aainstitutetext: Institute for Gravitation and the Cosmos, The Pennsylvania
State University, University Park, PA 16802, USAbbinstitutetext: Department of
Physics, The Pennsylvania State University, University Park, PA, 16802, USA

# Classifying the non-time-local and entangling dynamics of an open qubit
system

Sean Prudhoe<EMAIL_ADDRESS>Sarah Shandera<EMAIL_ADDRESS>
###### Abstract

We study families of dynamical maps generated from interactions with varying
degrees of symmetry. For a family of time-independent Hamiltonians, we
demonstrate the relationship between symmetry, strong-coupling, perfect
entanglers, non-Markovian features, and non-time-locality. We show that by
perturbing the initial environment state, effective time-local descriptions
can be obtained that are non-singular yet capture essential non-unitary
features of the reduced dynamics. We then consider a time-dependent
Hamiltonian that changes the degree of symmetry by activating a dormant degree
of freedom. In this example we find that the one-qubit reduced dynamics
changes dramatically. These results can inform the construction of effective
theories of open systems when the larger system dynamics is unknown.

††arxiv: 2201.07080

##  1 Introduction

The evolution of quantum systems coupled to unobserved or unobservable degrees
of freedom can be much more complex than the evolution of closed systems
OpenQS . Information may flow back and forth between the observed and
unobserved parts of the system, leading to equations of motion that may not be
local in time, and that give rise to non-unitary, non-Markovian evolution
Rivas2014 . While formal expressions for the evolution of open systems exist,
and exact expressions can be derived in particular cases where the unobserved
physics is known, there is not yet a procedure for systematically constructing
effective theories of open systems that can encapsulate the full range of
possible phenomena.

A particular class of complex open systems appears in cosmology, where the
volume of the universe accessible to a single observer is generically bounded.
An open-systems approach in cosmology has been advocated by some researchers
for decades CalzettaHu1989 ; Koks1997 , and recent theoretical developments
have led to a resurgence of interest in these ideas. For example, the
possibility of detecting primordial non-Gaussianity in the statistics of the
cosmic microwave background fluctuations Achucarro:2022qrl  motivated to a
comprehensive study of the effects of interacting fields during or after
inflation. Those interactions can couple Fourier modes of different
wavelengths in different ways. The finite extent of the observable universe
means that very long wavelength modes are irretrievably in the ‘environment’.
Modes with very short wavelength are practically unobservable as well. Tracing
out either, or both, sets of unobservable modes generates an effective
description for the observable modes, which form an open system
Shandera:2017qkg . The appropriate framework for understanding the space of
models that generate the data is that of an open effective theory. Moreover,
the field or fields relevant for inflation are expected to be accompanied by
many other degrees of freedom which may move from passive to active as
inflation proceeds. Indeed, it is postulated that if the inflationary field
explores too large of a range, a large number of fields will become relevant,
limiting the validity of the original model as an effective description
Ooguri:2006in ; Garg:2018reu ; Ooguri:2018wrx . Using an open system effective
theory for the inflaton, rather than a traditional low-energy effective
theory, one can treat these light degrees of freedom as a time-dependent
environment. These open systems are complex, since many results that are known
require non-perturbative techniques Burgess:2009bs  and the associated open
systems are frequently non-Markovian Agon2018 . However, one has symmetry
constraints to guide the effective theory Weinberg:2008hq . A similar story of
complexity restricted by symmetry applies to another well-studied system where
open effective theories are applicable, black holes Banks:1983by ;
Unruh:1995gn ; Burgess:2014eoa ; Burgess:2015ajz ; Kaplanek:2019vzj ;
Kaplanek:2020iay ; Burgess:2021luo . Cosmologists, then, would like to
understand how to construct effective theories of open systems that are non-
perturbative and non-Markovian, with time-dependent environments, but
constrained by symmetries.

To address these questions requires a non-perturbative understanding of the
evolution of the reduced system. For finite-dimensional quantum systems this
can be addressed using the dynamical map, the operator that governs the
dynamics of the reduced degrees of freedom. Of course, determining the
dynamical map is as difficult as solving the Schrödinger equation governing
the system/environment dynamics. Thus for this work we consider simple open
systems, constructing dynamical maps for a single system qubit. In this
setting the dynamical map can be determined as an analytic function of the
Hamiltonian parameters, for a generic initial environment state.

Due to their relative simplicity, much is understood about the general
structure of qubit dynamical maps nielsen00 ; Ruskai2001 ; Braun_2014 ;
Rodriguez06 . Such maps appear frequently in quantum computational settings,
where they model noisy interactions of qubits with the environment. Optimal
dynamical maps have been found for performing communication tasks that
leverage qubit entanglement to transmit quantum information securely Niu_1999
; King_2001 ; Berry_2005 ; https://doi.org/10.48550/arxiv.2006.16366 ;
Ghosal_2021 . So, while interacting qubits are far simpler systems than those
found in cosmology, it is an appropriate starting point to connect the
questions of interest to cosmologists to the recent advances in the
understanding of open systems that have been driven by laboratory and quantum
computational considerations.

We begin by investigating a solvable example of coupled qubits to explore how
non-Markovianity and the non-time-local aspects of the master equation for one
qubit depend on the initial state of the unobserved qubit and on the
symmetries and coupling constants of the full Hamiltonian. The first
Hamiltonian we consider has (1) a conserved quantity, and (2) a block-diagonal
structure of two equal-size pieces, allowing additional symmetry structures at
special points in parameter space. Since non-time-local master equations can
be particularly difficult to work with, we examine how much of the parameter
space requires a time non-local equation, and the perturbations or
approximations in both the Hamiltonian or the state of the unobserved qubit
that will generate time-local equations of motion.

We then extend the study of the two qubit model, using a subset of the
Hamiltonian family and initial environment states that allow for the
appearance of additional degree of symmetry. We introduce a third qubit that
remains a spectator for a time and is later switched on using an interaction
that explicitly breaks the extra symmetry appearing in the two qubit
interaction. The study focuses on the radical change in the invertibility of
the dynamical map obtained by tracing out the two qubits that act as the
environment, indicating that a time-local master equation cannot be used after
symmetry breaking.

In the rest of the introduction we briefly review the formalism for open
system dynamics and master equations in the context of our goals and model.
Then, in Section 2, we introduce the example systems. In Section 3 we derive
the reduced dynamics for a single-qubit systems, tracing out the environment
qubit(s). Section 4 derives the conditions for non-time-local dynamics via the
non-invertibility of the dynamical map and discusses several features we use
to classify the dynamical map including the non-Markovianity, divisibility,
and symmetries of the Hamiltonian. We determine how these depend on
Hamiltonian parameters, in particular whether it is strongly coupled or not,
and what role the initial environment state plays in non-time-locality. In
Section 5 we use perturbed initial environment states to construct approximate
time-local master equations. And we conclude in Section 6.

###  1.1 Open system evolution

For some open systems, the master equation governing the evolution of the
density matrix for the observed system, $\rho_{\rm S}(t)$, is Lindblad ;
Vittorio:1976 ; Shibata:1977 ; Chaturvedi:1979

$\displaystyle\partial_{t}\rho_{\rm S}(t)$ $\displaystyle=-i[H_{\rm
free}(t)+H_{\rm open}(t),\rho_{\rm
S}(t)]+\sum_{k}\gamma_{k}(t)\left(L_{k}\rho_{\rm
S}(t)L^{{\dagger}}_{k}-\frac{1}{2}\\{L^{{\dagger}}_{k}L_{k},\rho_{\rm
S}(t)\\}\right)\,.$ (1)

Here $H_{\rm free}(t)+H_{\rm open}(t)=H_{\rm eff}(t)$ is the effective
Hamiltonian of the system, containing both the original system Hamiltonian
$H_{\rm free}(t)$ and a piece, $H_{\rm open}(t)$, generated by the coupling to
an environment. The $L_{k}$ are operators acting on the system, and the
$\gamma_{k}(t)$ are functions describing the flow of information between the
system and environment. The $\gamma_{k}(t)$, which control the subset of all
possible operators $L_{k}$ that appear with non-zero coefficients, depend on
the system-environment coupling and the state of the environment. $H_{\rm
eff}(t)$ and $\gamma_{k}(t)$ are given by environment correlation functions
calculated using the initial environment state $\rho_{\rm E}(0)$.

If the environment and full Hamiltonian are unknown, one might begin
constructing an effective theory for the system by writing all possible
$L_{k}$ and a generic $H_{\rm eff}$ from the complete set of operators that
act on the system. Then, the work in the effective theory comes in specifying
some structure for the dissipation functions $\gamma_{k}(t)$, determining any
approximations that may allow some possible terms in $H_{\rm eff}$ to be
discarded, and in determining consistency between effects captured in $H_{\rm
open}(t)$ and in the non-unitary part of Eq.(1). Some broad guidelines for
this process are known: the simplest choice would be all $\gamma_{k}\geq 0$
and constant, restricting the system to non-unitary however time-independent,
Markovian evolution. Time-dependent Markovian dynamics would be described by
$\gamma_{k}(t)\geq 0$ at all times. Finally a restricted set of non-Markovian
dynamics would be captured by considering generic functions $\gamma(t)$.

However, the most general case allows the master equation to be non-local in
time. Then, in addition to the time-local part one adds an integral term. This
is the Nakajima-Zwanzig equation 1958PThPh..20..948N ; 1960JChPh..33.1338Z ,

$\partial_{t}\rho_{\rm S}(t)=K_{\rm TL}(t)\rho_{\rm S}(t)+\int K_{\rm
NZ}(t,\tau)\rho_{\rm S}(\tau)d\tau\,,$ (2)

where the time-local piece, $K_{\rm TL}(t)$, generates the same action on
$\rho_{\rm S}(t)$ as given in Eq.(1). The integral is over the history of the
evolution $[t_{0},t]$ where $t_{0}$ is a time where the system and environment
are uncorrelated and $t$ is the time where one is interested in calculating
ensemble averages. In constructing a parameterized effective theory for the
open system dynamics, one would like to know how to systematically address
whether a non-local equation is necessary. In addition, which qualitative
aspects of information flow can be captured in either the $\gamma(t)$ or the
time non-local kernel $K_{\rm NZ}(t,\tau)$, and how they should be
implemented?

To address these questions in a simple case, we explore the relationship
between the full Hamiltonian for system and environment, together with the
initial state of the environment, to several features of the reduced dynamics.
We consider measures of non-Markovianity and the conditions under which time
non-local evolution is required. We do this by first computing the exact
reduced dynamics via the dynamical map. This is a non-unitary generalization
of the time evolution operator, a completely positive and trace preserving
(CPTP) map from the initial density matrix to the density matrix at a later
time $t$,

$\rho_{S}(t)=\Lambda(t,0)\circ\rho_{S}(0)\,.$ (3)

For qubit dynamical maps the complications arising from non-Markovianity are
less severe. Non-Markovian qubit dynamical maps can be tractably studied, more
so than their master equation counterparts. For example the divisibility (how
the time evolution can be broken into steps) and how it relates to non-
Markovianity has been exhaustively studied in the case of qubit dynamical maps
2019A ; 2019B ; 2019C ; Chru_ci_ski_2014 . For the model we consider it is
therefore possible to make detailed statements about the relation between
properties of the dynamical map and its non-Markovianity.

Dynamical maps do not have to be invertible; maps with ${\rm
Det}\Lambda(\tau_{i})=0$ for some times $\tau_{i}$ require either a time-local
description that diverges at each $\tau_{i}$ or a non-time-local integral
kernel, as written in Eq.(2) Chru_ci_ski_2010 . For a simple system, we will
use the non-invertibility of the dynamical map to derive the conditions on the
full Hamiltonian and the environment that make a non-time-local master
equation necessary. We find that, independent of the Hamiltonian, there are
always a set of initial environment states which support time-local reduced
dynamics. This allows a time-local, approximate, master equation to be
constructed by shifting the initial state of the environment. For an initial
environment state, $\rho_{\rm E}(t_{0})$, and dynamics that requires a non-
time-local piece, there exist environment states nearby in trace distance
norm, $\\{\rho^{{}^{\prime}}_{\rm E}(t_{0})\\}$, which can be used to define a
master equation of the form

$\begin{split}\partial_{t}\rho_{S}(t;\rho_{\rm E}(t_{0}))&=K_{\rm
TL}(t;\rho^{{}^{\prime}}_{\rm E}(t_{0}))\rho_{S}(t;\rho_{\rm E}(t_{0}))\\\
&+\int K_{\rm NTL}(t,\tau;\delta\rho_{\rm E})\rho_{\rm S}(\tau;\rho_{\rm
E}(t_{0}))d\tau\,.\end{split}$ (4)

Here the non-time-local component is linear in $\delta\rho_{\rm E}=\rho_{\rm
E}(t_{0})-\rho^{{}^{\prime}}_{\rm E}(t_{0})$, and the integral is over the
evolution history $[t_{0},t]$.

