.154\;t+3$ (Red); $e^{\sqrt{3t}-\frac{1}{7}t^{2}}$ (Light Blue);
$20\sin(\frac{5}{6}\;t^{2})$ (Orange);
$\frac{\cos(t^{2})\;e^{t}}{t^{2}+11\;t\;+\;9}$ (Strong Blue); $0.57\;t+7$
satisfying (88) (Yellow) .

One clearly sees in the right plot of Fig. 10, showing the time derivatives of
the six $\theta$-orbit solutions in an exemplary time duration of their
Purely-QMM-UE: $\dot{\theta}_{t}\;;\;t\in[\;3a\>,27a\;]$, that the five
profile-non-invariant solutions all realize, precisely as the Subsection 5.2.1
has prescribed, one of the two unitary-state-histories (87,88) as their curve-
attractor. As we see, the time-derivate $\dot{\theta}_{t}$ of every one of the
five profile-non-invariant solutions has developed, after a while, a profile
of monotonically-shrinking oscillations which swiftly converge to _either_ the
slope of the profile-invariant yellow solution which satisfies (88), _or_ to
its sign-flipped mirror root. Indeed, the statement that every $\theta$-orbit
solution, $\theta_{t}$, has a curve-attractor being a profile-invariant
linear-in-time, $\theta^{\star}_{t}$, is translated to the statement that the
time-derivates of the solutions, $\dot{\theta}_{t}$, have fixed-point
attractors whose values are the roots $\alpha$ of the equation (88).
Accordingly, the right plot is a numerical manifestation of both the derived
equation (91) and the ever-shrinking simple-profile oscillations as
fluctuations around it, realized by all the $\theta$-orbit unitary state-
histories.

Moreover, we observe in the left plot of Fig. 10 that even within the first
few length-$a$ steps as of the outset of Purely-QMM-UE, all the five profile-
non-invariant solutions have effectively taken profiles which approximate
their dynamic attractors: the yellow (or its mirrored) solution. It must be
further highlighted that we observe similar rapid tendencies, and the same
curve-attractor behavior of $\theta$-orbit unitary state-histories shown in
the right plot, using sufficiently-many other choices of the initiation
functions for $\theta_{t\leq a}$, and in other sufficiently-late-times
durations of their Purely-QMM-UE.

Now, we come to the numerical investigation of the leading order
‘$\theta$-orbit-confined’ perturbations over the profile-invariant unitary
state-history (87,88). The aim is to investigate the asymptotic dynamics of
_general_ leading order perturbation modes, $\theta_{t}^{(1)}$, as derived
from equation (LABEL:whl). That is, we are interested in the late-time
behavior of the solutions to the delay-differential equation (LABEL:whl) which
are resulted from general time-dependent functions, $f^{(1)}_{t}$, taken as
their initial histories: $\theta_{t\leq a}^{(1)}$.

Figure 11: Evolving $\theta_{t}^{(1)}$ modes = excitons of the profile-
invariant unitary state-history on the $\theta$-orbit, solving (LABEL:whl)
with $(\lambda,a)=(-2,3.5)$ s.t. $\alpha\sim 0.783$, and five initiations:
$\theta^{(1)}(t\leq a)=f^{(1)}(t)=$ $\frac{1}{20}(t^{3}-14t^{2}+33t)$ (Green);
$\frac{2}{5}t\cos(t^{3}-t^{2}+5t)$ (Red); $\frac{1}{7}\sinh(t)$ (Blue);
$\sin(\cosh(t))$ (Orange); $\frac{1}{2}(t^{\;\frac{1}{1+t}}+\tanh(t))$
(Magenta). For better visibility, the five curves have received appropriate
amounts of relative vertical shifts.

Fig. 11 shows the solutions for $\theta_{t}^{(1)}$ modes evolving under
(LABEL:whl), assuming the five initial histories $f^{(1)}_{t}$, specified in
the caption, with diversely-different defining profiles. As one sees in Fig.
11, regardless of their initial histories, all $\theta_{t}^{(1)}$ modes have
developed fixed-point attractors. The same behavior is observed from any other
initial-history function taken for the simulation. Hence, as promised in
Subsection 5.2.1, we observe that all $\theta_{t}^{(1)}$ excitons are
asymptotically static and preserve the original slope of the profile-invariant
solutions (87,88): the $\theta$-orbit perturbations cannot destabilize the
characteristic profile invariance of the unitary state-histories (87,88).

We highlight that _quite exceptionally_ in this subsection, given the purposes
of the above two numerical simulations, the quantum-memory pools were filled
by means of initial histories chosen independent of Kicker Hamiltonians. That
is, sufficient numbers of temporal functions with distinctive defining
profiles were chosen as initial histories, to compare the $\theta$-orbit
Purely-QMM unitary state-histories ‘initiated’ quite differently.
Mathematically viewed, in this way, we have effectively explored the (almost-)
_entire moduli spaces_ of solutions to the delay-differential equations (85)
and (LABEL:whl). These moduli spaces contain as their (proper or improper)
sub-spaces the solutions generated by Kicker Hamiltonians. Such, we have come
up with confirming and completing the analytical results of Subsection 5.2.1
on the $\theta$-orbit one-qubit Purely-QMM-UEs, within an encompassing
numerical approach.

###  6.4 The Phase Diagram of One-Qubit Wavefunctions Evolving Under A (2,3)
Purely-QMM Hamiltonian

We now set to understand how the first-to-come higher order monomial
interactions between $N$ quantum memories impact the phase diagram of the one-
qubit QMM-UE. To this aim, the same analyses is performed for $(N=2,L=N+1=3)$
Purely-QMM system: we activate cubic links between a chosen pair of state-
history-members of the one-qubit system, $(\rho_{t-a},\rho_{t})$, and map the
evolution phases generated by the QMM-H.

Specifically, we classify the Purely-QMM-UE phases generated by the
$H^{(2,3)}_{\text{QMM}}(t)$ given in (LABEL:avn). Such unitary evolution of
one-qubit states is described by the QMM Schrödinger equations
(4.2,LABEL:ate:ma,LABEL:ate:mb,LABEL:SE23POLAR). This delay-differential
system contains three couplings: $\lambda^{R}$ (whose value is kept zero for
simplicity while maintaining a rich phase diagram), $\lambda^{I}$ and $\eta$
whose values we run over sufficiently-large domains to monitor the effects of
the cooperation or competition between the second and third order interactions
of the two quantum memories. In our numerical simulations, we safely choose
the very same initial conditions as the (2,2) case, for comparison purposes.
We highlight that the real coupling $\eta$ sets the strength of the third
order interaction, $L=3$, between the chosen quantum memories.

Indeed, higher order linking of the QMs are impactful. Numerical exploration
proves that (2, 3) Purely-QMM-UE phases are physically richer. This is true
despite the fact that the one-qubit (2, 3) system essentially develops the
five principal phases, even when $|\frac{\eta}{\lambda^{I}}|$ is not small.
First, one-qubit Purely-QMM-UE in the co-presence of the third order
interactions develop _dynamical phase transitions_ (between those same five
phases) in non-narrow regions of the $(\lambda^{I},\eta)$-space. Moreover,
even without dynamical phase transition taking place, the third order
interactions can produce, without any fine-tuning, remarkably-deformed
versions of the five phases of the $N=2$ system.

In order to make the effects of the third-order interactions crystal clear, we
compare the numerically-constructed (2, 3) solutions with their (2,2)
‘reductions’. That is, for every one of the (2,3) numerical solutions, we
additionally construct the solution of the (2,2) system with the same values
of control parameters (aside from $\eta$), to make a more precise comparative
analysis. But, it suffices to showing the (2, 3) solutions and explaining
notable differences. To discern all the novel effects of the Purely-QMM third-
order interactions, we choose sufficiently-many values of $\lambda^{I}$, run
(for every one of them) the third-order interaction coupling $\eta$ from zero
to sufficiently-large values, construct numerical solutions to the delay-
differential system (4.2,LABEL:ate:ma,LABEL:ate:mb,LABEL:SE23POLAR), and
classify the phases of the one-qubit unitary evolutions in correspondence.

