# Unitary Evolutions Sourced By Interacting Quantum Memories: Closed Quantum
Systems Directing Themselves Using Their State Histories

Alireza Tavanfar<EMAIL_ADDRESS>Champalimaud
Research, Champalimaud Center for the Unknown, 1400-038 Lisboa, Portugal
Institute of Neuroscience, University of Oregon, Eugene, OR 97403, USA
Aliasghar Parvizi<EMAIL_ADDRESS>Department of Physics, University of
Tehran, 14395-547, Tehran, Iran  School of Particles and Accelerators,
Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531
Tehran, Iran  Marco Pezzutto<EMAIL_ADDRESS>Complex Systems and
Statistical Mechanics, Physics and Materials Science Research Unit, University
of Luxembourg, L-1511 Luxembourg

###### Abstract

We propose, formulate and examine novel quantum systems and behavioral phases
in which momentary choices of the system’s memories interact in order to
source the internal interactions and unitary time evolutions of the system. In
a closed system of the kind, the unitary evolution operator is updated, moment
by moment, by being remade out of the system’s ‘experience’, that is, its
quantum state history. The ‘Quantum Memory Made’ Hamiltonians (QMM-Hs) which
generate these unitary evolutions are Hermitian nonlocal-in-time operators
composed of arbitrarily-chosen past-until-present density operators of the
closed system or its arbitrary subsystems. The time evolutions of the kind are
described by novel nonlocal nonlinear von Neumann and Schrödinger equations.
We establish that nontrivial Purely-QMM unitary evolutions are ‘Robustly Non-
Markovian’, meaning that the maximum temporal distances between the chosen
quantum memories must exceed finite lower bounds which are set by the
interaction couplings. After general formulation and considerations, we focus
on the sufficiently-involved task of obtaining and classifying behavioral
phases of one-qubit pure-state evolutions generated by first-to-third order
polynomial QMM-Hs made out of one, two and three quantum memories. The
behavioral attractors resulted from QMM-Hs are characterized and classified
using QMM two-point-function observables as the natural probes, upon combining
analytical methods with extensive numerical analyses. The QMM phase diagrams
are shown to be outstandingly rich, having diverse classes of unprecedented
unitary evolutions with physically remarkable behaviors. Moreover, we show
that QMM interactions cause novel purely-internal dynamical phase transitions.
Finally, we suggest independent fundamental and applied domains where the
proposed ‘Experience Centric’ Unitary Evolutions can be applied natuarlly and
advantageously.

###### Contents

  1. 1 Introduction
  2. 2 Unitary State-History-Based Quantum Behavior:  The Hamiltonian Formulation
    1. 2.1 The Central Idea
    2. 2.2 Quantum-Memory-Made Hamiltonians: A General Formulation
    3. 2.3 A Polynomial Family of Quantum-Memory-Made Hamiltonians
    4. 2.4 Pure State Histories Composing QMM Unitary Evolutions
  3. 3 The Dynamical Equations Of Unitary State-History-Based Quantum Behavior
    1. 3.1 QMM von Neumann and Schrödinger Equations
    2. 3.2 Detailing A Family of QMM-Hs With Two Quantum Memories
    3. 3.3 The Robust Non-Markovianity of Purely-QMM Unitary Evolutions
  4. 4 Purely-QMM Unitary Evolutions of The One-Qubit Closed System
    1. 4.1 State-History Representations And Quantum Memory Characters
    2. 4.2 (1,1) QMM Unitary Evolution: The Simplest QMM-H Made Of One Quantum Memory
    3. 4.3 (2,2) QMM Unitary Evolution: QMM-H Made By Two System’s States With Monomial Interactions Up To Second Degree
    4. 4.4 (2,3) QMM Unitary Evolution: QMM-H Made By Two System’s States With Monomial Interactions Up To Third Degree
    5. 4.5 (3,3) QMM Unitary Evolution: A Specific QMM-H Made By Three System’s States With Third-Degree Monomial Interactions
  5. 5 QMM Unitary Evolutions of The One-Qubit Closed System: Analytical Results
    1. 5.1 The General Scope of Analytical Investigations
    2. 5.2 Purely-QMM Unitary State-Histories On The ‘$\theta$-Orbit’
      1. 5.2.1 The Profile-Invariant State-History, Stability, Dynamical Attractor
      2. 5.2.2 Generic Purely-QMM Unitary State-Histories On The $\theta$-Orbit
      3. 5.2.3 Determining The Thresholds Along The $\theta-$Orbit
      4. 5.2.4 The $\theta$-orbit Purely-QMM-UEs In Their Infancy
    3. 5.3 Purely-QMM Unitary State-Histories On The ‘$\phi$-Orbit’
      1. 5.3.1 The $\phi$-Orbit Thresholds
    4. 5.4 Hybrid Unitary Evolutions: General-Orbit Near-Markovian Solutions
  6. 6 Mapping The Phase Diagrams of One-Qubit Pure States Evolving Under Purely-QMM-Hs: Numerical Investigations
    1. 6.1 Generalities About The Numerical Analyses And The Results
    2. 6.2  The Phase Diagram of One-Qubit Wavefunctions Evolving Under The (1,1) Purely-QMM Hamiltonians
    3. 6.3 The Phase Diagram of One-Qubit Wavefunctions Evolving Under The (2,2) Purely-QMM Hamiltonians
      1. 6.3.1 The General-Orbit Phase Diagram
      2. 6.3.2 Supplementary Numerical Solutions Along The $\theta$-orbit
    4. 6.4 The Phase Diagram of One-Qubit Wavefunctions Evolving Under A (2,3) Purely-QMM Hamiltonian
    5. 6.5 The Phase Diagram of One-Qubit Wavefunctions Evolving Under A (3,3) Purely-QMM Hamiltonian
    6. 6.6 Selected General Highlights On One-Qubit Purely-QMM-UEs
  7. 7 QMM-UEs: Outlook, Open Questions And Visions On  ‘Experience Centric Quantum Theory’
    1. 7.1 A Conceptual Overview And Selected Elucidations
    2. 7.2 Forward Views: A Generalization, Selected Open Questions And The Envisioned Applications In Fundamental And Applied Domains
    3. 7.3 Conclusion
  8. 8 Dedication
  9. 9 Acknowledgments
  10. A Purely-QMM Kicker Hamiltonians

## Table of Acronyms

Acronym | Meaning
---|---
QM | Quantum Memory
QMM | Quantum Memory Made
QMM-UE | Quantum Memory Made Unitary Evolution
QMM-H | Quantum Memory Made Hamiltonian
QMD | Quantum Memory Distance
QM-TPF | Quantum Memory Two Point Function
QM-TPO | Quantum Memory Two Point Operator
QMM-MF | Quantum Memory Made Magnetic Field
LHS | Left Hand Side
RHS | Right Hand Side

##  1 Introduction

In conventional treatments of the unitarily-evolving quantum systems, it is
typically assumed that many-body interactions and the dynamical equation of
the system are (as physically postulated and mathematically formulated,)
independent of the system’s quantum-state history. In this discipline, the
total Hamiltonian generating the system’s unitary time evolution is a (time-
independent or time-dependent) Hermitian operator that is ‘implanted’ inside
the system, independent of which trajectory of quantum states the system has
been taking up until the present moment. But, this is not necessarily the most
general framework to be taken, as one can envision. In fact, one naturally
tends to investigate general formulations and concrete realizations of
_fundamental-or-emergent closed quantum many body systems which function
beyond this established discipline_. Intending this investigation in the
present work, and regarding intrinsic connections between the state-history of
a quantum system and the system’s quantum memories, we begin with a very
selective and brief overview on the conventional perspective and framework of
Non-Markovian dynamics in quantum many body systems.

Time evolution of a quantum system is typically described through one of three
main types of processes [1, 2, 3]: i) the unitary evolution of a closed system
generated by a Hamiltonian, be it time-independent or time-dependent subject
to temporal locality; ii) the non-unitary evolution which is induced by a
quantum measurement process; iii) the dissipative, irreversible dynamics of an
open system undergoing interactions with an environment, as for example
formulated by a Lindblad Markovian master equation. At least for two reasons,
processes of type ii) and iii) are not completely independent: on the one
hand, the action of a quantum measurement repeated a sufficient number of
times can lead to an effective description in terms of a master equation for
the reduced dynamics of the system; on the other hand, the action of an
environment interacting with the system can sometimes be interpreted as a
series of repeated measurements.

However, processes of type i) and iii) are opposite in characteristics. The
unitary evolution of every closed system under a time-independent Hamiltonian
(with discrete spectrum) is quasi-periodic, due to the quantum Poincaré
recurrence theorem [4], while an open system evolving under a Lindblad master
equation can eventually evolve to attractor asymptotic states independent of
the initial state, such that the system’s initial-state information is not
preserved. Furthermore, while the interaction with the environment brings
dissipative effects, and so irreversibility in the dynamics, it can also
produce coherent effects developed by modifications to the Hamiltonian
dynamics [1, 5, 6, 7]. Between these two opposite settings, there comes a
variety of more complex scenarios in which the interacting environment does
not simply act as a sink in which the information is lost. Instead, the
system-environment interactions and the internal dynamics of the environment
may be such that the information from earlier quantum states of the system can
be dynamically fed back to the system. This interplay results in Non-Markovian
profiles for the system’s time evolution, because of becoming by nature state-
history dependent. Indeed, memory dependencies in quantum dynamics have been
acknowledged since the birth of the theory of open quantum systems. A step
taken often in deriving master equations is the Born-Markov approximation in
which the environment quickly ‘forgets’ any trace of its interactions with the
system. Hence, the system’s time evolution is described by a local-in-time
differential equation which is independent of the system’s (long or short)
state-history. Such a decoupling can be reasonable only under certain
conditions, like for sufficiently-weak system-environment couplings, or if the
characteristic timescale of the environment dynamics is sufficiently shorter
than the one of the system.

However there are many important physical settings in which the approximation
of system’s memory-independent dynamics breaks down significantly, leading to
complex Non-Markovian evolutions. Despite their relevance and significance,
only in recent times a promising deal of coherent research has been conducted
on analyzing these much-more-challenging settings. Indeed, in an increasingly
more diverse spectrum of domains in open quantum systems, finding out the many
impacts of strong memory effects has turned into a doorway to many body
physics scenarios with remarkable novelties from the standpoints of
theoretical and experimental quantum physics, and even mathematics. See,
merely as few examples, [8, 9, 10]. In this realm, developing more inclusive
definitions, characterizations and measures of Non-Markovian quantum processes
is still an evolving endeavour, hosting a variety of approaches, proposals and
concrete formulations [11, 12, 13]. These distinct approaches are based upon a
variety of different but compatible standpoints such as, the backflow of
quantum information from the environment to the system [14, 15, 16], the
divisibility of the dynamics [17], the evolution of the geometric volume in
the state space accessible to the system [18], and several more. Increasing
efforts are devoted to develop a comprehensive framework which sharpens
similarities and discrepancies between all these approaches [19, 20, 21, 22,
23, 24, 25].

Most known methods of quantifying non-Markovianity are formulated from the
very perspective of the system’s dynamics alone: access to the reduced density
operator of the system during its time evolution should be sufficient to
accurately decode the required signatures of Non-Markovianity. This dominant
standpoint has the double advantage of being minimalistic in formulation,
while also more practical form both theoretical and experimental viewpoints,
but is incapable of satisfactorily capturing the root causes of the emergence
of Non-Markovianity and memory effects in open quantum systems. However,
manifesting directly in the formulation, the actual role of the system-
environment correlations in sourcing and mediating the memory effects is
attracting increasing attention [26, 27, 28, 29, 30]. To this aim, a variety
of scenarios have been developed in recent years where the interacting
environment and its interplays with the system are modelled explicitly [31,
32, 33], allowing us to understand better the physics of crossovers between
Markovian and Non-Markovian evolutions [34, 35, 36, 37, 38, 39, 40].

Finally, recent promising advancements in experimental techniques for
controlling and manipulating quantum systems have led to the first direct
observations of Non-Markovian dynamics [41, 42, 43, 44, 45, 46, 47], while its
impact in a whole variety of areas has been also addressed. These important
areas, to name a few, include topological phases of matter [48, 49, 50],
quantum optics [51, 52, 53, 54], thermodynamics [55, 56, 57, 58] and
entanglement dynamics [59, 60, 61].

_The present work_ , however, aims to take the study of quantum-memory effects
to the very extreme: _we develop an idealization of a Non-Markovian quantum
dynamics which is completely coherent, free from any dissipative effect, and
retains the totality of the specifying qubits of the system’s states at past
times_. Stated more specifically, we study the time evolution of general
closed quantum systems under a completely new kind of Hamiltonians which are
(functionally and explicitly) made from arbitrary choices of the system’s
states in the course of its history from an initial time up to the present
time. Therefore, the physical context is that of a unitarily-evolving quantum
many body system whose many-body interactions and Hamiltonian emerge, moment
by moment, in an intrinsically Non-Markovian way: at every present moment, the
closed quantum system itself reconstructs its internal interactions and its
Hamiltonian by ‘connecting’ or ‘linking’ (a number of) its quantum memories.
In effect, we develop an original general type of quantum dynamics which is
marked with two complementary faces, at once and consistently: _the system’s
evolution is Non-Markovian and Unitary._

We propose and work out the most direct definition and formulation of the
promised theories for generally-defined closed quantum systems. One first
envisions a unitary quantum state history of a closed system beginning from
some initial moment $t_{0}$ up to the present moment $t$. We now identify the
density operator of the system, $\rho(t^{\prime})$, at each moment
$t^{\prime}\in[t_{0},t]$, with the _quantum memory_ of the closed system at
that moment. Henceforth, the quantum state history of the system evolves
onwardly, on a moment-by-moment basis, using the available quantum memories.
In this way, an entire unitary quantum state history can be constructed from
$t_{0}$ all the way to $t=\infty$. Concretely speaking, the closed system
‘selects’ and combines its available quantum memories at every present moment
to construct a composite (generically nonlocal in time) state-history-
dependent Hamiltonian, being called _‘Quantum-Memory-Made Hamiltonian’
(QMM-H)_ , which generates its unitary time evolution operator.

But, mathematically formulating a QMM-H is one thing, establishing that the
resulted dynamical equations can develop solutions which are consistent
physically is another thing. In particular, physical consistency of the QMM
unitary evolution implies that the quantum memories which construct the
Hamiltonian polynomial are _not_ actually independent variables. That is, any
pair of system’s density operators at times $t^{\prime}$ and
$t^{\prime\prime}$ which come in the above proposed construction of the
Hamiltonian at the present time $t$, must be themselves obtainable from one
another under the accordingly-constructed time evolution operator in between
the times $t^{\prime}$ and $t^{\prime\prime}$.

The upshot becomes the statement that the physical solutions to the nonlocal-
in-time nonlinear Schrödinger equation resulted from the QMM-H must realize,
one by one, a unitarily self-consistent state-history for the system. After
proving the consistency of the theory, it is important to establish that the
physical moduli space of the QMM Schrödinger equations, namely that which
corresponds to its unitarily self-consistent solutions, is a large and rich
landscape which, compared to the conventional settings, accommodates novel
phases of unitary state-histories, and also novel phase transitions. _This
demanding task_ consists the focus of a large part of the present paper which,
by means of analytical and numerical analyses in combination, is done
successfully.

A brief outline of the paper is now presented with the selected highlights. We
commence with a general theory of ‘$(N,L)$ QMM-Hs’, formed as order-$L$
polynomials made of $N$ quantum memories, together with their highly-novel
Schrödinger equations. There, we also introduce the ‘Quantum Memory
Characters’, specially the ‘QMM Two-Point Functions’, in terms of which QMM
Unitary Evolutions (QMM-UEs) are reformulated and characterized naturally and
usefully. Moreover, the general (2,2) QMM-UEs are detailed for further
clarifications. Finally, we establish that all purely-QMM unitary dynamics are
‘Robustly Non-Markovian’, meaning that non-constant wavefunctions are
developed only when the maxima of the ‘quantum-memory distances’ exceed the
finite thresholds which depend on the QMM-H couplings.

In the course of Sections 4, 5 and 6, given total novelty and significant
theoretical complexity of QMM-UE, we focus on exploring the most elementary
case studies: the closed system of one qubit evolving under (1,1), (2,2),
(2,3) and (3,3) QMM-Hs. First, the dynamical equations of all these cases are
detailed with important explanations, in Section 4. Section 5 presents a
variety of analytic solutions and results for the (2,2) QMM-UE. In Section 6,
phase diagrams of one-qubit wavefunctions made purely by quantum memories are
analyzed for each one of the above QMM-Hs, finding wide classes of
qualitatively unprecedented unitary quantum state histories. Moreover, we
present several selected examples of Dynamical Phase Transitions which are
unitarily developed in the course of the evolutions generated by QMM-Hs (in
even the most elementary closed quantum system, namely) the one-qubit closed
system, as a result of the higher order interactions between the system’s
quantum memories. Moreover, we present highlighted lessons, selected from all
these analyses, to complete Section 6.

The Proposal is revisited, elucidated and also evaluated in Section 7. On one
hand, we highlight several interesting open questions to be addressed in the
future works. On the other hand, we briefly present our current visions on the
diverse categories where QMM-UEs can be realized and utilized advantageously,
in future.

##  2 Unitary State-History-Based Quantum Behavior:
The Hamiltonian Formulation

###  2.1 The Central Idea

The central mission of the present work is to address and investigate in
details the question phrased below, which is significant and intriguing, but
largely unexplored. The roadmap we use to answer it, is first presenting the
general formulation of the merging envisioned in it, followed by extensive
(analytical and numerical) explorations of the resulted novel quantum
behaviors, in the simplest specific models and in some general cases.

_Can one unify, in the context of a closed quantum system and in a consistent
way, the principle of system’s time evolution unitarity with one’s request for
the system’s many-body interactions and total Hamiltonian being (entirely or
partially) made from the system’s state history?_

We now begin with presenting a quite general formulation of self consistent
unitary evolutions whose generating Hamiltonians are being made (partially or
entirely) by the past-until-present quantum-state-history of a general closed
system, and subsequently realizing the main implied features of such proposed
systems upon undertaking diverse analyses. Visions related to (the inquiry of)
_what kinds of natural or artificial systems_ can be described by these novel
Hamiltonians is indeed an intriguing and important question which we postpone
to _Section_ 7.

On general grounds, the defining operator of the total Hamiltonian of a closed
quantum many body system can have a _dichotomic_ structure (effectively or
fundamentally) as follows,

$\begin{split}&H_{{\text{(The Most
General)}}}^{\text{(Total)}}(t)\;=\;H_{(\text{State-History
Independent})}(t)\;+\;H_{(\text{State-History Dependent})}(t)\;\equiv\;\\\
&\hskip 42.67912pt\equiv\;H_{(\text{Conventional})}(t)\;+\;H_{(\text{Quantum
Memory Made})}(t)\;\equiv\;H_{{\rm C}}(t)\;+\;H_{{\rm QMM}}(t).\end{split}$
(1)

Given a closed quantum many body system at hand, the conventional part
$H_{C}(t)$ in definition (1) is the most general Hamiltonian one can formulate
out of the complete set of operators whose definitions do not depend on the
system’s state history. The ‘Quantum-Memory-Made’ (QMM) contributions to
($\ref{H-CQS})$, collectively named $H_{{\rm QMM}}(t)$, is the most general
Hamiltonian one can formulate as a Hermitian composite operator whose
constituting components have intrinsic dependencies on the state history of
the system which is developed from an initial time to the present moment.
Conventional Hamiltonians, collectively denoted by $H_{C}(t)$, can be either
constant operators or time dependent operators. In contrast, QMM-Hs do always
have time dependencies which are induced by their being made from the ongoing
state histories of the closed systems. In what follows from now, we develop a
general formulation of $H_{{\rm QMM}}(t)$ made directly from momentarily-
chosen past-until-present quantum states of the closed system.