##  2 The system-environment Hamiltonians and the unitary dynamics

This section introduces both the family of two-qubit Hamiltonians considered,
as well as a more restrictive family of three-qubit interactions. The two-
qubit model has an associated parity symmetry, which splits the Hamiltonian
into two equal-sized blocks. Such models are interesting as they have an
intermediate level of symmetry: more than the class with no non-trivial
symmetries, however not as much as Hamiltonians that preserve the total
angular momentum of the two qubits. Physically, dynamics of the type we use
here describe a pair of non-interacting qubits, most clearly seen through a
change of meronomic frame Hulse:2019 . We also characterize features of the
non-Markovianity of the single-qubit evolution, according to parameter choices
in the full Hamiltonian.

###  2.1 Two-qubit family

The two-qubit Hamiltonian that we study is

$\begin{split}H=&H_{\rm free}+H_{\rm int}\\\ =&\omega_{\rm S}(Z_{\rm
S}\otimes\mathbb{1}_{\rm E})+\omega_{\rm E}(\mathbb{1}_{\rm S}\otimes Z_{\rm
E})+\kappa_{\rm SE}(Y_{\rm S}\otimes X_{\rm E})+\kappa_{\rm ES}(X_{\rm
S}\otimes Y_{\rm E})\,.\end{split}$ (5)

The free parameters $\omega_{\rm S}$ and $\omega_{\rm E}$ provide the time
scales associated to the free dynamics of each individual qubit (with
$\hbar=1$), and the parameters $\kappa_{SE}$ and $\kappa_{ES}$ are coupling
strengths. $H$ has a symmetry, $[H,Z_{\rm S}\otimes Z_{\rm E}]=0$, so the
eigenstates have definite parity associated to $P_{zz}=Z_{\rm S}\otimes Z_{\rm
E}$. This $\mathbb{Z}_{2}$ symmetry allows $H$ to be split into even and odd
blocks, where the even block is spanned by states with correlated spins (e.g.
$|\uparrow\uparrow\rangle$) and the odd block is spanned by states with anti-
correlated spins (e.g. $|\uparrow\downarrow\rangle$).

The block diagonalization is achieved by splitting $H$ into symmetric and
anti-symmetric parts under exchange of system and environment operators.
Defining new parameters,

$\begin{split}&2\Delta_{\pm}=\omega_{\rm S}\pm\omega_{\rm E}\\\
&2\kappa_{\pm}=\kappa_{\rm SE}\pm\kappa_{\rm ES}\,,\end{split}$ (6)

the Hamiltonian can be written as

$\begin{split}H=&\Delta_{+}(Z_{\rm S}\otimes\mathbb{1}_{\rm E}+\mathbb{1}_{\rm
S}\otimes Z_{\rm E})+\kappa_{+}\left[Y_{\rm S}\otimes X_{\rm E}+X_{S}\otimes
Y_{\rm E}\right]\\\ +&\Delta_{-}(Z_{\rm S}\otimes\mathbb{1}_{\rm
E}-\mathbb{1}_{\rm S}\otimes Z_{\rm E})+\kappa_{-}\left[Y_{\rm S}\otimes
X_{\rm E}-X_{\rm S}\otimes Y_{\rm E}\right]\\\
\equiv&H_{+}+H_{-}\,,\end{split}$ (7)

where the $\pm$ labels correspond to the $\mathbb{Z}_{2}$ (parity) eigenvalues
of each block. The block diagonalization of $H$ introduces a subspace
decomposition $\mathcal{H}=\mathcal{Q}_{+}\oplus\mathcal{Q}_{\rm-}$, where the
spaces $\mathcal{Q}_{\pm}$ are spanned by the eigenstates of $H_{\pm}$. As we
show below, there is also a tensor product decomposition for which the two
subsystems decouple.

Using $|0\rangle$ and $|1\rangle$ to label the eigenstates of $Z_{\rm S}$ and
$Z_{\rm E}$, the stationary states of $H$ are

$\begin{split}&|0_{+}\rangle=\cos{\frac{\phi_{+}}{2}}|0_{\rm S},0_{\rm
E}\rangle+i\sin{\frac{\phi_{+}}{2}}|1_{\rm S},1_{\rm E}\rangle\\\
&|1_{+}\rangle=\sin{\frac{\phi_{+}}{2}}|0_{\rm S},0_{\rm
E}\rangle-i\cos{\frac{\phi_{+}}{2}}|1_{\rm S},1_{\rm E}\rangle\\\
&|0_{-}\rangle=\cos{\frac{\phi_{-}}{2}}|0_{\rm S},1_{\rm
E}\rangle+i\sin{\frac{\phi_{-}}{2}}|1_{\rm S},0_{\rm E}\rangle\\\
&|1_{-}\rangle=\sin{\frac{\phi_{-}}{2}}|0_{\rm S},1_{\rm
E}\rangle-i\cos{\frac{\phi_{-}}{2}}|1_{\rm S},0_{\rm E}\rangle\,,\end{split}$
(8)

with eigenvalues

$\pm\omega_{\pm}=\pm 2\sqrt{\Delta_{\pm}^{2}+\kappa_{\pm}^{2}}\,.$ (9)

In Equation (8), the angles

$\phi_{\pm}=\arctan{\frac{\kappa_{\pm}}{\Delta_{\pm}}}$ (10)

indicate the relative size of the interaction and free Hamiltonian parameters,
and if either provide a dominate contribution to the energy eigenvalues
$\omega_{\pm}$. As long as at least one of the blocks is interacting i.e.
$\kappa_{\pm}\neq 0$, the stationary states in the subsystem decomposition
$\mathcal{H}=\mathcal{Q}_{\rm S}\otimes\mathcal{Q}_{\rm E}$ are entangled, and
maximally entangled as
$\frac{\kappa_{\pm}}{\Delta_{\pm}}\rightarrow\pm\infty$.

We will see below that the reduced dynamics may acquire a non-time-local
component if $\phi_{+}+\phi_{-}\geq\frac{\pi}{2}$. Since this condition
requires that one or both of $\kappa_{\pm}\geq\Delta_{\pm}$, the part of
parameter space where non-time-local master equations can be required
coincides with strong coupling, although not all strongly coupled Hamiltonians
will have non-time-local dynamics.

So far, we have defined subsystems assuming a laboratory-based notion of
locality for operations on qubits, established by the system/environment
labels. However, the block-diagonal structure of the Hamiltonian suggests that
we also consider a re-organization of the Hilbert space into degrees of
freedom that decouple. That is, we can define qubits $A$ and $B$, with
orthonormal basis states $\\{1_{\rm A},0_{\rm A}\\}$ and $\\{1_{\rm B},0_{\rm
B}\\}$, so that $\mathcal{H}=\mathcal{Q}_{\rm A}\otimes\mathcal{Q}_{\rm B}$.
This is a change of meronomic frame Hulse:2019 . Explicitly, a (non-unique)
mapping between these bases is given by

$\begin{split}|0_{+}\rangle=&|0_{\rm A},0_{\rm B}\rangle\\\
|1_{+}\rangle=&|1_{\rm A},1_{\rm B}\rangle\\\ |0_{-}\rangle=&|0_{\rm A},1_{\rm
B}\rangle\\\ |1_{-}\rangle=&|1_{\rm A},0_{\rm B}\rangle\,.\end{split}$ (11)

One finds $H=\omega_{+}\left(|0_{+}\rangle\langle 0_{+}|-|1_{+}\rangle\langle
1_{+}|\right)+\,\omega_{-}\left(|0_{-}\rangle\langle
0_{-}|-|1_{-}\rangle\langle 1_{-}|\right)$, it is straightforward to change to
the frame given in Eq.(11), where

$H_{AB}=\omega_{\rm A}Z_{\rm A}\otimes\mathbb{1}_{\rm B}+\omega_{\rm
B}\mathbb{1}_{\rm A}\otimes Z_{\rm B}\,.$ (12)

Here $\omega_{\rm A}=\frac{1}{2}(\omega_{+}+\omega_{-})$, $\omega_{\rm
B}=\frac{1}{2}(\omega_{+}-\omega_{-})$, and $Z_{\rm A}=|0_{\rm
A}\rangle\langle 0_{\rm A}|-|1_{\rm A}\rangle\langle 1_{\rm A}|$, etc.
Comparing to Eq.(7), the two terms in Eq.(12) are just the A/B frame
expressions for $H_{+}$ and $H_{-}$. An advantage of this frame is that it is
easy to characterize the regions of parameter space with extra symmetry. A
particularly useful region of parameter space is

$\;\;\;\;\;\;\;\;\omega_{+}=\omega_{-}\Leftrightarrow\omega_{\rm
B}=0\,\,\,{(\rm\bf the\,\,degenerate\,\,family)}\,$ (13)

which is symmetric under local rotations on the B qubit subsystem. Since
symmetries are clearest in the A/B frame, we will continue to use it for that
purpose in Table 4.2 below, where we classify the reduced system dynamics
possible with this Hamiltonian, Eq.(5).

Returning to the system/environment frame, we characterize the entangling
properties of the time evolution. Using the computational basis for system and
environment qubits, the time evolution operator, $U(t)$, is

$\begin{split}U(t)=e^{-iHt}=\begin{bmatrix}\alpha_{+}(t)&0&0&-\beta_{+}(t)\\\
0&\alpha_{-}(t)&-\beta_{-}(t)&0\\\ 0&\beta_{-}(t)&\bar{\alpha}_{-}(t)&0\\\
\beta_{+}(t)&0&0&\bar{\alpha}_{+}(t)\end{bmatrix}\\\ ,\end{split}$ (14)

where

$\begin{split}&\alpha_{\pm}(t)=\cos\omega_{\pm}t-i\cos\phi_{\pm}\sin\omega_{\pm}t\,,\\\
&\beta_{\pm}(t)=\sin\phi_{\pm}\sin\omega_{\pm}t\,,\end{split}$ (15)

and $\bar{\alpha}_{\pm}(t)$ is the complex conjugate of $\alpha_{\pm}(t)$. The
pair of functions from each doublet satisfy
$|\alpha_{\pm}(t)|^{2}+|\beta_{\pm}(t)|^{2}=1$.

The functions $\beta_{\pm}(t)$ are generated by the interaction between system
and environment, so they determine both the entanglement and, as we see below,
the invertibility of the reduced dynamics for the system.

###  2.2 Three-qubit model of dynamical symmetry breaking

We further study a model where the reduced dynamics is phase covariant.
Reduced dynamics is phase covariant or time translation symmetric if the
following constraints hold https://doi.org/10.48550/arxiv.2106.05295

  1. I

$[H_{\rm free},H_{\rm int}]=0$

  2. II

$[H_{\rm free},\rho_{\rm E}(0)]=0$ .

These constraints have thermodynamic implications. The first constraint is a
strict energy conservation condition i.e. energy is not built up between the
system and environment boundary. The second assumption is that the initial
environment state is a Gibbs’ state defined with respect to the free
environment Hamiltonian. If these assumptions hold, then the set of time
evolution operator generated by the free system Hamiltonian commutes with the
action of the dynamical map. Hence the designation as a time translation
symmetry.

To incorporate phase-covariance breaking into our model we consider the
Hamiltonian

$\begin{split}\tilde{H}(t)=&H_{\rm free}+H_{\rm int}(t)\\\ &\omega(Z_{\rm
S}\otimes\mathbb{1}_{\rm E}\otimes\mathbb{1}_{\rm E^{\prime}}+\mathbb{1}_{\rm
S}\otimes Z_{\rm E}\otimes\mathbb{1}_{\rm
E^{\prime}})+\omega\left[X_{S}\otimes Y_{\rm E}\otimes\mathbb{1}_{\rm
E^{\prime}}-Y_{\rm S}\otimes X_{\rm E}\otimes\mathbb{1}_{\rm
E^{\prime}}\right]\\\ &+\gamma\Theta(t-\tau)\left[X_{\rm
S}\otimes\mathbb{1}_{\rm E}\otimes X_{\rm E^{\prime}}\right]\,,\end{split}$
(16)

where $\Theta(t)$ is the Heaviside theta function. The choice of the time-
independent portion of the Hamiltonian satisfies strict energy conservation.
The second assumption is satisfied if the initial state of the E qubit is
restricted to the form $\rho_{\rm E}(0)=\frac{1}{2}(\mathbb{1}_{\rm E}+z_{\rm
E}(0)Z_{\rm E})$. For what follows $\tilde{H}_{t<\tau}$ is the Hamiltonian for
$t<\tau$ and $\tilde{H}_{t\geq\tau}$ is the Hamiltonian for $t\geq\tau$, as we
can note the Hamiltonian is a piece-wise define function.