(a)

(b)

(c)

Figure 12: The purely-QMM-UE described by
(4.2,LABEL:ate:ma,LABEL:ate:mb,LABEL:SE23POLAR) with parameters:
$(B_{Kicker}^{y};\theta_{0};\phi_{0}|\lambda^{I},\eta|a;t_{completion})=(7.5;1.001,0.089|5.00,2.00|1.73;70000a)$,
showing dynamical phase transition of the one-qubit closed system under the
(2,3) QMM-H: switching phase five to phase one.

One-qubit (2,3) phases are sensitive only to the sign of $\lambda^{I}$. We now
begin with $\lambda^{I}\in\mathbb{R}^{+}$. Numerical investigations confirm
that the range $\lambda^{I}\in[0,5]$ covers all the interesting phases of the
system. When $\lambda^{I}=5$ at vanishing $\eta$, solutions show the attractor
phase, as it must. Increasing $\eta$ slightly from zero, the phase 5 of the
(2,2) system begins to be formed. Interestingly enough, before the
establishment of the phase 5, in the whole range of $1.6\lesssim\eta\lesssim
2.3$ with $\lambda^{I}=5$, the (2,3) Purely-QMM-UE of the one-qubit states
develops a _dynamical phase transition_ which switches, in the course of time,
the two phases mentioned above. We must highlight the purely internal and
dynamical nature of this ‘unitary’ phase transition: it does not take any kind
of external influences, or any change in the couplings or in the initial data,
to occur. It happens all-internally to the one-qubit closed system, during its
Purely-QMM-UE, and is sourced by nothing but the (higher order) interactions
that the past-or-present density operators of the unitary-evolving qubit make.
Also, in sharp contrast with (2,2) system, we notice the non-narrow bound for
the formation of this dynamical phase transition: $\frac{\Delta\eta}{\eta}\sim
0.48$. Fig. 12 shows a solution of the kind during $[0,13000\;a]$, but its
numerical construction was performed for one order of magnitude longer time
span, that is $[0,70000\;a]$, for being ensured of the defining nature of this
unitary phase transition. It is worthwhile to additionally mention that the
time period for jumps between two stable states is almost the same as the
corresponding (2, 2) system, but number of jumps before going into attractor
phase is now larger. As we see in Fig. 12 for the given values of
$(\lambda^{I},\eta)$, the one-qubit Purely-QMM-UE moves after $\sim 4000a$
from phase 5 to phase 1.

(a)

(b)

(c)

Figure 13: The Purely-QMM-UE described by
(4.2,LABEL:ate:ma,LABEL:ate:mb,LABEL:SE23POLAR) with parameters:
$(B_{Kicker}^{y};\theta_{0};\phi_{0}|\lambda^{I},\eta|a;t_{completion})=(7.5;1.001,0.089|10.00,4.00|1.73;5000a)$.
It equals the ‘Phase 5’ of one-qubit (2,2)-Purely-QMM-UE.

Increasing $\eta$ above 2.3, we find an all-times-surviving two-state meta-
stable phase which is identical to the phase 5 characterized for the (2, 2)
system. A representative solution of this phase is presented in Fig. 13. The
time span of each meta-stable state equals its (2, 2) reduction, being $\sim
50\;a$, and $w^{2}_{t-at}$ variations are almost equal too.

One also encounters in the phase diagram of the one-qubit $(2,3)$ Purely-QMM-
UE altered versions of the standard phase 5. These are morphological
deformations of the binary-metastable QMM-UE which are interesting physically.
One such deformation is presented in Fig. 14: we see that the altered
evolution features the standard profile of the QM-TPF in the phase 5, but now
in such a way that the relative configurations of both metastable states
undergo robust simple oscillations. The long wavelengths of these relative
oscillations are determined primarily by the $\lambda^{I}$ and $\eta$
couplings. We explored these interesting deformations of the phase 5 in a
sufficiently-wide range of the couplings. The numerical solution of Fig. 14
corresponds to $(\lambda^{I},\eta)=(2.6,2.53)$ as a representative, where a
periodicity of about $T\sim 1000\;a$ is featured. For optimal clarity, the
wavefunction is shown in two time intervals of the QMM evolution. One sees
again that more complex phases are made because of the third order
interactions.

(a)

(b)

(c)

(d)

(e)

Figure 14: The Purely-QMM-UE described by
(4.2,LABEL:ate:ma,LABEL:ate:mb,LABEL:SE23POLAR) with parameters:
$(B_{Kicker}^{y};\theta_{0};\phi_{0}|\lambda^{I},\eta|a;t_{completion})=(7.5;1.001,0.089|2.6,2.53|1.73;8000a)$.
Oscillatory deformation of the ‘Phase 5’ of one-qubit (2,2)-Purely-QMM-UE is
featured.

Moreover, we find that the one-qubit (2,3) Purely-QMM-UE can form dynamical
phase transitions under which a metastable phase 3 is eventually switched with
an stable phase 5. It has the same purely-internal nature that its previously
mentioned counterpart has, where a dynamical switching from the phase 5 to the
phase 1 was realized. Fig.15 and Fig.16 present a representative solution of
the kind where the unitary dynamical phase transition occurs at
$t_{\text{dpt}}\simeq 1155\;a$, given the couplings
$(\lambda^{I},\eta)=(1.3.4.93)$. To clarify the defining criterion of this
novel phase transition, we have exposed, and zoomed into, two sufficiently-
distant time intervals as follows: $[\;300\;a,400\;a\;]$ and
$[\;2200\;a,2300\;a\;]$. Specially, one sees in our zooming into the QM-TPF
profiles in the pre-vs-post phases of the purely-internal transition, Fig. 16,
that the (2,3) one-qubit Purely-QMM-UE has ‘itself’ evolved from the phase-3
evolution to the phase-5 evolution. In the plots of Fig. 15, we see the
transformed behaviors of $(Re[\Psi^{\text{down}}_{t}],\Psi^{\text{up}}_{t})$,
by means of the dynamical phase transition.

(a)

(b)

(c)

(d)

Figure 15: The Purely-QMM-UE described by
(4.2,LABEL:ate:ma,LABEL:ate:mb,LABEL:SE23POLAR) with parameters:
$(B_{Kicker}^{y};\theta_{0};\phi_{0}\;|\;\lambda^{I},\eta\;|\;a;t_{completion})=(7.5;1.001,0.089\;|\;1.3,4.93\;|\;1.73;3000\;a)$.
Dynamical Phase Transition of the one-qubit (2,3) system switching from a
metastable ‘Phase 3’ to a stable ‘Phase 5’. Figure 16: The Purely-QMM-UE
described by (4.2,LABEL:ate:ma,LABEL:ate:mb,LABEL:SE23POLAR) with parameters:
$(B_{Kicker}^{y};\theta_{0};\phi_{0}\;|\;\lambda^{I},\eta\;|\;a;t_{completion})=(7.5;1.001,0.089\;|\;1.3,4.93\;|\;1.73;3000\;a)$.
Dynamical Phase Transition of the one-qubit (2,3) system switching from a
metastable ‘Phase 3’ to a stable ‘Phase 5’.

Sufficiently strengthening the third-order (over the second-order)
interaction, we see that the one-qubit Purely-QMM-UE wavefunctions show the
defining quality of the phase 3. For example, having $\lambda^{I}=5$ fixed and
raising $\eta$ above $\sim 10$ is sufficient to derive the system into the
phase 3.

