Happy New Year everyone! I hope everyone has a great, peaceful and eventual year!
I decided to do a list at end of every month of my empties. I have seen a lot of other blogs doing it and I really like seeing what other bloggers use daily.
1. Brighten your eyes - lots of R&R means less puff. To fake it, fill a sink with cold water, 15 ice cubes and ⅓ cup witch hazel. Dunk your bare face five times, keeping eyes closed. It helps instantly firm under-eye skin.
2. Bronze up - a real tan has a red undertone, so choose a bronzer with a coppery tint. Blend it up to temples with a fluffy brush and dust a bit on eyelids.
3. Get flushed - nightly, massage a serum that contains stimulating ginseng or peppermint into bare skin. Both ingredients boost blood flow for a healthy, rest-looking face.
4. Go tanning - apply a tinted gradual self-tanner for body, then do an extra coat following contours of muscles to define. Layering the color makes your faux not look it.
5. Smooth your skin - mimic sand's exfoliating effect by running a pad with glycolic acid over rough spots post - shower.
This combination of exfoliating sugar and moisturizing honey will really smooth skin. Stir together 1 cup sugar, ½ coconut oil [or olive oil], the juice 1 lemon, and 1 tbsp honey.
According to the product's website, this night cream promises to "maximizes your body's own nighttime renewal process for powerful, visible results". The night cream is chocked full of lavender, chamomile, marigold, horse chestnut, peptides, Vitamin C, E and B-5. The potent night cream softens, refreshes, hydrates and reduces the appearance of lines and wrinkles. The cream goes on smoothly and absorbs quickly. But with a hefty $100 price tag and packed to the brim with good-for-you ingredients, you would think I would be wowed with the results. I was using this night cream during the winter to make my skin less dry. Except my face got drier and rough and using this night cream didn't make my skin feel or look any better.
The Philip B hair mask is packed with l-amino acid complex and Persian katira extract. The lightweight paraben free formula promises to leave your hair soft, silky, and shiny. The formula has Jello consistency and it was really hard to work into my hair and most of the product fell in clumps in the tub. The first time I used the hair masque, my hair became really sticky, dull looking and unmanageable. The second time I used the mask my hair felt soft, bouncy, and body. I had mixed in some of my regular conditioner to make the masque's consistency a bit smoother and used a shower cap to keep the treatment in place. I am guessing I just got a bad batch or bottle. I did read somewhere that with gel masque to coat wet or dry hair from roots to ends, leave on until it hardens, then rinse out. I guess I will try the next time, if I decided to buy this mask again.
I recently tried the different types of texture nail polishes and I thought I should try the Rock Candy Glazes. I thought these would be like the jelly nail polishes but Marie wasn't very jelly like. It was a sheer high glossy barely there lavender color. Even with multiple coats -- the color was still opaque. The nail polished dried hard and didn't have that "jelly" feeling but the annoying thing was that even after the polish dried it still left a mark when it got dinged. Overall, not my most favorite as it was syrupy to applied but I wouldn't mind trying the other colors in the Rock Candy collection.
Happy Holidays! I hope everyone has a great and peaceful holiday! I wish you and your family an amazing time.
I was recently asked to try out this palette. I love trying new brands so I jumped at trying this palette and since it was a holiday theme it is a win - win. The palette had four eyeshadow and one blush. When I swatched the colors I noticed that the eyeshadows were gritty and bit difficult to apply; unlike other makeup that goes on smoothly and blends easy. The colors were not highly pigmented and doesn't last very long. While the palette had a tutorial that did go in detail, the colors didn't match up in the tutorial and in the set. The palette contains a nice tutorial, it felt a bit lacking. It promoted using their brushes, but it didn't recommend following up the smokey eyes with mascara or any lip product. Everyone knows, to make a good smokey eye you need layers upon layers of mascara! Overall, this was mediocre set for the $45. The consistency and quality reminds me of those cheap drugstore gifts set they have around the holiday. If the price was lower [$10] it would be a good set for a teen or a novice makeup user. I don't see myself ever purchasing this again but I might give their other products a try.
Note: This code does not expire and is good for 10% off your entire order on www.studiogearcosmetics.com.


This year marks a century since the 1918 flu pandemic killed an estimated 9,000 New Zealanders, and to shine a light on what happened a group led by ancestry.com has just released an online register of the victims affected.
The list, made available on the Auckland Libraries heritage blog today, includes 5,000 of those who passed.
Ancestry.com content acquisition manager Jason Reeve says an estimated 2,000 Māori people were killed by the outbreak.
He says distrust following the conscriptions of WWI meant that Māori didn't want to visit hospitals.
One whānau tragically affected was the Mutu family who lived in a bush settlement in Piha.
The father John Mutu, who was an engine driver, and his wife passed away on November 8, 1918. Within a week four or five of their children also passed.
Another case was when a serviceman came home from serving in WWI.
Reeve says the pandemic attacked healthy young people, as well as the elderly.
Reeve says the information can help those looking to find out more information about their ancestors and whether or not they were caught up in the pandemic.
He says the group aims to have the full list of as many of the 9,000 victims live on Ancestry.com early next year.


text:
\section{Introduction}
    \label{sec:introduction}
Quantum phenomena are deemed to be the main ingredients of the next technological breakthroughs. Quantum correlations, quantum coherences and quantum non-Markovianity are the key resources
to outperform classical protocols in many tasks, within the contexts of, e.g.,
communication \cite{PhysRevLett.69.2881,Bylicka2014}, teleportation \cite{Laine2014}, cryptography \cite{Pirandola:20}, metrology \cite{RevModPhys.89.041003} and thermodynamics \cite{Binder2018}, thus providing the pillars for future progresses in quantum technology \cite{Acin2018a}.
Even though the developments of quantum theory already started at the beginning of the last century, a deep and thorough understanding of the above-mentioned features in view of their operational exploitation is still being developed \cite{Guhne2009a,Breuer2016a,Streltsov2017a}. This is why simple, controllable models play an important role to learn how to manipulate and control the quantum resources.

In this article, we will focus on the analysis of a Markov property in the quantum setting and on the description of a class of open quantum system dynamics featuring memory effects and allowing for a phenomenological treatment. The Markov property is a concept from the theory of classical stochastic processes, where a clear definition of Markov process can be introduced in terms of conditional probability distributions. This notion is connected with the memorylessness of the process, i.e. the fact that the future of the process is independent of its history. As stochastic processes are used to model
reality in many different fields of research, as finance, biology, chemistry and social science, this is a highly relevant and often recurring concept \cite{TN_libero_mab22026897}. Stochastic processes naturally appear
in the description of (open) classical systems where, at least in principle, the stochasticity
can be always traced back to the lack of knowledge on the underlying total Hamiltonian and the initial conditions \cite{TN_libero_mab22026897,VanKampen1992}.
The extension of the classical formalism to the theory of open quantum systems is not straightforward, due to the invasive nature of the quantum measurements. As a consequence, many different, nonequivalent definitions of quantum Markov process were introduced, all of them aimed to reveal the occurrence of memory effects in quantum evolutions. In this respect, the notion of memory in the quantum realm still calls for a full physical interpretation. Some hints in this direction come from the framework of quantum thermodynamics, where, e.g., the connection between non-Markovianity and irreversible entropy production has been explored \cite{Popovic2018a,PhysRevE.99.012120}.

We will point out how the class of quantum renewal processes can be used as a phenomenological
tool
to describe dynamics with different non-Markovian behaviours. Our study complements other approaches, whose starting point is rather a microscopic description specifying a reference total Hamiltonian. In particular, strategies aimed at controlling the non-Markovianity
of the dynamics
have explored the manipulation of the system-environmental coupling \cite{Ma2014a,Man2015a, Man2015b}, or the modification of
the reduced system itself \cite{Brito2015a,Franco2015a}. The possibility of delaying the
occurrence of non-Markovianity \cite{Burgarth2021a}, and enhancing it by means of feedback control \cite{Zong2020a} has been also investigated.

 The existence of an underlying microscopical description of the evolution ensures that the reduced dynamics is indeed physical, i.e. the corresponding dynamical map $\Lambda_t$ which maps the initial reduced density operator $\rho(0)$ to a density operator at later time $t$: $\rho(t)=\Lambda_t[\rho(0)]$, is completely positive and trace preserving (CPTP)\footnote{This follows from the assumption that initially the reduced system and its environment are not correlated, i.e. the initial total state is factorised: $\rho_{tot}(0)=\rho(0)\otimes \rho_E(0)$.}. The density operator
 yields the probability distributions in quantum physics, so that trace preservation of the dynamics keeps the correct normalisation of the probabilities, while complete positivity takes into account the possible entanglement of the system state with other, otherwise not involved, degrees of freedom \cite{breuerbook},
 ensuring that joint probabilities are properly defined. On the other hand, if one chooses a more phenomenological approach and fixes the form of the dynamical map or of the corresponding evolution equation for $\rho(t)$, the so-called master equation,

the CPTP property of $\Lambda_t$ needs to be warranted. For Markovian semigroups this issue is well under control, i.e. one can specify the general structure of the corresponding master equation, of the so-called Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) form \cite{Gorini1976,lindblad1976,PhysRevA.89.042120}, which describes any proper quantum evolution obeying a semigroup composition law.
Its generalisation to the case of the so-called CP-divisible dynamics\footnote{The dynamical map $\Lambda_t$ is CP-divisible if the map $\Lambda_{t,s}$, defined as $\Lambda_t=\Lambda_{t,s}\Lambda_s$ is CPTP for all $0\leq s\leq t$.} obeying a more general composition law is also known \cite{Breuer2012a}.
However, a comparable result is still missing for general non-Markovian evolutions, although the topic has attracted a lot of interest \cite{PhysRevA.69.042107,PhysRevLett.101.140402,PhysRevA.70.010304,PhysRevA.71.020101,PhysRevLett.117.230401,PhysRevA.94.020103,Bassano2020,doi:10.1063/5.0036620,PhysRevA.91.042105,PhysRevA.75.022103,refId0,PhysRevE.79.041147,10.1007/978-3-030-24748-5_8,kossakowski2008}.
Remarkably, in the case of quantum renewal processes that we analyse here CPTP of the dynamical map is guaranteed by construction. This makes this class of open quantum system dynamics a valuable
tool for the phenomenological description of reduced dynamics.
In addition, despite their simplicity, quantum renewal processes can show a wide range of non-Markovian behaviours, which we will analyse in details in the following.