###  2.2 Quantum-Memory-Made Hamiltonians: A General Formulation

Suppose we have a completely arbitrary closed quantum system and want to
consider its Hamiltonian-generated unitary evolution all the way from an
initial time to infinity, $t\in[t_{0},\infty)$. One straightforward way to
represent the system’s state-history up to the present moment $t$ is the
continuos set of density operators which represent the system’s quantum states
at the corresponding times in the course of this history,

$\mathcal{P}_{[t_{0},t]}\;\equiv\;\\{\leavevmode\nobreak\
\rho(t^{\prime})\;,\;\forall\leavevmode\nobreak\
t^{\prime}\in[t_{0},t]\leavevmode\nobreak\
\\}\;\subset\;\lim_{t\to\infty}\mathcal{P}_{[t_{0},t]}\;\equiv\;\mathcal{P}_{[t_{0},\infty)}.$
(2)

It is clear that every single bit of quantum information which is contained in
each one $\rho(t^{\prime})\in\mathcal{P}_{[t_{0},t]}$, namely in every
system’s past-or-present state, is one ‘qubit memory’ of the system. In other
words, the qubit memories of a quantum system up until a present time $t$
correspond, in a one to one manner, to all the quantum information bits
necessary and sufficient to specify the system’s quantum-state-history
$\mathcal{P}_{[t_{0},t]}$ uniquely. Likewise, we refer to every density
operator in the quantum-state-history $\mathcal{P}_{[t_{0},t]}$ as _one
‘quantum memory’ of the system_. Note that by a slight extension of
terminology, the system’s momentary density operator is likewise to be called
a quantum memory. We, now highlight a point which is conceptually essential to
all the formulation to be put forth: a contrast between (2) for a closed
quantum system and its counterpart in an open quantum system. The quantum
systems whose state-histories $\eqref{densityOp}$ we address here are
isolated, undergoing time evolutions which are reversible by being unitary. By
this virtue, the totality of the quantum information by which the system’s
states, $\rho(t^{\prime})\in\mathcal{P}_{[t_{0},\infty)}\;\forall t^{\prime}$,
are uniquely specified, are enforced to stay inside the entirety of the
quantum system and be eternally preserved. Hence, what plays the role of an
‘storage’, containing all the memories to be utilized in the construction of
$H_{{\rm QMM}}$, is the entire quantum system itself. Re-emphasizing that the
closed quantum many-body system under consideration has an arbitrary nature in
terms of its defining degrees of freedom, and can have a wide range of
defining structures, we now move on to formulate the envisioned evolution.

_Our aim is_ to formulate a unitary evolution operator whose generating
Hamiltonian $H_{{\rm QMM}}(t)$, at each present moment $t$, emerges from a
chosen assembly of the system’s available qubit memories, being arbitrarily
selected. That is, $H_{{\rm QMM}}(t)$ is (re-)formed, moment by moment, as a
nonlocal-in-time Hermitian operator which features structural dependence on
the information content of some chosen elements of
$\mathcal{P}_{[t_{0},t]}\;$. Therefore, the qubit memories contained in
system’s density operators at arbitrary past-to-present moments,
$\rho(t^{\prime}_{{\rm chosen}})\equiv\rho_{t^{\prime}_{{\rm
chosen}}}\in\mathcal{P}_{[t_{0},t]},\;\forall\;t^{\prime}_{{\rm
chosen}}\in[t_{0},t]$, participate in the momentary re-construction of the
Hamiltonian $H_{{\rm QMM}}(t)$. To make it more clear, we first determine a
subset of $\mathcal{P}_{[t_{0},t]}$, denoted by $\mathcal{P}^{{}_{{\rm
chosen}}}_{[t_{0},t]}$, which contains (nothing but all) the system’s
‘participant quantum memories’ out of which $H_{{\rm QMM}}(t)$ is to be made,

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{P}^{{\rm
chosen}}_{[t_{0},t]}\;\equiv\;\\{\leavevmode\nobreak\ \rho_{t^{\prime}_{{\rm
chosen}}}\;,\;\forall\leavevmode\nobreak\ t^{\prime}_{{\rm
chosen}}\in[t_{0},t]\leavevmode\nobreak\
\\}\;\subset\;\mathcal{P}_{[t_{0},t]}.$ (3)

We now state the general definition of a _QMM Hamiltonian (QMM-H)_ as follows,

$H_{{\rm QMM}}(t)\;=\;\mathcal{O}[\;\mathcal{P}^{{\rm
chosen}}_{[t_{0},t]}\;].$ (4)

In definition (4), $\mathcal{O}=\mathcal{O}^{\dagger}$ denotes a Hermitian
operator and the brackets (in the R.H.S of it) mean _‘explicit structural
dependence’_. Overall, the definition (4) states that a QMM-H at present time
$t$ is a Hermitian operator with explicit structural dependence on a chosen
subset of the past-to-present quantum-state-history $\mathcal{P}^{{\rm
chosen}}_{[t_{0},t]}$. Obviously, there are infinitely-diverse ways to
introduce operators $\mathcal{O}$ which realize (4), mainly based on: a) how
one extracts the qubit memories contained in the chosen quantum memories
$\rho_{t^{\prime}_{{\rm chosen}}}\;\forall t^{\prime}_{{\rm chosen}}$, and b)
how one assembles those qubit memories to construct a Hermitian operator as
the QMM-H. Among all possibilities, in what follows, we choose an infinitely-
large family of QMM unitary evolutions (QMM-UEs) whose QMM-Hs are formulated
by taking the straightforward way to realize (4). _We introduce QMM-Hs which,
as Hermitian composite operators, are functions, specially polynomials, of the
density operators which consist $\mathcal{P}^{{\rm chosen}}_{[t_{0},t]}$_ (3).
That is, $\mathcal{O}[\mathcal{P}^{{\rm chosen}}_{[t_{0},t]}]$ is a nonlocal-
in-time Hermitian operator whose constituting operators _include_ all the
chosen past-or-present density operators of the system, namely all the chosen
quantum memories themselves: $\rho_{t^{\prime}_{{\rm chosen}}}\;\forall
t^{\prime}_{{\rm chosen}}$. Furthermore, we can let $\mathcal{P}^{{\rm
chosen}}_{[t_{0},t]}$ be itself a countable or uncountable set. For
concreteness, we take the former possibility in what follows, returning to the
latter possibility in Section 7.

###  2.3 A Polynomial Family of Quantum-Memory-Made Hamiltonians

Having made all these specifications, one concrete family of QMM-Hs is now
presented according to the following _recipe_.

The recipe begins with choosing a doublet of positive integers, $(N,L)$. $N$
equals $|\mathcal{P}^{{\rm chosen}}_{[t_{0},t]}|$ and is the total number of
the system’s past-or-present density operators out of which the present-time
QMM-H, $H_{{\rm QMM}}(t)$, is constructed. $L$ is the maximum algebraic length
of the QMM operator monomials, made of the $\mathcal{P}^{{\rm
chosen}}_{[t_{0},t]}$ elements, which combine linearly to form $H_{{\rm
QMM}}(t)$. Now, given $N$ and looking backwardly from the present moment $t$,
we take a choice of $N$ ‘quantum-memory distances’ (QMDs),
$\Delta^{(t)}_{i=1\cdots N}$, which uniquely identify all the Hamiltonian-
maker quantum memories as the density operators of the system at $N$ past-to-
present moments given by $t_{i}\equiv t-\Delta^{(t)}_{i}$:
$\\{\rho_{t_{i}}\\}$. The chosen QMDs are constrained and ordered as follows,

$(\Delta^{(t)}_{1}\;,\;\cdots\;,\;\Delta^{(t)}_{N})\;\;\;;\;\;\;0\;\leq\;\Delta^{(t)}_{N}\;<\;\cdots\;<\;\Delta^{(t)}_{1}\;\leq\;(t-t_{0}).$
(5)

We note that with the above selections of $N$ and the spectrum of QMDs in (5),
the QMM-H argument in (4) is completely fixed as follows,

$\mathcal{P}^{{\rm
chosen}}_{[t_{0},t]}=\\{\;\rho_{t_{1}}\;,\;\cdots\;,\,\rho_{t_{N}}\;\\}.$ (6)

The recipe continues with specifying the QMM monomials which combine linearly
to become the QMM-H. A Hamiltonian monomial is denoted by
$\Phi^{s_{i_{1}}\;\cdot\cdot\cdot\;s_{i_{r}}}_{i_{1}\;\cdot\cdot\cdot\;i_{r}}(t)$,
where $r_{\;\geq 1}^{\;\leq L}$ is the monomial’s length as a density-operator
string, and the lower and upper indices specify respectively the monomial’s
variables picked up from $\mathcal{P}^{{\rm chosen}}_{[t_{0},t]}$ and their
algebraic powers. Every QMM monomial
$\Phi^{s_{i_{1}}\;\cdot\cdot\cdot\;s_{i_{r}}}_{i_{1}\;\cdot\cdot\cdot\;i_{r}}(t)$
is structured as follows. First, we choose an $r$-tuple
$(i_{1},\cdots,i_{r})\subset(\\{1,\cdots,N\\})^{r}$ which fixes
$(t_{i_{1}},\cdots,t_{i_{r}})\equiv(t-\Delta^{(t)}_{i_{1}},\cdots,t-\Delta^{(t)}_{i_{r}})$.
Second, we have the ‘option’ to take a choice of $r+1$ _state-history-
independent_ operators $\mathcal{A}^{(r)}=(A^{(r)}_{1},\cdots,A^{(r)}_{r+1})$.
Third, we choose the monomial-forming powers
$(s_{i_{1}},\cdot\cdot\cdot,s_{i_{r}})$. Now, the _‘QMM-monomial operator’_
$\Phi^{s_{i_{1}}\;\cdot\cdot\cdot\;s_{i_{r}}}_{i_{1}\;\cdot\cdot\cdot\;i_{r}}(t)$
is conceived in the following form,

$\Phi^{s_{i_{1}}\cdots\;s_{i_{r}}}_{i_{1}\;\cdot\cdot\cdot\;i_{r}}(t)\;\equiv\;A^{(r)}_{1}\;(\rho_{t_{i_{1}}})^{s_{i_{1}}}A^{(r)}_{2}\;(\rho_{t_{i_{2}}})^{s_{i_{2}}}\;\cdots\;(\rho_{t_{i_{r-1}}})^{s_{i_{r-1}}}A^{(r)}_{r}\;(\rho_{t_{i_{r}}})^{s_{i_{r}}}A^{(r)}_{r+1}.$
(7)

Finally, we form a linear combination of these QMM monomials, following a
choice of complex-or-real (constant-in-time or time-varying) ‘coupling’
coefficients $\lambda_{i_{1}\cdots i_{r}}^{s_{i_{1}}\cdots s_{i_{r}}}(t)$, and
sum it up with its Hermitian conjugate. We now have, in result, a Hermitian
nonlocal-in-time operator integrating some degree-$l^{(\leq L)}$ monomials of
$N$ past-or-present system’s quantum states, chosen from a specified
$\mathcal{P}^{{\rm chosen}}_{[t_{0},t]}$ as (6). Therefore, the recipe outputs
a broad realization of QMM-Hs (4), denoted by $H^{(N,L)}_{({\rm
QMM}|\mathcal{A})}(t)$ with $\mathcal{A}\equiv\\{\mathcal{A}^{(r)},\forall
r\\}$, in the following form,

$H^{(N,L)}_{({\rm QMM}|\mathcal{A})}(t)\;\;\equiv\sum^{(\text{must
include}\;r=L)}_{\begin{subarray}{c}\text{all chosen}\;r\\\ \\\
1\;\leq\;r\;\leq\;L\end{subarray}}\;\sum^{(i_{1},\cdots,i_{r})\;\subset\;\\{1,\cdots,N\\}^{r}}_{\begin{subarray}{c}\text{all
chosen}\big{(}(i_{1},s_{i_{1}}),\;\cdots\;,(i_{r},s_{i_{r}})\big{)}\end{subarray}}\;\lambda_{i_{1}\cdots
i_{r}}^{s_{i_{1}}\cdots
s_{i_{r}}}\;\;\Phi^{s_{i_{1}}\;\cdot\cdot\cdot\;s_{i_{r}}}_{i_{1}\;\cdot\cdot\cdot\;i_{r}}(t)\;\;+\;\;h.c.$
(8)

We must highlight three important points about the QMM-Hs (8). _First_ , the
above $\lambda$-couplings can be non-constants, furthermore, they can have
explicit dependencies on _scalar quantities_ made from the elements of (3).
However, we shall assume that all $\lambda$-couplings are state-history-
independent constants. _Second_ , we obtain more complex QMM-Hs as linear
combinations of $H^{(N,L)}_{({\rm QMM}|\mathcal{A})}$’s with distinct choices
of integer powers, coupling profiles and the insertion operators in
$\mathcal{A}$. _Third,_ depending on the context, one chooses or must employ
as some $\mathcal{A}^{(r)}$ elements _nontrivial_ state-history-independent
operators. Examples are projection or swap operators corresponding to the
eigenstates of the relevant observables. However, we choose to focus on the
behavioral phases of the _unexplored and intriguing category in which the QMM-
Hs are ‘entirely’ made of the system’s state-history_ : we carefully analyze,
in Sections 4, 5 and 6, the intriguing subclass of QMM-Hs (8) given by
constant couplings and the uniform choice: $A^{(r)}_{l}={\mathbb{1}}$.

The resulted _Entirely-QMM-Hs_ , denoted by $H^{(N,L)}_{{\rm QMM}}(t)$, read
as follows,

$H^{(N,L)}_{{\rm QMM}}(t)\;\;\equiv\sum^{(\text{must
include}\;r=L)}_{\begin{subarray}{c}\text{all chosen}\;r\\\ \\\
1\;\leq\;r\;\leq\;L\end{subarray}}\sum^{(i_{1},\cdots,i_{r})\;\subset\;\\{1,\cdots,N\\}^{r}}_{\begin{subarray}{c}\text{all
chosen}\big{(}(i_{1},s_{i_{1}}),\;\cdots\;,(i_{r},s_{i_{r}})\big{)}\end{subarray}}\lambda_{i_{1}\cdots
i_{r}}^{s_{i_{1}}\cdots
s_{i_{r}}}\;(\rho_{t_{i_{1}}})^{s_{i_{1}}}\;\cdots\;(\rho_{t_{i_{r}}})^{s_{i_{r}}}\;\;+\;\;h.c.$
(9)

One sees that additional care must be taken concerning the unitary time
evolution in a sufficiently-early time interval which we call the ‘quantum-
memory storing period’. Given $N$ and a choice of the largest QMD in (5),
$\Delta^{(t)}_{1}$, we define $t_{*}$ to be the earliest moment as of which,
or right after which, all the $N$ participant quantum memories in (6), have
become available for the system. For example, if one selects a constant value
$a$ for the largest QMD, $\Delta^{(t)}_{1}=a$, we have: $t_{*}=t_{0}+a$.
However, if one chooses a time-dependent linear profile,
$\Delta^{(t)}_{1}=c_{1}\;(t-t_{0})$ with $|c_{1}|\leq 1$, then: $t_{*}=t_{0}$.
In any case, the ‘quantum-memory storing period’ is always the time interval
from $t_{0}$ to $t_{*}$. This is because before going past this period, the
system has not yet developed enough state-history resources to build up
$H_{({\rm QMM}|\mathcal{A})}^{(N,L)}(t)$ as defined in (8). Independently,
even when $t_{*}=t_{0}$, because the initial-time Hamiltonian
$H^{(N,L)}_{({\rm QMM}|\mathcal{A})}(t_{0})$ becomes a polynomial in only
$\rho_{t_{0}}$, no nontrivial dynamics is generated by it, as the system is at
state $\rho_{t_{0}}$. Therefore, unless $t_{0}=-\infty$, we must employ an
_alternative_ Hamiltonian to be at work in the quantum-memory storing period.
This initial ‘Kicker Hamiltonian’, $H^{({\rm Kicker})}(t)$, i) generates a
nontrivial dynamics at the initial time of the state-history, $t_{0}$, and ii)
builds up dynamically a ‘quantum-memory pool’, extended from $t_{0}$ to the
moment $t_{*}$, using which the closed system can hand all its afterward
evolution to the QMM-H (8). As such, the total Hamiltonian of the closed
quantum system is given by,

$H^{({\rm Total})}_{({\rm
QMM}|\mathcal{A})}(t)\;\equiv\;\theta(t_{*}-t)\;H^{({\rm
Kicker})}(t)\;+\;\theta(t-t_{*})\;H^{(N,L)}_{({\rm QMM}|\mathcal{A})}(t).$
(10)

It is worth to remark the following points regarding the Kicker Hamiltonian.
_First_ , it is easily possible to construct a large variety of Kicker
Hamiltonians $H^{({\rm Kicker})}(t)$ which themselves are (Purely) QMM, as we
detail in Appendix A. _Second_ , as explained above, there are infinitely many
time dependent choices of the largest QMD, that is: $a\,=\,a(t)$, for which
the quantum-memory storing period vanishes, $t_{*}=t_{0}$, such that no Kicker
Hamiltonian is necessary. _Third_ , choosing $H^{({\rm Kicker})}=H_{{\rm C}}$
as was characterized in (1), namely a state-history independent operator as
the Kicker Hamiltonian, is sufficiently good for us in investigating the
examples of Sections 4, 5 and 6.