(a)

(b)

(c)

Figure 17: The Purely-QMM-UE described by
(4.2,LABEL:ate:ma,LABEL:ate:mb,LABEL:SE23POLAR) with parameters:
$(B_{Kicker}^{y};\theta_{0};\phi_{0}|\lambda^{I},\eta|a;t_{completion})=(7.5;1.001,0.089|-5,40.30|1.73;3000a)$.
Dynamical Phase Transition of one-qubit (2,3)-Purely-QMM-UE: ‘Phase 3 to Phase
2’.

We now explore the distinct category of negative $\lambda^{I}$ in regard with
the one-qubit phases of the (2,3)-Purely-QMM-UE. Setting $\lambda^{I}=-5$, we
increase $\eta$ from zero, with steps of $\sim 0.1$ or $0.01$, up to
sufficiently large values, to identify and classify the corresponding phases.
In the whole range of $\eta$ taking values from zero up to $\sim 50$, we
observe the phase 2. At $\eta\sim 50$, where the third-order interaction has
become dominant about one order of magnitude, the one-qubit purely-QMM-UE
makes a crossover to irregularity: the phase 3 takes place beyond this soft
border. Along the way to its crossover to irregularity, the one-qubit (2,3)
system makes a unitary and purely-internal dynamical phase transition from a
long-time-surviving phase 3 to an ultimate phase 2. In fact, we observe this
remarkable behavior in the whole range:
$\lambda^{I}=-5,\;40\lesssim\eta\lesssim 50$, corresponding to the bound
$\frac{\Delta\eta}{\eta}\sim 0.1$. Fig. 17 shows a clear representative
numerical solution which features this dynamical phase transition. We
highlight that, as it comes about in the next subsection, a phase-
interpolating behavior of the same kind is seen in the one-qubit (3,3) system.
This suggest that this dynamical phase transition is a signature of the third-
order Purely-QMM interactions.

(a)

(b)

(c)

Figure 18: The Purely-QMM-UE described by
(4.2,LABEL:ate:ma,LABEL:ate:mb,LABEL:SE23POLAR) with parameters:
$(B_{Kicker}^{y};\theta_{0};\phi_{0}|\lambda^{I},\eta|a;t_{completion})=(7.5;1.001,0.089|-1.30,6.5|1.73;3000a)$.
‘Phase 4’ of one-qubit (2,3)-Purely-QMM-UE.

One-qubit (2,3) Purely-QMM-UE with negatively-signed second-order interactions
can also form the phase 4. For example, realizations of phase 4 is obtained in
the entire range: $\lambda^{I}=-1.3$ and $5.5\lesssim\eta\lesssim 8$.
Moreover, because now the third-order interactions are present in the QMM-H,
quite non-generic versions of phase 4 are easily realizable. In these non-
generic and refined versions, the structured modules of phase 4 feature
significantly-enhanced orders in their patterns. A representative solution of
the kind is given in Fig. 18 where, for visual clarity, only the window
$[200\;a,300\;a]$ is presented. Furthermore, in narrow bounds of
$(\lambda^{I},\eta)$, dynamical phase transitions from phase 3 to phase 4 is
observed. Naturally, we expect that these alternative dynamical phase
transitions can likewise occur without any fine-tuning of the couplings, in
the presence of higher order interactions: $L>3$.

###  6.5 The Phase Diagram of One-Qubit Wavefunctions Evolving Under A (3,3)
Purely-QMM Hamiltonian

Now we construct and investigate one-qubit state-histories for $(3,3)$-Purely-
QMM-UE. We focus on the specific Hamiltonian (15), a simplified version of the
most general case (14) one could take, as only one complex coupling is now
turned on: $(\kappa^{I},\kappa^{R})$. However, the simplified (3,3) QMM-H (15)
is sufficiently complex, leading us to a whole variety of novel phases. The
coupling $\kappa$ is the strength of _the cubic hyper-link, $L=3,$ between
three specific quantum memories, $N=3$_. The quantum memories, reselected at
every present moment $t$, correspond to these three wavefunctions of the one-
qubit closed system: one is the qubit’s pure state at the very now,
$\rho_{t}$, one is qubit’s pure state in its past moment with an unchanging
temporal distance $b\in\mathbb{R}^{+}$ from the now, $\rho_{t-b}$, one is the
third qubit’s pure state, $\rho_{t-a}$, with $a>b$ being a free constant
specifying a moment further in the past from the time now. Hence, besides the
$(B^{y}_{\text{Kicker}};\theta_{0},\phi_{0})$ triplet of initial data, the
one-qubit $(3,3)$-QMM-UE generated by the QMM-H (15) has four more real valued
control parameters which we identify as follows: the couplings
$(\kappa^{I},\kappa^{R})$, the largest QMD $a$, and the ratio of the two past-
time QMDs: $\frac{a}{b}\equiv r\in[1,\infty]$. In fact, we numerically
investigate and establish that this _seventh control parameter_ $r$, despite
being totally independent of the couplings of third order interactions
$(\kappa^{I},\kappa^{R})$, plays significant role in deciding what particular
phase of the phase diagram is formed by a given one-qubit $(3,3)$-Purely-QMM-
UE. Moreover, as there are three quantum memories participating in the
generation of unitary time evolution, each one state-history is marked with
three QM-TPFs (LABEL:ctmcs): $w_{t-a,t-b}$, $w_{t-b,t}$ and $w_{t-a,t}$.

The QMM Schrödinger equation (27) encoding the above unitary evolution is (81)
for the dynamics of the one-qubit Bloch vector, or equivalently is the coupled
system of delay-differential equations (82,83) for the dynamics of the one-
qubit Bloch angles. As before, we construct the unitary state-histories under
consideration by solving the polar version of the QMM Schrödinger equation,
namely (82,83) by means of the same numerical discipline explained and used in
previous subsections. However, given that the matching of the two methods have
been established in the previous examples, it suffices here to implementing
the first method. We have seen by extensive numerical investigations that the
complete phase diagram of $(3,3)$-Purely-QMM-UE happens to be _much more
complex_ than that of its counterpart with only two quantum memories, even in
as elementary quantum setting as the one-qubit closed system. Because of this,
the forthcoming presentation of one-qbit unitary state-histories with $N=3$
and $L=3$ is cut down to a sufficiently-rich variety of numerical solutions to
(82,83) which we have found to be most remarkable qualitatively.

Besides few exceptions, for every one-qubit QMM unitary state-history of the
$(3,3)$ theory which is shown in this subsection, these _four plots_ are
presented: $\text{Re}[\Psi_{t}^{\text{down}}],\Psi_{t}^{\text{up}}$ and also
$w_{t-a\;t}^{2}$, as were presented for all the QMM unitary state-histories
with $N=1$ and $N=2$, together with only one (among the two) of the other QM-
TPFs: $w_{t-b\;t}^{2}$. Note that there still remains one more QM-TPF,
$w_{t-a\;t-b}^{2}$, whose plots we ignore to show. This is mainly because it
develops the same qualitative profile as the two other QM-TPFs, in all the
many numerical solutions we have constructed. This also prevents further
enlargement of the volume of the paper, and the population of the plots, which
are both already large enough. In addition to several numerical solutions
singled out for demonstration in figures, we present brief explanations about
all the phases of the one-qubit (3,3)-Purely-OMM-UE which we have seen in the
course of our explorations of its (large and complex) phase diagram. We first
explain the method with which we have organized our extensive (but not
exhaustive) exploration of the phase diagram.