The rest of the article is organized as follows. In Sect.~\ref{sec:nm} we introduce the concept of non-Markovianity for stochastic processes. After this, we describe a possible definition of quantum non-Markovianity based on the monotonicity of the trace distance between two reduced states, which we adopt in the whole article. In Sect.~\ref{sec:qrp} we continue with the presentation of the renewal processes in the classical and the quantum domain, while
Sect.~\ref{sec:qnmRP} is devoted to the trajectory picture of the reduced dynamics and to the description of the different
elements that influence the non-Markovianity of the quantum renewal process: the time-continuous part of the dynamics, the type of jumps and the waiting time distributions governing the whole stochasticity of the jumps' times. In Sect.~\ref{sec:nmqrp}
we analyse the impact on the non-Markovianity measure and, more generally, on the main features of the trace distance evolution, such as the number and instants of its revivals. Finally, we summarise our findings in the last Sect.~\ref{sec:concl}.

\section{Memory effects in quantum dynamics}\label{sec:nm}

We say that a stochastic process $X(t)$, $t\geq 0$, taking values in a discrete set $\{x_i\}_{i\in \mathbb{N}}$ is Markov if the corresponding conditional probability distributions satisfy for any finite $n$ the following inequalities
\begin{align}
p_{1|n}(x_{n+1},t_{n+1}|x_{n},t_{n};\ldots;x_{1},t_{1})=p_{1|1}(x_{n+1},t_{n+1}|x_{n},t_{n}),
\end{align}
where the times are ordered: $t_{n+1} \geq t_{n} \geq \ldots \geq t_1\geq 0 $,
i.e.,
once we know the value $x_n$ of the stochastic process at time $t_n$, the past history prior to $t_n$
does not affect the predictions about the value of the process at any future time $t_{n+1}$.
Due to the invasive nature of quantum measurements, the extension of this definition to the quantum regime is not straightforward \cite{Vacchini_2011} and many different, non-equivalent definitions of quantum Markovianity have been introduced \cite{Rivas_2014,RevModPhys.88.021002,LI20181}. In most of them Markovianity is a property of the dynamical map $\Lambda_t$ itself, such as
(C)P-divisibility \cite{Wolf2008a,PhysRevLett.105.050403,PhysRevA.89.042120}, the change of the volume of accessible states \cite{PhysRevA.88.020102} and monotonicity of the trace distance as a quantifier of state distinguishability \cite{PhysRevLett.103.210401, PhysRevA.81.062115};
the latter is the one we will adopt here. On the other hand, other approaches, such as the process matrix formalism \cite{PhysRevA.97.012127,Giarmatzi2021a},
ground the notion of quantum Markovianity on conditional probabilities
associated with sequences of measurements,
going beyond the single-time description of the open system dynamics and calling for multi-time correlations.

The definition of non-Markovianity we use here is based on the change of
distinguishability between system states, quantified in the original paper \cite{BLP} by means of the
trace distance between two reduced states in the course of the evolution. The trace distance between two quantum states $\rho$, $\sigma$ is defined as
\begin{align}
\mathcal{D}(\rho,\sigma)=\frac{1}{2}\text{Tr}|\rho-\sigma|=\frac{1}{2}\sum\limits_i |v_i|,
\end{align}
where $v_i$ are the eigenvalues of the operator $\rho-\sigma$.
The quantum dynamics fixed by the map $\Lambda_t$
is non-Markovian if and only if the trace distance is not a monotonous function of time, i.e.,
there exist a couple of initial states $\rho(0)$ and $\sigma(0)$ and a time $t>0$ for which
\begin{align}\label{eq:NM}
\frac{d}{dt}\mathcal{D}(\rho(t),\sigma(t))>0,
\end{align}
where $\rho(t)=\Lambda_t\rho(0)$ and $\sigma(t)=\Lambda_t\sigma(0)$.
 \begin{center}
 \minipage{1.0\textwidth}%

    \includegraphics[width=\columnwidth]{Figures/InfFig2}
   \endminipage
  \captionof{figure}{Sketch of the information backflow in an open quantum system dynamics, which is at the basis
   of the notion of quantum non-Markovianity used in this paper: initially the reduced states
    $\rho$, $\sigma$
    approach each other since the information is flowing out of
    the reduced system, to the environment or to the correlations between the system and the environment (left); on the other hand, an
    information backflow makes the two states
    diverge from each other at a later time (right), as can be witnessed
    via proper state distinguishability quantifiers}.
   \label{fig:InfFlow}
     \end{center}
Importantly, since the trace distance is contractive under the action of any (C)PTP map $\phi$:
\begin{align}
\mathcal{D}(\phi(\rho),\phi(\sigma))<\mathcal{D}(\rho,\sigma),
\end{align}
(C)P-divisibility \cite{Wolf2008a,PhysRevLett.105.050403,PhysRevA.89.042120,HelstromPdiv} implies monotonicity of the trace distance and thus Markovianity according to the definition above, while the converse does not hold \cite{PhysRevA.81.062115}.
The trace-distance based definition of non-Markovianity provides a clear-cut interpretation in terms of the information flow between the open quantum system and the environment as the key element possibly leading to the occurrence of memory effects in the dynamics.
In addition, this picture allows us to trace back the exchange of information between the open system
and the environment to the correlations established by their mutual interaction \cite{Laine2010b,Mazzola2012,Smirne2013,Cialdi2014,Campbell_2019,Smirne2021}, see Fig.~\ref{fig:InfFlow}. Initially the whole information is contained in the reduced system, however, due to the system-environment interaction, some information gets transferred to external degrees of freedom in the course of the evolution. Such information can be stored both in the environment and in the system-environmental correlations. In Markovian dynamics the information flow is unidirectional, i.e. the information is always flowing from the open system to the outside world and any couple of reduced states get closer and closer with the passing of time. On the other hand, for non-Markovian evolutions some information backflow occurs, which is witnessed by an
increase of the distance between pairs of reduced states on certain intervals of time.
Let us stress that this viewpoint was recently strengthened, as it was shown that also different distinguishability measures between two quantum states, including entropic quantities, can be used to quantify the information flow; it appears in particular that the quantum Jensen-Shannon divergence is a natural entropic quantifier of information backflow \cite{megier2021entropic}. Additionally, a connection between monotonic contractivity of a generalisation of the trace distance and P-divisibility exists \cite{Chruscinski2011a,HelstromPdiv}, providing a common background to these approaches to non-Markovianity,
which however goes beyond the scope of this work.

Relying on the trace distance, it is then possible to define a measure of the degree of non-Markovianity
of a quantum dynamics. The idea is to integrate over all the revivals of the trace distance over the duration of the
dynamics, i.e., to quantify the overall amount of information flown back to the reduced system.
In addition, since we want the non-Markovianity measure to be a property of the dynamical map, while
the change of the trace distance, Eq.~\eqref{eq:NM}, will generally depend on the chosen initial
states $\rho(0)$ and $\sigma(0)$,
the non-Markovianity measure involves an optimisation over all the possible couples of initial states \cite{PhysRevLett.103.210401}:
\begin{align}\label{eq:NMm}
\mathcal{N}= \max\limits_{\rho(0), \sigma(0)} \int\limits_{d\mathcal{D}(\rho(s),\sigma(s))/ds>0} \frac{d}{ds}\mathcal{D}(\rho(s),\sigma(s))ds.
\end{align}
It was shown in \cite{PhysRevA.86.062108} that the optimal pair of states, i.e. the one achieving the maximum in the non-Markovianity measure, lies on the boundary of the states space and is made of orthogonal states. In particular, for qubit states this means that the optimal pair consists of pure states that can be represented as a pair of antipodal points on the Bloch sphere.

\section{Renewal processes: classical and quantum}\label{sec:qrp}

Here we investigate a class of open quantum system dynamics, quantum renewal processes, which are a generalisation of a classical concept. Firstly, we briefly review semi-Markov processes, of which renewal processes are a subset, and then provide a formulation of the relevant notions in the quantum realm
\cite{Feller1968, Ross2003}.

As discussed in Sect.~\ref{sec:introduction}, the characterization of
a Markovian time evolution is essentially fixed by the GKSL theorem,
determining the expression of the generator of the dynamics. An
equivalent result for an arbitrary dynamics featuring non-Markovian
effects is not known, and only very specific results have been obtained. The
main difficulty lies in providing evolution equations whose solutions
are indeed CPTP maps. These so-called master equations can be recast in two
forms, either time local, i.e. with the functional expression \cite{breuerbook}
\begin{equation}
  \label{eq:1tcl}
  \frac{d}{dt}\rho(t)=\mathcal{L}(t)[\rho(t)],
\end{equation}
or time non-local, that is in the form \cite{nakajima,zwanzig}
\begin{equation}
  \label{eq:1nz}
  \frac{d}{dt}\rho(t)=\int\limits_{0}^t \mathcal{K}(t-s)[\rho(s)].
\end{equation}
The superoperators $\mathcal{L}(t)$ and
$\mathcal{K}(t)$ are generally related
\cite{Chruscinski2010a,Megier2020b,e22070796,Nestmann2021a},  though in a highly
non-trivial way. Moreover general conditions on their expression
warranting CPTP are not known, except for special cases. In this contribution we make reference to a large class of
well-defined evolutions obtained building on an analogy with
classical non-Markovian stochastic processes.