The QMM time evolution operator corresponding to a time interval
$[t^{\prime},t^{\prime\prime}]$, $U^{(\text{Total})}_{({\rm
QMM}|\mathcal{A})}(t^{\prime\prime},t^{\prime})$, generated by a continuous
sequence of momentary Hamiltonians $H^{\text{(Total})}_{({\rm
QMM}|\mathcal{A})}(t\in[t^{\prime},t^{\prime\prime}])$ defined in (10), is
unitary. The unitary evolution operator, $U^{(\text{Total})}_{({\rm
QMM}|\mathcal{A})}(t^{\prime\prime}\;,\;t^{\prime})$, equals the (limit)
product of infinitely many unitary operators corresponding to infinitesimal
time evolutions covering the entire interval $[t^{\prime},t^{\prime\prime}]$,
which is resummed in the form of a Dyson expansion using time ordering product
operator, $\mathcal{T}$, as follows,

$\begin{split}&U^{(\text{Total})}_{({\rm
QMM}|\mathcal{A})}(t^{\prime\prime}\;,\;t^{\prime})\;\;=\;\lim_{M\to\infty}\;\prod_{m=1}^{M}\;U_{m}^{(\text{Total})}\bigg{(}t^{\prime}+\frac{t^{\prime\prime}-t^{\prime}}{M}\;m\;,\;t^{\prime}+\frac{t^{\prime\prime}-t^{\prime}}{M}\;(m-1)\bigg{)}\equiv\\\
&\hskip
85.35826pt\;\equiv\;\lim_{M\to\infty}\;\prod_{m=1}^{M}\;\exp\bigg{[}\;i\;\frac{t^{\prime\prime}-t^{\prime}}{M}\;H_{({\rm
QMM}|\mathcal{A})}^{(\text{Total})}\Big{(}t^{\prime}+\frac{t^{\prime\prime}-t^{\prime}}{M}\;(m-1)\Big{)}\;\bigg{]}=\\\
&\hskip
85.35826pt\;=\,\lim_{M\to\infty}\;\prod_{m=1}^{M}\;\bigg{[}\;\mathbb{1}\;+\;i\;\frac{t^{\prime\prime}-t^{\prime}}{M}\;H_{({\rm
QMM}|\mathcal{A})}^{(\text{Total})}\Big{(}t^{\prime}+\frac{t^{\prime\prime}-t^{\prime}}{M}\;(m-1)\Big{)}\;\bigg{]}=\\\
&\hskip
85.35826pt\;=\;\mathcal{T}\exp\bigg{(}-i\,\int_{t^{\prime}}^{t^{\prime\prime}}d\tau\;H_{({\rm
QMM}|\mathcal{A})}^{(\text{Total})}(\tau)\;\bigg{)}.\\\ \end{split}$ (11)

As such, the system’s quantum states, which themselves are the constructional
resource for the QMM-H, must evolve consistently-and-unitarily as described
by,

$\rho_{t^{\prime\prime}}=U^{(\text{Total})}_{({\rm
QMM}|\mathcal{A})}(t^{\prime\prime}\;,\;t^{\prime})\;\rho_{t^{\prime}}\;U^{(\text{Total})\dagger}_{({\rm
QMM}|\mathcal{A})}(t^{\prime\prime}\;,\;t^{\prime}).$ (12)

###  2.4 Pure State Histories Composing QMM Unitary Evolutions

The QMM-Hs (8) can be made from arbitrary mixed quantum states. But, from now
on, it suffices to focus on investigating the QMM-UEs which are made by _the
pure states_ of the system, on the account of their sufficient richness.
Assuming state purity at the initial moment, given that the whole QMM time
evolution is ensured to be unitary (11, 12), the corresponding state-history
of the system (2) will be developed as a continual sequence of pure states: it
will be a ‘pure-state-history’. Being so, as of now, we only focus on pure-
state-histories of closed systems, that is,

$\begin{split}&\mathcal{P}_{[t_{0},\infty)}\;=\;\\{\leavevmode\nobreak\
\rho_{t^{\prime}}\;=\;|\Psi_{t^{\prime}}\rangle\langle\Psi_{t^{\prime}}|\;\;\;,\;\quad\forall\leavevmode\nobreak\
t^{\prime}\in[t_{0},\infty)\leavevmode\nobreak\ \\},\\\ &\hskip
37.84221pt\;\;\;\rho_{t^{\prime}}\;=\;\rho^{s}_{t^{\prime}}\;\;\;,\;\;\;\forall
s\in\mathbb{N}\;,\;\quad\forall\leavevmode\nobreak\
t^{\prime}\in[t_{0},\infty).\end{split}$ (13)

Now, merging (9) and (LABEL:hps), we arrive at the following family of _pure-
state-history QMM-Hs_ , denoted by $H^{(N|L)}_{{\rm QMM}}(t)$,

$\begin{split}&H^{(N,L)}_{\text{QMM.PureStateHistory}}(t)\;\equiv\;\\\
&\equiv\;H^{(N|L)}_{{\rm QMM}}(t)\;=\;\sum^{(\text{must
include}\;r=L)}_{\begin{subarray}{c}\text{all chosen}\;r\\\ \\\
1\;\leq\;r\;\leq\;L\end{subarray}}\;\;\sum^{(i_{1},\cdots,i_{r})\;\subset\;\\{1,\cdots,N\\}^{r}\;;\;i_{l}\neq
i_{l+1},\forall\;l}_{\begin{subarray}{c}\text{all
chosen}\big{(}(i_{1},s_{i_{1}}),\;\cdots\;,(i_{r},s_{i_{r}})\big{)}\end{subarray}}\;\;\;\big{[}\;\lambda_{i_{1}\cdots
i_{r}}\;(\;\rho_{t_{i_{1}}}\;\cdots\;\rho_{t_{i_{r}}}\;)\;+\;\\\ &\hskip
90.47955pt\hskip 142.26378pt\hskip
56.62114pt\;\;\;\;\;+\;\lambda^{\star}_{i_{1}\cdots
i_{r}}\;\;(\;\rho_{t_{i_{r}}}\;\cdots\;\rho_{t_{i_{1}}}\;)\;\big{]}.\end{split}$
(14)

The QMM-UEs generated by the above family of Hamiltonians will be the very
subject of all our analytical and numerical findings in the rest of this work.
Some results are general: they hold for the whole family (14), either
precisely or at least qualitatively. But, many exact results are the outcomes
of detailed analyses in a number of chosen concrete examples. Specifically, in
what follows, we write down the explicit choices of the QMM-Hs (14) whose one-
qubit-system realizations we will detail in Section 4, and map the phase
diagrams of their QMM wavefunctions in Section 6,

$\begin{split}&H^{(1|1)}_{{\rm QMM}}(t)\;=\;\mu\;\rho_{t-a}\hskip
56.62114pt\hskip
142.26508pt\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Delta^{(t)}_{1}=a,\;\;\;\;\;\;\;\;\hskip
11.3527pt\;\hskip 4.6945pt\\\ &H^{(2|2)}_{{\rm
QMM}}(t)\;=\;\mu\;\rho_{t-a}\;+\;\lambda\;\rho_{t-a}\;\rho_{t}+\lambda^{\star}\;\rho_{t}\;\rho_{t-a}\;\;\;\hskip
85.07388pt\;\;\;\;\;(\Delta_{2}^{(t)},\Delta_{1}^{(t)})=(0,a),\;\;\hskip
0.46931pt\\\ &H^{(2|3)}_{{\rm
QMM}}(t)\;=\;\lambda\;\rho_{t-a}\;\rho_{t}+\lambda^{\star}\;\rho_{t}\;\rho_{t-a}\;+\;\eta\;\rho_{t-a}\;\rho_{t}\;\rho_{t-a}\;\;\;\;\hskip
52.92258pt\;\;\;(\Delta_{2}^{(t)},\Delta_{1}^{(t)})=(0,a),\;\;\hskip
4.6945pt\\\ &H^{(3|3)}_{{\rm
QMM}}(t)\;=\;\kappa\;\rho_{t-a}\;\rho_{t-b}\;\rho_{t}+\kappa^{\star}\;\rho_{t}\;\rho_{t-b}\;\rho_{t-a}\;\;\;\;\;\;\;\hskip
53.49132pt(\Delta_{3}^{(t)},\Delta_{2}^{(t)},\Delta_{1}^{(t)})\;=\;(0,b,a).\;\;\;\;\end{split}$
(15)

The QMM-Hs (14) together with their QMM Schrödinger equations (to be coming
very soon) and their QMM evolving wavefunctions, can be recast in _an
alternative way_ which we highlight specially for its elegance and
effectiveness. One defines a family of ‘Quantum-Memory Characters and
Operators’ which are natural to the proposed systems. Given one’s choice of
all the participant quantum memories shown in (6), these quantum-memory
characters are as follows: the _‘Quantum-Memory Two-Point Functions’_
$m_{t_{i_{l}}t_{i_{r}}}$ (QM-TPFs); all their QM $n$-point-function
descendants $m_{t_{i_{r_{1}}}\cdots t_{i_{r_{n}}}}$; the QM Two-Point
Observables, in other words, the pairwise _QM-Fidelities_ ,
$w_{t_{i_{l}}t_{i_{r}}}^{2}$. They are defined as such,

$\begin{split}&m_{t_{i_{l}}t_{i_{r}}}\equiv\;\braket{\Psi_{t_{i_{l}}}}{\Psi_{t_{i_{r}}}}\;\;\;;\;\;\;m_{t_{i_{1}}t_{i_{2}}\cdots
t_{i_{r-1}}t_{i_{r}}}\equiv\;m_{t_{i_{1}}t_{i_{2}}}m_{t_{i_{2}}t_{i_{3}}}\cdots
m_{t_{i_{r-1}}t_{i_{r}}}=(\;m_{t_{i_{r}}t_{i_{r-1}}\cdots
t_{i_{2}}t_{i_{1}}}\;)^{\star},\\\
&w_{t_{i_{l}}t_{i_{r}}}^{2}\equiv|m_{t_{i_{l}}t_{i_{r}}}|^{2}={\rm
Tr}[\;\rho_{t_{l}}\;\rho_{t_{r}}\;]=\big{(}\text{Tr}\big{[}(\sqrt{\rho_{t_{l}}}\;\rho_{t_{r}}\;\sqrt{\rho_{t_{l}}})^{\frac{1}{2}}\big{]}\big{)}^{2}=F(\;\rho_{t_{l}}\;,\;\rho_{t_{r}}\;).\end{split}$
(16)

Moreover, we define the _‘Quantum-Memory Two-Point Operators’_ (QM-TPOs),
$M_{t_{i_{r}}t_{i_{l}}}$,

$\displaystyle\;\;\;M_{t_{i_{r}}t_{i_{l}}}\;$
$\displaystyle\equiv\;\ket{\Psi_{t_{i_{r}}}}\bra{\Psi_{{t_{i_{l}}}}}\;=\;M^{\dagger}_{t_{i_{l}}t_{i_{r}}}\;\;\>;\;\;\;\;M_{t_{i_{r}}t_{i_{r}}}\;=\;\rho_{t_{i_{r}}}\;\;\;;\;\;\>\text{Tr}[\;M_{t_{i_{r}}t_{i_{l}}}\;]=m_{t_{i_{l}}t_{i_{r}}}=m^{\star}_{t_{i_{r}}t_{i_{l}}}$

Now, in terms of the above quantum-memory measures and operators, the QMM-Hs
formulated in (14) are recast as follows,

$\begin{split}&H^{(N|L)}_{{\rm QMM}}(t)\;=\;\\\ &\hskip
35.56593pt=\;\sum^{(\text{must include}\;r=L)}_{\begin{subarray}{c}\text{all
chosen}\;r\\\ \\\
1\;\leq\;r\;\leq\;L\end{subarray}}\;\;\sum^{(i_{1},\cdots,i_{r})\;\subset\;\\{1,\cdots,N\\}^{r}\;;\;i_{l}\neq
i_{l+1},\forall\;l}_{\begin{subarray}{c}\text{all
chosen}\big{(}(i_{1},s_{i_{1}}),\;\cdots\;,(i_{r},s_{i_{r}})\big{)}\end{subarray}}\;\;\;\big{(}\;\lambda_{i_{1}\cdots
i_{r}}\;m_{t_{i_{1}}t_{i_{2}}\cdots
t_{i_{r-1}}t_{i_{r}}}\;M_{t_{i_{1}}t_{i_{r}}}\;+\;\\\ &\hskip
265.55238pt\;+\;\lambda^{\star}_{i_{1}\cdots
i_{r}}\;\;m_{t_{i_{r}}t_{i_{r-1}}\cdots
t_{i_{2}}t_{i_{1}}}\;M_{t_{i_{r}}t_{i_{1}}}\;\;\big{)}.\end{split}$ (18)

Although obvious, it would be worth emphasizing that the preservation of the
total measurement probabilities does hold in the course of QMM-UE, that is:
${\rm Tr}[\rho(t)]=1$, or for the pure states: $<\psi_{t}|\psi_{t}>=1$, if the
system’s state at $t_{0}$ is normalized to one. This dynamical preservation is
an automatic and direct implication of the _unitarity_ of the system’s
evolution under the QMM-Hs (8,14), manifested in (10,11,12). Moreover,
detailing $H^{(2|2)}_{{\rm QMM}}$ in Subsection 3.2, we even explicitly
demonstrate that $|\ket{\Psi_{t}}|$ is the invariant of QMM-UEs, using their
QMM Schrödinger equations (27).

Finally, we highlight an important point. Apart from the QMM-H with
$(N,L)=(1,1)$, in examples (15) to be analyzed in Sections 4, 5 and 6, we have
taken the smallest QMD to be zero, such that $\rho_{t}\in\mathcal{P}^{{\rm
chosen}}_{[t_{0},t]}$. But, this is optional: a ‘working-frame’ choice. That
is, in definition (8), we do _not_ demand that the present moment state of the
system, $\rho_{t}$, should be necessarily used in the making of $H_{{\rm
QMM}}(t)$. Indeed, whether or not the chosen $\mathcal{P}^{{\rm
chosen}}_{[t_{0},t]}$ contains $\rho_{t}$ is an arbitrary option under which
the formulation of $H_{{\rm QMM}}(t)$ remains fully covariant.

##  3 The Dynamical Equations Of Unitary State-History-Based Quantum Behavior

###  3.1 QMM von Neumann and Schrödinger Equations

As (10,11) demonstrate, evolution under QMM-Hs (7,8), or its pure-state-
history version (14), is unitary. The time-integrated evolution of the states
is given by (12) which combined with (11,14) leads to a highly complex
structure for the differential dynamical equation corresponding to it. The
complexity originates from the intrinsic quality of QMM-UEs: that, _states
evolve under states_ , and that for this process to run, there must be a
consistent interplay between the system’s states at distinct moments. This
interplay imposes a nonlocal-in-time nonlinearity on the differential
dynamical equations of QMM-UEs. Here we derive these equations and write their
expressions in alternative useful forms. Before moving on, let us note that we
employ conventional Hamiltonians for the Kicker Hamiltonian which is at work
during $t\leq t_{*}$, throughout the paper. Hence, the unitary evolution of
the states, given by (10,11,12), leads to a known conventional Schrödinger
equation during $[t_{0},t_{*}]$. Being so, we only need to work out the QMM
delay-differential dynamical equation which is on role as of having the
quantum-memory pool filled up, namely for $t\geq t_{*}$.

The equations (11) and (12) once combined immediately result at the well-known
general structurally-unpacked form of quantum dynamical equation for the
system’s states. The (Liouville–)von Neumann equation for QMM-H
$(\ref{MMHparts})$ during $t\in[t_{*},\infty)$, is given by,

$\dot{\rho}_{t}\;\;=\;i\;[\;\rho_{t}\;,\;H^{(N|L)}_{{\rm QMM}}(t)\;].$ (19)

Now, applying the explicit expression of $H^{(N|L)}_{{\rm QMM}}(t)$ in (14) to
general equation (19), the dynamical equations for the corresponding family of
pure-state-history QMM-UEs is yield as the following von Neumann equation,

$\begin{split}&i\;\dot{\rho}_{t}=\sum^{(\text{must
include}\;r=L)}_{\begin{subarray}{c}\text{all chosen}\;r\\\ \\\
1\;\leq\;r\;\leq\;L\end{subarray}}\;\sum^{(i_{1},\cdots,i_{r})\;\subset\;\\{1,\cdots,N\\}^{r}\;;\;i_{l}\neq
i_{l+1},\forall\;l}_{\begin{subarray}{c}\text{all
chosen}\big{(}(i_{1},s_{i_{1}}),\;\cdots\;,(i_{r},s_{i_{r}})\big{)}\end{subarray}}\;\;\big{(}\;\lambda_{i_{1}\cdots
i_{r}}\leavevmode\nobreak\ [\leavevmode\nobreak\
\rho_{t_{i_{1}}}\;\cdots\;\rho_{t_{i_{r}}}\;,\;\rho_{t}\leavevmode\nobreak\
]\;\;+\;\\\ &\hskip 100.72256pt\hskip
113.52664pt\;\;\;\;\;\;\;\;\;+\;\lambda_{i_{1}\cdots
i_{r}}^{\star}\leavevmode\nobreak\ [\leavevmode\nobreak\
\rho_{t_{i_{r}}}\;\cdots\;\rho_{t_{i_{1}}}\;,\;\rho_{t}\leavevmode\nobreak\
]\;\;\big{)}.\;\;\;\;\;\;\;\;\;\end{split}$ (20)

We have highlighted that it is insightful and useful to have the physics of a
QMM-UE also expressed in terms of the corresponding QM-TPFs together with
their descendants, and QM-TPOs, defined in (LABEL:ctmcs,2.4). Now, applying
those alternative expressions of the QMM-Hs (18) to (19), we obtain an
alternative form of the von Neumann equations (20),

$\begin{split}&i\;\dot{\rho}_{t}=\sum^{(\text{must
include}\;r=L)}_{\begin{subarray}{c}\text{all chosen}\;r\\\ \\\
1\;\leq\;r\;\leq\;L\end{subarray}}\;\sum^{(i_{1},\cdots,i_{r})\;\subset\;\\{1,\cdots,N\\}^{r}\;;\;i_{l}\neq
i_{l+1},\forall\;l}_{\begin{subarray}{c}\text{all
chosen}\big{(}(i_{1},s_{i_{1}}),\;\cdots\;,(i_{r},s_{i_{r}})\big{)}\end{subarray}}\big{(}\;\lambda_{i_{1}\cdots
i_{r}}\;m_{t_{i_{1}}t_{i_{2}}\cdots
t_{i_{r-1}}t_{i_{r}}}[\;M_{t_{i_{1}}t_{i_{r}}}\;,\;\rho_{t}\;]\;+\\\ &\hskip
237.12221pt+\;\lambda^{\star}_{i_{1}\cdots
i_{r}}\;m_{t_{i_{r}}t_{i_{r-1}}\cdots
t_{i_{2}}t_{i_{1}}}\;[\;M_{t_{i_{r}}t_{i_{1}}}\;,\;\rho_{t}\;]\;\big{)}.\end{split}$
(21)

The above equation can be rewritten in a nicer form. Remembering that
$\rho_{t}=\ket{\Psi_{t}}\bra{\Psi_{t}}$ and using (LABEL:ctmcs) and (2.4), we
immediately validate the following relations,

$m_{t_{a}...t_{b}}\;m_{t_{b}t}\;=\;m_{t_{a}...t_{b}t},$ (22)

and,

$[\;M_{t_{i_{1}}t_{i_{r}}}\;,\;\;\rho_{t}\;]\;=\;m_{t_{i_{r}}t}\;M_{t_{i_{1}}t}\;-\;m_{tt_{i_{1}}}\;M_{tt_{i_{r}}}.$
(23)

Upon applying (22,23) to (21), the _‘QMM von Neumann equations’_ are recast
as,

$\begin{split}&i\;\dot{\rho}_{t}\;=\;i\;\dot{M}_{tt}=\\\ &=\sum^{(\text{must
include}\;r=L)}_{\begin{subarray}{c}\text{all chosen}\;r\\\ \\\
1\;\leq\;r\;\leq\;L\end{subarray}}\;\sum^{(i_{1},\cdots,i_{r})\;\subset\;\\{1,\cdots,N\\}^{r};i_{l}\neq
i_{l+1},\forall\;l}_{\begin{subarray}{c}\text{all
chosen}\big{(}(i_{1},s_{i_{1}}),\;\cdots\;,(i_{r},s_{i_{r}})\big{)}\end{subarray}}\lambda_{i_{1}\;...\;i_{r}}\;(m_{t_{i_{1}}\;\cdot\cdot\cdot\;t_{i_{r}}t}\;M_{t_{i_{1}}t}-m_{tt_{i_{1}}\;\cdot\cdot\cdot\;t_{i_{r}}}\;M_{tt_{i_{r}}})\;+\\\
&\;\hskip 88.71558pt\hskip 56.62114pt\hskip
56.62114pt+\;\lambda^{\star}_{i_{1}\;...\;i_{r}}\;(m_{t_{i_{r}}\;\cdot\cdot\cdot\;t_{i_{1}}t}\;M_{t_{i_{r}}t}-m_{tt_{i_{r}}\;\cdot\cdot\cdot\;t_{i_{1}}}\;M_{tt_{i_{1}}}).\end{split}$
(24)