Geometrically, the phase diagram of the unitary state-histories whose
generator is the $(3,3)$-QMM-H (15) lives in one dimension higher than that
corresponding to the $(2,2)$-QMM-H (66). One can identify this additional
dimension with geometrization of the ratio of the two larger QMDs, $b$ and
$a$, that is (as introduced before) with the extra control parameter $r$. All
the rest of their control parameters are in a one-to-one correspondence. In
particular, the defining coupling constants of both QMM-Hs form
$\mathbb{C}=\mathbb{R}^{2}$. Moreover, by purity of the unitary state-
histories (LABEL:hps), and via a mapping
$(\hat{\mu}_{t-a},\lambda^{I})\leftrightarrow(\kappa^{R},\kappa^{I})$, the
total phase diagram of the $(2,2)$ theory corresponds to the following
codimension-one hypersurfaces of the phase diagram of the $(3,3)$-theory: one
is the boundary hypersurface $r=1$, and the other is the asymptotic boundary
$r=\infty$. In conclusion, $r\in\mathbb{S}^{1}$. Given geometric
reconstruction as such, and knowing the five $N=2$ phases, we have extensively
explored the phases of the $(3,3)$-Purely-QMM-UE. Focusing in turn on every
one of the five phases identified in Subsection 6.3, we first obtain a handful
of solutions to the QMM Schrödinger equations (82,83) at $r=1$, belonging to
that phase. Next, we gradually raise the (3,3)-parameter $r$ above $r=1$,
having kept
$(B^{y}_{\text{Kicker}}|\theta_{0};\phi_{0}|\kappa^{R};\kappa^{I})$ and the
largest QMD $a$ fixed, re-solve the system (82,83), and identify what
qualitatively-distinct phases are formed by the resulted $r$-deformed
solutions. We go on with this procedure up until hitting a sufficiently-large
$r$ which (with acceptable approximation) reproduces the initial solution with
$r=1$. Put differently, we systematically generate and single out the distinct
phases of the $N=3$ Purely-QMM theory as the higher-dimensional
$r$-deformations of the five phases of the counterpart $N=2$ theory.

_Mapping the $r$-deformations of the Phase 1_
Numerical investigations establish that the phase $1$ of $N=2$ is invariant
under the (3,3) $r$-deformations. All Purely-QMM unitarily-evolving one-qubit
wavefunctions which solve the $(3,3)$ dynamical system (82,83) within the
coupling range $0\leq|\frac{\kappa^{R}}{\kappa^{I}}|\ll 1$, develop at
sufficiently-late times _fixed-point attractors_.

_Mapping the $r$-deformations of the Phases 2,3,4_
The (3,3) phases found by applying the $r$-deformation on the phases 2,3 and 4
of the one-qubit $N=2$ theory, are essentially accommodated in the total
landscape of the $r$-deformations of the phase 5. A rich landscape to which we
now turn.

_Mapping the $r$-deformations of The Phase 5_
Now, we simulate and classify the (3,3) solutions, by moving on the $S^{1}$
deformation, from within the original phase $5$ of the one-qubit $N=2$ theory.
We know that at $r=1$, the initial point of the deformations, the behavior of
the one-qubit Purely-QMM-UE is the everlasting switching, under sharp
transitions, between two meta-stable states. Concretely, we classify the
numerical solutions obtained by varying the $r$ parameter along its $S^{1}$,
with the other control parameters being chosen and kept fixed as follows:
$(B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\lambda^{I},\hat{\mu}_{t-a}|a)=(35.5;1.001,0.089|5.124,3.185|3.67)$.
Beginning from $r=1$, increasing $r$-deformations are implemented in
sufficiently-small steps, and the system (82,83) is solved numerically. With
the choice of $r=1$: $\Delta^{(t)}_{1}=\Delta^{(t)}_{2}=a$, and we do obtain a
numerical solution whose profile is of the form presented in Fig. 9. We expect
that within a not-too-wide band above $r=1$, the system must stay in the phase
$5$. Indeed, numerical solutions confirm the stability of the phase $5$ under
$r$-deformation in the range: $1\leq r\lesssim 1.15$. Within this range, the
only effect of increasing $r$-deformations is that all the three simultaneous
stages of $w_{t_{i}t_{j}}=1$ shorten monotonically. Moreover, one sees that
the defining feature of bistate metastability is taken away gradually.

Making a _crossover_ at $r\sim 1.15$, one-qubit Purely-QMM-UEs enter a
_special phase_. This special phase, being formed everywhere in the following
$r$-deformation range: $1.15\lesssim r\lesssim 1.632$, is a _finely-balanced
blend of the phases $5$ and $4$_. A representative unitary state-history in
this special phase is presented in Fig. 19. See the distinctive orders, both
in the oscillations of the one-qubit wavefunctions, and in the clear patterns
of the QM-TPFs. It marks a very non-generic sub-class of the phase $4$, in the
sense of its built-in structures. Being formed as a phase-$5$ deformation,
however, the norms of the three QM-TPs $(w_{t-a\;t},w_{t-b\;t},w_{t-a\;t-b})$
feature the following pattern. Generically, between the neighbouring pairs of
the $w_{t-a\;t}=w_{t-b\;t}=w_{t-a\;t-b}=1$ stages, which are shortened by now
about two orders of magnitude compared to the corresponding stage in the $r=1$
phase $5$, one narrow bump is formed which contains three sharp extrema: two
unequal minima and one maximum. Interestingly enough, the dynamics of the
three QM-TPFs are finely synchronized, such that the $w_{t_{i}t_{j}}=1$ stages
and the aforementioned three extrema all happen simultaneously.

(a)

(b)

(c)

(d)

Figure 19: Numerical Solution of The QMM-UE system (82,83) with parameters:
$r=1.173$,
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\kappa^{I},\kappa^{R}|a;t_{\text{completion}}$)=($35.5;1.001,0.089|5.124,3.185|3.67;6600\;a$).
A special phase of one-qubit (3,3)-Purely-QMM-UE: blending phases $5$ and $4$
of the one-qubit $N=2$ Theory.

Surpassing $r\gtrsim 1.632$, the $(3,3)$ Purely-OMM-UE crossovers to the
general realm of the phase 2, and stays there up until $r\lesssim 3.23$. One
sees, in a considerable sub-interval of this range, two deformed variants of
the phase 2, in particular. Qualitatively, they are two possible
‘amplifications’ of the phase 2: _the ‘semi-frozen’ and ‘fully-frozen’
versions of phase 2_. In these remarkable variants, one, two or all of three
$w_{t_{i}t_{j}}$s become totally frozen, with the wavefunction oscillations
becoming perfectly regular. See Fig. 20 for a representative solution.
Moreover, depending on the value of $r$, other non-generic versions of the
phase $2$ are formed.

Figure 20: Numerical Solution of The QMM-UE system (82,83) with parameters:
$r=2.79$,
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\kappa^{I},\kappa^{R}|a;t_{\text{completion}}$)=($35.5;1.001,0.089|5.124,3.185|3.67;3518\;a$).
A special phase of one-qubit (3,3)-Purely-QMM-UE: fully-frozen version of
phase $2$.

More remarkably, one observes that at $r\sim 1.725$, within the above range,
the $(3,3)$ Purely-QMM-UE experiences a dynamical phase transition under which
the one-qubit wavefunction changes a long-lasting phase 3 with the stable
fully-frozen phase 2. The phenomenon is of the same nature observed and
described in Subsection 6.4: a unitary purely-internal dynamical transition,
resembling a phase-interpolating time-like kink, caused by the third-order
interactions between the chosen QMs of the closed system. The six plots of
Fig. 21 present a representative solution of the corresponding QMM Schrödinger
equations, given by (82,83). In particular, to further enhance clarity, one of
the QM-TPs has been zoomed in within two largely-separated time windows. As we
see, the dynamical phase transition has occurred in $\sim 920\;a$ after the
outset of the (3,3) Purely-QMM-UE.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 21: Numerical Solution of The QMM-UE system (82,83) with parameters:
$r=1.725$,
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\kappa^{I},\kappa^{R}|a;t_{\text{completion}}$)=($35.5;1.001,0.089|5.124,3.185|3.67;6000\;a$).
A phase-interpolating state-history of one-qubit (3,3)-Purely-QMM-UE.