$\tilde{H}_{t\geq\tau}$ is chosen to break the phase covariance of the reduced
dynamics, but in the process also breaks the parity symmetry. But other global
symmetries appear for $t\geq\tau$, which we can exploit to determine the
stationary states when the interaction changes. Using the Pauli anti-
commutation relations, it is not difficult to see
$[\tilde{H}_{t\geq\tau},Z_{\rm S}\otimes Z_{\rm E}\otimes Z_{\rm
E^{\prime}}]=0$. But from the block structure in the eigendecomposition of the
Hamiltonian, one can note the presence of another symmetry. We find that the
observable,

$\begin{split}\mathcal{O}_{2}=Y_{\rm S}\otimes\mathbb{1}_{\rm E}\otimes Y_{\rm
E^{\prime}}-Z_{\rm S}\otimes X_{\rm E}\otimes Y_{\rm E^{\prime}}\,\end{split}$
(17)

commutes with $\tilde{H}_{t\geq\tau}$. Therefore $\tilde{H}_{t\geq\tau}$
splits into four $2\times 2$ blocks with definite values of $\mathcal{O}_{1}$
and $\mathcal{O}_{2}$

Thus a familiar path may be taken to determine the time evolution generated by
$\tilde{H}_{t\geq\tau}$, which we denote by $\tilde{U}_{t\geq\tau}(t)$. We
begin with the eigenvalues which are found to be

$\begin{split}\pm\Omega_{\pm}=\pm\sqrt{\gamma^{2}\pm
2\sqrt{2}\gamma\omega+4\omega^{2}}\end{split}$ (18)

and are doubly degenerate. We shall start the computation in the eigenbasis of
$\mathcal{O}_{2}$ (called the polarized basis), and then go back to the
computational basis. To this end define the angles,

$\tan\psi_{\pm}=1\pm 2\sqrt{2}\frac{\omega}{\gamma}$ (19)

from which we obtain the time dependent functions (note the similarity to the
previous section),

$\begin{split}&\tilde{\alpha}_{\pm}(t)=\cos\Omega_{\pm}t-i\cos\psi_{\pm}\sin\Omega_{\pm}t\\\
&\tilde{\beta}_{\pm}(t)=\sin\psi_{\pm}\sin\Omega_{\pm}t\end{split}$ (20)

With these functions one can construct the unitary operator
$\tilde{U}_{t\geq\tau}(t)$ in the polarized basis. In the computational basis,
the following combinations of functions will be most useful

$\begin{split}&a_{\pm}(t)=\frac{1}{2}\Re{(\tilde{\alpha}_{+}\pm\tilde{\alpha}_{-})}\\\
&b_{\pm}(t)=\frac{1}{2}\Im{(\tilde{\alpha}_{+}\pm\tilde{\alpha}_{-})}\\\
&g_{\pm}(t)=\frac{1}{2}(\tilde{\beta}_{+}\pm\tilde{\beta}_{-})\\\
&h_{\pm}(t)=\frac{1}{2}(\tilde{\alpha}_{+}\pm\bar{\tilde{\alpha}}_{-})\end{split}\,.$
(21)

We have all we need to compute the total time evolution operator, which is
given by the following time ordered exponential

$\tilde{U}(t,0)=\mathcal{T}\exp{\left[-i\int_{0}^{t}\tilde{H}(\tau)d\tau\right]}\,.$
(22)

Breaking the time interval into $N$ pieces, and using the Baker-Campbell-
Hausdorff relation, one can show that in the limit $N$ goes to infinity the
time ordered exponential approaches

$\begin{split}\tilde{U}(t,0)=&\left[\Theta(t)-\Theta(t-\tau)\right]\tilde{U}_{t<\tau}(t)+\Theta(t-\tau)\tilde{U}_{t\geq\tau}(t-\tau)\tilde{U}_{t<\tau}(\tau)\end{split}$
(23)

where we have just seen how to compute $\tilde{U}_{t\geq\tau}(t)$, and we can
use the results of the previous section to determine $\tilde{U}_{t<\tau}(t)$.

##  3 The reduced dynamics

The reduced dynamics of the system is obtained from the full dynamics by
tracing out the environment. Restricting to factorized initial states ensures
that the reduced dynamics is completely positive Lindblad . In that case, the
density matrix of the system qubit alone, at time $t$, is

$\rho_{\rm S}(t)=tr_{\rm E}[U(t)\rho_{\rm S}(0)\otimes\rho_{\rm
E}(0)U^{{\dagger}}(t)]\,.$ (24)

The state of the system qubit at any time can be found using the dynamical
map, represented by a matrix acting on the vectorization of the reduced
density matrix Rivas2014 . For a qubit density matrix this is the Bloch
representation

$\rho_{\rm S}(0)=\frac{1}{2}(\mathbb{1}_{\rm S}+\vec{r}_{\rm
S}(0)\cdot\vec{\sigma}_{\rm S}),$ (25)

where $\vec{r}_{\rm S}(0)$ is a real, 3-dimensional vector with Euclidean norm
$||\vec{r}_{\rm S}(0)||\leq 1$. . The evolved density matrix can then be
written as

$\rho_{\rm S}(t)=\Lambda(t,0)\circ\rho_{\rm S}(0).$ (26)

where $\Lambda(t,0)$ is a $4\times 4$ matrix. Note that as the Hamiltonian is
time independent, the dynamical map depends on the initial and final times
only through the difference $t-t_{0}$, so for what follows we can simply write
$\Lambda(t)$ and suppress the initial time dependence. It is helpful to
further define a $3\times 3$ matrix with components $T^{ij}=\Lambda^{ij}$ and
a 3-vector $d^{i}=\Lambda^{0i}$ Ruskai2001 , so that the action of the
dynamical map can be written

$\begin{split}\rho_{\rm S}(t)=&\frac{1}{2}(\mathbb{1}_{\rm S}+\vec{r}_{\rm
S}(t)\cdot\vec{\sigma}_{\rm S})\\\ =&\frac{1}{2}\left(\mathbb{1}_{\rm
S}+\left(\vec{\vec{T}}(t)\circ\vec{r}_{\rm
S}(0)+\vec{d}(t)\right)\cdot\vec{\sigma}_{\rm S}\right)\,.\end{split}$ (27)

This presentation has the advantage that the trace fixing requirement is
immediate. For $\Lambda$ to be physical, $\vec{\vec{T}}$ and $\vec{d}$ must be
real so that $\rho_{\rm S}(t)$ is Hermitian. Unlike unitary maps, dynamical
maps need not be divisible. For example, one is generally unable to split the
time evolution as $\Lambda(t+\tau)=\Lambda(\tau)\Lambda(t)$. The reduced
dynamics further differs from unitary dynamics by exhibiting hallmark features
of open systems, including purity change and decoherence.

As with ordinary operators, we can define components of the dynamical map with
respect to an operator basis on $\mathcal{Q}_{\rm S}$,

$\Lambda^{ab}(t)=\frac{1}{2}tr_{\rm S}[\sigma^{a}_{\rm
S}\Lambda(t)\circ\sigma^{b}_{\rm S}]=\frac{1}{2}tr[(\sigma^{a}_{\rm
S}\otimes\mathbb{1}_{\rm E})U(t)(\sigma^{b}_{\rm S}\otimes\rho_{\rm
E}(0))U^{{\dagger}}(t)]$ (28)

where $a$ and $b$ can take the values $0,x,y,z$ where
$\sigma^{0}_{S}=\mathbb{1}_{\rm S}$.

###  3.1 Two-qubit family

Carrying out this calculation, we find the non-zero dynamical map components,
in terms of the environment qubit’s initial state, $\vec{r}_{E}$(0), and the
functions $\alpha_{\pm}(t)$ and $\beta_{\pm}(t)$ appearing in the time
evolution operator, Eq.(15), to be

$\begin{split}\Lambda^{z0}(t)=&\frac{1}{2}\left[|\alpha_{+}(t)|^{2}-|\beta_{+}(t)|^{2}-|\alpha_{-}(t)|^{2}+|\beta_{-}(t)|^{2}\right]z_{E}(0)\\\
\Lambda^{xx}(t)=&\Re[\alpha_{+}(t)\alpha_{-}(t)-\beta_{+}(t)\beta_{-}(t)]\\\
\Lambda^{xy}(t)=&\Im[\alpha_{+}(t)\alpha_{-}(t)+\beta_{+}(t)\beta_{-}(t)]\\\
\Lambda^{xz}(t)=&\Re[\alpha_{+}(t)\beta_{-}(t)+\alpha_{-}(t)\beta_{+}(t)]x_{E}(0)-\Im[\alpha_{+}(t)\beta_{-}(t)-\alpha_{-}(t)\beta_{+}(t)]y_{E}(0)\\\
\Lambda^{yx}(t)=&-\Im[\alpha_{+}(t)\alpha_{-}(t)-\beta_{+}(t)\beta_{-}(t)]\\\
\Lambda^{yy}(t)=&\Re[\alpha_{+}(t)\alpha_{-}(t)+\beta_{+}(t)\beta_{-}(t)]\\\
\Lambda^{yz}(t)=&-\Im[\alpha_{+}(t)\beta_{-}(t)+\alpha_{-}(t)\beta_{+}(t)]x_{E}(0)-\Re[\alpha_{+}(t)\beta_{-}(t)-\alpha_{-}(t)\beta_{+}(t)]y_{E}(0)\\\
\Lambda^{zx}(t)=&-\Re[\alpha_{+}(t)\beta_{+}(t)+\alpha_{-}(t)\beta_{-}(t)]x_{E}(0)+\Im[\alpha_{+}(t)\beta_{+}(t)-\alpha_{-}(t)\beta_{-}(t)]y_{E}(0)\\\
\Lambda^{zy}(t)=&-\Im[\alpha_{+}(t)\beta_{+}(t)+\alpha_{-}(t)\beta_{-}(t)]x_{E}(0)-\Re[\alpha_{+}(t)\beta_{+}(t)-\alpha_{-}(t)\beta_{-}(t)]y_{E}(0)\\\
\Lambda^{zz}(t)=&\frac{1}{2}\left[|\alpha_{+}(t)|^{2}-|\beta_{+}(t)|^{2}+|\alpha_{-}(t)|^{2}-|\beta_{-}(t)|^{2}\right]\,.\end{split}$
(29)

Note the difference in how the components of the initial environment state
enter, with $z_{\rm E}(0)$ appearing separately from $x_{\rm E}(0)$ and
$y_{\rm E}(0)$. It is instructive to see how the symmetry of the full
Hamiltonian simplifies the dynamical map dependence on the initial state of
the environment qubit, $\vec{r}_{\rm E}$(0). To that end, define the partial
components of the dynamical map by

$\begin{split}\Lambda^{ab}&=\frac{1}{2}\Lambda^{abc}r^{c}_{\rm E}(0)\\\
\Lambda^{abc}&=tr[(\sigma^{a}_{\rm S}\otimes\mathbb{1}_{\rm
E})U(t)(\sigma^{b}_{\rm S}\otimes\sigma^{c}_{\rm
E})U^{{\dagger}}(t)]\,.\end{split}$ (30)

The parity symmetry $P_{zz}=Z_{\rm S}\otimes Z_{\rm E}$ essentially halves the
number of non-zero $\Lambda^{abc}$, and separates the $xy$ and $z$ components
of initial environment state in the dynamical map.