A semi-Markov process is a  continuous time random jump process, for which the jump probabilities are possibly site dependent but independent from each other. The probability distribution of the time between the jumps is called waiting time distribution (WTD) and provides a probability density over the positive real line
\begin{equation}
  \label{eq:1}
  f(s)\geq 0 \qquad \int\limits_0^\infty ds f(s)=1.
\end{equation} If the WTD is exponentially distributed, then the semi-Markov process reduces to a continuous time Markov chain. Otherwise, for general distributions, the memory about the time already spent in the state affects the subsequent statistics of the process, which is then non-Markovian.
The transition probabilities $T_{nm}(t)$ from the state $m$ to the state $n$ obey the equation \cite{GILLESPIE197722}
\begin{align}
   \label{eq:trans}
 \frac{d}{dt} T_{mn} (t) = \int_0^t d\tau \sum_k
 \Big[ W_{mk} (\tau) T_{kn} (t-\tau)
 - W_{km} (\tau)T_{mn} (t-\tau) \Big],
\end{align}
where the matrices $W_{mk}(t)$ are fixed by the (possibly state-dependent) WTDs $f_i(t)$,
along with the corresponding survival probabilities $g_i(t)$ defined by
\begin{align}
g_i(t)=1-\int\limits_0^t ds f_i(s),
\end{align}
and the semi-Markov matrix $\underline{\Pi}$ whose entries are the jump probabilities between sites;
in particular, denoting as ${\tilde x}(u)$ the Laplace transform of $x(t)$, one has
$\tilde{W}_{mk}(u) = { \Pi }_{mk} {\tilde f}_k(u){\tilde g}^{-1}_k(u)$.
Moreover, note that a semi-Markov process can also be seen as the merging of a renewal process and a Markovian jump process. In a renewal process, the events, here the transitions among states, occur randomly in time and the time intervals between successive events are independent. Accordingly, the evolution depends only on the current site and the time elapsed since arriving at it.
In the case of the standard renewal process all waiting times are identical, while for a so-called modified process the first $k$ waiting time distributions are different.

The notion of \textit{trajectory} is one of the basic concepts in the description of classical stochastic processes.
Indeed, in abstract terms a stochastic process can always be characterized by a suitable measure over a sample space of trajectories. Recovering a notion of trajectory is less straightforward in the quantum case, where the object of interest is the reduced density matrix $\rho(t)$,
but this can actually be done in the context of open quantum systems \cite{Barchielli_1991,breuerbook}. More specifically, the dynamics we are considering allow for an interpretation in terms of an average over trajectories in the space of operators. All the trajectories start in the same initial state $\rho(0)$, and then in each trajectory the times at which the system state undergoes discontinuous changes, the so-called \textit{jumps}, are random variables. Accordingly, the reduced density operator of the open quantum system can be obtained by a weighted sum of all possible trajectories, corresponding to fixed jump times. We will see that this point of view also helps us understand the dependence of the non-Markovianity on the specific parameters.

Quantum renewal processes are a subclass of quantum semi-Markov processes \cite{breuer2008,Vacchini_2011,Chruscinski2016a,PhysRevLett.117.230401,e22070796}, for which the time evolution reads
\begin{eqnarray}\label{eq:semiMarkGen}
\rho(t)= p_0(t) \mathcal{F}_0(t) \rho(0)+  \sum\limits_{n=0}^\infty \int\limits_0^t dt_n\ldots \int\limits_0^{t_2} dt_1  p_n(t; t_n,\ldots,t_1) \mathcal{F}_n(t-t_n)\mathcal{E}_n \ldots.\mathcal{F}_2(t_2-t_1) \mathcal{E}_1 \mathcal{F}_1(t_1) \rho(0),
\end{eqnarray}
where the CPTP maps $\mathcal{E}_n$ describe the jumps, while the CPTP maps $\mathcal{F}_n(t)$ give the time-continuous evolutions between the jumps, and
$p_n(t; t_n;\ldots,t_1)dt_n \ldots dt_1$ is the probability that the jumps occur (solely) around fixed times $t_1,\ldots,t_n$. Note the close analogy to the classical description recalled above. In the case of the standard process the jump times are independent and identically distributed, i.e. each waiting time has the same probability distribution and they are all mutually independent. In a modified process, instead, the probability distributions for the first jumps can differ from each other and the following ones.

To obtain the quantum renewal processes from the general quantum semi-Markov processes one fixes the time evolution between the jumps to be of GKSL form \cite{Vacchini2020a}. What is more, one also introduces only two kinds of jumps: anterior $\mathcal{J}$ and subsequent $\mathcal{E}$ with respect to the time continuous evolution. Consequently, in quantum renewal processes one focuses on the stochastic distribution of the jumps, as in the case of classical
renewal processes. Accordingly, we obtain the following form of the open quantum system density operator at time $t$:
\begin{eqnarray}\label{eq:RenGen}
\rho(t)= p_0(t) e^{\mathcal{L}t} \rho(0)+  \sum\limits_{n=0}^\infty \int\limits_0^t dt_n\ldots \int\limits_0^{t_2} dt_1  p_n(t; t_n,\ldots,t_1)   e^{\mathcal{M}(t-t_n)} \mathcal{E} e^{\mathcal{L}(t_n-t_{n-1})} \mathcal{J}\ldots\mathcal{E} e^{\mathcal{L}t_1} \mathcal{J}\rho(0).
\end{eqnarray}
Here, we use a ``left-ordering'', as explained in \cite{PhysRevLett.117.230401}, since a particular ordering of the operators has to be chosen in order to construct the quantum evolution from the classical counterpart. We also set in the following $\mathcal{M}=\mathcal{L}$ (the time continuous evolution is always the same) and $\mathcal{J}=\id$. With this, the above mentioned trajectories correspond to the dynamical maps $e^{\mathcal{L}(t-t_n)} \mathcal{E} e^{\mathcal{L}(t_n-t_{n-1})}\ldots\mathcal{E} e^{\mathcal{L}t_1}$, which contribute to the overall evolution with weights $p_n(t; t_n,\ldots,t_1)dt_n \ldots dt_1$.

For the standard quantum renewal process the same WTD $f(t)$ governs the whole stochasticity of the jumps' times,
\begin{eqnarray*}
p_n(t; t_n,\ldots,t_1)=g(t-t_n)\ldots f(t_2-t_1)f(t_1),
\end{eqnarray*}
where $g(t)$ is the corresponding survival probability.
When the renewal process is modified the first $k$ WTDs can be different,
\begin{align}
p_n(t; t_n,\ldots,t_1)&=g_{n+1}(t-t_n)f_{n}(t_n-t_{n-1})\ldots f_1(t_1), && n \leq k\\
p_n(t; t_n,\ldots,t_1)&=g(t-t_n)\ldots f(t_{k+1}-t_{k}) f_k(t_k-t_{k-1})\ldots f_1(t_1), && n > k.
\end{align}

Here we will investigate how the non-Markovianity of the dynamics, in terms of the monotonicity of the trace distance, depends on the choice of the involved operators, describing intermediate continuous evolutions and jumps, as well as the chosen probability distribution for the jumps. We will observe a rich variety of possible behaviours and analyse the influence of particular parameters to control the strength, time of occurrence and precise manifestation of quantum non-Markovianity.

\section{Trajectory picture}\label{sec:qnmRP}

In general there exist infinitely many different decompositions of a reduced dynamics, i.e. different ways to
write the reduced density operator in the form
\begin{align}
\rho(t)=\sumint\limits_{i \in I} p_i(t) \rho_i(t),
\end{align}
where $I$ can be countable or uncountable set. In this representation the prefactors $p_i(t)$ can be interpreted as probabilities or probabilities densities, i.e. they are positive and normalized, and the operators $\rho_i(t)$ are themselves proper density operators, i.e. trace one and positive semi-definite.
If the operators can be obtained by the action of CPTP maps $\Lambda^i_t$ applied on the very same initial state $\rho(0)$, each $\rho_i(t)$ can be associated to a different trajectory, whose occurrence probability is indeed given by the corresponding $p_i(t)$.
There exist two main types of decompositions directly linked to a trajectory picture of the dynamics: time-continuous, as exemplary quantum state diffusion \cite{TN_libero_mab21458774, PhysRevA.58.1699,PhysRevLett.120.150402}, and so called jump unravelings \cite{PhysRevLett.68.580,dariusz2020open}.
As recalled above, also quantum renewal processes have a direct decomposition in terms of trajectories, which are defined at the level of the density operators,
see in particular Eq.~\eqref{eq:RenGen}. Finally, note that an important question connected with the trajectory description of the reduced dynamics
is the existence of a continuous measurement interpretation associated with it \cite{Barchielli_1991,RevModPhys.70.101,PhysRevLett.100.080401,PhysRevLett.101.140401,PhysRevLett.124.190402,PhysRevResearch.2.043376}.

The construction of a particular trajectory can take place in two different ways. In the first method one firstly fixes the time interval $[0,T]$
of interest and then draws the jumps' times according to the WTDs. After each drawing if the sum of waiting times exceeds $T$ one terminates the process.
Then the generation of the trajectory is obtained by inserting the jumps at the given times. In the second method the generation of the trajectory and drawing of the jump times take place in parallel. The time interval $[0,T]$ is divided into small intervals of length $\Delta t$, and at each intermediate midpoint one determines randomly if the jump takes place or not, with the probability fixed by the corresponding waiting time distribution.
In this second approach, fixing the  time interval $[0,T]$ in advance is in principle not necessary as one can decide along the trajectory when to stop the evolution. Note, that for a modified renewal process only the first method is applicable for the case in which the last $k$ waiting time distributions are different, a situation which was introduced in \cite{Bassano2020} under the name of inverse time operator ordering. The same is true when the last time-continuous evolution is different from the preceding ones, $\mathcal{M}\neq \mathcal{L}$ in Eq.~\eqref{eq:RenGen}, or in processes starting with a jump rather than with a time continuous evolution, $\mathcal{J}\neq \id$ in Eq.~\eqref{eq:RenGen}. In all these situations one has to know beforehand, i.e. before one starts to generate the trajectory, how many jumps occur in the investigated time interval $[0,T]$,
to know which waiting time distribution or which time evolution has to be used to generate the trajectory at a particular point of time. In this paper, for simplicity, we restrict ourselves to cases where both methods to generate the trajectory can be implemented.  We will see that the trajectory point of view in describing the evolution let us better understand the influence of the particular parameters on the non-Markovianity of the corresponding dynamical map.