Moreover, given the imposed purity of the system’s quantum-state-history
(LABEL:hps), one recasts the original form of Schrödinger equation for the
unitary evolution under $H^{(N|L)}_{{\rm QMM}}(t)$,

$i\;\ket{\dot{\Psi}_{t}}\;=\;H^{(N|L)}_{{\rm QMM}}(t)\;\ket{\Psi_{t}}.$ (25)

It can be recast by extraction from (24), or applying (18) to (25), and using,

$M_{t_{a}t_{b}}\ket{\Psi_{t}}\;=\;m_{t_{b}t}\;\ket{\Psi_{t_{a}}}.$ (26)

Either way, one finally arrives at ‘ _QMM Schrödinger Equation’ for unitary
evolution generated by $H^{(N|L)}_{{\rm QMM}}(t)$_,

$\begin{split}i\;\ket{\dot{\Psi}_{t}}\;=&\sum^{(\text{must
include}\;r=L)}_{\begin{subarray}{c}\text{all chosen}\;r\\\ \\\
1\;\leq\;r\;\leq\;L\end{subarray}}\;\;\sum^{(i_{1},\cdots,i_{r})\;\subset\;\\{1,\cdots,N\\}^{r}\;;\;i_{l}\neq
i_{l+1},\forall\;l}_{\begin{subarray}{c}\text{all
chosen}\big{(}\;(i_{1},s_{i_{1}}),\;\cdots\;,(i_{r},s_{i_{r}})\big{)}\end{subarray}}\;\;\;\big{(}\;\;\lambda_{i_{1}\cdots
i_{r}}\;\;m_{t_{i_{1}}\;\cdots\;t_{i_{r}}t}\;\ket{\Psi_{t_{i_{1}}}}\;\;\\\
&\hskip 113.52664pt\hskip 102.40143pt\;+\;\lambda^{\star}_{i_{1}\cdots
i_{r}}\;\;m_{t_{i_{r}}\;\cdot\cdot\cdot\;t_{i_{1}}t}\;\ket{\Psi_{t_{i_{r}}}}\;\big{)}.\end{split}$
(27)

Equation (27) is one of the main results of the present paper in three
complementary ways. First, it encodes the physical content of (14,11,12), all
in all. Second, it is a mathematically novel-and-elegant nonlinear delay-
differential equation with extremely complex and resourceful structure and
indeed moduli space. Third, all the rest of this paper is devoted to exploring
and understanding the novel physics, specially the phase diagrams, of the most
elementary closed system under the simplest QMM-UEs, upon solving the
corresponding Purely-QMM Schrödinger Equations (27). Furthermore, we highlight
in here three characteristic features of (27). First, notice that solving this
equation is also determining $H^{(N|L)}_{{\rm QMM}}(t)$. Because before that,
the QMM-H (18) is an unidentified operator, as it is made out of the yet-
unsolved state-history of the closed system (4). This is intrinsically unlike
conventional closed systems where the Hamiltonian is an identified operator
and we only solve the Schrödinger equation for the wavefunctions. But for QMM-
UEs, the triple equations (18,11,12) are to be solved jointly, upon solving
(27), with solutions identifying the system’s unitary time evolution operators
and the corresponding quantum-state-histories, all together.

Second, QMM Schrödinger equation is nonlocal-in-time. The ‘temporal
nonlocality’ of (27) is a direct implication of $H^{(N|L)}_{{\rm QMM}}(t)$
being made of the states at different times. In fact, nonconstant solutions to
(27) are developed only if $\Delta^{(t)}_{1}$ exceeds some finite threshold:
the temporal nonlocality of (27) cannot be even ‘softened’ arbitrarily.

Third, QMM Schrödinger equation is inherently nonlinear. Physically, the
nonlinearity is caused by the interactions which the closed system must turn
on between some of its quantum memories, to develop unitarity evolution based
on (18). Mathematically, it is sourced by all the QM-TPFs
$m_{t_{i}t_{j}}=\braket{\psi_{t_{i}}}{\psi_{t_{j}}}$ which enter into (27).
Hence, the temporal nonlocalities and the nonlinearities of (27) are merged to
the core, and in a specific form. Indeed, QMM Schrödinger equation (27) and
their solutions (to be shown) are, to the best of our knowledge [81, 82, 83,
84, 85], physically and mathematically distinctive from all nonlinear
Schrödinger equations available in the literature.

###  3.2 Detailing A Family of QMM-Hs With Two Quantum Memories

We now unpack the detailed structures of the QMM-H and its QMM Schrödinger
equation for an arbitrary closed system whose pure-state-history realizes the
unitary evolution generated by the QMM-H (14) based on two quantum memories.
The momentary Hamiltonian of the system, $H^{(2|L)}_{{\rm QMM}}(t)$, is a
Hermitian polynomial of arbitrary degree $L$ in the system’s states at the two
past-and-present moments identified with the following choice of the QMDs:
$(\Delta^{(t)}_{2},\Delta^{(t)}_{1})=(0,a)$ where $a$ is an arbitrary positive
constant. As before, $H^{({\rm Kicker})}(t)=H_{C}$, such that we only focus on
$H^{(2|2)}_{QMM}$ and its QMM Schrödinger equation during $t>t_{*}$. Given the
purity of the state history, the most general QMM-H characterized with $N=2$,
arbitrary $L$, and the above-made choices of QMDs is formed with the following
structure of QMM monomials,

$\begin{split}&H^{(2|L)}_{{\rm QMM}}(t)\;\equiv\;H^{(2|L)}_{({\rm
odd})}(t),\;+\;H^{(2|L)}_{({\rm even})}(t),\\\ &H^{(2|L)}_{({\rm
odd})}(t)\;\equiv\;\big{(}\;\mu_{1}\;\rho_{t-a}\;+\;\mu_{3}\;\rho_{t-a}\rho_{t}\rho_{t-a}\;+\;\mu_{5}\;\rho_{t-a}\rho_{t}\rho_{t-a}\rho_{t}\rho_{t-a}\;+\;\cdots\;\big{)}_{{\rm
up\;to\;degree}\;=\;L}\\\ &\hskip 198.88492pt\hskip
151.65323pt;\;\mu_{r=1,3,5,\cdots}\in\mathbb{R},\\\ &H^{(2|L)}_{({\rm
even})}(t)\equiv\big{(}\;\lambda_{2}\;\rho_{t-a}\rho_{t}+\lambda_{4}\;\rho_{t-a}\rho_{t}\rho_{t-a}\rho_{t}+\lambda_{6}\;\rho_{t-a}\rho_{t}\rho_{t-a}\rho_{t}\rho_{t-a}\rho_{t}+\cdots+\\\
&\hskip
37.84482pt\;\;\;\;\;+\lambda_{2}^{\star}\;\rho_{t}\;\rho_{t-a}+\lambda_{4}^{\star}\;\rho_{t}\rho_{t-a}\rho_{t}\rho_{t-a}+\lambda_{6}^{\star}\;\rho_{t}\rho_{t-a}\rho_{t}\rho_{t-a}\rho_{t}\rho_{t-a}+\cdots\big{)}_{{\rm
up\;to\;degree}\;=\;L}.\end{split}$ (28)

Notice that all the monomials beginning with the $\rho_{t}$ factor have been
ignored in the ‘odd’ part of the QMM-H (28). This is because, by the purity of
the states, all such monomials commute with the system’s density operator at
the present time, $\rho_{t}$. Thus, the system’s state gains no dynamics under
their inclusion in $H^{(2|L)}_{{\rm QMM}}(t)$.

For $N=2$, and with the chosen spectrum of the QMDs
$(\Delta^{(t)}_{2},\Delta^{(t)}_{1})=(0,a)$, the only QM-TPFs and nonlocal-in-
time QM-TPOs are: $m_{t-a,t}=m_{t,t-a}^{\star}=\braket{\Psi_{t-a}}{\Psi_{t}}$,
$M_{t-a,t}=M_{t,t-a}^{\dagger}=\ket{\Psi_{t-a}}\bra{\Psi_{t}}$, respectively.
Moreover, the central observable to probe the dynamics of the wavefunctions of
$N=2$ QMM-UE is the norm of the QM-TPF,

$w_{t-a\;t}\;\equiv\;w_{(a|t)}\;\equiv\;|m_{t-a,t}|\;=\;|\braket{\Psi_{t-a}}{\Psi_{t}}|\;=\;(\text{Tr}[\;\rho_{t-a}\;\rho_{t}])^{\frac{1}{2}}.$
(29)

Now, we introduce the following twin polynomials in terms of the QM observable
$w_{(a,t)}$ whose coefficients are given by the Hamiltonian couplings $\mu$’s
and $\lambda$’s, respectively,

$\begin{split}&P_{L}^{(\mu)}(w_{(a|t)})\;\;\;\equiv\sum_{s}^{\;\;\;\;s\;\in\\{1,\cdots,\frac{L+1}{2}\\}}\mu_{2s-1}\;w_{(a|t)}^{2(s-1)},\\\
&P_{L}^{(\lambda)}(w_{(a|t)})\;\;\;\equiv\sum_{s}^{\;\;\;\;\;\;\;s\;\in\\{1,\cdots,\frac{L}{2}\\}}\lambda_{2s}\;w_{(a|t)}^{2(s-1)}.\end{split}$
(30)

By means of the polynomials (30), and the QM-TPs and QM-TPOs themselves, the
QMM-H (28) takes its final expression as follows,

$\begin{split}&H^{(2|L)}_{{\rm
QMM}}(t)\;=\;P_{L}^{(\mu)}(w_{(a|t)})\;\rho_{t-a}\;+\;P_{L}^{(\lambda)}(w_{(a|t)})\;m_{t-a,t}\;M_{t-a,t}\;+\;P_{L}^{(\lambda^{\star})}(w_{(a|t)})\;m_{t-a,t}^{\star}\;M_{t-a,t}^{\dagger}.\end{split}$
(31)

Now, applying (31) to (25), or by working out the right hand side of (27), the
QMM Schrödinger equation corresponding to $H^{(2|L)}_{{\rm QMM}}(t)$ is
organized in an efficient way in terms of two particular quantum-memory
functions $\mathcal{S}^{(2|L)}_{R}$ and $\mathcal{S}^{(2|L)}_{N}$, as follows,

$\begin{split}&i\leavevmode\nobreak\
|\dot{\Psi}_{t}\rangle\;=\;\mathcal{S}^{(2|L)}_{R}\ket{\Psi_{t-a}}\;+\;\mathcal{S}^{(2|L)}_{N}\ket{\Psi_{t}},\\\
&\hskip
46.94687pt\;\mathcal{S}^{(2|L)}_{R}\;\equiv\;P_{L}^{(\mu+\lambda)}(w_{(a|t)})\;\braket{\Psi_{t-a}}{\Psi_{t}}\;=\;(\;P_{L}^{(\mu)}(w_{(a|t)})\;+\;P_{L}^{(\lambda)}(w_{(a|t)})\;)\;\braket{\Psi_{t-a}}{\Psi_{t}},\\\
&\hskip
46.94687pt\;\mathcal{S}^{(2|L)}_{N}\;\equiv\;P_{L}^{(\lambda^{\star})}(w_{(a|t)})\;w_{(a|t)}^{2}.\end{split}$
(32)

The above QMM functions $\mathcal{S}^{(2|L)}_{R}$ and
$\mathcal{S}^{(2|L)}_{N}$ are denoted here as the ‘Retarded’ and ‘Now’
(Shrödinger equation) Functions of the $(2,L)$ QMM-UE, respectively. Finally,
let us further write down the fully-unfolded forms of these two functions for
the choices of $L=2$ and $L=3$, given that the one-qubit-wavefunction
solutions to (32) will be analyzed later for both of these cases. Upon
renaming the couplings according to (15), here they come,

$\begin{split}&L\;=\;2\;:\;\;\;\mathcal{S}^{(2|2)}_{R}\;=\;(\mu+\lambda)\;\braket{\Psi_{t-a}}{\Psi_{t}}\;\;\;;\;\;\;\mathcal{S}^{(2|2)}_{N}\;=\;\lambda^{\star}\;|\braket{\Psi_{t-a}}{\Psi_{t}}|^{2},\\\
&L\;=\;3\;:\;\;\;\mathcal{S}^{(2|3)}_{R}\;=\;(\mu+\lambda+\eta\;|\braket{\Psi_{t-a}}{\Psi_{t}}|^{2})\;\braket{\Psi_{t-a}}{\Psi_{t}}\;\;\;;\;\;\;\mathcal{S}^{(2|3)}_{N}\;=\;\lambda^{\star}\;|\braket{\Psi_{t-a}}{\Psi_{t}}|^{2}.\end{split}$
(33)

Before closing this subsection, we would like to explicitly check and make
sure that the QMM Schrödinger equation (32) preserves the norms of its
wavefunction solutions. This explicit confirmation would be done here despite
that the norm-conservation of the wavefunctions $\ket{\Psi_{t}}$ is guaranteed
by the very fact that the QMM time evolution operator is unitary (11). But it
is still worth showing, as a mere consistency check, that the corresponding
QMM Schrödinger equation also by itself realizes this feature. Indeed, it
would be enough to do this check for the case of $L=2$. Applying the $(R,N)$
Schrödinger operators (33) to (32), one can easily check that,

$\begin{split}\frac{d}{dt}\braket{\Psi_{t}}{\Psi_{t}}=2\;\text{Re}\big{[}\braket{\Psi_{t}}{\dot{\Psi}_{t}}\big{]}=2\;\text{Im}\big{[}\bra{\Psi_{t}}\mathcal{S}^{(2|L)}_{R}\ket{\psi_{t-a}}+\bra{\Psi_{t}}\mathcal{S}^{(2|L)}_{N}\ket{\Psi_{t}}\big{]}\;=\;0.\end{split}$
(34)

Being so, the QMM Schrödinger equation does by itself secure that the total
probability of the measurements is preserved to be one by the wavefunction
solutions which evolve under the corresponding QMM-H. In fact, this
preservation remains always true in the entire course of the state-history of
the closed quantum system $\mathcal{P}_{[t_{0},\;\infty)}$,

$\hskip 37.84221pt{\rm
Tr}[\;\rho_{t}\;]=1\;\;\;,\;\;\;\forall\;\rho_{t}\in\mathcal{P}_{[t_{0},\;\infty)}.$
(35)

###  3.3 The Robust Non-Markovianity of Purely-QMM Unitary Evolutions

One important aspect in understanding the physics of a memory-dependent
quantum many body system is probing its ‘Near-Markovian limit’, namely a limit
of the control parameters where QMM contributions to the system’s time-
evolution and observables can be perturbatively computed as some sufficiently-
small modifications to a Markovian version of the system. Clearly, the exact
identification of this limit and the formulation of the perturbation scheme
with mathematical and physical rigour can vary depending on the specifics of
the system. We now formulate and probe the Near-Markovian limit of the QMM-UE
(11,12,14), for a general closed system whose entire evolution (as of $t_{*}$)
is generated by its interacting quantum memories, uncombined with any
conventional Hamiltonian. This is surely not the general case, as the system’s
total Hamiltonian can be in its hybrid form (1). These hybrid settings will be
also briefly analyzed in the present paper, but the main idea of our analyses
in this work is that one naturally begins with exploring and understanding the
unprecedented aspects of the most extreme case of Unitary Non-Markovian
physics, that is, closed quantum systems whose unitary evolutions are
completely made of their quantum memories.

Fortunately the definition of the Near-Markovian limit of QMM-UE is
straightforward. This is because, the Hamiltonian $H^{(N|L)}_{{\rm QMM}}(t)$
is by construction a direct composite of the system’s quantum memories which
define its state-history $\mathcal{P}_{[t_{0},t]}$ (LABEL:hps). Being so, all
the QMDs $\Delta_{i=1,\cdots,N}^{(t)}\equiv t-t_{i}$ (5), by which the
building-block density operators are singled out, form a set of variables to
which the QMM-H (14) explicitly depend as a function. Given this feature, the
above-mentioned Near-Markovian limit of $H^{(N|L)}_{{\rm QMM}}(t)$ corresponds
to taking the largest QMD to be a sufficiently small parameter,
$\Delta^{(t)}_{1}\ll 1$, in terms of which all the observables of the system
can develop perturbative expansions. With this definition, recollecting that
the full content of the equations (11,12,14) is contained in the QMM
Schrödinger equation (27) or in its equivalent form (20), the central question
we are addressing in this subsection is the following:

In a perturbative scheme based on the sufficient smallness of the largest QMD
$\Delta^{(t)}_{1}\equiv a$, $a\ll 1$, what perturbative solutions
$\rho_{t}^{({\rm pert})}$ to the QMM dynamical equations (20) are developed by
a closed quantum system which unitarily evolves under $H^{(N|L)}_{{\rm
QMM}}(t)$? Are the solutions to (20) analytic functions of $a$, specially at
the Markovian point $a=0$, and of all the other $\Delta^{(t)}_{i}$s likewise?
Moreover, one wants to know from a more general mathematical perspective, what
functions of the QMDs the typical solutions to the equations (20) or
equivalently to QMM Schrödinger equations (27) are.

The solutions to the $H^{(N|L)}_{{\rm QMM}}$ dynamical equation (20) can be
generally thought as functions $\rho(t,a)\equiv\rho_{t}(a)$, with the QMD $a$
playing the role of an independent variable. We now look for the near-
Markovian dynamical solutions $\rho^{({\rm pert})}(t,a)\equiv\rho_{t}^{({\rm
pert})}(a)$ to (20) which, as functions of a sufficiently small variable $a\ll
1$, are analytic at the Markovian point $a=0$. In other words, we look for the
solutions $\rho^{({\rm pert})}(t,a)$ which can be recast as a full Taylor
expansion over the second variable $a$ at the point $a=0$,

$\hskip 56.87685pt\rho^{({\rm
pert})}(t,a)\;=\;\sum_{j=0}^{\infty}\;a^{j}\rho^{(j)}(t)\;\equiv\;\sum_{j=0}^{\infty}\;a^{j}\rho_{t}^{(j)}\;\;\;\;\;\;\;;\;\;\;\;\;\;\rho_{t}^{(j)}:a-\text{independent}.\;$
(36)

Now, to capture the near-Markovian solutions to (20) in a way which is
consistent order by order in the perturbation parameter $a$, we must also
implement the QMM ‘temporal’ Taylor expansions, as follows. For every past-
time quantum memory that comes into the making of $H^{(N|L)}_{{\rm QMM}}(t)$,
namely for every state corresponding to $\Delta^{(t)}_{i}>0$, the right hand
side of the dynamical equation (20) contains one time-shifted density
operator, $\rho_{t-\Delta^{(t)}_{i}}$. Given the perturbation bound $a\ll 1$
and the analyticity assumption, every such entry $\rho^{({\rm
pert})}(t-\Delta^{(t)}_{i},a)$ additionally admits a temporal Taylor expansion
corresponding to its tiny time shift, $-\Delta^{(t)}_{i}$, away from the
present moment $t$,

$\begin{split}&\rho^{({\rm
pert})}(t-\Delta^{(t)}_{i},a)=\sum_{m=0}^{\infty}\frac{(-1)^{m}}{m!}\;[\Delta^{(t)}_{i}]^{m}\;\frac{d^{m}}{dt^{m}}\rho^{({\rm
pert})}(t,a)\;=\;\\\ &\hskip
85.35826pt\;=\;\sum_{m=0}^{\infty}\;\sum_{n=0}^{\infty}\;\frac{(-1)^{m}}{m!}\;a^{n}\;[\Delta^{(t)}_{i}]^{m}\;\rho_{t}^{(n|m)}\;\;\;\;\;\;;\;\;\;\rho_{t}^{(n|m)}\;\equiv\;\frac{d^{m}}{dt^{m}}\;\rho_{t}^{(n)}.\end{split}$
(37)

Merging (20,36,37) and organizing the result order by order in the
perturbation parameter $a$, we are led to a hierarchically-coupled system of
differential equations for the unknown functions $\rho_{t}^{(j)}$. The
outcomes are the time-dependent operator-coefficients of the Taylor expansion
(36) which yield the Near-Markovian analytic solutions to (20). We now work
out the details and the result of this procedure for one specific case which
is nevertheless sufficient to clearly draw the general result which holds for
all the $(N,L)$ QMM-Hs defined in (14).