Then, since $P_{zz}$ commutes with $U(t)$ and satisfies
$P^{2}_{zz}=\mathbb{1}$, we see that

$\begin{split}\Lambda^{abc}&=tr[P^{2}_{zz}(\sigma^{a}_{\rm
S}\otimes\mathbb{1}_{\rm E})U(t)(\sigma^{b}_{\rm S}\otimes\sigma^{c}_{\rm
E})U^{{\dagger}}(t)]\\\
&=(-1)^{\pi_{a}+\pi_{b}+\pi_{c}}tr[P_{zz}(\sigma^{a}_{\rm
S}\otimes\mathbb{1}_{\rm E})U(t)(\sigma^{b}_{\rm S}\otimes\sigma^{c}_{\rm
E})U^{{\dagger}}(t)P_{zz}]\\\
&=(-1)^{\pi_{a}+\pi_{b}+\pi_{c}}\Lambda^{abc}\end{split}\,,$ (31)

where the $\pi_{d}$ are defined such that,

$P_{zz}(\sigma^{e}_{\rm S}\otimes\sigma^{f}_{\rm
E})=(-1)^{\pi_{e}+\pi_{f}}(\sigma^{e}_{\rm S}\otimes\sigma^{f}_{\rm
E})P_{zz}\,.$ (32)

Considering the cases where either e=0 or f=0, allows one to speak of the
parity of local system and environment operators. For example the parity of
the operator $\mathbb{1}_{\rm S}\otimes\sigma^{f}_{\rm E}$ is determined by
$\pi_{f}$. The condition $\pi_{a}+\pi_{b}+\pi_{c}=1$ (mod 2) if satisfied
implies that $\Lambda^{abc}$ vanishes. This is equivalent to the following:
given $\Lambda^{ab}$ if the parity of $(\pi_{a}+\pi_{b})$ is even (odd) then
the only contributing partial components have $\pi_{c}$ even (odd).
$\Lambda^{zzz}$ also vanishes as a consequence of the parity symmetry. However
this is obviously not a consequence of the previous argument. Instead it
follows using the $P_{zz}$ symmetry and that $tr_{\rm S}(Z_{\rm S})=0$.

Several additional partial components vanish, although not enforced by
Eq.(31). For example,

$\begin{split}&\Lambda^{xtx}=\Lambda^{xty}=0\\\
&\Lambda^{ytx}=\Lambda^{yty}=0\end{split}$ (33)

which imposes that the shift $\vec{d}$ must be parallel to the z axis.
Finally,

$\Lambda^{xxz}=\Lambda^{xyz}=\Lambda^{yxz}=\Lambda^{yyz}=0\,.$ (34)

Important to mention is that the shift of the dynamical map is restricted to
having only a non-zero z-component. Thus the dynamical map is unital if
$z_{rmE}(0)=0$. A dynamical map is unital if

$\Lambda(t)\mathbb{1}_{\rm S}=\mathbb{1}_{\rm S}\,,$ (35)

which requires $\vec{d}=\vec{0}$. These maps have the distinct property that
they never increase the initial purity ($tr\rho_{\rm S}^{2}$) of any state.
When restricting to qubit dynamical maps, they can always be Kraus decomposed
using unitary Kraus operators. That is the reduced dynamics generated by
unital channels may be constructed by averaging over unitary channels.

It is worth noting that for the 2-qubit model there is a connection between
the free Hamiltonian and the magnitude of the shift. The magnitude of the
shift is zero iff the state $\Omega_{\rm E}=\frac{1}{2}(\mathbb{1}_{\rm
S}\otimes\rho_{\rm E}(0))$ and $H_{\rm free}$ are orthogonal operators i.e.
Tr($\Omega_{\rm E}H_{\rm free}$)=0. But as we see in the next section, there
are special choices of Hamiltonian parameters that make the dynamical map
unital independent of the choice of initial environment state.

###  3.2 Three-qubit model of dynamical symmetry breaking

The reduced dynamics for the three qubit model is computed from

$\tilde{\Lambda}^{ab}(t,0)=\frac{1}{4}tr\left[\sigma^{a}_{\rm
S}\otimes\mathbb{1}_{\rm E}\otimes\mathbb{1}_{\rm
E^{\prime}}\tilde{U}(t,0)\sigma^{b}_{\rm S}\otimes(\mathbb{1}_{\rm E}+z_{\rm
E}(0)Z_{\rm E})\otimes\rho_{\rm
E^{\prime}}(0)\tilde{U}^{{\dagger}}(t,0)\right]$ (36)

where we assume that the second qubit is in an initial state compatible with
phase covariance. For this reason we do not include initial correlations
between subsystems $E$ and $E^{\prime}$, although this is not necessary for
the reduced dynamics to be CP.

For $t<\tau$, we claim that the dynamical map is phase covariant. Using the
results of the previous section we find

$\tilde{\Lambda}_{t<\tau}(t)=\begin{bmatrix}1&0&0&0\\\ 0&\cos^{2}2\omega
t&-\cos{2\omega t}\sin{2\omega t}&0\\\ 0&\cos{2\omega t}\sin{2\omega
t}&\cos^{2}{2\omega t}&0\\\ z_{\rm E}\sin^{2}{2\omega t}&0&0&\cos^{2}{2\omega
t}\end{bmatrix}$ (37)

If we now focus on the unital action of $\tilde{\Lambda}_{t<\tau}$

$\tilde{T}_{t<\tau}(t)=\begin{bmatrix}\cos{2\omega t}&-\sin{2\omega t}&0\\\
\sin{2\omega t}&\cos{2\omega t}&0\\\
0&0&1\end{bmatrix}\begin{bmatrix}\cos{2\omega t}&0&0\\\ 0&\cos{2\omega t}&0\\\
0&0&\cos^{2}{2\omega t}\end{bmatrix}\,,$ (38)

it is evident that any rotation about the $z$-axis will commute with
$\tilde{T}_{t<\tau}(t)$. Since the shift is left fixed by rotations about the
$z$-axis, it follows that

$\tilde{\Lambda}_{t<\tau}(t)[e^{-i\omega tZ_{\rm S}}\rho_{\rm S}e^{i\omega
tZ_{\rm S}}]=e^{-i\omega tZ_{\rm S}}[\tilde{\Lambda}_{t<\tau}(t)\rho_{\rm
S}]e^{i\omega tZ_{\rm S}}\,,$ (39)

i.e. the reduced dynamics is phase covariant as it has a time-translation
symmetry generated by $\omega Z_{\rm S}$. The phase-covariant map presented
here is known as the generalized amplitude damping channel, which appears in
quantum information to describe dissipative processes nielsen00 .

For $t\geq\tau$, where we introduce the phase covariance breaking into the
interaction, the dynamical map is found to be

$\tilde{\Lambda}_{t\geq\tau}(t)=\begin{bmatrix}1&0&0&0\\\
\tilde{d}_{x}(t)&0&0&0\\\ \tilde{d}_{y}(t)&0&0&0\\\
\tilde{d}_{z}(t)&0&0&0\end{bmatrix}\,,$ (40)

where

$\begin{split}\tilde{d}_{x}(t)&=\frac{z_{\rm E}x_{\rm
E^{\prime}}}{\sqrt{2}}\Bigg{[}\bigg{(}g_{+}g_{-}-\Re{[b_{-}g_{+}]}-\Im{[g_{-}h_{-}]}+\Re{[a_{+}]}\Re{[h_{-}]}-\Re{[b_{-}]}\Im{[h_{-}]}\bigg{)}\\\
&+\bigg{(}g_{+}g_{-}-\Im{[g_{+}h_{+}+g_{-}h_{-}]}+\Re{[h_{+}h_{-}}]\bigg{)}\cos{4\omega\tau}\\\
&+\bigg{(}\Im{[h_{+}h_{-}]}-\Re{[g_{+}h_{+}+g_{-}h_{-}]}\bigg{)}\sin{4\omega\tau}\Bigg{]}\\\
\tilde{d}_{y}(t)&=\frac{z_{\rm E}x_{\rm
E^{\prime}}}{\sqrt{2}}\Bigg{[}\bigg{(}\Re{[g_{-}h_{-}]}-\Re{[a_{+}g_{+}]}+\Re{[b_{-}]}\Re{[h_{-}]}+\Re{[a_{+}]}\Im{[h_{-}]}\bigg{)}\\\
&+\bigg{(}\Re{[g_{+}h_{+}]}+\Re{[g_{-}h_{-}]}+\Im{[h_{+}\bar{h}_{-}]}\bigg{)}\cos{4\omega\tau}\\\
&+\bigg{(}-g_{+}g_{-}+\Im{[g_{+}\bar{h}_{+}]}+\Im{[g_{-}h_{-}]}+\Re{[h_{+}h_{-}}]\bigg{)}\sin{4\omega\tau}\Bigg{]}\\\
\tilde{d}_{z}(t)&=\frac{z_{\rm
E}}{2}\Bigg{[}\bigg{(}|a_{+}|^{2}-|b_{-}|^{2}+g^{2}_{-}+2\Im{[g_{+}h_{-}]}\bigg{)}+\bigg{(}g^{2}_{-}-|h_{+}|^{2}-2\Im{[g_{+}h_{-}]}\bigg{)}\cos{4\omega\tau}\\\
&+2\bigg{(}\Re{[g_{-}h_{+}]}-\Re{[h_{-}]}\Im{[h_{-}]}\bigg{)}\sin{4\omega\tau}\Bigg{]}\end{split}\,$
(41)

and $a_{\pm}$, $b_{\pm}$, $g_{\pm}$, and $h_{\pm}$ are defined in Eq.(21), and
are all evaluated at $t-\tau$. The dynamical map after phase covariance
breaking is a time-dependent SWAP operation, mapping all initial states to
$\vec{\tilde{d}}(t)$. The transition of the reduced dynamics is continuous iff
$\tau=\frac{\pi}{2\omega}(n+\frac{1}{2})$ for some positive integer $n$; i.e.
only if $\tau$ coincides with a time when $\Lambda_{t<\tau}$ is non-
invertible. If $\tau$ is not finely tuned, the image of
$\Lambda_{t<\tau}(\tau)$ will suddenly collapse to the point
$\vec{\tilde{d}}(\tau)=z_{\rm E}(0)\hat{z}$.

##  4 Features of the reduced dynamics

This section reviews important non-unitary features of the dynamical map
families that we have found in the previous section. We begin with a look at
the invertibility of the dynamical maps, as this determines the structure of
possible master equations used to generate the reduced dynamics. We also look
at the classes of dynamical maps for the 2-qubit model, which we characterize
by their degree of symmetry.

###  4.1 Invertibility

Of paramount interest is the invertibility structure, which determines when
time-local master equations are viable for use as generators. In the case of
the 3-qubit example it is clear that the map is always non-invertible as
$\tilde{T}_{t\geq\tau}(t)=0$.

But the invertibility of the dynamical map for the two-qubit is non-trivial,
and we show in this section how it depends on the Hamiltonian and the state of
the environment. We find that there are in fact three independent ingredients
that play a role in defining the invertibility: the strength of the
system/environment coupling, the ratio between eigenenergies, and the initial
environment state. The fact that these are distinct criteria is seen from the
special row structure of $\Lambda$

${\rm Det}\,\Lambda(\tau)={\rm
Det}\,\vec{\vec{T}}(\tau)=|\vec{T}_{z}(\tau)|^{2}$ (42)

where $\vec{T}_{z}$ is the third row of $\vec{\vec{T}}$. This now leads to
three conditions that determine when the dynamical map is non-invertible as we
have

${\rm Det}\,\Lambda(\tau)=0\Leftrightarrow\vec{T}_{z}(\tau)=\vec{0}\,.$ (43)

Starting with the $z$-component, $\Lambda^{zz}(\tau)=0$ implies

$\sin^{2}\phi_{+}\sin^{2}(\omega_{+}\tau)+\sin^{2}\phi_{-}\sin^{2}(\omega_{-}\tau)=1\,.$
(44)

Eq.(44) may only be satisfied at a discrete set of times, and only if
$\sin^{2}\phi_{+}+\sin^{2}\phi_{-}\geq 1$. The set of $\phi_{\pm}$ that
satisfy this condition constitute the strong coupling regime of parameter
space. Figure 1 shows that this region of parameter space partially overlaps
with the space where $U(t)$ is a perfect entangler.

The $(\phi_{-},\phi_{+})$ plane($\frac{\pi}{2},\frac{\pi}{2})$

Figure 1: The $(\phi_{+},\phi_{-})$ plane split into regions based on the non-
local properties of the dynamics. The light gray region contains Hamiltonians
that are not perfect entanglers, whereas the other regions (black+dark grey)
are perfect entanglers. The entire gray region supports time local dynamics.
The black region is the strong coupling region and contains the only
Hamiltonians that may have ${\rm Det}\Lambda=0$.