The quantum renewal processes, due to the non-trivial interplay between the operatorial and stochastic contributions, can manifest a wide range of non-Markovian behaviours. However, if one assumes that all WTDs coincide, i.e. the quantum renewal process
is unmodified, and are given by an exponential distribution
\begin{align}
f(t)=\mu e^{-\mu t},
\end{align}
where $\mu$ is the corresponding rate,
the issue simplifies significantly.
In this case, a simple connection between the WTD $f(t)$ and the associated survival probability $g(t)$ exists: $f(t)=\mu g(t)$. As shown in \cite{Vacchini2020a}, the corresponding memory kernel in the Laplace picture reads
\begin{align}
\tilde{\mathcal{K}}(u)= \mathcal{L}+ (\mathcal{E}  - 1) \tilde{f}(u-\mathcal{L}) \tilde{g}^{-1}(u-\mathcal{L}).
\end{align}
Accordingly, in time domain we obtain for this case
\begin{align}
\mathcal{K}(t)=\delta(t) [\mathcal{L}+\mu (\mathcal{E}  - 1)],
\end{align}
no matter what the generator $\mathcal{L}$ and the jump operator $\mathcal{E}$ are. This memory kernel corresponds to a quantum dynamical semigroup, and, accordingly, the underlying evolution is Markovian.

To go beyond this case, we analyse how the time continuous dynamics, type of jumps and waiting time distributions influence qualitatively and quantitatively the non-Markovianity of the corresponding process. We focus on qubit evolutions, so that the trace distance between two quantum states equals the half of the  Euclidian distance of these states when depicted on the Bloch ball. Recall that any qubit state can be written as
\begin{align}
\rho=\frac{1}{2}(\mathbbm{1}+\vec{r} \cdot\vec{\sigma}),
\end{align}
with the vector $\vec{\sigma}$ consisting of the Pauli matrices, $\vec{\sigma}^T=(\sigma_1,\sigma_2,\sigma_3)$,
and $\vec{r}^T=(x,y,z)$ defining the Bloch vector representation of the state $\rho$.
Accordingly, the trace distance between two qubit states evolving via a quantum renewal process can be calculated as
\begin{align}
\label{td}
\mathcal{D}(\rho^1(t),\rho^2(t))=\frac{1}{2} \lim\limits_{N\rightarrow \infty}\sqrt{\left(\frac{1}{N}\sum\limits_{n=1}^N \Delta^x_n(t)\right)^2 + \left(\frac{1}{N}\sum\limits_{n=1}^N \Delta^y_n(t)\right)^2 +\left(\frac{1}{N}\sum\limits_{n=1}^N \Delta^z_n(t)\right)^2 },
\end{align}
where the sums are running over realisations of the stochastic process governed by the associated WTDs, and $\Delta^i_n(t)$ corresponds to difference of the $i$-coordinates in $n$-th realisation, e.g.
\begin{align}
\label{coord}
\Delta^x_n(t) = x^1_n(t)-  x^2_n(t),
\end{align}
which we call an $x$-trajectory. {Consequently, the trace distance between two states is not an average trace distance between the corresponding random trajectories and calculating the trace distance has to occur after generating the whole set of trajectories.
}
Note that to have non-monotonicity in the trace distance, a non-monotonicity of the absolute value of at least one of the coordinates is necessary. This is the case not only when one of the coordinates is non-monotonous, but also when it changes its sign. This can only happen when some realisations of the trajectories include a sign change. This is however not a sufficient condition, as we will see in the following.

We now set the different elements of the quantum renewal processes fixing the resulting trajectories and average dynamics.

\subsection{Intermediate evolutions}

We choose the time continuous evolution to be unital
\begin{align}
\label{L}
\mathcal{L}[\rho]=\sum \limits_{k=1}^3 \frac{1}{2}\gamma_k (\sigma_k \rho \sigma_k - \rho),
\end{align}
with $\gamma_j \geq 0$ and
\begin{align}
\label{lambda}
e^{\mathcal{L}t}[\sigma_i]=e^{-t\lambda_i} \sigma_i  &&\lambda_i=\gamma_j + \gamma_k, && \text{for } i \neq j \neq k.
\end{align}
Choosing a unital dynamical map does not affect the trace distance measure of non-Markovianity, which is anyhow insensitive to translations of the Bloch sphere \cite{Rivas_2014,PhysRevA.87.042103,megier2021entropic}. As the time-continuous evolution introduced above describes a monotonic contraction of the Bloch sphere, we do not expect that it introduces any memory effects. Indeed, the dynamical map $e^{\mathcal{L}t}$ is not only Markovian according to the distinguishability criterion introduced in \cite{BLP}, but it is a CP-divisible semigroup.
We will see that a greater "strength" of this dephasing evolution - corresponding to larger values of the $\lambda$'s introduced in Eq.~\eqref{lambda} - will result in smaller non-Markovianity of the associated quantum renewal process.

\subsection{Jumps}

As said above, the quantum non-Markovianity will not occur if for all realisations of the stochastic process the coordinates, Eq.~\eqref{coord}, are monotonic and do not change sign. Accordingly, a jump channel which only consists of a contraction (and possibly translation, which, however, cannot be detected by the trace distance condition - see comment above) will necessarily lead to a Markovian dynamics. An example of such a channel is the amplitude damping (AD) channel $\mathcal{E}_{\mathrm{AD}}$, with Kraus operators
\begin{align}
K_0= \begin{pmatrix}
1  & 0\\
0 & \sqrt{1-\gamma}
\end{pmatrix},
&&
K_1= \begin{pmatrix}
0  & \sqrt{\gamma}\\
0 & 0
\end{pmatrix},
\end{align}
which shrinks the Bloch ball and translates it along the z-axes by
factors determined by the decay rate $\gamma$.
Consequently, no non-Markovianity is detected, no matter what probability distribution drives the stochasticity of the jumps' times. In particular, also for a choice of classically non-Markovian waiting time distributions, as Erlang distributions {introduced later,} one still obtains Markovian evolution, according to the trace distance criterion.

Consequently, the next step is to choose a jump channel that results in changing the sign of the trajectories. We have chosen the $x$-Pauli channel composed with AD:
$$\mathcal{E}_{x-\mathrm{AD}}=\mathcal{E}_{x}\circ \mathcal{E}_{\mathrm{AD}},$$
with
\begin{align*}
\mathcal{E}_{x}[\rho]= \sigma_x \rho \sigma_x.
\end{align*}
The Pauli channels describe a $\pi$ rotation about the corresponding axis and in particular we focus here on the composition of the AD channel with the $x$-PC. {For this jump channel we will, indeed, manage to detect non-Markovianity, depending on the choice of parameters determining the dynamics.}

{Note, that, as the superoperators $\mathcal{E}_{x}$ and $\mathcal{E}_{\mathrm{AD}}$ do not commute, the jump channels $\mathcal{E}_{x-\mathrm{AD}}$ and $\mathcal{E}_{\mathrm{AD}-x}=\mathcal{E}_{\mathrm{AD}} \circ\mathcal{E}_{x}$ are different. Generally speaking, the latter possibility leads to a slightly greater non-Markovianity measure, as the jumps occur before the disruptive AD channel. Nonetheless, the qualitative behaviour for both of the choices is similar, and for simplicity we restrict here to $\mathcal{E}_{x-\mathrm{AD}}$.}

\subsection{Waiting time distributions}

{As noticed earlier, when the underlying WTDs are exponentials and the process is unmodified, i.e. all WTDs are the same, the evolution is Markovian, independently of the choice of the jump channel. This is the case even if the trajectories are non-monotonic and the sign changes take place, so, in particular, for the channel $\mathcal{E}_{x-\mathrm{AD}}$ investigated by us. However, the situation drastically changes if we allow for a modified quantum renewal process. Even if all the WTDs are exponentials, but the first $k$-th of them have different rates, we can observe a high variety of different behaviours. In particular, the number of revivals strongly depends on the choice of the rates.}

There is, however, no need to restrict our choice of WTDs to exponentials. To go beyond this case,
we also analyse the quantum renewal process dynamics where the stochasticity of the jumps is governed by the Erlang WTD, which reads in the Laplace domain \cite{cox}\footnote{In book \cite{cox} the Erlang distribution is called the special Erlangian distribution.}:
\begin{align}
\label{erlang}
\tilde{f}_r(u)=\left(\frac{\mu}{\mu + u }\right)^r,
\end{align}
from which one can see that it is the convolution of $r$ exponential
distributions with the same rate parameter $\mu$. The ratio
${r}/{\mu}$ fixes the mean waiting time
while the variance reads ${r}/{\mu^2}$. Accordingly, for the Erlang WTDs the mean value and the variance can be independently varied, as contrasted with the exponential WTD, where the mean waiting time $1/\mu$ fixes the variance.

We will see that in the case of Erlang WTDs even the unmodified process can lead to non-Markovianity.

   \begin{center}
   \minipage{0.45\textwidth}
             \includegraphics[align=c,width=\textwidth]{Figures/XPADOPa}
             \endminipage
         \hspace{1cm}
         \minipage{0.45\textwidth}
   \includegraphics[align=c,width=\textwidth]{Figures/XPADTR}
   \endminipage
   \captionof{figure}{
   Left: Value of the non-Markovianity
    measure of a quantum renewal process as given in Eq.~\eqref{eq:NMm} in the dependence on the choice of initial orthogonal pure states, identified by the extremes of a diameter in the Bloch sphere; it clearly appears that optimal pairs lie on a vertical equator. Right: $y$-component for the trajectory in a particular realization of the process corresponding to an initial  optimal pair $\ket{\phi_{1/2}}=\frac{1}{\sqrt{2}}(\ket{0} \pm i\ket{1})$. We are here considering a $\mathcal{E}_{x-\mathrm{AD}}$ jump channel and parameters $\gamma=0.3$, $\mu=1$, $\mu_1=10$, $\lambda_{1}=\lambda_{2}=\lambda_{3}= 0.9$. Here and in the following we work in arbitrary units.
}
  \label{Fig:2}
 \end{center}

\section{Non-Markovianity of quantum renewal processes}\label{sec:nmqrp}

As mentioned in Sect.~\ref{sec:nm}, occurrence and strength of
memory effects depend on the chosen pair of initial states.
This is clarified in Fig.~\ref{Fig:2},
left, where the value of the non-Markovianity measure for the case of
the jump operator {$\mathcal{E}_{x-\mathrm{AD}}$} is plotted as a
function of the direction identifying a pair of pure orthogonal
states, corresponding to points on the Bloch sphere. It clearly appears
that the maximum is attained for states $\ket{\phi_{1/2}}=\frac{1}{\sqrt{2}}(\ket{0} \pm i\ket{1})$. We will
therefore in the following consider always this pair of initial states
lying on the $y$-axes. Note that for this choice
$\Delta_x(t)=\Delta_z(t)=0$, corresponding to the fact that
the $x$ and $z$ components of the Bloch vector of the two evolving states remain equal, so that the only relevant parameter in
the continuous time evolution is the rate
$\lambda_2$. This behaviour is due to our particular choice of the jump channel, leading to a rotation about the $x$-axis. A typical trajectory of the $y$-component
of the Bloch vector is depicted in
Fig.~\ref{Fig:2}, right, characterized by sign changes
which determine possible non monotonicity of the trace distance obtained as in Eq.~\eqref{td}. In our analysis we will not only investigate the mere change of the non-Markovianity measure, but also the way the trace distance evolution is altered: number of revivals, times of revivals and other qualitative features.