Walking the direction of $r$-deformations ahead, the one-qubit closed system
makes crossover to ‘ordered behavioral patterns’: the general realm of the
phase $4$ of the N=2 theory. Right before it, we see a special profile in a
narrow band about $r=3.235$: Purely-QMM-UEs are simple synchronized
oscillations. Fig. 22 shows clearly the highly nongeneric, regular, simple
configurations which are formed in the aforementioned range. As we see in the
representative solution presented in Fig. 22, all the three system’s QM-TPFs,
and its wavefunction experience stable, almost-perfectly simple oscillations.
Moreover, the oscillations of $(w_{t-a\;t},w_{t-b\;t})$ are oppositely
synchronized, as precise examination of their maxima/minima shows. After this,
the system moves into the ordered-but-complex oscillations of the phase 4, up
until $r\lesssim 3.3$.

(a)

(b)

(c)

(d)

Figure 22: Numerical Solution of The QMM-UE system (82,83) with parameters:
$r=3.235$,
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\kappa^{I},\kappa^{R}|a;t_{\text{completion}}$)=($35.5;1.001,0.089|5.124,3.185|3.67;6600\;a$).
Special one-qubit (3,3)-Purely-QMM-UE, featuring Synchronized Simple
Oscillations.

A remarkable family of one-qubit (3,3) Purely-QMM-UEs which structurally are
highly ‘intricate and exotic’ are formed, within a narrow band of
$r$-deformations at the verge of system’s crossovering from the phase $4$ to
the phase $5$. Fig. 23 shows a numerical solution which represents this
intricately-patterned family of unitary state-histories, corresponding to
$r=3.299$. One can examine and confirm that the QM-TPFs $w_{t-a\;t}$ and
$w_{t-b\;t}$ are, once more, oppositely synchronized: their local extrema
inside every structural module, being a handful in population, happen in
reverse orders at almost simultaneous moments.

(a)

(b)

(c)

(d)

Figure 23: Numerical Solution of The QMM-UE system (82,83) with parameters:
$r=3.299$,
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\kappa^{I},\kappa^{R}|a;t_{\text{completion}}$)=($35.5;1.001,0.089|5.124,3.185|3.67;3300\;a$).
It represents various intricately-patterned forms of one-qubit (3,3)-Purely-
QMM-UEs, right before making the crossover from Phase $4$ to phase $5$.

Finally, for $r\gtrsim 3.5$, the one-qubit closed system develops $(3,3)$
Purely-QMM-UEs which feature ongoing two-state-switching metastability,
corresponding to the phase $5$, including _morphological deformations_. See
the representative of Fig. 24 at $r=3.77$.

(a)

(b)

(c)

(d)

Figure 24: Numerical Solution of The QMM-UE system (82,83) with parameters:
$r=3.77$,
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\kappa^{I},\kappa^{R}|a;t_{\text{completion}}$)=($35.5;1.001,0.089|5.124,3.185|3.67;3900\;a$).
One-qubit (3,3)-Purely-QMM-UE corresponding to morphologicaly-deformed phase
$5$.

Likewise, we have analyzed the resulted $r$-deformations of the $r=1$ phase
$5$ for a handful of independent examples, classified their phases, and found
them qualitatively equivalent with those obtained based on the solutions
above. But, we highlight that our extensive explorations do not necessarily
exhaust the entirety of the large-and-complex spectrum of all possible
$r$-deformations of the phase $5$. Nevertheless, we regard that the presented
spectroscopy of one-qubit (3,3) Purely-QMM-UEs is inclusive enough.

###  6.6 Selected General Highlights On One-Qubit Purely-QMM-UEs

Having known the distinct phases of the $(N\leq 3,L\leq 3)$ Purely-QMM-UE of
the one-qubit closed system, we now want to highlight few general remarks
about the phase diagram. Although these points were essentially brought about
in the previous subsections given the context, these five remarks can serve
the presented results as a supplement.

One-qubit Purely-QMM-UEs in which the wavefunction experiences ongoing
disordered or structured oscillations are realizable _if_ (at least) one
_Hermitian_ QMM monomial is turned on in the Hamiltonian. Hence,
$(1,1)$-or-$(N=2)_{|_{\lambda^{I}=0}}$ QMM-Hs are sufficient.

One-qubit Purely-QMM-UE attains fixed-point attractor, along with the
everlasting behavior of switching between two metastable attractors, the
phases ‘one and five’, _on the condition_ that one turns on interaction
operators given by _anti-Hermitian_ QMM monomials. The simplest Hamiltonian to
realize those two phases is (66) in which the _commutators_ of two quantum
memories, $\mathcal{O}^{(\dagger)}_{a}=[\rho_{t-a},\rho_{t-a}]^{(\dagger)}$,
run the above-mentioned interactions. Moreover, being intuitive, the formation
of the two-attractor meta-stable dynamics of the phase five demands that the
strengths of the couplings $\lambda^{I}$ and $\hat{\mu}_{t-a}$, which compete
to drive the system to oscillatory versus fixed-point-attractor behavior, are
made comparable to one another.

The only kind of transitions that one observes between the distinct phases of
one-qubit Purely-QMM-UEs, as a consequence of varying the QM-couplings
$(\mu_{t-a\;t};\lambda^{I},\lambda^{R};\eta;\kappa)$, are _crossovers_.
Turning on flows of QM-couplings causes continuous changes. Fig. 25 shows a
clear example of the inter-phase crossovers. Plot (a) shows a solution in
phase four. With $\lambda^{I}=-2$ and all other parameters fixed,
$\hat{\mu}_{t-a\;t}$ is lowered from $3.01$ to $2.992$. The new solution is in
phase two. Gradually making the above shift, $\delta\hat{\mu}_{t-a\;t}=0.18$,
and classifying the solutions, we find that the crossover has occurred at
$\hat{\mu}_{t-a\;t}\cong 3$.

(a)

(b)

Figure 25: Crossover by the numerical solution from Phase 4 (a) to Phase 2 (b)
of the one-qubit (2,2)-Purely-QMM-UE, demonstrated by $w^{2}_{t-a\;t}$, upon a
shift: $\delta\hat{\mu}_{t-a}=0.018$.
(a):
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\lambda^{I},\hat{\mu}_{t-a}|a;t_{\text{completion}}$)=($7.5;1.001,0.089|-2,3.01|1.73;3600\;a$).
(b):
($B^{y}_{\text{Kicker}};\theta_{0},\phi_{0}|\lambda^{I},\hat{\mu}_{t-a}|a;t_{\text{completion}}$)=($7.5;1.001,0.089|-2,2.992|1.73;3600\;a$).

Purely-QMM-UEs realize dynamical phase transitions which, resembling time-like
kinks, switch between a pair of the one-qubit phases. A generic solution of
the kind has a long lasting stay in one phase, but makes a sharp transition
which stabilizes it in one other phase. Interestingly, they can switch between
various (unstructured or structured) oscillatory phases and the (fixed-point
or multi) attractor phases. They take place without external influences,
coupling variations, or quenched disorders. These unitary transitions are
caused internally, by the higher order interactions, $L\geq 3$, between the
$N\geq 2$ QMs which compose the Hamiltonian. Finally, these unitary dynamical
phase transitions are realized, in diverse forms, in various regions of the
interaction-coupling space whose volumes increase by raising the degree of QM
monomial interactions, but are considerable even at $L=3$.

Overall, the one-qubit closed system endowed with first, second or third order
monomial interactions between its past-or-present quantum states can develop
five principle kinds of unitary time evolutions, along with time-like kink
behavior interpolating between them: fixed-point attractor behavior (phase
one), three distinct kinds of everlasting oscillatory behavior (phases two,
three and four), and everlasting swapping behavior between two meta-stable
attractors (phase five). That, the Purely-QM-UEs of the most elementary
quantum setup is capable of such surprising behavior is remarkable to us.
These findings seem to be truly promising for the visions we bring about in
Section 7.