One can see that it is sufficient to address the most minimal case:
$(N,L)=(1,1)$ specified in (15), because similar patterns develop, leading to
the same end result, in the cases with larger $N$ or $L$, but only more
involved algebraically. For this take, the QMM-H and the QMM dynamical
equation (20) resulting from it read as follows,

$H^{(1|1)}_{{\rm
QMM}}(t)=\mu\;\rho_{t-a}=\mu\;\rho(t-a,a)\;\;\;;\;\;\;i\;\frac{\partial}{\partial
t}\rho(t,a)\;=\;\mu\;[\;\rho(t-a,a)\;,\;\rho(t,a)\;].$ (38)

The near-Markovian solutions of (38) are of the form (36), in which we take
$a$ to be an arbitrary constant, for mere simplicity. For $N=1$, there can be
only one past-time quantum memory, and therefore one time-shifted density
operator entering in (38). Being so, (37) is given as follows,

$\begin{split}&\rho^{({\rm
pert})}(t-a,a)\;=\;\sum_{m=0}^{\infty}\;\sum_{n=0}^{\infty}\;\frac{(-1)^{m}}{m!}\;a^{n+m}\;\rho_{t}^{(n|m)}.\end{split}$
(39)

Applying (36,39) to (38), we obtain the following system of differential
equations, with $j\;\geq\;0$ denoting the perturbation order in terms of the
QMD $a$,

$i\;\dot{\rho}^{(j)}_{t}=i\;\rho_{t}^{(j|1)}=\mu\;\sum_{m,n=0}^{m+n=j}\;\frac{(-1)^{m}}{m!}\;\;[\;\rho_{t}^{(n|m)}\;,\;\rho_{t}^{(j-n-m)}].$
(40)

Now we look for the functions $\rho^{(j)}_{t}$ which solve the system (40)
order by order in the Near-Markovian perturbation. At the zeroth order, $j=0$,
we find,

$i\;\dot{\rho}_{t}^{(0)}\;=\;0.$ (41)

That is, $\rho^{(0)}_{t}$ is constant. At the first order of perturbation, we
extract the differential equation for $\rho_{t}^{(1)}$ from (40), and apply
the zeroth-order result to it. Thus, we obtain,

$i\;\dot{\rho}_{t}^{(1)}=\;\mu\;[\;\rho^{(0)}_{t},\dot{\rho}^{(0)}_{t}\;]\;=\;0.$
(42)

Therefore, $\rho^{(1)}_{t}$ is also found to be constant. Next, extracting
from (40) the equation for the second-order entry $\rho^{(2)}_{t}$ given in
(36), and using the constancy of $\rho^{(0)}_{t}$ and $\rho^{(1)}_{t}$,
yields,

$\begin{split}i\;\dot{\rho}_{t}^{(2)}&=-\mu\;(\;[\;\dot{\rho}^{(0)}_{t},\rho^{(1)}_{t}\;]-\dfrac{1}{2}\;[\;\ddot{\rho}^{(0)}_{t},\rho^{(0)}_{t}\;]\;+[\;\dot{\rho}^{(1)}_{t},\rho^{(0)}_{t}\;]\;)\;=\;0.\end{split}$
(43)

That is, $\rho^{(2)}_{t}$ is also a constant solution of time. Continuing this
procedure to all the higher orders in the perturbation over the QMD variable
$a$, we obtain,

$\begin{split}&\dot{\rho}_{t}^{(m\;\geq\;0)}=0\;\;\;\Longrightarrow\;\;\;\rho^{({\rm
pert})}(t,a)\;=\;\text{constant}.\end{split}$ (44)

Therefore, one concludes as follows. The only solutions to the $(1,1)$-QMM
dynamical equations (38) which are analytic at the Markovian point $a=0$ are
constant-in-time density operators. In other words, the Near-Markovian limit
of the equations (38) does not admit any time-dependent perturbative
solutions.

It is easy to see that by following the aforementioned procedure, detailed for
the $(1,1)$ QMM-UE, in the case of larger values of $N$ and $L$, we are lead
to the same conclusion. For each case, the computation based on (20,36,37) is
straightforward and similar, but just becomes increasingly more involved, with
the final result remaining the same. Robust Non-Markovianity is stable under
interactions between quantum memories.

_The purely-QMM-UEs are ‘Robustly non-Markovian’_. They are robustly non-
Markovian in the following senses: i) Their Near-Markovian limits, namely when
the largest QMD becomes sufficiently small, are perturbatively trivial,
developing only constant-in-time states as perturbative solutions to their QMM
Schrödinger equations. ii) Given any $H^{(N|L)}_{{\rm QMM}}(t)$, the analytic
and nontrivial quantum-state-histories are developed only when the largest
QMD, $\Delta_{1}^{(t)}\equiv a$, surpasses (or is not less than) some finite
threshold $a^{\star}$ which depends on the couplings. That is, collectively
denoting the couplings $\lambda_{i_{1}\cdots i_{r^{(\leq N)}}}$ introduced in
(14) by $\vec{\lambda}$, there is the following ‘Robust Non-Markovianity
Condition’ to be satisfied for developing state-histories which are
dynamically non-trivial and solve the QMM Schrödinger equation (27),

$a\;\geq\;a^{*}=a^{*}(\vec{\lambda}).$ (45)

Three highlights are in order. First, typical state-histories solving the QMM
Schrödinger equations of $H^{(N|L)}_{{\rm QMM}}(t)$s are highly dynamical and
develop giant landscapes which are extremely rich qualitatively. The many
solutions of the QMM Schrödinger equations (27) which we shall obtain
analytically and numerically in the next sections establish the fact.
Moreover, for each one of those solutions, we will confirm that the Robust
Non-Markovianity Condition (45) is validated.

Second, the threshold $a^{\star}$ of the purely-QMM-UE of is generally a
complicated function of the $\lambda_{i_{1}\cdots i_{r}}$ couplings. In some
elementary examples, we compute $a^{\star}$, or lower bounds to it,
analytically. For example, for a specially interesting class of unitary state-
histories developed by $H^{(2|2)}_{{\rm QMM}}(t)$ in (15), the right hand side
of (45) is realized as,

$a^{\star}\>\geq\;\frac{1}{|\lambda|}.$ (46)

Third, hybrid QMM-UEs are threshold-free, as we will show later. For them, the
conventional parts of the hybrid QMM-Hs add source terms to (20), such that
$\rho^{(m)}_{t}$s in (36) become non-constant, yielding dynamical solutions
for arbitrarily small $a$.

##  4 Purely-QMM Unitary Evolutions of The One-Qubit Closed System

###  4.1 State-History Representations And Quantum Memory Characters

Having formulated a very general theory of QMM-UEs, we now focus on
chracterizing, solving and analyzing, the unitarily-evolving wavefunctions of
the closed system of a single qubit, to be called the ‘one-qubit closed
system’, as generated by its quantum memories, in the concrete forms specified
in (15). Physically and mathematically, the one-qubit closed system is the
most fundamental setting to be explored in the novel context of QMM-UE, for
three (two very basic and one rather surprising) reasons. First, its two
dimensional Hilbert space is the smallest state-space capable of hosting
genuinely-quantum features. As a result, given a choice of $H^{(N,L)}_{{\rm
QMM}}$, the corresponding QMM Schrödinger equations (27) which are very
complex and unprecedented (highly nonlocal-in-time, highly-nonlinear)
differential equations find their simplest possible mathematical profiles in
this very setting. Despite significant simplifications, we will see that
handling, analyzing and solving the one-qubit QMM Schrödinger equations is
still a very demanding task, not only analytically, but also numerically.
Second, one finds many physical situations in which the actual larger (even
infinite dimensional) quantum system can be effectively or temporarily
approximated by the one-qubit closed system, given that the quantum degrees of
freedom which are dynamically active, or are relevant to the observables we
are interested in, become effectively that of a qubit. Third, we will find out
that the unitary-evolving one-qubit wavefunctions which solve the QMM
Schrödinger equations develop phase diagrams which are remarkably rich, in the
two cases of purely-QMM-UE and hybrid-QMM-UE. In fact, as we will observe in
Section 6, these phase diagrams host universality classes of closed-system
quantum states which (to the best of our knowledge) are qualitatively
unprecedented, moreover are likely useful for various intriguing applications.
Being so, the mathematical forms and physical aspects of the purely-QMM
Schrödinger equations (27) of the one-qubit closed system unitarily evolving
under (15) are worked out in Section 4 in all details. The solutions and phase
diagrams will be obtained in Sections 5 and 6.

The one-qubit Hilbert space, spanned by the states
$\\{|-\rangle,|+\rangle\\}$, is isomorphic to $\mathbb{CP}^{1}$. Qubit pure
states, parametrised by the Bloch angles $\theta_{t}\in[0,\pi]$ and
$\phi_{t}\in[0,2\pi)$ are,

$\begin{split}&\rho_{t}=\ket{\Psi_{t}}\bra{\Psi_{t}}=\sin^{2}\frac{\theta_{t}}{2}\;A_{-}+\frac{1}{2}\sin\theta_{t}e^{-i\phi_{t}}\;C_{-}+\frac{1}{2}\sin\theta_{t}e^{i\phi_{t}}\;C_{+}+\cos^{2}\frac{\theta_{t}}{2}\;A_{+},\\\
&C_{\pm}\;\equiv\;|\mp\rangle\langle\pm|\;\;\;;\;\;\;A_{\pm}\;\equiv\;|\pm\rangle\langle\mp|,\\\
&|\Psi_{t}\rangle=\sin\dfrac{\theta_{t}}{2}\;e^{i\phi_{t}}\;|-\rangle\;+\;\cos\dfrac{\theta_{t}}{2}\;|+\rangle.\end{split}$
(47)

Representing by matrices, $(\ket{-},\ket{+})\equiv(\begin{pmatrix}0\\\ 1\\\
\end{pmatrix},\begin{pmatrix}1\\\ 0\\\ \end{pmatrix})$, the one-qubit pure
states (47) are rewritten as follows,

$\ket{\Psi_{t}}\;=\;\begin{pmatrix}\cos\frac{\theta_{t}}{2}\\\
\sin\frac{\theta_{t}}{2}\;e^{i\phi_{t}}\\\
\end{pmatrix}\;\;\;;\;\;\;\rho_{t}\;=\;\frac{1}{2}\;\begin{pmatrix}1+\cos\theta_{t}\;&\;\sin\theta_{t}\;e^{-i\phi_{t}}\\\
\sin\theta_{t}\;e^{i\phi_{t}}\;&\;1-\cos\theta_{t}\\\ \end{pmatrix}.$ (48)

In spin representation where one employs the three $SU(2)$ Pauli matrices
$\vec{\sigma}=2\vec{S}$, the basis states $\ket{\pm}$ are mapped to the
eigenstates of $\sigma^{3}$ with eigenvalues $\pm 1$, respectively. Moreover,
we have,

$\sigma^{1}\ket{\mp}\;=\;\ket{\pm}\;\;\;;\;\;\;A_{\pm}\;=\;\frac{\mathbb{1}\pm\sigma^{3}}{2}\;=\;\frac{\mathbb{1}}{2}\pm
S^{3}\;\;\;;\;\;\;C_{\mp}\;=\;\frac{\sigma^{1}\;\pm\;i\sigma^{2}}{2}\;\equiv\;\frac{\sigma^{\pm}}{2}\;=\;S^{\pm}.$
(49)

Now, upon introducing a time-dependent Bloch vector of norm one,
$\vec{r}_{t}$, the one-qubit pure states (48) are represented in terms of the
Pauli matrices $\sigma^{\mu}\equiv(\mathbb{1},\vec{\sigma})$ as follows,

$\begin{split}&\rho_{t}\;\equiv\;\frac{1}{2}\;r^{\mu}_{t}\;\sigma^{\mu}\;=\;\frac{1}{2}\;(\;\mathbb{1}\;+\;\vec{r}_{t}\;.\;\vec{\sigma}\;),\\\
&\text{Pure-State
Condition},\;\forall\;t\;:\;\;\;\;|\vec{r}_{t}|\;=\;1\;\;\;\text{s.t.}\;\\\
&\vec{r}\;\equiv\;(x_{t}\;,\;y_{t}\;,\;z_{t})\;=\;\;(\sin\theta_{t}\;\cos\phi_{t}\;,\;\sin\theta_{t}\;\sin\phi_{t}\;,\;\cos\theta_{t}).\\\
\end{split}$ (50)

Alternatively, the one-qubit closed system can be reformulated fermionically,
a la Schwinger. One begins with introducing two constrained fermion fields,
say each one corresponding to either of the coloured flavours: red and green,
$(f_{r},f_{r}^{\dagger})$ and $(f_{g},f_{g}^{\dagger})$. The one-qubit Hilbert
subspace is identified with the sub-Hilbert space of this pair of fermions for
which the following constraint of total flavour-number conservation holds:
$n_{r}+n_{g}=1$. In effect, one has the following correspondence,

$\sigma_{z}\;\longleftrightarrow\;n_{g}\;\;\;;\;\;\;\sigma^{-}\;\longleftrightarrow\;f_{r}^{\dagger}f_{g}\;\;\;;\;\;\;\sigma^{+}\;\longleftrightarrow\;f_{g}^{\dagger}f_{r}.$
(51)

Applying this map to (50), and introducing a pair of related coordinates
$x_{t}^{\pm}\equiv x_{t}\pm iy_{t}$, one finds the following Schwinger
fermionic representation of the one-qubit pure states,

$\rho_{t}\;=\;\dfrac{1-z_{t}}{2}\;\mathbb{1}\;+\;z_{t}\;n_{g}\;+\;\frac{x^{+}_{t}}{2}\;f^{\dagger}_{r}f_{g}\;+\frac{x^{-}_{t}}{2}\;f^{\dagger}_{g}f_{r}.$
(52)

Now, representations of the state-histories of the one-qubit closed system are
clear. Given the unitarity of the evolution, every continuous-in-time
succession of the above representations of the one-qubit pure quantum states,
(50) or (52), collected from the initial time $t_{0}$ to the present moment
$t$, must be identified with a representation of one (in-principle) possible
pure-state-history $\mathcal{P}_{[t_{0},t]}$ of the one-qubit closed system
(LABEL:hps). That is, given continuous functions
$(f^{\theta}(t),f^{\phi}(t))\equiv(f^{\theta}_{t},f^{\phi}_{t})=(\theta_{t},\phi_{t})$,
the map is,

$\;\\{\;(\;\sin(f^{\theta}_{t})\cos(f^{\phi}_{t})\;,\;\sin(f^{\theta}_{t})\sin(f^{\phi}_{t})\;,\;\cos(f^{\theta}_{t})\;)\;\equiv\;\vec{r}_{t}\;;\;\forall\;t\in[t_{0},t]\;\\}\;\longleftrightarrow\;\mathcal{P}_{[t_{0},t]}.$
(53)

Now, we compute the QM assemblers, that is, QM-TPF (LABEL:ctmcs) and QM-TPO
(2.4), for any pair of moments $(t^{\prime},t^{\prime\prime})$ in the pure-
state-history of the one-qubit closed system. It is natural to begin with
computing the fundamental QM observable, that is, the (square of the) norm of
the QM-TPF as defined in (29),

$w_{t^{\prime}t^{\prime\prime}}^{2}\;=\;|m_{t^{\prime}t^{\prime\prime}}|^{2}\;=\;|\braket{\Psi_{t^{\prime}}}{\Psi_{t^{\prime\prime}}}|^{2}\;=\;\frac{1}{2}\;\text{Tr}[\;\\{\rho_{t^{\prime}}\;,\;\rho_{t^{\prime\prime}}\\}\;].$
(54)

Using the representation (50) and the algebra of Pauli matrices, we obtain,

$\\{\rho_{t^{\prime}}\;,\;\rho_{t^{\prime\prime}}\\}\;=\;\frac{1}{2}\;[\;(1+\vec{r}_{t^{\prime}}.\vec{r}_{t^{\prime\prime}})\;\mathbb{1}+(\;\vec{r}_{t^{\prime}}+\vec{r}_{t^{\prime\prime}}\;).\;\vec{\sigma}\;].$
(55)

Therefore, from (54) and (55), one finds as follows the fundamental QM
observable of the one-qubit closed system,

$\begin{split}&w_{t^{\prime}t^{\prime\prime}}^{2}\;=\;\frac{1}{4}\;|\vec{r}_{t^{\prime}}+\vec{r}_{t^{\prime\prime}}|^{2}\;=\;\frac{1}{2}\;(1\;+\;\vec{r}_{t^{\prime}}\;.\;\vec{r}_{t^{\prime\prime}})\;=\;\\\
&\;\;\;\hskip
16.84389pt=\;\frac{1}{2}\;\big{(}\cos\theta_{t^{\prime}}\cos\theta_{t^{\prime\prime}}\;+\;\sin\theta_{t^{\prime}}\sin\theta_{t^{\prime\prime}}\cos(\phi_{t^{\prime}}-\phi_{t^{\prime\prime}})\;+\;1\big{)}.\end{split}$
(56)

Indeed, the norm of the one-qubit QM-TPF reaches its maximum value,
$w_{t^{\prime}t^{\prime\prime}}=1$, when
$\vec{r}_{t^{\prime\prime}}=\vec{r}_{t^{\prime}}$, for which
$\rho_{t^{\prime\prime}}=\rho_{t^{\prime}}$, and its minimum value,
$w_{t^{\prime}t^{\prime\prime}}=0$, when
$\vec{r}_{t^{\prime\prime}}=-\vec{r}_{t^{\prime}}$, corresponding to
$\rho_{t^{\prime\prime}}=1-\rho_{t^{\prime}}$. Now, the one-qubit QM-TPF
$m_{t^{\prime}t^{\prime\prime}}$ differs from its norm
$w_{t^{\prime}t^{\prime\prime}}$ by a phase factor which is removable by
unobservable phase transformations on the QM wavefunctions. Therefore, the
dynamical equations (20) and (24) do not depend on the to-be-determined phase
of $m_{t_{i}t_{j}}$, because density operators are by construction insensitive
to the phase redundancies of the wavefunctions. But, the total QM-TPFs,
including their phases, do enter into the QMM Schrödinger equations (27) which
are written in terms of the state-history wavefunctions. Using definition
(LABEL:ctmcs) and the Bloch parametrization of the wavefunctions (47), we
obtain the one-qubit QM-TPF,

$m_{t^{\prime}t^{\prime\prime}}=\cos\frac{\theta_{t^{\prime}}}{2}\;\cos\frac{\theta_{t^{\prime\prime}}}{2}\;+\;\sin\frac{\theta_{t^{\prime}}}{2}\;\sin\frac{\theta_{t^{\prime\prime}}}{2}\;e^{i(\phi_{t^{\prime\prime}}-\phi_{t^{\prime}})}.$
(57)

Finally, we must compute the QM-TPO $M_{t^{\prime}t^{\prime\prime}}$ which can
be done in different ways. Using representation (50) and also the algebra of
Pauli matrices, one obtains the following expression for the product of any
two one-qubit density operators,

$\rho_{t^{\prime}}\rho_{t^{\prime\prime}}\;=\;\frac{1}{4}\;[\;(1+\vec{r}_{t^{\prime}}.\vec{r}_{t^{\prime\prime}})\;\mathbb{1}+(\;\vec{r}_{t^{\prime}}+\vec{r}_{t^{\prime\prime}}+\;i\;\vec{r}_{t^{\prime}}\times\vec{r}_{t^{\prime\prime}}\;).\;\vec{\sigma}\;].$
(58)

Now, using (LABEL:ctmcs,2.4), the one-qubit QM-TPO is obtained as follows,

$M_{t^{\prime}t^{\prime\prime}}\;=\;\frac{1}{m_{t^{\prime}t^{\prime\prime}}}\;\rho_{t^{\prime}}\rho_{t^{\prime\prime}}=\begin{pmatrix}\cos\frac{\theta_{t^{\prime}}}{2}\;\cos\frac{\theta_{t^{\prime\prime}}}{2}\;&\;\cos\frac{\theta_{t^{\prime}}}{2}\;\sin\frac{\theta_{t^{\prime\prime}}}{2}\;e^{-i\phi_{t^{\prime\prime}}}\\\
\sin\frac{\theta_{t^{\prime}}}{2}\;\cos\frac{\theta_{t^{\prime\prime}}}{2}\;e^{i\phi_{t^{\prime}}}\;&\;\sin\frac{\theta_{t^{\prime}}}{2}\;\sin\frac{\theta_{t^{\prime\prime}}}{2}\;e^{i(\phi_{t^{\prime}}-\phi_{t^{\prime\prime}})}\\\
\end{pmatrix}.$ (59)

In conclusion, for any pair of moments $(t_{i},t_{j})$ in the pure-state-
history of the one-qubit closed system, its QM-TPO $M_{t_{i}t_{j}}$ is
determined by putting together the results (57), (58) and (59).