\subsection{Exponential WTD - general results}
\label{sec:Exp1}

Here, we focus on the behaviour of the trace distance in the case of exponential WTDs. Accordingly, beside the dependence on the dephasing rate of the continuous time evolution
$\lambda_2$ and the decay rate $\gamma$ corresponding  to the strength of AD jumps, the non-Markovianity is also influenced by the rates $\mu_i$ fixing the exponential WTDs.

\begin{center}
  $\vcenter{\hbox{
  \includegraphics[width=.40\textwidth]{Figures/3WTDNMax1b}}}$
     $\vcenter{\hbox{   \includegraphics[width=.36\textwidth]{Figures/4WTDNMax1b}}}$
     \hspace{0.1cm}
        $\vcenter{\hbox{    \includegraphics[width=.12\textwidth]{Figures/Legend}}}$
      \captionof{figure}{The number of revivals of the trace distance for a quantum renewal process with $\mathcal{E}_{x}$ jump channel
      in its dependence on the value of the rates fixing the WTDs. The maximal number of revivals for the modified quantum renewal process with $k$ exponential WTDs equals $k-1$ (here we take $k=3$ and $k=4$ from left to right; $\lambda_2=0.9$, $\mu=1$ and $\mu_3=3$ (right panel) in arbitrary units). The white lines mark the boundaries between parameter regions corresponding to processes whose trajectories exhibit different number of jumps.}
  \label{Fig:3}
\end{center}

	\begin{center}
   \minipage{0.31\textwidth}
      \includegraphics[width=\textwidth]{Figures/Exp1b}
      \endminipage

       \minipage{0.31\textwidth}
      \includegraphics[width=\textwidth]{Figures/Exp2b}
      \endminipage

        \minipage{0.31\textwidth}
      \includegraphics[width=\textwidth]{Figures/Exp3b}
      \endminipage
    \captionof{figure}{
   The trace distance, testifying non-Markovianity when showing a non-monotonic behavior, for a quantum renewal process with exponential WTDs. In the left panel jumps are realized by means of a $\mathcal{E}_{x-\mathrm{AD}}$ jump channel, and one can appreaciate the reduction of the revivals for growing damping. In the middle panel jumps are given by $\mathcal{E}_{x-\mathrm{AD}}$ and stronger dephasing in the intermediate time evolution again suppresses non-Markovianity. The right panel, with jump operator $\mathcal{E}_{x}$, shows how a  larger  number  of  revivals  does  not  necessarily  lead  to  a  higher non-Markovianity measure. Across the panels $\lambda_2=0.9$, $\mu=3$ and $\mu_1=13$, apart from the last panel with $\mu_1=30$.}
  \label{Fig:4}
\end{center}

  \begin{center}
     \minipage{0.45\textwidth}
        \includegraphics[width=\textwidth]{Figures/Traj1a}
        \endminipage
    \hspace{1cm}
       \minipage{0.45\textwidth}
    \includegraphics[width=\textwidth]{Figures/Traj2a}
    \endminipage
  \captionof{figure}{
  Examples of jump trajectories with  parameters as in Fig.~\ref{Fig:4}, left and middle, respectively.
  In the left panel we vary the damping rate $\gamma$, while in the right panel we vary the decay strength $\lambda_2$ associated to the intermediate time evolution.}
  \label{Fig:5}
\end{center}

The number of revivals, i.e. time intervals where the trace distance
grows, strongly depends on the number of different WTDs and on the
corresponding rates. It can be
observed that for a process with $k$-WTDs the maximal number of
revivals is $k-1$ and can only be reached if the following relation
between the rates is satisfied:
\begin{align}
\label{relation}
\mu_1>\mu_2>\ldots>\mu_{k-1}>\mu.
\end{align}
This fact is investigated in Fig.~\ref{Fig:3}, where we report the number of revivals for a modified process with $\mathcal{E}_{x}$ jump channel and with 3 WTDs (left) or 4 WTDs (right) in dependence on the rate values. Note that throughout the manuscript we work in arbitrary units.
The different coloured areas correspond to different numbers of
revivals, clearly growing with the number of WTDs and depending on
the corresponding rates.
The presence of amplitude damping in the jump decreases the parameter range
corresponding to higher number of revivals. At the same time the AD reduces
the value of the non-Markovianity measure. This is put into evidence
in Fig.~\ref{Fig:4}, left, where the behaviour of the trace
distance is plotted together with the estimate for the associated
non-Markovianity measure, corresponding to the sum of the revival
heights. A similar effect is obtained by
increasing the strength of the dephasing rate
$\lambda_2$ describing the time
continuous dynamics, as shown in Fig.
\ref{Fig:4}, middle, where only {$\mathcal{E}_{x}$}
determines the jumps.

We further stress that a higher number of revivals does not
necessarily lead to a higher non-Markovianity measure, see Fig.
\ref{Fig:4}, right. Non-Markovianity is enhanced when the rate of the first WTD is much larger than the rate of the following one, {$\mu_1  \gg  \mu_2$}, allowing for a larger revival.
Subsequent rates play a less relevant role, since, on average, the dephasing has become more effective by the time the corresponding jump occurs.

The different role of $\gamma$ and
$\lambda_2$ is visible by comparing Fig.
\ref{Fig:4}, left and Fig.
\ref{Fig:4}, middle, noticing that only
$\gamma$ affects the value of the (first) revival time.
Their different influence at the level of the trajectories is visualised in Fig.
\ref{Fig:5}. As one can observe, an increase of the decay rate implies that the height of the jumps decreases, while it does not affect the previous time continuous dynamics. This is different in the case of varying $\lambda_2$, where both the extension and the starting point of the jumps is changed and the influence on the revival time after averaging over all trajectories is wiped out.

One can also understand the necessity of the hierarchy
given in \eqref{relation} to have the maximal
number of revivals, as well as their maximum number $k-1$.
When the condition is satisfied, then (approximately) the first, second, \ldots, $k-1$ jumps do not influence each other. With this we mean that the $n$-th jump occurs when $n-1$ jumps have already taken place in most of the trajectories. Accordingly, the $k-1$ first jumps are connected with the revival of the trace distance, while the following jumps do not result in the revivals. The reason is that for the exponential WTDs the mean value and the variance cannot be modified independently and are such that for an unmodified process the trace distance is monotonically decreasing, as was shown in Sect.~\ref{sec:qnmRP}. This will be different in the case of Erlang WTD, which we discuss in Sects \ref{sec:Erl1} and \ref{sec:Erl2}.  When the condition \eqref{relation} is not satisfied, the number of the revivals for a modified process with $k$ different WTDs will be smaller than $k-1$.

All revivals depicted till now started when the trace distance assumed value zero, i.e. when at the
associated time the evolved states are the same. This can be seen as
a special realization of non-Markovian behaviour, since in this case the dynamical map is neither invertible nor divisible. This is, however, not always the case. We observe that a revival occurs for larger values of the trace distance when the condition $\mu_2 \gtrapprox \mu_1 \gg \mu$ (3-WTDs process) is satisfied, see Fig.~\ref{1}, left, where the $\mu_2$ is varied, and right, where $\mu$ is altered. The mean waiting time of the second jump is small
enough with respect to the first jump to prevent the trace distance to reach zero, and the following third jumps occur too late to change this tendency. Note, that in this case the maximal number of revivals, $k-1$, cannot be reached.

  \begin{center}
       \minipage{0.47\textwidth}
    \includegraphics[width=\textwidth]{Figures/1a2}
    \endminipage
    \hspace{1cm}
         \minipage{0.45\textwidth}
      \includegraphics[width=\textwidth]{Figures/1b2}
      \endminipage
  \captionof{figure}{The plots show the dependence of the times at which revivals take place on the rates of the exponential WTDs.
  We consider the $\mathcal{E}_{x-\mathrm{AD}}$ jump channel together with damping rate $\gamma=0.3$ and relaxation rate $\lambda_2=0.1$. Overall $\mu_1=15$, with fixed $\mu=1$ in the left panel and $\mu_2=12$ in the right panel.}
  \label{1}
\end{center}

\subsection{Exponential WTD - purely jump dynamics}
\label{sec:Exp2}
As already elaborated, in the considered case the time continuous
dynamics between the jumps does not strongly affect the qualitative
picture of non-Markovianity. It is therefore of interest to consider
the effect of jumps and modified waiting time distributions alone, setting $\mathcal{L}(t)=0$, see Eq.~\eqref{eq:RenGen}. In this case the density operator follows the evolution:
\begin{align}\label{noCont}
\rho(t)=\sum\limits_{n=0}^\infty p_n(t)\mathcal{E}^n \rho(0),
\end{align}
where $p_n(t)$ is the probability of having exactly $n$ jumps till
time $t$, i.e. no statements about the times of the particular jumps
are made as contrasted with $p_n(t; t_n,\ldots,t_1)$ in Eq.~\eqref{eq:RenGen}. As the influence of AD jump was also mainly in decreasing
the non-Markovianity measure, with the same
argument we take $\mathcal{E} \rightarrow \mathcal{E}_x$ so that
we have idempotency of the jump transformation $\mathcal{E}^2=\id$. Accordingly, the sum in Eq.~\eqref{noCont} can be split in two terms, one with even $n$ and one with odd $n$, see \cite{Vacchini_2011} for an analogous discussion with the $z$-Pauli channel:
\begin{align}
\rho(t)=( p_{even}(t)+   p_{odd}(t)\mathcal{E})\rho(0).
\end{align}
The difference between the matrices $\rho_1(t)$ and $\rho_2(t)$ then
simply becomes
\begin{align}
\rho_1(t)-\rho_2(t)=\begin{pmatrix}
\Delta_{11}&  p_{even}(t) \Delta_{10} - p_{odd}(t) \Delta^*_{10}\\
p_{even}(t) \Delta^*_{10} - p_{odd}(t) \Delta_{10}\ & -\Delta_{11}
\end{pmatrix},
\end{align}
where $\Delta_{ij}$ gives the difference of the associated components of the operators $\rho_1(t)$ and $\rho_2(t)$ in the $\sigma_z$ basis.
With the choice of the optimal states, $\ket{\phi_{1/2}}=\frac{1}{\sqrt{2}}(\ket{0} \pm i\ket{1})$, we obtain for the trace distance
\begin{align}
\label{pe-po}
\mathcal{D}(\rho_1(t),\rho_2(t))=| p_{even}(t)- p_{odd}(t)|=|q(t)|,
\end{align}
which is the absolute value of the difference between the probability of the even number of jumps and odd number of jumps. As distinct from investigations in \cite{Vacchini_2011}, here we take into account also case of modified processes, where first $k$ WTDs are different from the following one. The quantities $p_{even}(t)$ and $p_{odd}(t)$ take then in Laplace picture the form