##  7 QMM-UEs: Outlook, Open Questions And Visions On
‘Experience Centric Quantum Theory’

###  7.1 A Conceptual Overview And Selected Elucidations

All fundamental laws of nature as we have discovered are successfully
formulated by means of microscopic interaction operators, Lagrangians and
Hamiltonians which are unchangeably ‘enforced’ on closed systems, in the sense
that they are, and so remain, intrinsically indifferent to the accumulating-
and-evolving ‘experiences’ that the total closed system and its constituent
parts develop in the course of their time evolutions. Mathematically, they are
given by local-in-time operators on the Hilbert space of a closed system that
are defined with no dependencies on the system’s past-or-present quantum
states. Indeed, a discipline as such establishes a strictly-one-way
relationship between states and their evolutions. Hamiltonians and their
constituting interactions make the states moment-by-moment evolve unitarily,
however, do not ‘tolerate’ (any structural kind of) back reaction from them:
they receive no impact, hence no update, from the past-or-present states of
the closed system. The standard model (= theory) of particle physics is our
best theoretical and experimented validation of how successful the assumption
of the one-way relationship between the closed-system states and their unitary
evolution has (perhaps astonishingly) turned out to be.

Nevertheless, a paradigm shift is necessary if we aim for a general sensible
theory of _quantum intelligence_. What we mean by quantum intelligence is the
ability of a system to generate, organize and output a sufficiently-rich
spectrum of behaviors which are general quantum behavior on the one hand, and
general intelligent behavior on the other hand. In other words, if one aims to
bring about a general synthesis and merge of quantum behavior and intelligent
behavior, within a sound and successful theoretical framework, a novel
approach, featuring the alteration of some conventional (and ‘less’
fundamental) assumptions, must be adapted. Specially, it is in the outset
crystal clear that the aforementioned one-way relationship between unitary
evolutions and the system’s quantum states is at complete odds with _the most
(or one of the most) fundamental characteristic(s) of intelligent behavior_.
Any possible form of intelligent system must output manifold behaviors whose
dynamical profiles undergo recurrent remarkable updates which are directly
originated from its state-history experiences. In particular, the very time
evolution of the states must receive moment-by-moment revisitations, and
recurrent updates, based on the relevant sub-histories of the system. For
closed systems which are the focus of the present work (as we shall argue
soon), the mere possibility of intelligent behavior requires that the
Hamiltonian generating the unitary evolution operator, and hence the system’s
internal interactions which source it, can-and-must-typically receive
_adequate ‘structural’ updates_ which are driven by the system’s quantum-
state-history. In other words, _already in the primary context of closed
evolving systems, we need quantum memory effects which manifest themselves as
compositional nonlocal-in-time dependencies of the unitary time-evolution
operator and its sourcing interactions on the (relevant information content of
the) state-history_ , as one necessary condition for the system’s capability
to develop behavior which are, at the same time, quantum and intelligent.

In this regards, it is highlighted that the quantum memory effects induced or
mediated by the system-environment coupling are ubiquitous and cause a
landscape of Non-Markovian time evolutions in open quantum systems. These
effects are quantified, and also measured indirectly, using independent
characterizations such as information backflow or lack of CP-divisibility.
_But_ , environmental Non-Markovianities come generically with dissipation,
decoherence, and time evolution irreversibility: they source evolutions which
are _Non-Unitary_. Moreover, the strength-or-largeness of such memory effects
_follows dynamically_ the strength-or-largeness of the overall system-
environment coupling, and hence generically the correlated strength-or-
largeness of the induced Non-Unitarity of the system’s evolution.

_On the contrary_ , as various thought experiments can easily show, _no such
monotonic or dynamical binding between the Non-Markovianity and the Non-
Unitarity of the time evolution should-or-is required by quantum intelligent
systems in the general context_. Clearly enough, general intelligent behaviors
neither should, nor actually do, bound themselves to any inevitable-or-
monotonic dynamical non-unitarity cost _in exchange for_ required memory
dependencies as their experience resource. We must highlight that the above
statement stays valid even when the initial formation of learning and
intelligence would require coupling the system to an interacting environment
for sufficiently-long times. Indeed, an intelligent system which ‘lives’ in an
interactive environment and respond to it, undergos generically some form of
Non-Markovian evolution of the non-unitary kind. Nevertheless, if the system
with (a degree, kind or level of) _emerged_ intelligent behavior becomes (even
if nearly) isolated, its time evolution should become (nearly) unitary, but
now in some non-conventional form which has sufficiently-strong dependencies
on its in-the-making state-history. The three points in (conceptual and
technical) relation with the above highlight are now summarized:

_(1)_ Firstly, there is no fundamental argument against the remarkable
possibility that, even in the total absence of initial environmental
interactions, a sufficiently-complex sufficiently-aged system can _self-
organize in isolation_ (some degrees, kinds, or levels of) _emergent
intelligent goal-oriented behavior_. _(2)_ Even if we assume that the
collective development of intelligence, as a sustainable behavioral _emergent_
quality of a complex (closed or open) system across a proper subset of all its
phases, needs an initial history of sufficient environmental interactions,
once the environmental couplings are made negligible, the corresponding
(dominantly-) unitary evolution must be generated by a nonlocal-in-time
Hamiltonian operator which is state-history-based ‘constructionally’. _(3)_
Now, if we consider _open_ intelligent systems coupled to an interactive
environment, the intelligence-required Non-Markovianity of the system’s
evolution does not need to follow a corresponding scale of evolution Non-
Unitarity. That is, the general dynamics of open intelligent systems should
not necessarily be of the conventional Non-Markovian kind, namely, of the kind
deducible from the conventional Hamiltonian evolution of an encompassing
closed system. Indeed, as in the case of conventional quantum systems, a
general deduction should instead follow a ‘closed $\to$ open’ direction as
follows. _General Non-Unitary Non-Markovian evolutions of intelligent open
systems_ are deducible from Unitary Non-Markovian evolutions, such as those in
the present work, of encompassing intelligent closed systems.

_Above all, it is within the broader perspective of intelligent quantum
behavior_ that we have proposed, formulated and primarily investigated a
complementary distinctive way of memory participations in the system’s
internal interactions and unitary evolution. Quantum memory roles and effects
come with three distinctive qualities in this novel way: _(a)_ they can be
(completely or partially) _Non-Environmental_ , namely, _Internal_ ; _(b)_
they are _‘constructive’_ , serving the (re-)organization of the system’s
behavior, by being (some of the) direct building-block operators of the
updating Hamiltonian; _(c)_ In principle, their compositional role in the
system’s evolution are pushable all the way to the _extreme_ , that is, the
Hamiltonian operator can be entirely made from the system’s density operators
as developed in its state-history. Specifically, the focus of our work has
been the central role that the quantum-state-history of a closed system can
play in sourcing, updating and directing its internal interactions and unitary
evolution.

The novel family of Non-Markovian quantum processes proposed by the present
work are completely coherent: their quantum-state-history dependencies are
developed free from any dissipative effects (at least ideally) and they remain
so, therefore, preserving the quantum information in the entire state-history-
based evolution. Indeed, despite being Non-Markovian, the proposed evolutions
are (by construction) _perfectly unitary_. The perfect unitarity of these
evolutions is directly implied by the built-in _Hermicity_ of a moment-by-
moment Quantum-Memory-Made operator which generates it, namely
$H_{t}^{(\text{QMM})}$ which serves the closed system as its nonlocal-in-time
Hamiltonian. Moreover, the unitarity of these QMM processes has been shown
crystal clearly in _the explicit construction_ of their time-evolution
operator presented in (11), using (the continual sequence of) momentary QMM-Hs
generally defined in (4) and formulated in (7,8) specifically. Furthermore, we
remark that in all one-qubit realizations presented in the Sections 4, 5 and
6, the time-evolution unitarity has been tested and re-confirmed in several
independent ways.