However, strong coupling between system and environment is not a sufficient
for the map to be non-invertible. Eq.(44) also depends on the energy
eigenvalues. For example consider the boundary between the dark gray and black
regions of Figure 1 where $\sin^{2}\phi_{+}+\sin^{2}\phi_{-}=1$ (excluding the
corners where either $\phi_{+}=0$ or $\phi_{-}=0$). In this region Eq.(44) is
satisfied iff,

$\sin^{2}\omega_{+}\tau=\sin^{2}\omega_{-}\tau=1\,.$ (45)

In order for this to be possible integers $k$ and $l$ must exist such that,

$\frac{\omega_{+}}{\omega_{-}}=\frac{2k+1}{2l+1}\,.$ (46)

Defining $\nu$ such that $\omega_{+}=(2k+1)\nu$ and $\omega_{-}=(2l+1)\nu$;
for any positive integer n a non-invertiblity can appear in the reduced
dynamics at times given by,

$\tau_{n}=\frac{(2n+1)\pi}{2\nu}\,.$ (47)

So on this boundary only Hamiltonians with commensurate eigenenergies require
time-non-local dynamics (although incommensurate Hamiltonians on the boundary
can generate dynamical maps with small determinant). On the other hand, away
from the boundary, farther into the blue region, most frequency pairs will
generate dynamics with non-time-locality for some initial environment states.

So far, we have considered only one of the conditions for non-invertibility,
$\Lambda^{zz}(\tau)=0$, which imposes conditions on the Hamiltonian that must
be satisfied for the map to be non-invertible. However, constraining the
Hamiltonian to be in the strong-coupling regime is not sufficient: the initial
environment state also plays a role. The class of initial states of the
environment that lead to non-time-local dynamics is found by setting the
remaining components of $\vec{T}_{z}(\tau)$ to zero,

$\begin{split}&\Lambda^{zx}(\tau)=\Lambda^{zxx}(\tau)x_{E}+\Lambda^{zxy}(\tau)y_{E}=0\,,\\\
&\Lambda^{zy}(\tau)=\Lambda^{zyx}(\tau)x_{E}+\Lambda^{zyy}(\tau)y_{E}=0\,.\end{split}$
(48)

We can characterize the set of initial states for which non-invertibility will
occur by finding the vector $\vec{\eta}(\tau)$ in the $xy$ plane associated to
the environment qubit state for which both

$V(\tau_{i})\vec{\eta}(\tau)=\begin{bmatrix}\Lambda^{zxx}(\tau)&\Lambda^{zxy}(\tau)\\\
\Lambda^{zyx}(\tau)&\Lambda^{zyy}(\tau)\end{bmatrix}\vec{\eta}(\tau)=0\,$ (49)

and ${\rm Det}\,V(\tau)=0$ at some fixed $\tau$. The solution is

$\vec{\eta}(\tau)=\Lambda^{zyy}(\tau)\hat{x}-\Lambda^{zyx}(\tau)\hat{y}\,,$
(50)

and the orthogonal direction is

$\vec{\eta}^{\perp}(\tau)=\Lambda^{zxx}(\tau)\hat{x}+\Lambda^{zxy}(\tau)\hat{y}\,.$
(51)

A non-invertibility at time $\tau$ can be removed by shifting the initial
state to contain a component in the direction $\vec{\eta}^{\perp}(\tau)$.
Furthermore, there can only be a discrete set of times, $\tau_{i}<T$, where
the condition $\Lambda^{zz}(\tau_{i})=0$ can be satisfied. Assume there are
$N$ non-invertible times,

$0<\tau_{1}<...<\tau_{\rm N}<T,$ (52)

with the associated initial environment states that preserve the non-
invertibility $\\{\hat{\eta}_{1},...,\hat{\eta}_{N}\\}$. Single out
$\hat{\eta}_{1}$ and note that so long as
$\hat{\eta}^{\perp}_{1}\cdot\hat{\eta}^{\perp}_{k}\neq 0$, then the non-
invertibility at $\tau_{k}$ is eliminated by the presence of
$\hat{\eta}^{\perp}_{1}$ in $\vec{r}_{\rm E}(0)$. The remaining directions all
must satisfy $\hat{\eta}^{\perp}_{1}\cdot\hat{\eta}^{\perp}_{k}=0$. Since
these initial environment states lie in a two dimensional space, all the
remaining $\hat{\eta}^{\perp}_{i}$ must equal $\hat{\eta}_{1}$. The question
of removing all non-invertibilities up to $T$ is then equivalent to finding a
$\theta$ such that,

$\eta^{\perp}_{i}\cdot(\cos\theta\hat{\eta}^{\perp}_{1}+\sin\theta\hat{\eta}_{1})\neq
0,$ (53)

for all $i\in\\{1,N\\}$.

We have demonstrated that certain off diagonal components (coherences) of the
initial environment state in the eigenbasis of the free environment
Hamiltonian ($H_{\rm E;\rm free}$) control the appearance of time-non-locality
in the reduced dynamics. In Section 5 we use this knowledge of the time-local
environment states in order to construct exact non-local master equations as
well as approximate time-local master equations.

###  4.2 Dynamical map families

We saw above how the invertibility of the dynamical map can be dramatically
changed by symmetry breaking during the dynamics. On the other hand, for the
time-independent Hamiltonian, non-invertibility only occured in a subset of
the parameter space. In this section we classify all the dynamics possible
with the Hamiltonian in Eq.(5). This classification illustrates the
relationship between symmetry, non-invertibility, and two other key properties
of open system dynamics: non-Markovianity and unitality.

Table 4.2 lists the various dynamical map families contained in this model,
together with each family’s characteristics that we derive below. The largest
family is the non-commensurate family ($\mathcal{N}$), with two independent
frequencies. If we consider the time evolution in the Bloch ball, the
trajectories generated are dense for any initial state $\vec{r}_{\rm S}(0)$.
As mentioned previously, we use the (AB) frame to discuss the symmetries as
they are simplest in this frame. For example the symmetries present in the
entire family of Hamiltonians are $Z_{\rm A}\otimes\mathbb{1}_{\rm B}$ and
$\mathbb{1}_{\rm A}\otimes Z_{\rm B}$. Of course these are equivalent to $H$
and $P_{zz}$, however we find that this frame compresses the discussion of
Hamiltonian families that have additional symmetries beyond these two.

The Hamiltonians which generate the family $\mathcal{D}$ have more symmetries
than those corresponding to $\mathcal{N}$; these Hamiltonians commute with all
rotations performed in the $B$ subsystem. The apparent symmetry between $A$
and $B$ in Eq.(12) is broken by the relationship to the $S$/$E$ frame. The
presence of more symmetries simplifies the time-dependence of the dynamical
map, thus less complicated trajectories are generated and the non-Markovian
measures are periodic. The witnesses of non-Markovianity become periodic even
in the case that the eigenfrequencies of the Hamiltonian are commensurate
($\mathcal{C}$). $\Lambda\in\mathcal{D}$ are not structurally different than
those in $\mathcal{N}$ i.e. no additional components or partial components
vanish. $\mathcal{D}$ does not have conserved quantities at the level of
reduced dynamics.

There are special subfamilies of $\mathcal{D}$ that support conserved
quantities at the level of reduced dynamics. These families
$\mathcal{D}_{\pm}$ contain dynamical maps that are phase damping channels,
which are unital channel with the additional condition that $\lambda_{x}=1$
and $\lambda_{y}=\lambda_{z}=\lambda$. They are generated from the
Hamiltonians with parameters such that $\phi_{+}=\pm\phi_{-}$. These families
are simple to study and allow us to easily determine the set of initial
environment states that yield invertible dynamics. For example in the family
$\mathcal{D}_{+}$, there is only 1 distinct direction $\vec{\eta}(\tau)$ that
allows non-invertibility namely $\vec{\eta}=\hat{y}$.

Restricting further the allowed set of initial environment states uncovers a
set of Markovian families $\mathcal{M}_{\pm}$. These families contain the maps
in $\mathcal{D}_{\pm}$ generated using $\vec{r}_{\rm E}(0)=\hat{x}$
($\mathcal{M}_{+}$) and $\vec{r}_{\rm E}(0)=\hat{y}$ ($\mathcal{M}_{-}$). This
is simply explained by looking at the Kraus decomposition of these channels.
To illustrate this take an element of $\mathcal{D}_{+}$. The minimal Kraus
decomposition has the form

$\Lambda_{+}\rho_{\rm S}=\frac{1+x_{\rm E}}{2}U_{+}\rho_{\rm
S}U^{{\dagger}}_{+}+\frac{1-x_{\rm E}}{2}(Z_{\rm S}U_{+}Z_{\rm S})\rho_{\rm
S}(Z_{\rm S}U^{{\dagger}}_{+}Z_{\rm S})$ (54)

thus when the initial environment state is pure and in the $x$-direction the
channel becomes unitary. We say the dynamics is Markovian as the dynamical map
is unitary and thus generated by a time-dependent Lindblad equation with non-
negative rates. Note that unitary reduced dynamics is the only kind of
Markovian dynamics possible, since the environment Hilbert space is of finite
dimension, so necessarily has a free Hamiltonian that is bounded from below
Wolf_2008 . As in all Markovian open systems, the initial environment state
remains fixed under the time evolution in $\mathcal{M}_{\pm}$.

So far we have not mentioned the family of maps that are phase covariant i.e.
the generalized amplitude damping channels $\mathcal{A}_{\pm}$, often referred
to by the acronym GADC. These families do not neatly fit within the previous
families, but instead have non-zero overlap with $\mathcal{N}$, $\mathcal{C}$,
and $\mathcal{D}$. These maps have singular values satisfying
$\lambda_{z}=\lambda_{x}^{2}=\lambda^{2}_{y}$, and are non-unital with the
only non-zero component given by $d_{z}=z_{\rm E}(1-\lambda_{z})$. For $z_{\rm
E}=1$ the maps are simply known as amplitude damping channels. These maps are
generated when either $\Delta_{-}=\kappa_{+}=0$ ($\mathcal{A}_{+}$) or
$\Delta_{+}=\kappa_{-}=0$ ($\mathcal{A}_{-}$) and $\vec{r}_{\rm E}(0)=z_{\rm
E}(0)\hat{z}$.

It is worth noting that the families $\mathcal{D}_{\pm},\mathcal{A}_{\pm}$,and
$\mathcal{M}_{\pm}$ consist entirely of extreme channels; i.e. channels that
live in the boundary of the set of qubit channelsRuskai2001 . The extreme
channels form a tetrahedron, and consist of minimal rank Kraus operators,
where maps on the interior have non-minimal Kraus rank. The families
$\mathcal{M}_{\pm}$ are contained in the corner representing the trivial
channel. The families $\mathcal{D}_{\pm}$ live on an edge connected to the
trivial channel. And the families $\mathcal{A}_{\pm}$ live on a face of the
tetrahedron.

Dynamical Map Families
---
Family |  Time Evolution |  Extra Symmetries |  Unital
Non-Commensurate ($\mathcal{N}$)
$\omega_{+}\neq q\omega_{-}$ |  Aperiodic
Non-Markovian |  None |  If $z_{\rm E}(0)=0$
Commensurate ($\mathcal{C}$)
$\omega_{+}=q\omega_{-}$ |  Periodic
Non-Markovian |  None |  If $z_{\rm E}(0)=0$
Degenerate ($\mathcal{D}$)
$\omega_{+}=\omega_{-}$ |  Periodic
Non-Markovian |  $\\{\mathbb{1}_{\rm A}\otimes X_{\rm B}$, $\mathbb{1}_{\rm A}\otimes Y_{\rm B}\\}$ |  If $z_{\rm E}(0)=0$
Phase Damping($\mathcal{D}_{\pm}$)
$\phi_{+}=\pm\phi_{-}$ |  Periodic
Non-Markovian |  $\\{\mathbb{1}_{\rm A}\otimes X_{\rm B}$, $\mathbb{1}_{\rm A}\otimes Y_{\rm B}\\}$ |  Any $\vec{r}_{\rm E}(0)$
Markovian ($\mathcal{M}_{\pm}$)
$\vec{r}_{\rm E}(0)$ = $\hat{x}$(+) or $\hat{y}$(-) |  Markovian |  $\\{\mathbb{1}_{\rm A}\otimes X_{\rm B}$, $\mathbb{1}_{\rm A}\otimes Y_{\rm B}$} |  Always
Amplitude Damping
($\mathcal{A}_{\pm}$) |  Aperiodic NM ($\mathcal{A}_{\pm}\cap\mathcal{N}$)
Periodic NM ($\mathcal{A}_{\pm}\cap\mathcal{D}$) |  None
$\\{\mathbb{1}_{\rm A}\otimes X_{\rm B}$, $\mathbb{1}_{\rm A}\otimes Y_{\rm B}$} |  If $z_{\rm E}(0)=0$

The method we devise later for constructing effective master equations relies
on there being a set of fundamental time scales that can be used to determine
all non-invertible times of the reduced dynamics. Such a task is especially
simple for the families $\mathcal{D}_{\pm}$ and
$\mathcal{A}_{\pm}\cap\mathcal{D}$, where only one such time scale exists. But
for the set of Hamiltonians where the parameters are not carefully chosen
($\mathcal{N}$), such a construction is unfavorable as the non-invertible
times behave as essentially random.