\begin{align}
 \tilde{p}_{even}(u)&= \tilde{g}_1(u) +
                      \tilde{f}_1(u)\tilde{f}_2(u)\tilde{g}_3(u)+\ldots \nonumber  \\
                    &\hspace{1truecm}+\tilde{f}_1(u)\ldots\tilde{f}_k(u)
 \left(\frac {1+\tilde{f}(u)}2-(-)^k\frac {1-\tilde{f}(u)}2\right)
                     \frac{1}{1-\tilde{f}^2(u)} \tilde{g}(u), \label{eq1}\\
                     \tilde{p}_{odd}(u)&=\tilde{f}_1(u)\tilde{g}_2(u)+
                     \tilde{f}_1(u)\tilde{f}_2(u)\tilde{f}_3(u)\tilde{g}_4(u)+\ldots \nonumber \\
                    &\hspace{1truecm}+\tilde{f}_1(u)\ldots\tilde{f}_k(u)
\left(\frac {1+\tilde{f}(u)}2+(-)^k\frac {1-\tilde{f}(u)}2\right)
                    \frac{1}{1-\tilde{f}^2(u)} \tilde{g}(u). \label{eq2}
\end{align}
In the case of the exponential WTDs we can accordingly go beyond the
Markovian case of an exponential distribution corresponding to $q(t)=e^{-2 \mu t}$. For the simplest case of 2 WTDs one obtains
\begin{align}
q(t)&=\frac{2 (\mu -\mu_1) e^{ -t \mu_1}+\mu_1 e^{-2 \mu  t}}{2 \mu -\mu_1}.
\end{align}
The expression of $q(t)$ for a larger number of WTDs retains the same form, i.e. a weighted sum of $k$ exponentials $e^{-2 t \mu }$, $e^{-t \mu _1}$,\ldots, $e^{-t \mu _k}$.
Non monotonicity of the absolute value of the function $q(t)$ can
arise in two ways: non monotonicity of $q(t)$ itself or its
sign change. Note that these are not independent, as $q(t)$
convergences to zero for $t \rightarrow \infty$. Accordingly, with
every sign change at least one local maximum or minimum has to
follow. On the other hand, a local maximum (minimum) can occur without
sign change, but then need to be followed by a minimum (maximum).

For the case of two waiting time distributions one can analytically
verify that the maximal number of revivals is one, and that revivals take place at
\begin{align}
t=-\frac{1}{2\mu-\mu_1}\ln\frac{2(\mu_1-\mu)}{\mu_1},
\end{align}
where the condition $\mu_1>\mu$ has to be satisfied. This corresponds to
the requirement obtained for the dynamics considered in Sect.~\ref{sec:Exp1}, Eq.~\eqref{relation},
which, however, could feature an intermediate time continuous evolution and a
jump transformation containing AD.
Note that the time $t$ is smaller than the mean jump time of the first jump ${1}/{\mu_1}$ for $\mu_1>2\mu$, otherwise it is larger. For larger $k$ in general no closed-form formula for the number or the times of revivals can be given, as the exponential function is transcendental. Nonetheless, thanks to the Descartes' rule of signs the maximal number of revivals ($q(t)=0$) equals the number of the sign changes of the prefactors of the exponential functions, where the rates are put in ascending (or descending) order \cite{10.2307/40378610}. The sign change can happen maximally $k-1$ times for $k$-terms, which explains the observation we have made earlier in Sect.~\ref{sec:Exp1}. Note, that the same argument could be used for the derivative of $q(t)$, connected with the occurrence of local maxima/minima. However, the maximal number of revivals $k-1$ can only happen when all of the revivals are at zero distance, as the non-monotonicity of $q(t)$ without sign change involves one minimum and one maximum per revival. Note that consequently for a process with 2 WTDs the revival can only occur because of the sign change of $q(t)$, i.e. at zero trace distance.

\subsection{Erlang WTD - general results}
\label{sec:Erl1}

Considering WTDs that can lead to non-Markovianity for unmodified
processes, the maximal possible number of revivals can get larger. This
can be observed by taking into account an Erlang distribution, whose WTD is given by Eq.~\eqref{erlang}, governing the
randomness of the jump times. For Erlang distributions with fixed mean
value, the higher the shape parameter $r$ or the larger the rate $\mu$, the
narrower the distribution. Accordingly, with growing $r$ or $\mu$ the revivals of the trace distance
can be seen more and more like independent phenomena. In this case the
jumps do not "destructively interfere" with each other and the time
intervals of the jumps are almost disjoint. This explains the increase
of the non-Markovianity measure with higher shape parameter $r$ or larger
rate $\mu$, as one can see in the simulations in Fig.~\ref{Erl1}, left. {This slightly influences also the time of the revivals, and the higher the shape parameter, the closer this time is to the mean value of the first WTD.}

  \begin{center}
       \minipage{0.3\textwidth}
    \includegraphics[width=\textwidth]{Figures/ErlR1a3}
    \endminipage
    \hspace{0.5cm}
         \minipage{0.3\textwidth}
      \includegraphics[width=\textwidth]{Figures/ErlR2a3}
      \endminipage
      \hspace{0.5cm}
           \minipage{0.3\textwidth}
      \includegraphics[width=\textwidth]{Figures/ErlR3a3}
  \endminipage
  \captionof{figure}{Behavior of the trace distance for the case of a quantum renewal process with a $\mathcal{E}_{x-\mathrm{AD}}$ jump channel and WTDs given by Erlang distributions. In the left and right panel we see that revivals increase with the shape parameter $r_1$ ($\mu=4$, $r=2$, $r_1/\mu_1=1/2$ left and $\mu=12$, $r=6$, $r_1/\mu_1=2/3$ right). In the middle panel we see dependence on the parameter $r$ with fixed
  $\mu_1=3$, $r_1=2$ and $r/\mu=1/2$.}
  \label{Erl1}
\end{center}

Also in the case of the Erlang WTDs the trace distance revivals do not
necessarily occur when the trace distance takes the value zero. This
behaviour was observed for modified renewal processes. The small $r$ of the first WTD and the large $r$ of the subsequent WTD
boost the phenomenon, see Fig.~\ref{Erl1}, middle and right.
Note, that contrary to the case of the
exponential WTDs, here the revival can occur at non-zero trace
distance also for the simplest modified process, i.e. with 2 distinct
WTDs.

\subsection{Erlang WTD - purely jump dynamics}
\label{sec:Erl2}

For the limiting case of no time continuous evolution in between the jumps $\mathcal{E}_{x}$, relying on Eq.~\eqref{pe-po} for the trace distance between the optimal pair of states, one can analytically show that an infinite number of revivals is possible. The difference of the probability of the even and odd number of jumps for an unmodified process is given in Laplace domain by
\begin{align}
\tilde{q}(u)=\frac{(\mu +u)^r-\mu ^r}{u \left(\mu ^r+(\mu +u)^r\right)}.
\end{align}
In particular, for $r=2$, so for WTD given by a convolution of two exponential functions with the same rate, we obtain
\begin{align}
q(t)=e^{-\mu t }(\sin(\mu t) +\cos(\mu t) ),
\end{align}
which obviously leads to an infinite number of revivals, always occurring at the zero trace distance.
For the modified process, with two different WTD and when both shape parameters equal two, $r=r_1=2$, one obtains
\begin{align}
q(t)=\frac{1}{\left(2 \mu ^2-2 \mu _1 \mu +\mu _1^2\right){}^2}
\left(2 \left(\mu _1-\mu \right) e^{-\mu_1  t} \left(\mu _1^3-3 \mu _1^2 \mu
+2 \mu _1 \mu ^2 -2 \mu ^3
+t\mu _1 \left(\mu _1^3-3 \mu _1^2 \mu
+4 \mu _1 \mu ^2 -2 \mu ^3\right)
\right) \right.
\nonumber \\ \left.
-\mu _1^2 e^{-\mu  t} \left( ((2\mu-\mu_1)^2-2\mu^2) \cos(\mu t) - (2\mu^2-\mu_1^2)  \sin(\mu t) \right)\right).\label{eq:qErl}
\end{align}
Accordingly, we have a term characterised by an oscillation, which is damped with a damping rate $\mu$, and a polynomial of the first order in $t$, damped with a damping rate $\mu_1$. From Fig.~\ref{Erl2} we see that for $\mu=1$ and $r=r_1=2$, if the rate of the first waiting time distribution $\mu_1$ is between zero and a value close to one, no revivals take place. This can be understood from Eq.~\eqref{eq:qErl}, since
if the rate $\mu$ is larger than the rate of the first WTD, the oscillatory part is strongly suppressed. However, for this regime the polynomial part stays always positive, and no revivals occur. Otherwise, we have an infinite number of revivals.

  \begin{center}
       \minipage{0.4\textwidth}
    \includegraphics[width=\textwidth]{Figures/Erl2a2}
  \endminipage
  \captionof{figure}{Plot of the function $q(t)$ as in Eq.~\eqref{eq:qErl}. The non-monotonicity of this function determines non-Markovianity in the model, see Eq.~\eqref{pe-po}.  The function $q(t)$ corresponds to the difference between the probability of having an even or an odd number of jumps as a function of time and WTDs' rates. We consider two Erlang WTDs with shape parameters $r=r_1=2$ and $\mu=1$, so that the vertical black line corresponds an unmodified renewal process.  Note the periodic change of values along the vertical axis determining an infinite number of revivals.}
  \label{Erl2}
\end{center}

\section{Conclusions and outlooks}
\label{sec:concl}

In this work,
we have analysed a simple and versatile class of quantum dynamics, the quantum renewal processes,
focusing on the different kinds of non-Markovian behavior that can be obtained by controlling their
defining properties.