###  4.2 (1,1) QMM Unitary Evolution: The Simplest QMM-H Made Of One Quantum
Memory

We naturally begin with the most minimal QMM-UE ever possible, the choice:
$N=1$. Upon this choice, the purity of the state-history formulated in
(LABEL:hps) implies that $L=1$ is the longest length that a monomial string in
the QMM-H can take. That is, $\mu\;\rho_{t-\Delta^{(t)}_{1}}$, with $\mu$
being an arbitrary real valued coupling, can be the only monomial which the
Hamiltonian can contain. Furthermore, setting the QMD to be an arbitrary
constant parameter denoted by $a$, we conclude the QMM-H specified in (15):
$H^{(1,1)}_{\text{QMM}}=\mu\;\rho_{t-a}$. Finally, picking the ‘spin’
representation of the one-qubit density operators given in (50), and upon
removing the dynamically-trivial term $\propto\mathbb{1}$, the QMM-H becomes,

$\bar{H}^{(1,1)}_{\text{QMM}}(t)\;=\;\frac{1}{2}\;\mu\;\vec{r}_{t-a}\cdot\vec{\sigma}.$
(60)

In useful analogy with conventional one-qubit Hamiltonians, it is natural to
define a formal magnetic field in terms of which the QMM-H (60) can be re-
expressed as the Hamiltonian of a magnetic field. The magnetic field, however,
is purely made from the quantum memory of the one-qubit closed system. We call
this nonlocal-in-time QMM field, the ‘QMM Magnetic Field’ (QMM-MF). The QMM-MF
written in terms of the QM Bloch vector, and the re-expressed QMM-H (60) are,

$\vec{\mathcal{B}}^{(1,1)}_{\text{QMM}}(t)\;\equiv\;\mu\;\vec{r}_{t-a}\;\;\;;\;\;\;\bar{H}^{(1,1)}\;=\;\vec{\mathcal{B}}^{(1,1)}(t)\cdot\vec{S}.$
(61)

Having defined (61), the QMM-H (60) and the dynamical equations (20), or (24)
for the one-qubit closed system unitarily evolving under (60), take the
following forms,

$\begin{split}&\dot{\vec{r}}_{t}\;=\;\vec{\mathcal{B}}_{\text{QMM}}^{(1,1)}(t)\times\vec{r_{t}}\;=\;\mu\;\vec{r}_{t-a}\times\vec{r}_{t}.\end{split}$
(62)

Equivalently, the QMM Schrödinger equation (27) for the QMM-H (60) is given
by,

$i\ket{\dot{\Psi}_{t}}\;=\;2\;\mu\;m_{t-a\;t}\ket{\Psi_{t-a}}.$ (63)

Now, we present the system of _delay-differential equations_ or difference-
differential equations [62, 63, 64, 65] for the evolution of Bloch angles
$(\theta,\phi)$ of a qubit unitarily evolving under
$H^{(1,1)}_{\text{QMM}}(t)$. It can be obtained by extracting it from the
vector equations (62), or by directly applying the QM-TPF $m_{t-a\;t}$ given
in (57) to (63). The result is,

$\dot{\theta}_{t}=\mu\;\mathcal{G}_{t-a,t}\;\;\;;\;\;\;\sin\theta_{t}\;\dot{\phi}_{t}=\mu\;\mathcal{F}_{t-a,t}.$
(64)

in which,

$\displaystyle\mathcal{F}_{t-a,t}$
$\displaystyle=\cos\theta_{t-a}\sin\theta_{t}-\sin\theta_{t-a}\cos\theta_{t}\cos(\phi_{t}-\phi_{t-a}),$
$\displaystyle\mathcal{G}_{t-a,t}$
$\displaystyle=-\sin\theta_{t-a}\sin(\phi_{t}-\phi_{t-a}).$ (65)

The one-qubit unitary state-histories which solve (64,4.2) will be
investigated in the next sections. Moreover, we will present their phase
diagram in Subsection 6.2.

###  4.3 (2,2) QMM Unitary Evolution: QMM-H Made By Two System’s States With
Monomial Interactions Up To Second Degree

We now work out in full details the Purely-QMM-UE of the one-qubit closed
system made by two interlinked quantum states, one of them being the present
moment state, $\rho_{t}$, and the other the state a distance $a$ back in time,
$\rho_{t-a}$, forming strings of length one (linear in QMs) and two (quadratic
in QMs) in the Hamiltonian. That is, we formulate a unitary two-state system
whose evolution is re-fabricated moment by moment by nothing but a pair of its
quantum memories, corresponding to
$(\Delta^{(t)}_{2},\Delta^{(t)}_{1})=(0,a)$, building up by their monomial
interactions the $(N,L)=(2,2)$ version of the QMM-H (14). We spelled out in
(15) this QMM-H for any closed system. Here we move on to presenting the
mathematical and physical structures of the QMM-H, and its dynamical
equations, once applied to the one-qubit closed system. In doing so, however,
we shall add a dynamically effectless term linear in $\rho_{t}$ to the QMM-H
(15) which gives more symmetric forms to some of the quantities and equations
to come. We highlight that this Hamiltonian term which commutes with
$\rho_{t}$ will make no observable effect on the final dynamical equations
apart from unphysical deformations which only have to do with unobservable
phases of the wavefunctions $\ket{\Psi_{t}}$ given in (47). Therefore,
denoting the real and imaginary parts of the $\lambda$ coupling in (15) by
$(\lambda^{R},\lambda^{I})$ and upon renaming the past-QM $\mu$ coupling as
$\mu_{t-a}$, we now consider an isolated qubit whose Hamiltonian is made in
the following form by two of its quantum memories,

$H^{(2,2)}_{\text{QMM}}(t)\;=\;\mu_{t-a}\;\rho_{t-a}\;+\;\mu_{t}\;\rho_{t}\;+\;\lambda^{R}\;\\{\;\rho_{t-a},\rho_{t}\;\\}\;+\;i\lambda^{I}\;[\;\rho_{t-a}\;,\;\rho_{t}\;].$
(66)

Applying the representation (50) of the one-qubit pure density operator, using
the elementary properties of the Pauli matrices, and introducing the deformed
$\mu$ couplings:
$(\hat{\mu}_{t-a},\hat{\mu}_{t})\equiv(\mu_{t-a}+\lambda^{R}\;,\;\mu_{t}+\lambda^{R})$,
the QMM-H (66) takes the following form,

$\begin{split}&H^{(2,2)}_{\text{QMM}}(t)\;=\;\\\
&=\frac{1}{2}\;[\;(\mu_{t-a}+\mu_{t}+\lambda^{R}+\lambda^{I}\;\vec{r}_{t-a}\;\cdot\;\vec{r}_{t}\;)\;\mathbb{1}+\;(\hat{\mu}_{t-a}\;\vec{r}_{t-a}+\hat{\mu}_{t}\;\vec{r}_{t}-\lambda^{I}\;\vec{r}_{t-a}\;\times\;\vec{r}_{t}\;)\cdot\vec{\sigma}\;].\end{split}$
(67)

It is natural to introduce a QMM-MF
$\vec{{\mathcal{B}}}^{(2,2)}_{\text{QMM}}(t)$ which in every moment $t$ is
remade from the two interacting quantum memories of the qubit
$(\rho_{t-a},\rho_{t})$,
$\vec{{\mathcal{B}}}^{(2,2)}_{\text{QMM}}(t)=\vec{{\mathcal{B}}}^{(2,2)}_{\text{QMM}}(\vec{r}_{t-a},\vec{r}_{t})$.
Being so, dropping the term $\propto\mathbb{1}$ in (67) and renaming its
dynamically relevant part as $\bar{H}^{(2,2)}_{\text{QMM}}(t)$, the (2,2)
QMM-H of the one-qubit closed system is re-expressed as follows,

$\vec{{\mathcal{B}}}^{(2,2)}_{\text{QMM}}(t)\;\equiv\;\hat{\mu}_{t-a}\;\vec{r}_{t-a}\;+\;\hat{\mu}_{t}\;\vec{r}_{t}\;-\;\lambda^{I}\;\vec{r}_{a}\times\vec{r}_{t}\;\;\;;\;\;\;\bar{H}^{(2,2)}_{\text{QMM}}(t)\;=\;\vec{{\mathcal{B}}}^{(2,2)}_{\text{QMM}}(t)\cdot\vec{S}.$
(68)

Accordingly, the dynamical equations (20) or (24) which are resulted from the
above QMM-H come to the following familiar Bloch-vectorial form,

$\begin{split}&\hskip
4.6945pt\dot{\vec{r}}_{t}\;=\;\vec{\mathcal{B}}^{(2,2)}_{\text{QMM}}(t)\times\vec{r_{t}}\;\;\;\Longrightarrow\;\;\;\dot{\vec{r}}_{t}\;=\;\lambda^{I}\;\vec{r}_{t-a}\;-\;\lambda^{I}\;(\vec{r}_{t-a}\cdot\vec{r}_{t})\;\vec{r}_{t}\;+\;\hat{\mu}_{t-a}\;\;\vec{r}_{t-a}\times\vec{r}_{t}.\end{split}$
(69)

One now highlights, as an alternative, the insightful formulation of the (2,2)
QMM-UE of the one-qubit closed system by means of Schwinger fermions. The
purely-fermionic representation of the QMM-H, $H^{(2,2)}_{\text{QMM}}(t)$, can
be done in two ways: either we work it out independently by going back to
(66), applying the fermionic representation (52) to it and doing the algebra,
or we simply apply the mapping (51) to the spin representation (67). We shall
write the final result only. First, introducing the following compact
notations are useful,

$\begin{split}&v\;\equiv\;1-z\;\;\;;\;\;\;(a\wedge
b)_{t-a\;t}\;\equiv\;a_{t-a}\;b_{t}\;-\;a_{t}\;b_{t-a}\;,\;\forall\;(a,b).\end{split}$
(70)

Now, the $(2,2)$ QMM-H of the one-qubit closed system finds the following
alternative representation as a QMM purely-fermionic system,

$\begin{split}&H^{(2,2)}_{\text{QMM}}(t)\;=\;C_{1}(t)\;\mathbb{1}\;+\;\bar{H}^{(2,2)}_{\text{QMM}}(t),\\\
&2\;C_{1}(t)\;=\;\mu_{t-a}\;v_{t-a}\;+\;\mu_{t}\;v_{t}\;+\;\lambda^{I}\;(x^{-}_{t-a}\;x^{+}_{t})^{I}\;+\;\lambda^{R}\;[\;(x^{-}_{t-a}\;x^{+}_{t})^{R}\;+\;v_{t-a}\;v_{t}\;],\\\
&\bar{H}^{(2,2)}_{\text{QMM}}(t)\;=\;\zeta_{\text{QMM}}(t)\;n_{g}\;+\;\tau_{\text{QMM}}^{\star}(t)\;f^{\dagger}_{r}f_{g}\;+\;\tau_{\text{QMM}}(t)\;f^{\dagger}_{g}f_{r}.\end{split}$
(71)

As we see in (71), the dynamically relevant part of the Hamiltonian,
$\bar{H}^{(2,2)}_{\text{QMM}}(t)$, takes the following couplings: a nonlocal-
in-time real coupling $\zeta_{\text{QMM}}(t)$ which is similar to a QMM
chemical potential, and a nonlocal-in-time complex coupling
$\tau_{\text{QMM}}^{\star}(t)$ which is similar to a QMM tunneling amplitude.
In terms of the Bloch coordinates of the system’s states at $t-a$ and $t$,
these QMM couplings are derived as follows,

$\begin{split}&\;\;\;\zeta_{\text{QMM}}(t)\;=\;\hat{\mu}_{t-a}\;z_{t-a}\;+\;\hat{\mu}_{t}\;z_{t}\;-\;\lambda^{I}\;(x^{-}_{t-a}\;x^{+}_{t})^{I},\\\
&2\;\tau_{\text{QMM}}(t)\;=\;\hat{\mu}_{t-a}\;x^{-}_{t-a}\;+\;\hat{\mu}_{t}\;x^{-}_{t}\;+\;i\;\lambda^{I}\;(z\wedge
x^{-})_{t-a\;t},\\\
&2\;\tau_{\text{QMM}}^{\star}(t)\;=\;\hat{\mu}_{t-a}\;x^{+}_{t-a}\;+\;\hat{\mu}_{t}\;x^{+}_{t}+i\;\lambda^{I}\;(z\wedge
x^{+})_{{t-a}\;t}.\end{split}$ (72)

Given (71,LABEL:hubbardcouplingsare), the $(2,2)$-QMM one-qubit closed system
physically takes the form of a (nonlocal-in-time version of) Hubbard model
[86] with two sites, each one site for one of the Schwinger fermions. The
unconventionality of the above Hubbard model comes about by the QMM structure
of the couplings: the tunneling amplitude and chemical potential are remade,
moment by moment, by the state-history history of the system. We also obtain
an ‘unconventional-Hubbard-model’ representation of the $(1,1)$ one-qubit
closed system by turning off the $\lambda$ coupling in
(LABEL:hubbardcouplingsare). Likewise, the $n$-qubit closed system yield a
nonlocal-in-time $2n$-site Hubbard model in which all tunneling amplitudes,
density-density interactions and chemical potentials are QMM. Finally, we note
that unconventional fermionic-or-spin systems similar to
(71,LABEL:hubbardcouplingsare) or (68), but albeit with increasingly more
complex QMM many-body interactions, represent the $(N,L)$-QUEs of all finite-
dimensional closed quantum systems.

Now using either of the equivalent representations
(71,LABEL:hubbardcouplingsare) or (68), one can obtain the differential
equations for the time evolution of the one-qubit Bloch coordinates. For
example, these equations can be extracted from the dynamical equation (69), by
using the angular parametrization of the two QM Bloch vectors
$(\vec{r}_{t-a},\vec{r}_{t})$, given in (50). Alternatively, they can be
obtained by applying the QM-TPF data (57) and (54), together with the Bloch-
angle parametrizations (47), to the (2,2) QMM Schrödinger equation given in
(32,33). The result is the following coupled system of nonlocal-in-time
differential equations,

$\displaystyle\dot{\theta}_{t}\;=\;\hat{\mu}_{t-a}\;\mathcal{G}_{t-a,t}\;-\;\lambda^{I}\;\mathcal{F}_{t-a,t},$
$\displaystyle\sin\theta_{t}\;\dot{\phi}_{t}\;=\;\hat{\mu}_{t-a}\;\mathcal{F}_{t-a,t}\;+\lambda^{I}\;\mathcal{G}_{t-a,t}.$
(73)

The two QMM functions $(\mathcal{F}_{t-a,t},\mathcal{G}_{t-a,t})$ are the same
as given in (4.2). In next sections, some analytic solutions to the above
equations are obtained and analyzed. Moreover, we numerically work out the
total phase diagram of the unitarily-evolving wavefucntions which solve (4.3)
is Subsection 6.3.