About the relation between standard quantum mechanics and the present work,
namely our generalizing framework and the novel systems formulated within it,
there are some clarifying points which we now come to highlight. First and
foremost, the QMM-UEs defined in (4) and specified in (7,8) and in their next-
subsection generalizations, _do not necessitate any underlying form of the so-
called ‘nonlinear quantum mechanics’_. The statement holds despite the fact
that all QMM-UE must be unavoidably nonlinear and nonlocal-in-time, due to
their compositional dependencies on the past-to-present state-history of the
closed system, as QMM Schrödinger equations manifest. The point is, as
explained above, the QMM-Hs ((4);(7,8);(132)) are proposed _foremostly_ as
_emergent Hamiltonians_ in any complex system that describe the (sufficiently-
developed ‘higher-level’) _phases which possess (the primary characteristics
of) quantum intelligent behavior_. The system itself, however, can be a
completely-standard quantum system at its microscopic level of formulation,
and likewise at all its ‘lower’ levels. In fact, the system does not even need
to be necessarily quantum-mechanical at all: it can be a classical complex
many-body-interacting system which develops at functional collective levels a
form of emergent quantum behavior. Moreover, _intelligence-aside_ , QMM-UEs
and their non-unitary open-system descendents provide effective descriptions
of new emergent phases of microscopically-standard quantum systems whose Non-
Markovian behaviors feature significantly more coherent quantum memory
effects. Nevertheless, one does not exclude, either, the intriguing
possibility of _the ‘orthogonal’ branch_ of the physical relevance of QMM-UEs,
which deduction-wise goes the other way around: one can propose these
generalized quantum evolutions to define more fundamental quantum theories,
for example in the context of quantum gravity, out of which both standard
quantum mechanics and geometric spacetime emerge. We shall present more
detailed explanations on _the above-mentioned points_ in Subsection 7.2.

The proposed unification of time evolution Unitarity and Non-Markovianity for
closed quantum systems led us to observe, already in the most minimal context
of a one-qubit closed system evolving by means of the lowest-order
interactions between its quantum memories, a rich landscape of unprecedented
behavioral phases and dynamical phase transitions. Mathematically, they arise
from the inseparably-entangled nonlinearity and nonlocality-in-time of the
corresponding QMM Schrödinger and Von Neumann equations presented in Sections
3 and 4. The physical origin of these novel kinds of quantum behavior and
purely-internal behavioral transitions, however, is _the merge between state-
history participations and a ‘renewable’ operator of evolution Unitarity._ The
surprising characteristics of the behavioral landscape of the one-qubit QMM-
UEs which were detailed and explained in Section 6, do clearly indicate to us
that these novel systems can naturally find promising applications in several
envisioned domains and disciplines which shall be mentioned in the next
subsection.

_Independently_ , we have advanced towards a speculative idealized quantum
scheme in which _states, interactions and evolutions are all unified_. The
extreme message of such a unified scheme is most clearly conveyed by the
scenario (132). Taking the subsystem decomposition in the recipe (132) all the
way down to its ultimate level of resolution, a radical perspective of a
consistently-circular interplay manifests. It speculates a closed universe in
which all many-body interactions and all evolutionary processes are remade
continually from within its internal experience(s), _and vice versa_.

###  7.2 Forward Views: A Generalization, Selected Open Questions And The
Envisioned Applications In Fundamental And Applied Domains

QMM-UEs can naturally find promising applications in several distinct domains,
as we envision and conjecture. Clearly, there are also many directions to
advance the work of the present paper, and there are many interesting open
questions to be addressed. We now elaborate, on a selective basis, on some
open questions, future advancements, and envisioned applications of QMM-UEs.

Firstly, we present an alternative more general recipe to formulate the QMM-
UEs for a general closed system, denoted by $\mathcal{S}$. Indeed, it is
within the _very same_ paradigm we conceptualized in Section 3 that this
second recipe is now proposed. Nevertheless, it generalizes the formulation in
(7,8). Consider a partitioning of $\mathcal{S}$ into $M$ interacting
subsystems $\\{\mathcal{S}_{i}\\}_{i=1}^{M}$, mirroring an arbitrary Hilbert
space factorization,
$\mathcal{H}^{(\mathcal{S})}=\otimes_{i=1}^{M}\mathcal{H}_{i}$. _The second
recipe_ is to define the system’s total QMM-H,
$H_{(\mathcal{S})}^{\text{QMM}}$, directly based on the chosen subsystem
decomposition, using the past-or-present states of the subsystems
$\rho_{i}(t_{i})$, defined by the complementary-space traces
$\rho_{i}(t_{i})\equiv\text{Tr}_{\mathcal{H}_{\bar{i}}}[\rho^{(\mathcal{S})}(t_{i})]$.
One chooses a set of QMM $r$-body operators interlinking the $M$
$\mathcal{S}_{i}$’s, $\mathcal{O}^{\text{QMM}}_{i_{1}\cdots
i_{r}|t_{i_{1}}\cdots
t_{i_{r}}}[\;\rho_{i_{1}}(t_{i_{1}}),\cdots,\rho_{i_{r}}(t_{i_{r}})\;]$, and
packs them into a Hermitian $H_{(\mathcal{S})}^{\text{QMM}}$. For example, one
can form such a $H_{(\mathcal{S})}^{\text{QMM}}$, by introducing
$\mathcal{O}^{\text{QMM}}_{i_{1}\cdots i_{r}|t_{i_{1}}\cdots t_{i_{r}}}$ with
$r\leq 3$, together with the corresponding collections of state-history-
independent operators and the $\lambda$-couplings $(\mathcal{A},\Lambda)$,
likewise (7,8), as,

$\begin{split}&H_{(\mathcal{S})}^{\text{QMM}}(t)\;\equiv\;H_{(\text{Polynomial},\;\text{Exponential},\;\cdots)}^{(\text{Hermitian})}[\;\\{\;\mathcal{O}_{i|t_{i}}^{\text{QMM}}\;\\}\;,\;\\{\;\mathcal{O}^{\text{QMM}}_{ij|t_{i}t_{j}}\;\\}\;,\;\\{\;\mathcal{O}^{\text{QMM}}_{ijk|t_{i}t_{j}t_{k}}\;\\}\;|\;(\mathcal{A}\;,\;\Lambda)\;],\\\
&\mathcal{O}_{i|t_{i}}^{\text{QMM}}\equiv\rho_{i}(t_{i})\otimes\mathbb{1}_{\bar{i}}\;\;;\;\;\mathcal{O}^{\text{QMM}}_{ij|t_{i}t_{j}}\equiv\rho_{i}(t_{i})\rho_{j}(t_{j})\otimes\mathbb{1}_{\overline{ij}}\;\;;\;\;\mathcal{O}^{\text{QMM}}_{ijk|t_{i}t_{j}t_{k}}\equiv\rho_{i}(t_{i})\rho_{j}(t_{j})\rho_{k}(t_{k})\otimes\mathbb{1}_{\overline{ijk}},\\\
&\rho_{i}(t_{i})=\text{Tr}_{\mathcal{H}_{\bar{i}}}[\rho^{(\mathcal{S})}(t_{i})]\;;\;\mathcal{H}^{(\mathcal{S})}=\mathcal{H}_{i}\otimes\mathcal{H}_{\bar{i}}\;,\;\hskip
18.49411pt\forall\;t,t_{i},t_{j},t_{k}\in\;[t_{*},t]\;,\;\forall\;i,j,k\in\;1,\cdots,M.\end{split}$
(132)

For $M=1$ and Polynomial QMM-Hs, the above recipe reduces to the recipe
presented in (7,8). Nevertheless, they represent two distinctive scenarios to
form QMM-UEs. In the first scenario the QMM-UE of a closed system is directly
driven from the history of its total-or-‘global’ quantum states. In the second
scenario, however, a closed system interconnects the state-histories of its
chosen subsystems, to structure its QMM-UE.