Quantum renewal processes naturally allow for a representation of the dynamics in terms of an average over stochastic trajectories
and we have here investigated the influence that the time-continuous part of the trajectories, the type of the jumps and the waiting time distributions have on the quantitative and qualitative features of the trace distance evolution.
In particular, we focused not only on the measure of non-Markovianity, but also on relevant modifications of the trace distance
evolution, as the number, times of occurrence and extension of the revivals.
Among others, the revivals of the trace distance
can be significantly altered or even
enhanced when dealing with
modified renewal processes, where there is a difference between a certain number of initial waiting time distributions and the subsequent ones,
or if one considers Erlang waiting time distributions, which are classically non-Markovian and can lead to higher number of revivals than the exponential ones.

Our analysis shows that
the trajectory picture of quantum renewal processes yields a deeper insight into how to manipulate the trace distance
evolution,
for a varied class of
dynamics built on the analogy with classical stochastic processes.
Indeed, it will be of interest to explore to which extent the trajectory viewpoint can be a convenient starting
point to engineer non-Markovianity also in more complex and general quantum dynamics, pointing to different
features of the evolution that can be addressed to enhance or suppress the
presence of
memory effects.

\begin{acknowledgments}
{This research was funded by the UniMi Transition Grant H2020. N.M. was funded by the Alexander von Humboldt Foundation in form of a Feodor-Lynen Fellowship.}
\end{acknowledgments}

\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


Emmy®-award winning casting director Meredith Layne has worked in live-action and animation on both feature films and television. On staff with Nickelodeon Animation until 2018, she cast TEENAGE MUTANT NINJA TURTLES (2013 Emmy Award), KUNG FU PANDA: LEGENDS OF AWESOMENESS (2012 Emmy Award), PENGUINS OF MADAGASCAR (2011 Emmy award) and many others. She has also worked with Disney Feature Animation and Pixar Animation, and has taught voice-over for DreamWorks Animation.
Now a freelance Casting and Voice Director, Meredith is working on CASTLEVANIA for Netflix, BEE & PUPPYCAT and BRAVEST WARRIORS for VRV, and several unannounced projects for Netflix and Amazon.


Morriva is a variable sized cow girl. She was designed as a pleasure slave so she has some pretty noticeable curves. In these pics. She's only 8 ft tall which is her normal size. But we get to see the mini gts Morriva spending some time with her boyfriend Terry. Terry belongs to Tinyguy01.
Terry is a white tabby/tiger hybrid and stands about 5'4".
She's scared of snakes? And dressed as a nurse?
Well I guess being dressed as Indiana Jones was a bit too obvious for a pun.
FoxChess wrote: She's scared of snakes? And dressed as a nurse?
Lol I didn't even think of that joke actually. Probably because she is NOTHING like Indi... but that would have been pretty good. Though I'm sure an outfit like his wasn't really meant for a figure like hers either.
They look so cute together. Plus, her response to those costumes is adorable.
Yeah they fit well together. And Morriva may be big, but she's definitely a softy... not just physically either.
Now a few more pictures.
First, Kat and Aryanna have a habit of talking their friends into weird things. Sometimes its simple, like a quick costume change. Kat likes talking the new guys into trying on her outfit. Somewhat out of vanity, somewhat because she just likes being able to stare at other women in skimpy clothes, also because she just likes to see how it makes the less secure ones uncomfortable. So we see they've talked Morriva and Sonnya into trying on Kat's outfit. Jet, the speedster is wearing his special speed safety suit, which Kat, impersonating a police officer at the time, tricked him into a while back.
Next... what's better than Morriva, or Emily outgrowing a building? How about BOTH at once?!
I do love that first panel with giant Emily, a wonderful point of view of that wonderful "little" mouse girl. But it's nice to know Morriva's a softie, we need more big softies like her.
Who's the most remarkable, extraordinary fellow?
Yep Morriva's soft in both appearance and actions. Just part of who/what she is.
Good to see Emily's still popular. Guess people just like seeing the little mouse become a big mouse. pretty funny though. I thought for sure Morriva with her curves and clothes shredding growth would be the most popular of this one. But it seems Emily's subtle charms still make her stick out.
Emily the mouse has no inherent growth abilities. But her best friend Misty, the magician, is actually magic, she just doesn't notice because as a magician, she believes she's just doing tricks. So Misty ACCIDENTALLY used her magic to make Emily grow, meaning her clothes and everything grew with her because of magic.
Morriva though... she's just growing, since her clothes are not magic they have no such luck.


I have a lot to be grateful for this Thanksgiving. I am grateful that many of the readers who started this blog with me back in the early days are still with me. I am grateful for the healing that has taken place with my mother and my sister, even though it was painful and disturbing at times. I am grateful for the love and support of my husband Rich, who created a cocoon of love and safety for me and without whom none of this work could have been possible. I am grateful for the wisdom that my challenges have brought me, and for the prosperity that has been the result.
Gratitude is a powerful exercise. It opens the heart and creates space for new possibilities. It has the power to transform the way we view our lives and the trials in which we may find ourselves. Over the past few months I have grown very cranky – no doubt the result of the pressure from Saturn, Uranus and Pluto and an extremely difficult personal situation. Normally I have a far easier time finding the silver lining to any cloud but right now I am having to make the awareness of the gifts in my life a daily practice.
So at this time I wish you the happiest of Thanksgivings, and the joy of giving thanks. And thank you for sticking with me for all of these years!
Happy Blogday, Lynn! Sorry for all the troubles you went through this year, and hope the way is cleared for a sweeter path going forward. Thanks for all your wonderful posts, wherever they reside!
Thanks to all of you for not giving up on this blog when the writing becomes sporadic… And for always sharing your own insights and wisdom.
Lynn, Happy Thanksgiving and blogiversary. I want to thank you for sticking with all of us. A lot of people would have set their gift aside to deal with "their stuff" but you kept in contact the best you could so we could see why and where our stuff was coming from. As You know these times they are a changing and the way we go forward is exactly how you have been doing it. I keep saying that old saying " this too shall pass" ….it does and another unfinished thing shows up!
Thanks again for all that you are.
So true and well said Nancy! I'll ditto that.
I love reading your posts when they come out. Isn't it great to have the tools of knowledge that astrology affords us in understanding what is going on in our daily lives. I have studied astrology for the last 40 years and still learn something new every day. I think I would be looney tunes without it!
I hope you have a relaxing Thanksgiving day!
Thanks for all the wonderful and inspiration writing you do. Very much appreciated.
Thanks so much for all you do Lynn. I find your blog to repeatedly offer solid insights that none of the other astrologer's I read (quite a few) seem to give me. And not forcing yourself to write something when it's not there for you makes things even more genuine!
Happy Blogiversary Lynn! Hope your Thanksgiving Day brought you lots of love, smiles, and good family memories. This blog was such a great reminder that gratitude truely does work wonders. Thank you for this and all of your posts, and for sharing your feelings on life. They have helped more than you could ever know.


text:
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redpajama_set_name: RedPajamaCommonCrawl


text:
Women's Opens League Play With Home Loss
Jack Murray
This Family Weekend the women's ice hockey team began the league play portion of their season with two home contests, the first against the St. Lawrence Saints and the second against the Clarkson Golden Knights. The Raiders unfortunately ended the weekend with two tough losses, falling 3-0 to St. Lawrence and 2-0 to Clarkson. The Raiders played a strong first period against the Saints, having a great opportunity to score early in the frame when senior Jocelyn Simpson connected with junior Miriam Drubel in the left faceoff circle. The Saints' goalie managed to push Drubel's shot to the side to keep the game scoreless.
Just 1:33 into the period, the Saints got on the scoreboard when Mel Desrochers found an opening from the top of the left faceoff circle. The Raiders attempted to tie it up when senior Taylor Volpe fired a shot on net with 12 minutes to go, but the Saints came up with another save to maintain a 1-0 lead. To close the first period, St. Lawrence gained a 2-0 lead when Rylee Smith shot glove side past sophomore goalie Ashlynne Rando. After the Saints were called for checking in the final minutes of the first period, the Raiders started the middle frame on a power play, but St. Lawrence was able to execute a tight defense and block all of Colgate's attacks on goal despite their man-up opportunity. Soon afterwards, junior Melissa Kueber broke free towards the net, but St. Lawrence goalie Carmen MacDonald made another great stop to maintain their 2-0 lead.
In the final minutes of the second period, the Saints were awarded a man-up opportunity after a checking call on the Raiders. After Dayle Wilkinson's shot was initially blocked, junior relief goalie Susan Allen was unable to fall on the puck before Alex Moore made the shot into the back of the net, putting the score at 3-0 heading into the final period.
Colgate had two man-up opportunities in the third period, but was unable to make good on either of them despite very strong attempts on goal. Junior defender Katie Case had the last opportunity to score with under two minutes remaining in the contest. She took a shot after receiving the puck from Kueber, but MacDonald had yet another big save to secure St. Lawrence's 3-0 victory. The next day, Colgate faced the No. 5 team in the country, the Golden Knights. Despite a 2-0 loss, the Raiders played a phenomenal defensive game, with junior goalie Brittney Brooks making a career-high 44 saves in her first start of the year.
Clarkson started off the first period with four shots within the first two minutes, all of which were blocked by Brooks. A few minutes later, senior Stephanie Giannopoulos had a good chance on net, but the Knights goalie Erica Howe was there to make the save. 8:11 into the period, Clarkson intercepted the puck at center ice, ultimately giving Jamie Lee Rattray a shot past Brooks, giving the Golden Knights the first goal of
the game. Clarkson started off the second period on a power play after Colgate was called for slashing in the final minute of the first period, but the Raiders managed to hold off the Golden Knights and deny them a goal despite their man-up advantage. First-year Cat Quirion had a big opportunity on goal after Simpson skated down the right wing with the puck, but Howe was able to make another big save to deny the Raiders a goal.
Six minutes into the second period, the Golden Knights had another man-up opportunity after the Raiders were called for tripping, but Brooks stood strong and deflected any real chance of goal. A few minutes later, Giannapoulos intercepted the puck and had a shot on net, but Erin Ambrose slid in at the last second to make the save. After Colgate was called for two more quick penalties, Clarkson was given a 5-on-3 opportunity, but Colgate had an incredible defensive effort led by sophomore Nicole Gass to kill off Clarkson's attempts on goal, ending the second period with the score still 1-0. About four minutes into the final period, Vanessa Plante scored a goal for Clarkson after a scramble in front of the net, giving the Golden Knights a 2-0 lead. The Raiders were given one final man-up advantage after the Golden Knights were called for interference with about nine minutes remaining in the game, but unfortunately nothing came of the opportunity. Despite a strong effort from Colgate, Clarkson ended the day with the 2-0 win. Next up, the Raiders will hit the road to face Quinnipiac and Princeton in two away contests. Colgate will play Quinnipiac on Friday, November 1 at 7 p.m. and Princeton on Saturday, November 2 at 4 p.m.
Contact Jack Murray at [email protected]