###  4.4 (2,3) QMM Unitary Evolution: QMM-H Made By Two System’s States With
Monomial Interactions Up To Third Degree

Now, we consider, and later analyze and solve, the QMM-UE with $N=2$ and
$L=3$. The motivation is more than just analyzing a QMM-H with more complex
structure: the aim is to understand the effect of turning on higher-order
monomial interactions (in this case: third order), while keeping the number of
quantum memories fixed (in this case: two). The corresponding Hamiltonian,
$H^{(2,3)}_{\text{QMM}}(t)$, is specified in (15), with linear terms being
ignored in it, because (as we have seen) the linear coupling $\mu_{t-a}$ can
be absorbed in a deformation of the coupling of the second-degree monomial
interaction: $\lambda^{R}\to\hat{\mu}_{t-a}=\mu_{t-a}+\lambda^{R}$. To this
aim, as motivated before, we restrict to the fundamental example: the one-
qubit closed system under QMM-UE generated by $H^{(2,3)}_{\text{QMM}}(t)$ in
(15). For the one-qubit closed system, using (66,56,50), the $(2,3)$ QMM-H in
(15) is,

$\begin{split}&H^{(2,3)}_{\text{QMM}}(t)\;=\;H^{(2,2)}_{\text{QMM}}(t)_{|_{(\mu_{t-a},\mu_{t})=(0,0)}}\;+\;H^{(2,3|\Delta
L=1)}_{\text{QMM}}(t),\\\ &H^{(2,3|\Delta
L=1)}_{\text{QMM}}(t)\;=\;\eta\;\rho_{t-a}\;\rho_{t}\;\rho_{t-a}\;=\;\eta\;w_{t-a\;t}^{2}\;\rho_{t-a}\;=\\\
&\;\;\;\;\;\hskip
52.63777pt=\;\frac{\eta}{2}\;(\;1+\vec{r}_{t-a}\cdot\vec{r}_{t}\;)\;\rho_{t-a}\;=\;\frac{\eta}{4}\;(\;1+\vec{r}_{t-a}\cdot\vec{r}_{t}\;)\;(\mathbb{1}+\vec{r}_{t-a}\cdot\vec{\sigma}).\end{split}$
(74)

As in (1,1) and (2,2) cases, we now define a QMM-MF
$\vec{{\mathcal{B}}}^{(2,3)}_{{\text{Q}MM}}(t)$. Introducing
$\bar{H}^{(2,3)}_{\text{QMM}}(t)$ by setting aside the term
$\propto\mathbb{1}$ in (LABEL:avn), and remembering (68), we immediately find,

$\begin{split}&\bar{H}^{(2,3)}_{\text{QMM}}(t)\;=\;\vec{{\mathcal{B}}}^{(2,3)}_{{\text{Q}MM}}(t)\;\cdot\;\vec{S},\\\
&\vec{{\mathcal{B}}}^{(2,3)}_{{\text{Q}MM}}(t)\;=\;\lambda^{R}\;\vec{r}_{t}\;+\;[\;\lambda^{R}\;+\;\dfrac{\eta}{2}\;(\;1\;+\;\vec{r}_{t-a}\cdot\vec{r}_{t}\;)\;]\;\vec{r}_{t-a}-\;\lambda^{I}\;\vec{r}_{t-a}\times\vec{r}_{t}.\end{split}$
(75)

Being so, the resulted dynamical equations (20) or (24) are given as follows,

$\begin{split}&\dot{\vec{r}}_{t}\;=\;\vec{{\mathcal{B}}}^{(2,3)}_{{\text{Q}MM}}(t)\times\vec{r}_{t}\;\;\;\Rightarrow\;\\\
&\dot{\vec{r}}_{t}\;=\;\lambda^{I}\;\vec{r}_{t-a}\;-\;\lambda^{I}\;(\vec{r}_{t-a}\cdot\vec{r}_{t})\;\vec{r}_{t}\;+\;[\;\lambda^{R}+\dfrac{\eta}{2}\;(1+\vec{r}_{t-a}\cdot\vec{r}_{t})\;]\;\vec{r}_{t-a}\times\vec{r}_{t}.\end{split}$
(76)

Finally, either by applying (57, 54; 47) to QMM Schrödinger equation (27) for
$H^{(2,3)}_{\text{QMM}}(t)$, or by working out (76) in Bloch polar
coordinates, a coupled system of nonlocal-in-time differential equations
describing the time evolution of the Bloch angles under one-qubit $(2,3)$ QMM-
UE is obtained. Defining functions
$(\mathcal{A}^{\theta}_{t-a,t},\mathcal{A}^{\phi}_{t-a,t})$ as,

$\begin{split}&\mathcal{A}^{\phi}_{t-a,t}\;=\;\frac{1}{4}\bigg{(}-\cos(\phi_{t}-\phi_{t-a})\Big{(}\sin(2\theta_{t})\sin^{2}(\theta_{t-a})\cos(\phi_{t}-\phi_{t-a})\\\
&\;+\;2\cos(\theta_{t})\sin(\theta_{t-a})+\cos(2\theta_{t})\sin(2\theta_{t-a})\Big{)}\\\
&\;+\;\sin(2\theta_{t})\cos^{2}(\theta_{t-a})+2\sin(\theta_{t})\cos(\theta_{t-a})\bigg{)}.\end{split}$
(77)
$\begin{split}&\mathcal{A}^{\theta}_{t-a,t}\;=\;-\frac{1}{2}\;\;\sin(\theta_{t-a})\sin(\phi_{t}-\phi_{t-a})\;\times\\\
&\;\times\;\Big{(}\sin(\theta_{t})\sin(\theta_{t-a})\cos(\phi_{t}-\phi_{t-a})+\cos(\theta_{t})\cos(\theta_{t-a})+1\Big{)}.\end{split}$
(78)

and recalling $(\mathcal{F}_{t-a,t},\mathcal{G}_{t-a,t})$ in (4.2), the
dynamical equations are presented as follows,

$\begin{split}&\dot{\theta}_{t}\;=\;\eta\;\mathcal{A}^{\theta}_{t-a,t}\;+\;\lambda^{R}\;\mathcal{G}_{t-a,t}\;-\;\lambda_{t-a,t}^{I}\;\mathcal{F}_{t-a,t},\\\
&\sin\theta_{t}\;\dot{\phi}_{t}\;=\;\eta\;\mathcal{A}^{\phi}_{t-a,t}\;+\;\lambda^{R}\;\mathcal{F}_{t-a,t}\;+\;\lambda_{t-a,t}^{I}\;\mathcal{G}_{t-a,t}.\end{split}$
(79)

The phase diagram of the unitarily-evolving wavefunctions which solve the
equation (LABEL:SE23POLAR) is explored in Section 6.4.

###  4.5 (3,3) QMM Unitary Evolution: A Specific QMM-H Made By Three System’s
States With Third-Degree Monomial Interactions

Finally, we detail the QMM-UE of the one-qubit closed system with
$(N,L)=(3,3)$. With this choice of (N,L), at every present moment $t$, the
system’s Hamiltonian $H^{(3,3)}_{\text{QMM}}(t)$ must be made of three time-
varying quantum memories which interact in the form of monomials of maximum
degree three. Furthermore, we fix these interacting quantum memories by
choosing the following spectrum of QMDs:
$(\Delta_{3}^{(t)},\Delta_{2}^{(t)},\Delta_{1}^{(t)})=(0,b,a)$, with $a$ and
$b<a$ being arbitrary real numbers. That is, the three quantum memories are
identified with the system’s states at the present time $t$, at a past moment
$t-b$, and at a far past moment $t-a$, respectively:
$(\rho_{t-a},\rho_{t-b},\rho_{t})$.

Finally, it suffices to include only interaction monomials of degree three in
defining the corresponding QMM-H, given the sufficient complexity and richness
of this choice, as will be seen later in the resulted phase diagram of the
one-qubit closed system. Therefore, given the assumed purity of the states,
the most general $(3,3)$ QMM-H with this criteria is
$H^{(3,3)}_{\text{QMM}}(t)$ specified in (15). As in all the previous cases,
the one-qubit QMM-H is naturally expressed in terms of a QMM-MF which in every
moment $t$ is remade by the identifying Bloch vectors of the three interacting
states: $(\vec{r}_{t-a},\vec{r}_{t-b},\vec{r}_{t})$. That is, using the
representation (50) and defining the dynamically relevant part of the QMM-H
$\bar{H}^{(3,3)}_{\text{QMM}}(t)$ which differs from the original Hamiltonian
given in (15) only by a dynamically irrelevant term $\propto\mathbb{1}$, one
has,

$\begin{split}&H^{(3,3)}_{\text{QMM}}(t)\;=\;\kappa\;\rho_{t-a}\;\rho_{t-b}\;\rho_{t}\;+\;\kappa^{\star}\;\rho_{t}\;\rho_{t-b}\;\rho_{t-a}\;=\;C(\vec{r}_{t-a},\vec{r}_{t-b},\vec{r}_{t})\;\mathbb{1}\;+\;\bar{H}^{(3,3)}_{\text{QMM}}(t),\\\
&\bar{H}^{(3,3)}_{\text{QMM}}(t)\;=\;\vec{\mathcal{B}}^{(3,3)}_{\text{QMM}}(t)\cdot\vec{S},\\\
&\vec{\mathcal{B}}^{(3,3)}_{\text{QMM}}(t)\;=\;\dfrac{\kappa^{R}}{2}\;\big{[}\;(\vec{r}_{t-a}+\vec{r}_{t-b}+\vec{r}_{t})+(\vec{r}_{t-a}\cdot\vec{r}_{t-b})\vec{r}_{t}-(\vec{r}_{t-a}\cdot\vec{r}_{t})\vec{r}_{t-b}+(\vec{r}_{t-b}\cdot\vec{r}_{t})\vec{r}_{t-a}\;\big{]}-\\\
&\;\;\;\;\;\;\;\hskip
28.45274pt-\dfrac{\kappa^{I}}{2}\;(\;\vec{r}_{t-a}\times\vec{r}_{t-b}\;+\;\vec{r}_{t-b}\times\vec{r}_{t}\;+\;\vec{r}_{t-a}\times\vec{r}_{t}\;).\end{split}$
(80)

Dynamical equations (20) or (24) which correspond to the $(N,L)=(3,3)$ QMM-UE
of the one-qubit closed system under the above Hamiltonian (80) are now given
by,

$\begin{split}&\dot{\vec{r}}_{t}\;=\;\frac{1}{2}\;\kappa^{R}\;\big{[}\;(1+\vec{r}_{t-b}\cdot\vec{r}_{t})\;\vec{r}_{t-a}\times\vec{r}_{t}\;+\;(1-\vec{r}_{t-a}\cdot\vec{r}_{t})\;\vec{r}_{t-b}\times\vec{r}_{t}\;\big{]}+\\\
&\;\;\;\;+\;\frac{1}{2}\;\kappa^{I}\;\big{[}\;(1+\vec{r}_{t-b}\cdot\vec{r}_{t})\;\vec{r}_{t-a}\;+\;(1-\vec{r}_{t-a}\cdot\vec{r}_{t})\;\vec{r}_{t-b}\;-\;(\vec{r}_{t-a}\cdot\vec{r}_{t}\;+\;\vec{r}_{t-b}\cdot\vec{r}_{t})\;\vec{r}_{t}\;\big{]}.\end{split}$
(81)

Finally, we obtain a coupled system of nonlocal-in-time differential equations
describing the dynamics of the one-qubit wavefunction Bloch angles under (80).
As before, this can be done either by applying (57, 54) and (47) to QMM
Schrödinger equation (27) corresponding to the one-qubit Hamiltonian
$H^{(3,3)}_{\text{QMM}}(t)$ given in (80), or by detailing the vectorial
equation (76) in Bloch polar coordinates. The result is,

$\displaystyle\dot{\theta}_{t}$
$\displaystyle=\kappa^{R}\;\tilde{\mathcal{G}}_{t-a,t-b,t}\;-\;\kappa^{I}\;\tilde{\mathcal{F}}_{t-a,t-b,t},$
$\displaystyle\sin\theta_{t}\;\dot{\phi}_{t}$
$\displaystyle=\kappa^{R}\;\tilde{\mathcal{F}}_{t-a,t-b,t}\;+\;\kappa^{I}\;\tilde{\mathcal{G}}_{t-a,t-b,t}.$
(82)

The pair of nonlocal-in-time QMM functions
$(\tilde{\mathcal{F}}_{t-a,t-b,t}\;,\;\tilde{\mathcal{G}}_{t-a,t-b,t})$ in
equations (82) are specified as follows,

$\begin{split}&\tilde{\mathcal{F}}_{t-a,t-b,t}\;\;=\;\frac{1}{2}\;\big{[}-\sin(\theta_{t-a})\cos(\phi_{t}-\phi_{t-a})\cos(\theta_{t-b})\;+\;\cos(\theta_{t-a})\sin(\theta_{t-b})\cos(\phi_{t}-\phi_{t-b})\;+\\\
&\hskip
85.35826pt\;+\;\sin(\theta_{t})\cos(\theta_{t-a})\;-\;\cos(\theta_{t})\sin(\theta_{t-a})\cos(\phi_{t}-\phi_{t-a})\;+\\\
&\;\hskip
85.35826pt+\;\sin(\theta_{t})\cos(\theta_{t-b})\;-\;\cos(\theta_{t})\sin(\theta_{t-b})\cos(\phi_{t}-\phi_{t-b})\;\big{]},\\\
&\\\
&\tilde{\mathcal{G}}_{t-a,t-b,t}\;=\;\frac{1}{2}\;[\;-\cos(\theta_{t})\sin(\theta_{t-a})\sin(\phi_{t}-\phi_{t-a})\cos(\theta_{t-b})\\\
&\;\;\;\hskip
52.63777pt\;\;\;\;\;\;+\;\sin(\theta_{t-b})\cos(\phi_{t-b})\;(\sin(\theta_{t})\sin(\theta_{t-a})\sin(\phi_{t-a})-\sin(\phi_{t}))\;+\\\
&\;\;\;\hskip
52.63777pt\;\;\;\;\;\;+\;\cos(\theta_{t})\cos(\theta_{t-a})\sin(\theta_{t-b})\sin(\phi_{t}-\phi_{t-b})\;+\\\
&\;\;\;\hskip
52.63777pt\;\;\;\;\;\;+\;\sin(\theta_{t-b})\sin(\phi_{t-b})(\cos(\phi_{t})-\sin(\theta_{t})\sin(\theta_{t-a})\cos(\phi_{t-a}))\;+\\\
&\;\;\;\hskip
52.63777pt\;\;\;\;\;\;+\;\sin(\theta_{t-a})\sin(\phi_{t-a}-\phi_{t})\;].\end{split}$
(83)

The complex phase diagram of the QMM unitary state-histories which solve
equations (82) is probed in Section 6.5. Indeed, we will demonstrate that, in
comparison with its $N=2$ counterpart whose unprecedented phase diagram is
remarkably rich, $N=3$ QMM-UE of the one-qubit closed system does develop
several phases of QMM unitary state-histories which are distinctive and
unprecedented qualitatively.

##  5 QMM Unitary Evolutions of The One-Qubit Closed System:
Analytical Results

###  5.1 The General Scope of Analytical Investigations

In this section, a number of distinctive unitarily-evolving wavefunctions
which, exactly or perturbatively, solve the QMM Schrödinger equations of the
one-qubit closed system are obtained analytically. Moreover, we work out
analytically selected features of these solutions, because next to solving the
equations mathematically, we must develop an intuitively-clear understanding
of the qualitatively-novel physics of the QMM unitary state-histories which
are realized by them. To develop these analyses, it suffices to focus all
through this section on the $(2,2)$ QMM-UE of the one-qubit closed system
which was extensively detailed in Subsection 4.3. The main reason for this
sufficiency is twofold. First, QMM Schrödinger equations (4.3) are already
very challenging mathematically. Second, as we will find out, the solutions to
these equations do realize unitary-evolving one-qubit states which cannot be
developed by conventional Hamiltonian counterparts, already before considering
the more-involved $(N,L)$ QMM Schrödinger equations.

###  5.2 Purely-QMM Unitary State-Histories On The ‘$\theta$-Orbit’

In this subsection, the simplest (exact and ‘excitonic’) analytic wavefunction
solutions to the QMM Schrödinger equations corresponding to the QMM-H (66), in
the form presented in (4.3), are presented. Moreover, we analyze, determine or
constrain the $a^{\star}$ thresholds, which mark the Robust Non-Markovianity
of a family of related solutions.

####  5.2.1 The Profile-Invariant State-History, Stability, Dynamical
Attractor

We now introduce the first special ‘orbit’ of the state time evolution of the
one-qubit closed system, and then obtain and analyze several (here the
simplest) wavefunction solutions along it, realizing QMM unitary state-
histories in correspondence with (66). This special orbit, and its counterpart
introduced in the next subsection, are obtained by imposing the simplest
dynamically-consistent constraints on the time evolution of the state, and
also on the Hamiltonian couplings. The motivation for considering these orbits
is that they are instructionally insightful and useful, specially for one’s
analytic explorations. This is because the QMM Schrödinger equations are
simplified along these special orbits considerably, nevertheless, they do not
miss the main features of the general-orbit equations, neither mathematically
nor physically. Moving to Section 6, we solve equations (4.3) along generic
one-qubit orbits, using numerical tools.

The first special orbit, _the $\theta$-orbit_, refers collectively to all the
unitary time evolutions of the one-qubit wavefunctions for which, stated in
terms of the Bloch representation (47), the $\phi$ angle is kept constant, but
the $\theta$ angle is set free to evolve based on its interacting memories:
$\dot{\theta}_{t}=f(\theta_{t-a},\theta_{t})$. Along the $\theta$-orbit, where
$\forall\;t$: $\phi_{t}=\phi_{t_{0}}$, one has,

$\mathcal{G}_{t-a,t}=0\;\;\;;\;\;\;\mathcal{F}_{t-a,t}=\;\sin(\theta_{t}-\theta_{t-a}).$
(84)

Moreover, imposing the dynamical constraint $\dot{\phi}_{t}\equiv 0$ to
equations (4.3) enforces the following constraint on the couplings of the
QMM-H (66): $\hat{\mu}_{t-a}=\mu_{t-a}+\lambda^{R}=0$. These constraints,
together with (84), yield _the dynamical equation on $\theta$-orbit_,

$\dot{\theta_{t}}=-\lambda^{I}\sin\big{(}\theta_{t}-\theta_{t-a}\big{)}.$ (85)

The beautiful equation (85) is concise, but infinite dimensional. It must be
highlighted that the above equation inherits the essential characteristics of
equations (4.3). It is a nonlinear delay-differential equation [62, 63, 64,
65] with a simplified but similar form. Being so, our analytic analyses will
be mostly based on equation (85). Applying the defining $\theta$-orbit
constraint $\phi_{t}=\phi_{t_{0}}$ to (57), and using the corresponding
equation (85), one finds these expressions for the QM-TPFs of the unitary
state-histories on the $\theta$-orbit,

$m_{t-a\;t}=w_{t-a\;t}=\cos\big{(}\frac{\theta_{t}-\theta_{t-a}}{2}\big{)}\;\;\;;\;\;\;w^{2}_{t-a\;t}\;\big{(}1-w^{2}_{t-a\;t}\big{)}=\frac{\dot{\theta}_{t}^{2}}{4\;(\lambda^{I})^{2}}.$
(86)

It is natural to begin with looking for those exact solutions to the delay-
differential $\theta$-orbit equation (85) whose defining profiles, as
expressed in terms of finitely-many elementary functions, remain invariant in
the whole course of time evolution. These are solutions $\theta_{t}$ which
preserve their defining initial-history profile, chosen for $[t_{0},t_{0}+a]$,
ever after the turning-on of all the interacting quantum memories: namely all
through $[t_{0}+a,\infty)$. Obviously, compared with generic solutions whose
profiles can change in $a$-dependent intervals, the ‘profile-invariant
solutions’ are the first to be obtained.

First, we see that all the $(N,L)$ Purely-QMM-UEs of all closed quantum many-
body systems admit constant-in-time density operators as their trivial exact
solutions. This fact can be easily confirmed by noticing that upon
substituting: $\rho_{t_{l}}=\rho_{t},\forall\;t_{l}$ into (14), we get:
$H_{\text{QQM}}^{(N,L)}(t)\propto\rho_{t}$, resulting at:
$\dot{\rho}_{t}=\rho_{t_{0}}$. The perturbative of this static solution to
(85), $\theta_{t}=\text{const.}$, will be investigated later in what comes in
this section. Next, we ask: what is the simplest profile-invariant dynamical
solution to the $\theta$-orbit equation? Equation (85) is featured with the
special structure that the time-derivative of its variable is sourced by the
mere difference between the variable at (the quantum memory moments) $t-a$ and
$t$. Being so, the delay differential equation turns into a generically-
solvable algebraic equation upon substituting in it a linear-in-time ansatz.
Likewise, equation (85) realizes its the simplest dynamical solution as a one-
qubit unitary state-history (LABEL:hps) whose evolving density operators are
parametrized by a linear-in-time $\theta$ angle, modulo its identifications.
That is, we look for the state-history,

$\begin{split}\;&\hskip 52.63777pt\hskip
5.12128pt\big{(}\theta_{t}\;,\;\phi_{t}\big{)}\;=\;\big{(}\alpha\;t+\beta\;,\;\phi_{t_{0}}\big{)},\\\
&\;\;\;\;\;\;\;\;\;\hskip
7.11317pt\;\;\;\;\;\;m_{t-a\;t}\;=\;w_{t-a\;t}\;=\;\cos\big{(}\frac{a\;\alpha}{2}\big{)}.\end{split}$
(87)

Indeed, (87) satisfies the equation (85) for arbitrary $\beta$ if the slope
$\alpha$ is a real root of,

$\alpha+\lambda^{I}\sin(a\;\alpha)=0.$ (88)

The parameters of the above algebraic equation in terms of which the slope
$\alpha$ is fixed are $\lambda^{I}$, namely the Hamiltonian coupling
introduced in (66), and $a$, being the largest QMD. For example, as long as
$\lambda^{I}$ is taken positive, $\alpha=0$ is the only solution, but $a=1$
and $\lambda^{I}=-2$ yields the following three solutions: $\alpha=0$ and
$|\alpha|\approx 1.89549$. Indeed, one solution to (88) is always $\alpha=0$,
resulting at a static one-qubit state-history. But, even numbers of non-zero
$\alpha$ solutions to (88) appear (in $\pm$ pairs) only on a proper subspace
of the $(a,\lambda^{I})$ plane. Stated precisely, this subspace corresponds to
the $(\lambda^{I}<0,a>a^{\star})$ region for a threshold
$a^{\star}=a^{\star}(\lambda^{I})$ which will be determined in Subsection
5.2.3.