###  4.3 Entanglement generation and non-Markovianity

For the range of parameters possible in the two-qubit Hamiltonian, we can
characterize the flow of information between system and environment by
examining the non-Markovianty and the degree to which the system and
environment can become entangled.

The parameters required for the time-evolution to be (periodically) perfectly
entangling can be found using the criteria of Makhlin Makhlin:2000 ;
Zhang:2003zz , that the convex hull of the eigenvalues of the matrix
$m(U)=(Q^{\dagger}UQ)^{\rm T}Q^{\dagger}UQ$ contains zero, where $Q$ is the
operator that changes to the Bell basis. Evaluating the eigenvalues of $m(U)$,
the convex hull condition becomes (see details in Appendix A)

$\cos^{2}\varphi\beta_{+}^{2}(t)+\sin^{2}\varphi\beta_{-}^{2}(t)=\frac{1}{2}\,,$
(55)

where $\varphi\in[0,2\pi)$ parameterizes the remaining convex combinations.
This can be satisfied iff the largest of $\beta^{2}_{+}(t)$ and
$\beta^{2}_{-}(t)$ is greater than or equal to $\frac{1}{2}$, which will hold
at some times as long as ${\rm Max}(\phi_{+},\phi_{-})\geq\frac{\pi}{4}$. This
condition on the parameter space of the Hamiltonian is shown in Figure 1,
compared with other conditions we derive below related to properties of the
reduced dynamics.

For nearly all parameter values in the Hamiltonian, Eq.(7), and initial
states, the dynamical map is non-Markovian. This is expected since the system
and environment are the same (small) size. The non-Markovianity is diagnosed
by information back-flow into the system from the environment Laine2010 , with
standard indicators being non-monotonicity in the evolution of trace distance
and fidelity. For any two states on the reduced system, $\vec{r}_{1}(t)$ and
$\vec{r}_{2}(t)$, the trace distance ($\mathcal{D}$) and fidelity
($\mathcal{F}$) are Chen2002 ,

$\begin{split}&2\mathcal{D}(\vec{r}_{1}(t),\vec{r}_{2}(t))=||\vec{r}_{1}(t)-\vec{r}_{2}(t)||\\\
&2\mathcal{F}(\vec{r}_{1}(t),\vec{r}_{2}(t))=1+\vec{r}_{1}(t)\cdot\vec{r}_{2}(t)+\sqrt{(1-r^{2}_{1}(t))(1-r^{2}_{2}(t))}\,.\end{split}$
(56)

Generically these measures are oscillatory, and aperiodic unless
$\omega_{+}=q\omega_{-}$ for some $q\in\mathbb{Q}$.

The degenerate family, Eq.(13), always has periodic measures of non-
Markovianity and is a useful case to look at in more detail. Figure 2 shows
the trace distance and fidelity in the degenerate family, demonstrating the
non-Markovian character of the reduced dynamics. The dimensionless parameter
$\chi=\frac{\omega t}{\pi}$ ($\omega=\omega_{+}=\omega_{-}$) is used to
construct these plots, and the environment memory time-scale can be read off
as $\tau_{\rm NM}\sim$ $\mathcal{O}(\frac{\pi}{2\omega})$.

Figure 2: The fidelity and trace distance between time evolved reduced states
using the initial conditions $\vec{r}_{1}(0)=\hat{x}$ and
$\vec{r}_{2}(0)=\cos\theta\hat{x}+\sin\theta\hat{y}$ are plotted. The
parameters used are $\omega=\omega_{+}=\omega_{-}=\frac{\sqrt{5}}{2}$,
$\tan\phi_{+}=\tan\phi_{-}=2$, and $\vec{r}_{\rm E}(0)=\vec{0}$. The non-
Markovian nature of the reduced dynamics is evident from the oscillations of
$\mathcal{D}$ and $\mathcal{F}$.

Figure 2 is generated with $\phi_{+}=\phi_{-}$ allowing for interesting
features to appear in $\mathcal{F}$. If $\phi_{\pm}$ are perturbed so that
$\phi_{+}\neq\phi_{-}$, these interesting features also vanish. First note the
plateaus, which indicate there is a unitary phase (i.e.
$\mathcal{F}$=constant) if we further require that $\vec{r}_{\rm
E}(0)=\hat{x}$. Additionally, preceding the plateaus are discontinuities in
$\dot{\mathcal{F}}$. In some other finite open systems, such discontinuities
indicate dynamical phase transitions PhysRevB.102.220302 .

Non-Markovianity of the dynamics has been equated to the indivisibility of the
dynamical map into channels (CPTP maps) Laine2010 , however the actual
relationship is more complicated as non-Markovian dynamical maps can be CP
divisible 2019A ; 2019B ; 2019C . To study the divisibility of the dynamical
map one looks at the interweaving maps ($\Phi$), defined using two times
$\tau_{2}>\tau_{1}\geq 0$,

$\Lambda(\tau_{2})=\Phi(\tau_{2},\tau_{1})\Lambda(\tau_{1})\,.$ (57)

For invertible reduced dynamics the interweaving map is computed as
$\Phi(\tau_{2},\tau_{1})=\Lambda(\tau_{2})\Lambda^{-1}(\tau_{1})$, although
$\Phi$ may still be defined even when $\Lambda^{-1}$ does not exist 2019B .

The degree of positivity of $\Phi(\tau_{2},\tau_{1})$ determines the
divisibility class of the map $\Lambda(\tau_{2})$, as $\Lambda(\tau_{1})$ is
completely positive by construction. It is important to note that divisibility
is meant in a holistic sense i.e. the reduced dynamics is considered CP
divisible up to time $\tau_{2}$ only if $\Phi(\tau_{2},\tau_{1})$ is
completely positive for all $\tau_{1}<\tau_{2}$. A qubit channel is
(infinitesimally) P-divisible if ${\rm Det}\Lambda(t)\geq 0$ for all $t\geq 0$
Wolf_2008 . We have shown this to be true for the 2-qubit family studied. So
the dynamical maps presented here may be arbitrarily split into positive maps,
each made close to the identity channel.

The question of CP divisibility is in general more subtle. For qubit dynamical
maps of full rank, these maps are CP divisible if $s_{1}s_{2}s_{3}>0$, where
$s_{i}$ are the singular values of the Lorentz normal form of the channel
2019C . For unital channels, this is equivalent to requiring ${\rm
Det}\Lambda(t)>0$. We are able to find many examples of such maps. Take
$\Lambda(t)\in\mathcal{N}$ and $z_{\rm E}(0)=0$. As long as the initial
environment state is not pure, $\Lambda(t)$ will be a full Kraus rank unital
channel. Therefore it will be CP divisible iff it is invertible. Of course
these maps generate oscillation in non-Markovian measures, so do not
constitute Markovian reduced dynamics. In Appendix B, we study the CP
divisibility of deficient Kraus rank maps in $\mathcal{D}_{+}$. We find that
even when these maps are invertible, there typically exist time intervals s.t.
for $\tau\in[\tau_{a},\tau_{b}]$ the map $\Phi(\tau_{2},\tau)$ fails to be CP.

##  5 Master equations

We present the standard master equations that can be associated to any
$\Lambda(t,0)$, both local and non-local in time. We find that by changing
$\vec{r}_{\rm E}(0)$, different partitions of the singularity between non-
local and local terms can be achieved. We show that we can capture some non-
unitarity features of the reduced dynamics just using the local term.

###  5.1 Standard master equations

The dynamical map can be used to construct the time-local, although possibly
singular, generator for the master equation

$\partial_{t}\rho_{S}(t)=K_{\rm
TL}(t)\rho_{S}(t)=\dot{\Lambda}(t)\Lambda^{-1}(t)\rho_{S}(t)\,.$ (58)

Since the inverse dynamical map is

$\begin{split}\Lambda^{-1}(t)&=\frac{1}{T^{2}_{z}}\begin{bmatrix}T^{2}_{z}&0&0&0\\\
-d_{z}\vec{T}_{z}&\vec{T}_{y}\times\vec{T}_{z}&\vec{T}_{z}\times\vec{T}_{x}&\vec{T}_{z}\end{bmatrix}\end{split}\,,$
(59)

the time-local generator is expressed in terms of the dynamical map components
as

$\begin{split}K_{\rm TL}(t)=\frac{1}{T^{2}_{z}}\begin{bmatrix}0&0&0&0\\\
-d_{z}(\dot{\vec{T}}_{x}\cdot\vec{T}_{z})&\dot{\vec{T}}_{x}\cdot(\vec{T}_{y}\times\vec{T}_{z})&\dot{\vec{T}}_{x}\cdot(\vec{T}_{z}\times\vec{T}_{x})&\dot{\vec{T}}_{x}\cdot(\vec{T}_{z})\\\
-d_{z}(\dot{\vec{T}}_{y}\cdot\vec{T}_{z})&\dot{\vec{T}}_{y}\cdot(\vec{T}_{y}\times\vec{T}_{z})&\dot{\vec{T}}_{y}\cdot(\vec{T}_{z}\times\vec{T}_{x})&\dot{\vec{T}}_{y}\cdot(\vec{T}_{z})\\\
-d_{z}(\dot{\vec{T}}_{z}\cdot\vec{T}_{z})&\dot{\vec{T}}_{z}\cdot(\vec{T}_{y}\times\vec{T}_{z})&\dot{\vec{T}}_{z}\cdot(\vec{T}_{z}\times\vec{T}_{x})&\dot{\vec{T}}_{z}\cdot(\vec{T}_{z})\end{bmatrix}\,.\end{split}$
(60)

Fig (3) contains plots of ${\rm Det}\Lambda$ and ${\rm Tr}K_{\rm TL}$ using
$\Lambda(t)\in\mathcal{D}_{+}$, which demonstrate that singularities in
$K_{\rm TL}$ occur where ${\rm Det}\Lambda=0$.

(a) Time-local reduced dynamics

(b) Non-time-local reduced dynamics

Figure 3:  Tr$K_{\rm TL}$ (scaled by $\frac{\pi}{4\omega}$) and Det$\Lambda$
are plotted with $\vec{r}_{\rm E}(0)=\vec{0}$. a) Uses parameters
$\omega=\omega_{+}=\omega_{-}=\frac{\sqrt{17}}{4}$ and
$\tan\phi_{+}=\tan\phi_{-}=\frac{1}{2}$; the dynamics is invertible with
Det$\Lambda$ separated from zero. b) Uses parameters
$\omega=\omega_{+}=\omega_{-}=\sqrt{5}$ and $\tan\phi_{+}=\tan\phi_{-}=4$;
Det$\Lambda$=0 has solutions, where the dynamical map becomes a projection
onto the $\hat{Q}_{1}(\tau)$ direction in the Bloch ball.

The generator $K_{\rm TL}$ can be put into Lindblad form,

$\partial_{t}\rho_{S}(t)=-i[(\vec{H}_{\rm eff}(t)\cdot\vec{\sigma}_{\rm
S}),\rho_{\rm
S}(t)]+\sum_{i=1}^{3}\sum_{j=1}^{3}\gamma_{ij}(t)(\sigma^{j}_{\rm S}\rho_{\rm
S}(t)\sigma^{i}_{\rm S}-\frac{1}{2}\\{\sigma^{i}_{\rm S}\sigma^{j}_{\rm
S},\rho_{\rm S}(t)\\})\,,$ (61)

where $\vec{H}_{\rm eff}(t)=\omega_{\rm S}\hat{z}+\vec{H}_{\rm open}(t)$
generates unitary evolution on $\mathcal{Q}_{\rm S}$ with a contribution from
internal parameters and a portion that knows about the environment and
interaction, $\vec{H}_{\rm open}(t)$. The coefficients $\gamma_{ij}(t)$ are
the Lindblad coefficients which generate the non-unitarity that appears in the
reduced dynamics.