With many choices for low impact activity, it shouldn't be too diffi cult to fi nd a workout that suits your personality, is easy on your body and still works your heart and major muscle groups.
Effective low impact cardiovascular exercise should allow the heart rate of the participant to reach a level greater than 50 percent of its maximum. Low impact cardiovascular exercises generally have low physical risk and are effective for general fi tness.
To f ind low impact act ivit ies, look for anything that involves intense stretching and strengthening components. According to The National Institutes of Health, www.nlm.nih.gov, walking and swimming are both perfect examples. For most people, walking is a low impact exercise that may be performed at a slow pace or with an intense whole body motion, at speeds in excess of 5 mph; typical conventional walking is done at approximately 2.5-3 mph. This faster type is referred to as power walking. Swimming is a good example of an activity that is generally low impact on the entire body and yet can also be as intense as the swimmer makes it.
Yoga, Pilates, calisthenics and aerobics routines with no jumping are also low impact activities that you should be able to do while still getting a good cardiovascular workout. Cycling at a medium pace is also considered a low impact activity. Stresses created by cycling come from hard sprints or hill climbing, so with this type of cycling, the rider should remain in the seat for the entire activity. Mountain biking will also result in increased impact on the joints, due to the rougher riding surface.
Yoga can improve both your physical fi tness and your overall wellbeing. Regular yoga practice puts your body through a series of postures and breathing exercises, which help to develop strength, balance and fl exibility. It also lifts your mood.
This ancient Chinese art promotes mental and physical wellbeing. Movements are slow and controlled, so you won't improve your cardiovascular fi tness or get a calorie-burning workout, but you will improve your strength, fl exibility and balance.
This is a low impact activity, but it requires basic swimming ability as it's mostly done in water waist- high or deeper. Water workouts use a variety of techniques taken from studio aerobics, including walking or running backwards and forwards, jumping jacks, various arm movements and more.
This activity focuses on rebalancing the body and improving posture through slow, controlled movements and exercises.
Rowing offers a full body workout that is low impact and effective. It's a great aerobic alternative to cycling and running, boosting the cardiovascular system and working the abdomen, back and upper body. You can also use a rowing machine to warm-up for other activities.
OLD SCHOOL: Shin splints, ankle problems, knee pain and lower-back problems are prevalent with those who choose to run or jog. If you want to run, you've got to be willing to pay the price of chronic structural damage.
NEW THOUGHTS: Low impact favorites such as swimming, cycling, elliptical machines and rebounding (indoor mini trampoline) make it possible to get in shape and stay in shape at any age, free from the joint pain associated with high impact exercise methods.


CeraVe foaming facial cleanser gently removes dirt, excess oil and dead skin cells to reveal a clean, fresh face. The non-irritating formula maintains a healthy moisture balance, never stripping the skin, while eliminating shiny, oily areas. Replenishing hyaluronic acid and ceramides produce a healthy skin barrier so your face feels comfortable without any tightness or dryness.Areas of concern-Acne prone skin care.
This acne-fighting facial cleanser washes away excess oil, dirt and other impurities. It removes dead skin cells with tiny, smooth exfoliating beads. Micronized benzoyl peroxide used in this face wash is designed to quickly penetrate deep into pores to start to kill acne-causing bacteria on contact. The combination of cleansing and exfoliating helps to clear up existing acne blemishes and prevent new ones from forming . Directions -Use morning and night. Apply a small (dime-sized) amount to dampened skin and gently massage. Rinse thoroughly with warm water, pat dry. Follow with Revitalizing Toner.
EXTREME ANTIOXIDANT BENEFITS - With potent antioxidants like Vitamin C, and natural Organic Aloe, this facial wash works for both men and women to rejuvenate the skin. All skin types will see the nourishing benefits of this professional-grade formula.
BLEMISH DEFENSE - Our Vitamin C Cleanser is a gentle face wash that helps clear and unclog pores, breakouts, and blemishes. The ingredients in this acne face wash help to cleanse away dulling skin cells, fade the appearance of sun spots, discoloration, and damage that comes from prolonged sun exposure.
BALANCED COMPLEXION - Penetrating deep to keep the skin hydrated this oil cleanser uses a combination of soothing oils and moisturizers to leave the face with a boost of hydration and a clean feel for both morning and night.
EXFOLIATING AGE REVERSAL - Great to use alongside an exfoliating face wash such as our Glycolic Cleanser; this anti aging face wash acts as your skin's daily defense and ensure that you see youth restored to the skin.
BEST TREATMENT FOR CYSTIC ACNE: Our acne spot treatment is a fast-acting solution that fights pimples and cystic acne breakouts quickly. It works by unclogging pores, removing dead skin cells, and nourishing the trouble area to provide relief so that you won't have to worry every time a new blemish pops up. You can apply it anytime and reapply as often as needed. Leave it on overnight for extra strength results.
FIGHT CYSTIC ACNE WITH BENTONITE CLAY: Unlike other stubborn zit treatments, our spot treatment for zits contains bentonite clay, which is known to help draw out toxins from sebaceous glands, eliminate impurities, and soothe irritated skin. It's also known to provide oxygen to cells via drawing out excess hydrogen in the cell, allowing oxygen to replace it.
PREVENT ADULT & TEEN ACNE: Our spot acne treatment products not only gets rid of facial acne and bacne, it helps prevent them from coming back. Continued use means your skin continues to receive the nourishment needed for smooth, pimple-free skin. Our formulation helps balance oily skin production, thus cutting down the chances of another hormonal nodular breakout.
UNIQUE ANTI-ACNE TOPICAL FORMULA: Benzoyl peroxide has long been used as an anti-acne fighting ingredient, but it's been found to cause dryness, irritation, swelling, and inflammation. We want to provide only the best and safest in acne elimination. Our spot treatment contains natural anticyst antibacterial ingredients and botanicals that heal your skin without stressing it out. Enjoy the powerful calming benefits of teatree, witch hazel, pink grapefruit essential oil, spearmint, and aloe vera.
TEA TREE ACNE TREATMENT GUARANTEED TO BLOW YOU AWAY: TreeActiv is confident that the Cystic Acne Spot Treatment will be your new favorite skin care product. That's why we back it up with a 100% guarantee. If you're ever unsatisfied, simply message us and we will provide a full refund while also trying to find a different product that will work better for you.
Tough on makeup residue, but easy on your face, our balm can be used on all skin types - even ladies with dry or sensitive skin.
This product is crafted from naturally derived ingredients, and is not tested on animals. It is free of formaldehyde and artificial coloring.
Farmer Cultivated & Scientist Activated. We use cutting-edge science to harness the healing powers of ingredients found in nature. The result is farm-to-face skincare that lives up to its full potency and potential through clean, thoughtful formulas.
Massage into skin before rinsing thoroughly with water and patting skin dry.
loofah and help to remove makeup, dirt and excess oil; leaving your skin smooth, soft and perfectly prepped for your daily moisturizer, cosmetics or tanning lotion.
5) Leave your buff puf sponge pad to dry naturally for your next use.
THE FACIAL SPONGE THAT REALLY WORKS! Buff Puff regular style exfoliating pads are one the best-kept beauty secrets for daily cleansing and *deep* exfoliation. When used with your cleanser, puf pads can really make a difference to your skin condition helping to remove built-up dirt, makeup, and excess oil.
FOR A MORE YOUTHFUL & RADIANT LOOKING COMPLEXION, Daily exfoliation helps remove dead skin cells and aids in collagen reproduction. Buf Puff style pads are one of the most effective and gentle ways to assist in your daily cleansing routine. Our pads can be used with all skin types including normal, dry, combination and oily skin.
EXCELLENT VALUE MULTI PACK: Each pack comes with 10 pink single sided (.75'' x 4'' x 2.75'') exfoliating Buff Puf pads providing you with excellent value.
PROUDLY MADE IN THE USA! Our replacement filters are made in accordance with applicable medical-grade foam production standards, including ISO 13485:2003 and ISO 9001:2008, and are PROUDLY MADE IN THE USA!
Hold the pore-purifying power of charcoal in the palm of your hand with the Biore Pore Penetrating Charcoal Bar. Made to deeply cleanse and exfoliate both face and body, it s infused with natural charcoal to draw out and trap impurities. Smooth jojoba beads gently exfoliate dead skin cells and debris, and natural peppermint oil leaves a tingly fresh, cool feeling all over. Start your day with a refreshing deep clean that leaves pores 2.5 times cleaner than with a basic soap.
Wet hands with warm water and work bar into a white lather. Massage lather gently across face and/or body, avoiding eye area. Rinse thoroughly and pat dry. Use up to twice daily. If dryness occurs, reduce usage.


19th Century Antique Furniture | Page 131 | Interior Boutiques – Antiques for sale and mid century modern furniture, french furniture, antique lighting, retro furniture and danish furniture.
Condition Report : Later painted .
Condition Report : Original blue paint ,date 1826 original locks with working keys.
Condition Report : Original working locks and keys.
1950s Spanish commode with recent stylish paint.