Now, we advance to point out that the solution (87,88), despite being a
special one, has a very significant relation with all the solutions in the
moduli space of the $\theta$-orbit. Let us consider any state-history
$(\theta_{t}=\theta(t),\phi_{t}=\text{const.})$ which identifies a one-qubit
unitarily-evolving wavefuncion on the $\theta$-orbit, by solving (85).
Clearly, an important characteristic of any generic solution is its _late-time
asymptotics_ , also (getting to know) if this asymptotics by itself
corresponds to any specific exact solution of (85) as a (static or dynamical)
_attractor_ of the system. We now analyze analytically all these questions. To
this aim, we first introduce a natural dimensionless coordinate which is the
time measured in the units of the largest QMD, that is:
$\tau\equiv\frac{t}{a}$. Likewise, we introduce the dimensionless coupling
$\xi\equiv\lambda^{I}a$ of the QMM-H, $H^{(2,2)}_{\text{QMM}}(\tau)$ by which
the $\tau$-evolution is generated. Re-expressing the $\theta$-orbit dynamics
in terms of $(\tau,\xi)$, equation (85) takes the following form,

$\dot{\theta}_{\tau}\;=\;-\;\xi\;\sin\big{(}\theta_{\tau}-\theta_{\tau-1}\big{)}.$
(89)

The objective is obtaining the leading-order solution to (89) at sufficiently-
late times $\tau\gg 1$, which also is the general asymptotic solution to the
above equation for $\tau\to\infty$. Now, to work out the dominant solution at
late times by means of a well-defined Taylor series, it is good to do a second
change of variable from $\tau$ to $\tilde{x}\equiv\tau^{-1}$, such that the
asymptotics is captured by $\tilde{x}\ll 1$. Applying the change of variables,
the largest QMD $\Delta_{1}^{(t)}=a$ is mapped respectively to
$\Delta_{1}^{(\tau)}=1$ and
$\Delta_{1}^{(\tilde{x})}=-\tilde{x}^{2}+\text{O}[\tilde{x}^{3}]\approx-\tilde{x}^{2}$.
Therefore, the difference between the $\theta$ angle at the two quantum-memory
moments which appears in the right hand side of the above equation is recast,
to the leading order in the $\tilde{x}\ll 1$ Taylor expansion, as follows:
$\theta_{\tilde{x}}-\theta_{\tilde{x}-\Delta_{1}^{(\tilde{x})}}\approx-\tilde{x}^{2}\;\dot{\theta}_{\tilde{x}}$.
Being so, the late-time dynamics of the $\theta$-orbit equation is described
by the following first-order differential equation,

$\big{(}\tilde{x}^{2}\;\dot{\theta}_{\tilde{x}}\big{)}+\xi\;\sin\big{(}\tilde{x}^{2}\;\dot{\theta}_{\tilde{x}}\big{)}=a\;\big{[}\;\dot{\theta}_{t}+\lambda^{I}\sin\big{(}a\;\dot{\theta}_{t}\big{)}\;\big{]}=0.$
(90)

Finally, comparing (90) with (88) proves that the late-time dynamics is given
by,

$\dot{\theta}_{t}^{(\frac{t}{a}\gg
1)}=\text{constant}=\alpha(\lambda^{I},a)_{|_{\text{a real root
of}\;\eqref{eq:alphaII}}}.$ (91)

This completes the proof that the profile-invariant unitary state-history
(87,88) is _the dynamic-or-‘curve’ attractor_ of all one-qubit unitary state-
histories on the the $\theta$-orbit. Note that we did not demand the smallness
of $a$: the proof holds for arbitrary $a^{(\geq a^{\star})}$. Moreover, as
anticipated, generic solutions do not wait for very long times (in $a$ units)
to realize their dynamical attractor (87,88) with good approximations. In
Subsection 6.3.2, both for re-establishing the result (91), and for examining
the fast tendency of the $\theta$-orbit state-histories in approaching it, we
present numerical solutions to (85).

Now, we investigate the stability of the profile-invariant solution (87,88)
under general perturbations, noting that the stability analysis of the trivial
solution is also included. The very statement that the solution (87,88) is the
dynamical attractor for all $\theta$-orbit solutions indicates its stability
under the subclass of perturbations which are confined on the $\theta$-orbit.
Here in what follows analytically, joined with the numerical results of
Subsection 6.3.2, we directly examine the most general perturbations of this
solution. That is, we must look for sufficiently small deformations of the
$\theta$-orbit special solution (87,88) in the entire moduli space of
solutions to the general-orbit equations (4.3). Introducing a perturbation
parameter $\epsilon\ll 1$, and choosing any root $\alpha$ of (88) for a given
$(a,\lambda^{I})$, we apply to (85) the following ansatz,

$\begin{split}&\theta_{t}\;=\;\theta^{(0)}_{t}\;+\;\epsilon\;\theta^{(1)}_{t}\;+\;\cdots\;=\;(\alpha\;t+\beta)\;+\;\epsilon\;\theta^{(1)}_{t}\;+\;\cdots\\\
&\phi_{t}\;=\;\phi^{(0)}_{t}\;+\;\epsilon\;\phi^{(1)}_{t}\;+\;\cdots\;=\;\phi_{t_{0}}\;+\;\epsilon\;\phi^{(1)}_{t}\;+\;\cdots\end{split}$
(92)

Expansing the expressions to first order in $\epsilon$ yields two linear
first-order delay-differential equations for the time evolution of the
fluctuation modes. For the $\theta^{(1)}_{t}$ modes we obtain,

$\begin{split}&\dot{\theta}^{(1)}_{t}\;=\;-\lambda^{I}\;\cos(a\;\alpha)\;\big{(}\theta^{(1)}_{t}-\theta^{(1)}_{t-a}\big{)}.\end{split}$
(93)

The $\phi^{(1)}_{t}$ modes are evolved under an independent similar equation
which is nevertheless featured, in distinction with (LABEL:whl), with a time-
dependent prefactor,

$\begin{split}\dot{\phi}^{(1)}_{t}\;=\;\alpha\;\big{[}\cot(a\;\alpha)-\cot(\alpha\;t+\beta)\;\big{]}\;\big{(}\phi^{(1)}_{t}-\phi^{(1)}_{t-a}\big{)}.\end{split}$
(94)

The first thing to observe about the above equations is that, similar to their
nonlinear ancestor equation (85), the time-derivatives are sourced by the mere
differences of the functions at the quantum-memory moments $t-a$ and $t$.
Given this characteristics, it is clear that the solution for the mode ever
after the onset of quantum memories, namely: $t\geq t_{0}+a$, vanishes
identically, that is: $\theta^{(1)}_{t}=0$ or $\phi^{(1)}_{t}=0$, if the
initial-history profile of the perturbation during $t\in[t_{0},t_{0}+a]$ is
taken to be any constant-in-time function. Now, we must deal with the
solutions for the modes for time-dependent initial histories of the
perturbations. Given the decoupling of the dynamical equations for
$\theta^{(1)}_{t}$ and $\phi^{(1)}_{t}$, one can investigate the two moduli
spaces independently.

Let us begin with profile-invariant solutions to (LABEL:whl). It is immediate
to see that by inserting in (LABEL:whl) a linear-in-time profile for the mode,
we get $\theta^{(1)}_{t}=\theta^{(1)}_{c}$: a static mode. Now, we must
consider the exponential-in-time modes: $\theta^{(1)}_{t}\propto
e^{\gamma\;t}$ [62, 63, 64, 65]. The ‘characteristic equation’ which
determines these modes is found from (LABEL:whl) as follows,

$\gamma+\lambda\;\cos(a\;\alpha)\;(1-e^{-a\;\gamma})\;=\;0.$ (95)

It is easy to see that the transcendental equation (95) yields $\gamma=0$ as
its single root. First, $\gamma=0$ is always the trivial solution to the
equation. Second, the derivative of the right hand side of (95) with respect
to $\gamma$ never change its sign on the entire real $\gamma$ axis. Being so,
equation (95) can have one real root, but not more. Given the reality of the
modes, the complex roots of (95) must be purely imaginary and must organize
linear combinations with oscillatory profiles:
$\theta^{(1)}_{t}=c_{1}\sin(\gamma\;t+\beta)+c_{2}\cos(\gamma\;t+\beta)$. But
then it can be seen directly that demanding these profiles to solve the
$\theta^{(1)}_{t}$ equation (LABEL:whl) enforces $\gamma$ to vanish, again
giving us the static $\theta^{(1)}_{t}$ mode as the only solution. Similarly,
we get only static modes for all other types of profile-invariant
fluctuations, for example, those with arbitrary polynomial-in-time profiles.
The conclusion is that there cannot be any profile-invariant mode fluctuating
over the solution (87,88).

We now extend our analysis to the most general family of solutions to
(LABEL:whl): these are the solutions which are developed out of arbitrary
initial histories: $\theta^{(1)}_{t_{|_{(t_{0}\leq t\leq
t_{0}+a)}}}\equiv\theta_{t}^{(1|0)}$, and being so, cannot be profile-
invariant generically. Because general analysis of these generic solutions
cannot be done using the known analytical methods, we must utilize numerical
methods. This numerical investigation is to be presented in Subsection 6.3.2.
The upshot is that all the $\theta^{(1)}_{t}$ fluctuations which solve
(LABEL:whl) undergo ever-shrinking oscillations whose attractors,
$\theta^{(1)}_{c}$, are history-dependent constants: all $\theta^{(1)}_{t}$
modes are destined to become static. The values of these fixed-point
attractors are generically non-zero, nevertheless they correspond to
unphysical modes. This is because the sole effect of a static modes is a
perturbative shift in the $\beta$ parameter which appears in (87), an effect
which in turn can be gauged away by resetting $t_{0}$. Finally, given the
decoupling of the $\theta^{(1)}_{t}$ and $\phi^{(1)}_{t}$ modes, we conclude
that the profile-invariant solution (87,88) is stable against all the
perturbations which are confined to the $\theta$-orbit.

Next, we must probe generic $\phi^{(1)}_{t}$ modes. It is easy to find out
that the evolution of these modes is significantly different from that of
$\theta^{(1)}_{t}$ modes. The reason is that $\phi^{(1)}_{t}$ modes evolve
according to equation (94) whose structure, although similar to (LABEL:whl),
is marked by a diverging time-dependent factor in its right hand side. Indeed,
precisely due to this factor, generic $\phi^{(1)}_{t}$ modes are rapidly
driven to grow nonperturbatively, destabilizing their background
$\theta$-orbit solution (87,88). To see this fact concretely, we set a known
but arbitrary initial-history $\phi^{(1)}_{t_{|_{(t_{0}\leq t\leq
t_{0}+a)}}}\equiv\phi_{t}^{(1|0)}$ for the mode. Now, one traces the
$\phi^{(1)}_{t}$ evolution across time intervals of length $a$ consecutively,
using “method of steps” [62, 63, 64, 65]. Denoting the $m$th-step solutions by
$\phi^{(1|m)}_{t}$, every next-step solution $\phi^{(1|n+1)}_{t}$ satisfies
this ordinary differential equation in terms of the already-known
$\phi^{(1|n)}_{t}$,

$\dot{\phi}^{(1|n+1)}_{t}\;=\;\alpha\;\big{[}\cot(a\;\alpha)-\cot(\alpha\;t+\beta)\;\big{]}\;\big{(}\phi^{(1|n+1)}_{t}-\phi_{t-a}^{(1|n)}\big{)}.$
(96)

As we observe, with $t$ approaching any root of $\sin(\alpha\;t+\beta)$ = 0,
$\dot{\phi}^{(1|n+1)}_{t}$ is boosted up boundlessly, unless
$\phi^{(1|n+1)}_{t}=\phi_{t-a}^{(1|n)}$ exceptionally. For example, taking
$t_{0}=\beta=0$, the first divergence happens at $t=\frac{\pi}{\alpha}$ which,
given (88), is located in one of the time steps exceeding
$t=\frac{\pi}{|\lambda^{I}|}$. Supposing again $(a,\lambda^{I})=(1,-2)$, this
first divergence is reached at the very first time interval after the outset
of the quantum memories. Numerical simulations of the solutions to (94) re-
confirm the destabilizing behavior of the $\phi^{(1)}_{t}$ modes, due to the
role played by the time-dependent prefactor on its RHS.

This concludes the stability analysis of the profile-invariant state-history
(87,88) which is the dynamical attractor for all the state-histories
realizable on the $\theta$-orbit. The result is that this special state-
history is stable against perturbations confined to its orbit, but is
destabilized by ‘general orbit perturbations’, namely those launched by
turning on a non-constant initial quantum-history of the $\phi$-angle
fluctuations: $\phi^{(1)}_{t_{|_{(t_{0}\leq t\leq
t_{0}+a)}}}=\phi_{t}^{(1|0)}$. In fact, the above instability is qualitatively
very fruitful. This is so, because it is a cause for the phenomenon that the
state-histories which solve the equations (4.3) form a rich phase diagram,
featured with generic late-time behavior of unitarily-evolving wavefunctions
which is significantly different from the linear-in-time $\theta$-orbit
solution. In Section 6, using numerical methods, we numerically map the
remarkably-rich phase diagram of the one-qubit closed system under $(2,2)$
Purely-QMM unitary evolution.

####  5.2.2 Generic Purely-QMM Unitary State-Histories On The $\theta$-Orbit

Generic $\theta$-orbit state-histories, namely the solutions to the delay-
differential equation (85) which are developed out of a generic initial
quantum-history $\theta_{t_{|_{(t_{0}\leq t\leq
t_{0}+a)}}}\equiv\theta_{t}^{[0]}$ are ‘profile-changing’ (in the sense
defined in Subsection 5.2.1). As established, all such state-histories must
tend to (87,88), being the only profile-invariant state-history on the
$\theta$-orbit, as their dynamical attractor. Furthermore, in Subsection
6.3.2, we expose numerically examples of generic solutions to the
$\theta$-orbit equation (85). Nevertheless, it remains worthwhile to develop
an analytic understanding of these state-histories. It is the objective of the
present subsection to fill this gap.

To work out such solutions, as used in probing the $\phi_{t}^{(1)}$ modes, the
best method one utilizes is the method of steps to consecutively integrate the
delay-differential equation across time intervals of length $a$, beginning
from an initial quantum-history $\theta_{t}^{[0]}$ [62, 63, 64, 65]. The
implemented procedure is iterative: the solution at each new $(n+1)$th-step,
$\theta^{[n+1]}_{t}$, is obtained based on the already-obtained previous-step
solution $\theta_{t}^{[m]}$, by solving this nonlinear first-order ordinary
differential equation,

$\dot{\theta}^{[n+1]}_{t}\;=\;-\lambda^{I}\;\sin\big{(}\theta^{[n+1]}_{t}-\theta_{t-a}^{[n]}\big{)}.$
(97)

To integrate iteratively the chain of ordinary differential equations which
encode the full-time solution to a given delay-differential equation, one
takes, as initial condition, an input-function which serves as the initial
history in the time interval $[t_{0},t_{0}+a]$. Depending on what input-
function one chooses as the initial history, exact solutions to the resulted
differential equations at (some of the) subsequent length-$a$ time-steps may
be obtained analytically. In the case of the nonlinear delay-differential
equation (97), if we take a linear-in-time initial quantum history
$\theta_{t}^{[0]}\propto t$, being the simplest choice, then the first-step
differential equation for $\theta_{t}^{[1]}$ becomes integrable in terms of
trigonometric and inverse-trigonometric functions. However, because this
integrability does not further even to the second time-step for
$\theta_{t}^{[2]}$, it is not particularly instructional to write it down
explicitly. Indeed, it is well-known to mathematicians that nonlinear delay-
differential equations are very rarely integrable by available analytical
methods. Therefore, we need to take an alternative approach to carry out the
integration of the sequential differential equations (97) analytically. The
alternative is a perturbative approach in which the resulted integrability
comes with the following three merits. First, it works with very diverse
choices of the initial quantum history $\theta_{t}^{[0]}$. Second, it extends
up to arbitrary orders in the perturbation, in a way which is entirely
covariant. Third, it continues in a highly similar fashion to arbitrary
length-$a$ time-steps.

Let us consider an arbitrary function $h^{[0]}_{t\in[t_{0}+t_{0}+a]}$ and
marginalize it by the action of a sufficiently small prefactor $\epsilon\ll
1$. We identify the initial quantum history with this combination. Now, we aim
to construct each one of the fixed-step state-histories
$\theta_{t}^{[n\;\geq\;1]}$ in the orders of perturbation in terms of the
$\epsilon$ parameter. That is, we set a perturbative scheme as follows to
consecutively integrate the sequential differential equations (97),

$\hskip
37.84221pt\;\;\;\;\;\;\theta_{t}^{[0]}\;\equiv\;\epsilon\;h^{[0]}_{t}\;\;\;\;\;\;;\;\;\;\;\;\;\theta_{t}^{[n]}\;=\;\sum_{m=1}^{\infty}\;\epsilon^{m}\;\theta_{t}^{[n|m]}.$
(98)

Substituting (98) into the first-step equation (97) for $\theta_{t}^{[1]}$,
and organizing the resulted expression as a series expansion in powers of
$\epsilon$, we obtain a generically-inhomogeneous first-order differential
equation of the following form for every one of the functions
$\theta_{t}^{[1|m]}$,

$\begin{split}&\dot{\theta}_{t}^{[1|m]}\;+\;\lambda^{I}\;\theta_{t}^{[1|m]}\;=\;\lambda^{I}\;J_{t}^{[1|m]}\big{(}h^{[0]}_{t-a}\;,\;\theta_{t}^{[1|r<m]}\big{)}.\end{split}$
(99)

To be concrete, here are the explicit forms of the first few ‘source’
functions $J_{t}^{[1|m]}$,

$\begin{split}&J_{t}^{[1|1]}=h^{[0]}_{t-a}\;;\;J_{t}^{[1|2]}=0\;;\;J_{t}^{[1|3]}=\frac{1}{6}\;\big{(}\theta_{t}^{[1|1]}-h^{[0]}_{t-a}\big{)}^{3}\>;\;J_{t}^{[1|4]}=\frac{1}{2}\;\theta_{t}^{[1|2]}\;\big{(}\theta_{t}^{[1|1]}-h^{[0]}_{t-a}\big{)}^{2},\\\
&J_{t}^{[1|5]}\;=\;\frac{1}{2}\;\big{(}\theta_{t}^{[1|1]}-h^{[0]}_{t-a}\big{)}^{2}\;\theta_{t}^{[1|3]}\;+\;\frac{1}{2}\;\big{(}\theta_{t}^{[1|1]}-h^{[0]}_{t-a}\big{)}\;(\theta_{t}^{[1|2]})^{2}\;+\;\frac{1}{120}\;\big{(}\theta_{t}^{[1|1]}-h^{[0]}_{t-a}\big{)}^{5}.\end{split}$
(100)