(a) $\Re\gamma_{xy}=\Re\gamma_{yz}$

(b) $\Im\gamma_{xy}=\Im\gamma_{yz}$

Figure 4: Plotted are the real and imaginary parts of the off diagonal
Lindblad coefficients. Plots are generated using the parameters
$\omega_{+}=\omega_{-}=\frac{\sqrt{5}}{2}$ and $\tan\phi_{+}=\tan\phi_{-}=2$.
Included in the plots is the trace distance evaluated using the same
parameters and initial states $\vec{r}_{1}(0)=-\vec{r}_{2}(0)=\hat{x}$. We
include the trace distance to demonstrate that experimentally accessible
quantum information measures can constrain the parameters that appear in
effective master equations; in this case oscillation frequencies of the
Lindblad coefficients.

Obtaining the Lindblad form of the master equation amounts to a change of
basis, for example the effective Hamiltonian has components

$\begin{split}&H^{i}_{\rm
eff}(t)=\frac{1}{2}\epsilon^{ijk}K_{jk}(t)\end{split}\,,$ (62)

and similar expression exist for the $\gamma_{ij}(t)$, though we do not
include them here. Figure 4 shows how we can use the quantities such as the
$\mathcal{D}(t)$ to determine the oscillation timescales that appear in the
Lindblad coefficients, which is useful to set a scale for environment memory
timescales that will inevitably appear in effective master equations.

In the strong coupling region where the time-local description has singular
behavior, we instead use an integral master equation known as the Nakajima-
Zwanzig equation. The Hamiltonian being time independent allows the NZ
generator to take the form $K_{NZ}(t,\tau)=K_{NZ}(t-\tau)$. Thus the NZ
equation is a convolution,

$\partial_{t}\rho_{\rm S}(t)=\int_{0}^{t}K_{\rm NZ}(t-\tau)\rho_{\rm
S}(\tau)d\tau\,.$ (63)

The convolution kernel may be obtained from the dynamical map using the
Laplace transform of the dynamical map $\Phi(s)$ (as in Smirne2010 ),

$\tilde{K}_{\rm NZ}(s)=s\mathbb{1}_{4}-\Phi^{-1}(s)$ (64)

The form of the functions $\alpha_{\pm}(t)$ and $\beta_{\pm}(t)$ restrict the
dynamical map components to having few Fourier components. That is each
$\Lambda^{ab}$ has a Fourier decomposition of the form,

$\begin{split}\Lambda^{ab}(t)=C^{ab}+&F^{ab}_{1}e^{-i2\omega_{+}t}+F^{ab}_{2}e^{-i2\omega_{-}t}+F^{ab}_{3}e^{-i(\omega_{+}+\omega_{-})t}+F^{ab}_{4}e^{-i(\omega_{+}-\omega_{-})t}\\\
+&G^{ab}_{1}e^{i2\omega_{+}t}+G^{ab}_{2}e^{i2\omega_{-}t}+G^{ab}_{3}e^{i(\omega_{+}+\omega_{-})t}+G^{ab}_{4}e^{i(\omega_{+}-\omega_{-})t}\end{split}\,.$
(65)

If $ab$ is even (odd) then the component $\Lambda^{ab}$ involves only even
(odd) time dependent functions. It is now simple to go to the frequency
domain, where we can write the generic Laplace transformed components,

$\begin{split}\Phi^{ab}(s)=\frac{C^{ab}}{s}+&\frac{F^{ab}_{1}}{s+i2\omega_{+}}+\frac{F^{ab}_{2}}{s+i2\omega_{-}}+\frac{F^{ab}_{3}}{s+i(\omega_{+}+\omega_{-})}+\frac{F^{ab}_{4}}{s+i(\omega_{+}-\omega_{-})}\\\
+&\frac{G^{ab}_{1}}{s-i2\omega_{+}}+\frac{G^{ab}_{2}}{s-i2\omega_{-}}+\frac{G^{ab}_{3}}{s-i(\omega_{+}+\omega_{-})}+\frac{G^{ab}_{4}}{s-i(\omega_{+}-\omega_{-})}\end{split}\,.$
(66)

In principle one can use these formulae to construct the exact non-time-local
master equation associated to the reduced dynamics, although the expressions
involved are cumbersome. Instead for what follows we devise a method to expand
the NZ kernel,

$K_{\rm NZ}(t-\tau)=\delta(t-\tau)K_{\rm TL}(t)+K_{\rm NTL}(t,\tau)\,.$ (67)

Such an expansion is available for each time-local initial environment state
($\vec{r}_{\rm TL}$), where the relative importance of the time-local vs non-
time-local component is controlled by the magnitude of $\vec{r}_{\rm
TL}-\vec{r}_{\rm E}(0)$.

###  5.2 Effective master equation

An exact time local master equation without singularities exists as long as
either of the following conditions are met

  1. I

The Hamiltonian parameters are in the weak coupling region, i.e.
$\sin^{2}\phi_{+}+\sin^{2}\phi_{-}<1$.

  2. II

$\vec{r}_{\rm E}$(0) has off diagonal elements in the eigenbasis of $H_{\rm
E;\rm free}$ along certain directions described in Section IIIB.

However, when neither of the above holds only a non-local equation can capture
the exact time evolution of the reduced state. Such a master equation can be
found using reduced dynamics from the time-local region.

Consider the case where the Hamiltonian parameters are in the strong coupling
region and $\vec{r}_{\rm E}(0)=\vec{r}_{\rm NTL}$, yielding non-
invertibilities in the reduced dynamics. By choosing a shifted initial
environment state ($\vec{r}^{\,{}^{\prime}}_{\rm E}$), we can engineer the
time local component of a non-time local master equation. By using the exact
non-time local dynamical map we can also determine the corresponding non-time-
local component for the master equation. As the non-time-local piece is linear
in $\delta\vec{r}=\vec{r}_{\rm NTL}-\vec{r}^{\,{}^{\prime}}_{\rm E}$, the
choice of shifted initial state controls its relevance.

We can choose $\vec{r}^{\,{}^{\prime}}_{\rm E}$ by considering states near to
$\vec{r}_{\rm E}(0)$ that are shifted along the direction(s) $\eta^{\perp}$ as
defined in Eq.(51). The time derivative of $\rho_{\rm S}(t;\vec{r}_{\rm NTL})$
may be expanded as,

$\begin{split}\dot{\rho}_{\rm S}(t;\vec{r}_{\rm NTL})&=\partial_{t}\rho_{\rm
S}(t;\vec{r}^{\,{}^{\prime}}_{\rm
E}+\delta\vec{r})=\partial_{t}\left[\Lambda(t;\vec{r}^{\,{}^{\prime}}_{\rm
E})+(\Lambda(t;\delta\vec{r})-\Lambda(t;\vec{0}))\right]\rho_{\rm S}(0)\\\
&=K_{\rm TL}(t;\vec{r}^{\,{}^{\prime}}_{\rm E})\rho_{\rm
S}(t;\vec{r}^{\,{}^{\prime}}_{\rm
E})+\delta\vec{r}\cdot\left[\frac{\partial\dot{\Lambda}(t;\delta\vec{r})}{\partial\delta\vec{r}}\right]\rho_{\rm
S}(0)\\\ &=K_{\rm TL}(t;\vec{r}^{\,{}^{\prime}}_{\rm E})\rho_{\rm
S}(t;\vec{r}_{\rm
NTL})+\delta\vec{r}\cdot\left[\frac{\partial\dot{\Lambda}(t;\delta\vec{r})}{\partial\delta\vec{r}}-K_{\rm
TL}(t;\vec{r}^{\,{}^{\prime}}_{\rm
E})\frac{\partial\Lambda(t;\delta\vec{r})}{\partial\delta\vec{r}}\right]\rho_{\rm
S}(0)\,.\end{split}$ (68)

Note that since the dynamical map is linear in the initial environment state,
we can replace
$\frac{\partial\Lambda(t;\delta\vec{r})}{\partial\delta\vec{r}}$ in the above
equation with $\frac{\partial\Lambda(t;\vec{r}^{\,{}^{\prime}}_{\rm
E})}{\partial\vec{r}^{\,{}^{\prime}}_{\rm E}}$

$\begin{split}\dot{\rho}_{\rm S}(t;\vec{r}_{\rm NTL})&=K_{\rm
TL}(t;\vec{r}^{\,{}^{\prime}}_{\rm E})\rho_{\rm S}(t;\vec{r}_{\rm
NTL})+\delta\vec{r}\cdot\left[\frac{\partial\dot{\Lambda}(t;\vec{r}^{\,{}^{\prime}}_{\rm
E})}{\partial\vec{r}^{\,{}^{\prime}}_{\rm E}}-K_{\rm
TL}(t;\vec{r}^{\,{}^{\prime}}_{\rm
E})\frac{\partial\Lambda(t;\vec{r}^{\,{}^{\prime}}_{\rm
E})}{\partial\vec{r}^{\,{}^{\prime}}_{\rm E}}\right]\rho_{\rm
S}(0)\,.\end{split}$ (69)

This can be further simplified noting that
$\dot{\Lambda}(t;\vec{r}^{\,{}^{\prime}}_{\rm E})-K_{\rm
TL}(t;\vec{r}^{\,{}^{\prime}}_{\rm E})\Lambda(t;\vec{r}^{\,{}^{\prime}}_{\rm
E})=0$, yielding the final expression

$\begin{split}\dot{\rho}_{\rm S}(t;\vec{r}_{\rm NTL})&=K_{\rm
TL}(t;\vec{r}^{\,{}^{\prime}}_{\rm E})\rho_{\rm S}(t;\vec{r}_{\rm
NTL})+\delta\vec{r}\cdot\left[\frac{\partial K_{\rm
TL}(t;\vec{r}^{\,{}^{\prime}}_{\rm E})}{\partial\vec{r}^{\,{}^{\prime}}_{\rm
E}}\Lambda(t;\vec{r}^{\,{}^{\prime}}_{\rm E})\right]\rho_{\rm
S}(0)\,.\end{split}$ (70)

We have now explicitly isolated a time-local component which is evaluated at
the same Hamiltonian parameters as $\Lambda(t;\vec{r}_{\rm TNL})$, but uses
the initial environment state $\vec{r}^{\,{}^{\prime}}_{\rm E}$ instead of
$\vec{r}_{\rm NTL}$. How large $\delta\vec{r}$ is compared with $\vec{r}_{\rm
TNL}$ determines the maximum value of ${\rm Tr}K_{\rm
TL}(t;\vec{r}^{\,{}^{\prime}}_{\rm E})$, which in turn determines how relevant
the integral term is in capturing the non-invertibility.

An effective time local description is obtained keeping only the zeroth order
term,

$\partial_{t}\rho_{S}(t;\vec{r}_{\rm NTL})\approx K_{\rm
TL}(t;\vec{r}^{\,{}^{\prime}}_{\rm E})\rho_{S}(t;\vec{r}_{\rm NTL})\,$ (71)

Using this equation to generate the reduced dynamics is equivalent to
exchanging the dynamical map $\Lambda(t,\vec{r}_{\rm NTL})$ for the dynamical
map $\Lambda(t;\vec{r}^{\,{}^{\prime}}_{\rm E})$, so obviously defines a
completely positivity master equation.

##  6 Conclusions

For a two-qubit system with symmetry, we have classified how the non-
Markovianity, unitality, and time non-locality depend on the
system/environment Hamiltonian and the initial state of the environment. We
also considered a three-qubit example with a time-dependent interaction that
broke a dynamical symmetry.

The oscillatory character of the reduced dynamics demonstrates that there is a
continuous flow of information flow back and forth between system and
environment. For sufficiently symmetric Hamiltonians, special initial
environment states exist that support Markovian reduced dynamics, where the
information flow becomes one-way. While this is not surprising, it is a
reminder that the assumption of a Markovian master equations is not consistent
with typical interacting Hamiltonians. This is true in the cosmological
context as well Shandera_2018 ; Burgess:2022nwu .