Regarding analysis extensions, an obvious direction is studying the closed
system of $M$ interacting qubits, for small $M$ and $M\to\infty$, and $M$
quantum fields (of various kinds and symmetries) endowed with QMM many-body
interactions. We have focused on a single qubit, but the Hubbard-model-like
reformulation of the $N=2$ one-qubit system in (71,LABEL:hubbardcouplingsare),
manifests possible forms of two-body and higher $r$-body QMM interactions. It
is straightforward to extract these interactions out of the two recipes,
likewise one-qubit cases in Section 4. Investigating the behaviors of the QMM
variants of quantum spin systems, Hubbard models, and various QFTs can be
insightful.

Another interesting direction of furthering our analyses corresponds to QMM-
UEs sourced by interacting quantum memories which fill up a continuum
state-(sub)history. One re-introduces the $H_{\text{(QMM)}}^{(N;L)}$’s defined
in (7,8,14) in their $N\to\infty$ QM continua, integrating degree-$l^{(\leq
L)}$ monomials of the past-or-present density operators over entire
participating subsets of $[t_{*},t]$. _For example_ , choosing a ‘QM-width’
parameter $a\in[t_{*},t]$ and taking $L=2$, one intuitive QMM-H made by all
quantum states in the history window $[t-a,t]$,
$H_{\text{QMM}}^{(\infty|2|a)}(t)$, can be formulated as the following
reformation of (15),

$\begin{split}&H_{\text{QMM}}^{(\infty|2|a)}(t)\;\equiv\;\\\
&\equiv\;\int_{t-a}^{t}dt^{\prime}\;f_{t^{\prime}t}\;\big{[}\;\mu_{t^{\prime}}(\mathcal{P}^{{\rm
chosen}}_{[t-a,t]})\;\rho_{t^{\prime}}\;+\;i\;\lambda^{I}_{t^{\prime}t}(\mathcal{P}^{{\rm
chosen}}_{[t-a,t]})\;[\;\rho_{t^{\prime}}\;,\;\rho_{t}\;]\;+\;\lambda_{t^{\prime}t}^{R}(\mathcal{P}^{{\rm
chosen}}_{[t-a,t]})\;\\{\;\rho_{t^{\prime}}\;,\;\rho_{t}\;\\}\;\big{]}.\;\end{split}$
(133)

The temporal functions $f_{t^{\prime}t}$ multiply the QM-couplings
$\big{(}\mu_{t^{\prime}},\lambda_{t^{\prime}t}^{I},\lambda_{t^{\prime}t}^{R}\big{)}$
in order to weight the participation of those quantum memories. One can choose
a monotonically-decreasing tunable function $f_{t^{\prime}t}=f(t-t^{\prime})$.
Moreover, one can choose the QM-couplings
$(\lambda_{t^{\prime}t}^{I,R},\mu_{t^{\prime}})$ to be themselves functions of
the quantum-memory fidelities or various other kinds of state-history-
dependent scaler observables, incorporating higher-oder interactions and
quantum memory effects already at the $L=2$ level.

One can investigate statistical behavioral effects of ‘memory-participation
randomness’ in QMM-UEs, using the same general recipe of QMM-Hs formulated in
(5, 6,7,8).
For example, considering a probability distribution $\mathcal{A}$ with mean
value $a$, variance $\alpha^{2}$, and support
$D_{{}_{\mathcal{A}}}\subset\mathbb{R}^{+}$, a QM-random $H^{(2|2)}_{{\rm
QMM}}(t)$ is formulated by imposing that the participating past-states are
chosen moment by moment, based on a random-variable QM distance drawn from
$\mathcal{A}$. The QMM-H reads as,

$H^{(2|2)}_{{\rm
QMM}}(t|\mathcal{A})\;=\;\mu\;\rho_{t-a_{t}}+\;i\lambda^{i}\;[\;\rho_{t-a_{t}}\;,\;\rho_{t}\;]+\lambda^{R}\;\\{\;\rho_{t-a_{t}}\;,\;\rho_{t}\;\\}\;\;\;;\;\;\;a_{t}^{(\in\;\mathbb{R}^{+})}\;\sim\;\mathcal{A}(\;a,\alpha^{2}\;).$
(134)

Moreover, one can take some or all independent dimensionless couplings to be
drawn from probability distributions. Mapping disorder-averaged phase diagrams
of random QMM-UEs (134) and their generalizations is interesting. In the
context of their spin models, it leads us to various novel families of quantum
spin glasses endowed with QMM random interactions.

We find it natural, moreover theoretically elegant, to reformulate general
QMM-UEs in the ‘History Hilbert space’ frameworks such as [67, 68, 72], using
higher quantizations and the postulated algebras of quantum-memory fundamental
operators. It can serve as a useful abstract language to investigate and
establish general characterizations of QM many body interactions.
Independently, another advancement of our analyses is to explore QMM-UEs made
of general mixed quantum states, mapping their richer behavioral phases and
dynamical phase transitions.

The hybrid family of QMM-UEs, generated by combining conventional (time-
independent or local-in-time) Hamilonians and Purely-QMM Hamiltonians,
constitute novel closed quantum systems of special importance to be
investigated extensively. The importance is both theoretical and
observational. Because the limit of QMM-UEs with extreme memory dependency,
namely the case of Purely-QMM-Hs, was totally unexplored while quite
interesting and sufficiently complex, it constituted the bulk of our analyses
in the present work. However, the analysis of Subsection 5.4 shows that the
hybridization is qualitatively nontrivial: some principal features of the pure
limit do remain stable, other behavioral aspects undergo interesting
deformations. Hence, the phase diagram of hybrid QMM-UEs is a matter for
careful explorations. Phenomenologically, hybrid Hamiltonians which are
dominantly conventional, that is coming from perturbing the conventional
Hamiltonians by means of QMM operators, are specially interesting. This is
because, in the context of closed quantum systems in nature, even if one finds
some structural Non-Markovianity in their unitary evolutions, the
observational effects likely come from some very-weakly-coupled perturbative
QMM operators [87]. Nevertheless, because of aspects such as ‘mobility
phenomenon’ [69, 70, 71], perturbative couplings can be impactful. Hence, one
specially interesting question for future works is to pinpoint robust
fingerprints of perturbatively-coupled QMM operators on two-or-higher point
functions of observables in cosmological settings and laboratory-setup closed
quantum matter. Indeed, _investigating and characterizing such robust traces
should be a crucial advancement toward discovering phenomenological landmarks
of QMM-UEs_[87].

The capacity of our model to explore a wide range of regimes, from a
completely memory-less all the way to a heavily non-Markovian regime, places
our work naturally in the stream of research studying the crossover from a
Markovian to a non-Markovian dynamics, explored previously on the theory side
in works such as [77], [78] and [79], and on the experimental side in [80] in
which full control over the (non-)Markovian character of the dynamics could be
achieved in an all-optical setup. Moreover, the ability to access explicitly
the past states of the system can potentially shed more light on the
connection between memory in quantum dynamics and quantum correlations, a
topic already explored for example in [26, 27, 28, 29, 30]. The main focus of
these works is on the momentary system-environment correlations, while we can
look instead at time correlations of the system with itself. In the common
scenario where memory effects are induced by the system-environment
correlations, the environment plays the role of an intermediary, storing and
carrying over traces of the system’s history. In a way, our approach is more
fundamental, since we wish to look instead at a purely-internal source of
participating quantum memories: the closed quantum system itself.

_The experimental realization of the QMM-UEs_ defined in (4) is a subtle
exciting open question. One can first consider the scenarios (7,8,14) and
(132) which are two specific ways of structuring (4) as _the_ general
definition of QMM-UEs. In either scenario, a closed system must select and
connect its available ‘internal experiences’, as contained in its quantum-
state-history, to update its momentary unitary evolution. They however differ
in the way that this central point is structured. In (7,8,14), the available