In typical effective theories of open systems, then, we need some guidance for
how to parameterize the possible non-Makovian behavior. In the qubit model,
whether the flow of information between system and environment was periodic or
aperiodic depended on the degree of symmetry present in the Hamiltonian. There
was no reason to assume that any of the parameters in the Hamiltonian take
related values (for example, commensurate frequencies in the
system/environment free Hamiltonians). In some cosmological contexts, however,
there are effects that connect parameter values. For example, in inflation all
modes undergo a common squeezing driven by the background expansion. This
squeezing can determine a significant part of the time-dependence in the
dissipative coefficients $\gamma(t)$ Shandera_2018 , including the
periodicity. The background expansion may suggest the appropriate scale to use
in an effective open model.

A further wrinkle in open effective theories is that the master equations need
not be local in time. Most constructions in cosmology restrict to time-local
equations for simlicity Agon2018 ; Shandera_2018 ; Brahma:2022yxu ;
Colas:2022hlq  although work in the context of black holes has shown that non-
time-local evolution can be required Burgess:2021luo . Our work here explores
the conditions necessary for time-local dynamics in small qubit models. We
find that even in the regions of parameter space with the largest degrees of
symmetry, it can be necessary to use a time-non-local master equation. We make
this determination based on the non-invertibility of the dynamical map, which
depends both on the Hamiltonian and the initial environment state. Strong
coupling is a necessary condition for non-invertibility, however so is a
rather restricted set of initial environment states. Those states are defined
largely by the dynamics of the free Hamiltonian for the environment. Diagonal
initial environment states in the eigenbasis of $H_{\rm E;\rm free}$ are the
most non-local, as their dynamical maps are non-invertible in the presence of
strong coupling.

In cosmology, the Hamiltonian is generically time-dependent, so it is not
uncommon for the relevant (active) degrees of freedom and couplings to evolve
over time. We used a 3-qubit model to investigate how such a time-dependence
effects the nature of the dynamical map. We found that introducing a degree of
freedom that does not respect the symmetry of the initial interaction
drastically altered the reduced dynamics. The dynamical map became non-
invertible at all times after the activation of the spectator qubit. This
strongly suggests that the framework of open effective theories in cosmology
should include a time-non-local component.

###### Acknowledgements.

We thank Nishant Agarwal, Sarang Gopalakrishnan, Archana Kamal, and Sahin
Ozdemir for helpful discussions in the construction of this work. We further
would like to acknowledge the useful suggestions made by Unnati Akhouri,
Brenden Bowen, and Andrew Keefe. This work was supported by the National
Science Foundation under PHY-1719991.

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##  Appendix A Derivation of the maximally entangling parameter space

Following [43, 44], we find the essential non-local properties of $U$ by
changing to the Bell basis using the unitary operator,

$Q=\frac{1}{\sqrt{2}}\begin{bmatrix}1&0&0&i\\\ 0&i&1&0\\\ 0&i&-1&0\\\
1&0&0&-i\end{bmatrix}\,.$ (72)

The choice of Bell states is made so that the entanglement quadratic form
$\hat{Ent}$ maps to the identity matrix. The operator $\hat{Ent}$ is defined
as,

$DetI_{\rm
C}(|\psi\rangle)=\psi_{00}\psi_{11}-\psi_{01}\psi_{10}=|\psi\rangle^{T}\hat{Ent}|\psi\rangle\,,$
(73)

where $I_{\rm C}$ is the Choi isomorphism $I_{\rm C}:\mathcal{H}\rightarrow
Mat(2,\mathbb{C})$, and one sees upon further inspection that
$\hat{Ent}=-\frac{1}{2}Y_{\rm S}\otimes Y_{\rm E}$ and
$Q^{T}(\hat{Ent})Q=\frac{1}{2}\mathbb{1}_{\rm S}\otimes\mathbb{1}_{\rm E}$. At
the level of observables $Q$ takes the subalgebra of local observables
$\mathfrak{su}(2)_{\rm S}\oplus\mathfrak{su}(2)_{\rm E}$ to the generators of
4D rotations on the Bell space $\mathfrak{so}(4)_{\rm Bell}$.

The time evolution operator is expressed in the Bell basis as,

$U_{\rm
Bell}(t)=\begin{bmatrix}\cos\omega_{+}t&0&0&e^{i\phi_{+}}\sin\omega_{+}t\\\
0&\cos\omega_{-}t&-e^{i\phi_{-}}\sin\omega_{-}t&0\\\
0&e^{-i\phi_{-}}\sin\omega_{-}t&\cos\omega_{-}t&0\\\
-e^{-i\phi_{+}}\sin\omega_{+}t&0&0&\cos\omega_{+}t\end{bmatrix}$ (74)

The non-local properties of the time evolution in the reduction frame are
determined by the eigenvalues of $U_{\rm Bell}^{T}U_{\rm Bell}$ found to be,

$\begin{split}&u_{+}=\cos^{2}\omega_{+}t+(\cos
2\phi_{+}+i\sqrt{\sin^{2}2\phi_{+}+\sin^{2}\phi_{+}})\sin^{2}\omega_{+}t\\\
&u_{-}=\cos^{2}\omega_{+}t+(\cos
2\phi_{+}-i\sqrt{\sin^{2}2\phi_{+}+\sin^{2}\phi_{+}})\sin^{2}\omega_{+}t\\\
&v_{+}=\cos^{2}\omega_{-}t+(\cos
2\phi_{-}+i\sqrt{\sin^{2}2\phi_{-}+\sin^{2}\phi_{-}})\sin^{2}\omega_{-}t\\\
&v_{-}=\cos^{2}\omega_{-}t+(\cos
2\phi_{-}-i\sqrt{\sin^{2}2\phi_{-}+\sin^{2}\phi_{-}})\sin^{2}\omega_{-}t\end{split}\,.$
(75)

Assume we are given a linear combination of these eigenvalues
$au_{+}+bu_{-}+cv_{+}+dv_{-}$ such that $a,b,c,d\geq 0$ and $a+b+c+d=1$. For
this combination to be real we must have $a=b$ and $c=d$. Setting
$a=\cos^{2}\varphi$ and $c=\sin^{2}\varphi$ we have,

$\begin{split}0&=\cos^{2}\varphi(\cos^{2}\omega_{+}t+\cos
2\phi_{+}\sin^{2}\omega_{+}t)+\sin^{2}\varphi(\cos^{2}\omega_{-}t+\cos
2\phi_{-}\sin^{2}\omega_{-}t)\\\
&=\cos^{2}\varphi(|\alpha_{+}|^{2}(t)-\beta^{2}_{+}(t))+\sin^{2}\varphi(|\alpha_{-}|^{2}(t)-\beta^{2}_{-}(t))\end{split}$
(76)

A little bit of algebra and we find,

$\cos^{2}\varphi\beta_{+}^{2}(t)+\sin^{2}\varphi\beta_{-}^{2}(t)=\frac{1}{2}\,.$
(77)

This condition may only be satisfied iff the largest of $\beta^{2}_{+}(t)$ and
$\beta^{2}_{-}(t)$ is greater than or equal to $\frac{1}{2}$, or equivalently
Max$(\phi_{+},\phi_{-})\geq\frac{\pi}{4}$. That is maximally entangled states
may only be generated when $U$ has large enough off diagonal components, which
we see becomes one of the conditions that non-invertibilities appear in the
reduced dynamics.

##  Appendix B Divisibility in channels

In this appendix we establish the divisibility of the dynamical map family
$\mathcal{D}_{+}$. For simplicity we assume that $\vec{r}_{E}(0)=\vec{0}$
although the results derived apply even if $\vec{r}_{\rm E}(0)=x_{\rm
E}(0)\hat{x}$. We are interested in when the map $\Phi(\tau_{2},\tau_{1})$
fails to be completely positive, and how this depends on $\tau_{2}$ and
$\phi$. To that end, the dynamical map has the structure,

$\Lambda(t)=\begin{bmatrix}1&0&0&0\\\ 0&\Lambda_{xx}(t)&\Lambda_{xy}(t)&0\\\
0&\Lambda_{yx}(t)&\Lambda_{yy}(t)&0\\\ 0&0&0&\Lambda_{zz}(t)\end{bmatrix}$
(78)

with determinant Det$\Lambda=\Lambda^{2}_{zz}$ and inverse,

$\Lambda^{-1}(t)=\begin{bmatrix}1&0&0&0\\\
0&\frac{\Lambda_{yy}(t)}{\Lambda_{zz}(t)}&-\frac{\Lambda_{xy}(t)}{\Lambda_{zz}(t)}&0\\\
0&-\frac{\Lambda_{yx}(t)}{\Lambda_{zz}(t)}&\frac{\Lambda_{xx}(t)}{\Lambda_{zz}(t)}&0\\\
0&0&0&\frac{1}{\Lambda_{zz}(t)}\end{bmatrix}\,,$ (79)

and the (possibly singular) interweaving map is,

$\Phi(\tau_{2},\tau_{1})=\begin{bmatrix}1&0&0&0\\\
0&\frac{\Lambda_{xx}(\tau_{2})\Lambda_{yy}(\tau_{1})-\Lambda_{xy}(\tau_{2})\Lambda_{yx}(\tau_{1})}{\Lambda_{zz}(\tau_{1})}&\frac{-\Lambda_{xx}(\tau_{2})\Lambda_{xy}(\tau_{1})+\Lambda_{xy}(\tau_{2})\Lambda_{xx}(\tau_{1})}{\Lambda_{zz}(\tau_{1})}&0\\\
0&\frac{\Lambda_{yx}(\tau_{2})\Lambda_{yy}(\tau_{1})-\Lambda_{yy}(\tau_{2})\Lambda_{yx}(\tau_{1})}{\Lambda_{zz}(\tau_{1})}&\frac{\Lambda_{yy}(\tau_{2})\Lambda_{xx}(\tau_{1})-\Lambda_{yx}(\tau_{2})\Lambda_{xy}(\tau_{1})}{\Lambda_{zz}(\tau_{1})}&0\\\
0&0&0&\frac{\Lambda_{zz}(\tau_{2})}{\Lambda_{zz}(\tau_{1})}\end{bmatrix}\,.$
(80)

N.B. that the map $\Phi(\tau_{2},\tau_{1})$ can have a restricted domain,
where instead of the Bloch ball the interweaving map only acts on the image of
$\Lambda(\tau_{1})$.

The criterion in [21] asserts that $\Phi(\tau_{2},\tau_{1})$ is completely
positive if,

$\begin{split}&\left[\frac{\Lambda_{xx}(\tau_{2})\Lambda_{yy}(\tau_{1})-\Lambda_{xy}(\tau_{2})\Lambda_{yx}(\tau_{1})}{\Lambda_{zz}(\tau_{1})}\pm\frac{\Lambda_{yy}(\tau_{2})\Lambda_{xx}(\tau_{1})-\Lambda_{yx}(\tau_{2})\Lambda_{xy}(\tau_{1})}{\Lambda_{zz}(\tau_{1})}\right]^{2}\\\
\\\
&\leq\left[1\pm\frac{\Lambda_{zz}(\tau_{2})}{\Lambda_{zz}(\tau_{1})}\right]^{2}\,.\end{split}$
(81)

The + inequality is saturated at all times, so the components of $\Lambda$
satisfy the relation (recall $\Lambda_{yx}(t)=-\Lambda_{xy}(t)$),

$\Lambda_{xx}(\tau_{2})\Lambda_{yy}(\tau_{1})+2\Lambda_{xy}(\tau_{2})\Lambda_{xy}(\tau_{1})+\Lambda_{yy}(\tau_{2})\Lambda_{xx}(\tau_{1})=\Lambda_{zz}(\tau_{2})+\Lambda_{zz}(\tau_{1})\,.$
(82)

Thus $\Phi$ is completely positive if,

$(\Lambda_{xx}(\tau_{2})\Lambda_{yy}(\tau_{1})-\Lambda_{yy}(\tau_{2})\Lambda_{xx}(\tau_{1}))^{2}\leq(\Lambda_{zz}(\tau_{2})-\Lambda_{zz}(\tau_{1}))^{2}\,.$
(83)

We already see that if $\tau_{n}=\frac{n\pi}{2\omega}$, then
$\Phi(\tau_{n},\tau_{1})$ is CP as the above inequality reduces to a CP
condition satisfied by $\Lambda(\tau_{1})$. Therefore, special times
$\tau_{2}$ exist where the dynamics is CP divisible. This condition is not
dependent on what particular values are chosen for $\omega$ and $\phi$.
However for other values of $\tau_{2}$, the CP inequality will fail to be
satisfied for certain values of $\tau_{1}$. The size of this interval is not
dependent on $\omega$, but depends on $\phi$ and $\tau_{2}$.