# Quantum causal correlations and non-Markovianity of quantum evolution

Shrikant Utagi<EMAIL_ADDRESS>Poornaprajna Institute of Scientific
Research, Bengaluru - 562164, India Graduate Studies, Manipal Academy of
Higher Education, Manipal - 576104, India.

###### Abstract

A non-Markovianity measure for quantum channels is introduced based on
causality measure - a monotone of causal (temporal) correlations - arising out
of the pseudo-density matrix (PDM) formalism which treats quantum correlations
in space and time on an equal footing. Using the well known damped Jaynes-
Cummings model of a two-level system interacting with a bosonic reservoir at
zero temperature as an example, it is shown that breakdown of monotonicity of
the causality measure is associated with the revival of temporal (causal)
correlations hence with the negativity of the decay rate. Also, a note on the
comparison with other well known non-Markovianity measures is given.

Keywords: Pseudo-density matrix, quantum temporal (causal) correlations, non-
Markovianity

##  I Introduction

Quantum non-Markovian dynamics has received significant attention over the
last few years [1, 2]. Defining, characterizing and quantifying memory effects
in open quantum system dynamics has been a rich topic of interest [3,
shrikant2018non-Markovian]. A number of non-Markovianity measures, based on
distance [5, 6], fidelity [7], correlations [8, 9, 10], channel capacity [11],
canonical decay rate [12] and that based on deviation from semigroup structure
[13] have been proposed to name a few. However, equivalence of the measures is
still under intense investigation [14].

Lately, a number of approaches have been formulated to address quantum
correlations in space and time in a unified picture [15, 16]. A framework now
known as pseudo-density matrix (PDM) formalism was introduced in [17], in
which spatial and temporal correlations are taken into account in a unified
way. Through a suitable mapping, the PDM formalism has been shown to be
equivalent to a number approaches, e.g., process matrix formalism which deals
with quantum theory without a definite global causal order although obeying
quantum mechanics locally, for studying temporal correlations in non-
relativistic quantum mechanics [15].

Recently, non-Markovian dynamics has been studied using temporal steering and
quantified using temporal steerable weight [18]. Interestingly, a quantum
channel with memory, arising due to correlation between successive use of
channel does not violate causality: outputs of channel at $t$ do not depend on
the input of the channel at $t^{\prime}>t$ [19]. However, in the PDM
formalism, application of channels preserves causality, but the causal order
may be indefinite [17]. See, e.g., [15], where a tripartite PDM is shown to be
equivalent to quantum switch - the only known higher order linear operation
that gives rise to indefinite causal order [20, 21]. Interestingly, violation
of temporal Bell inequalities, i.e., temporal Tsirelsen bound, as an
indication of completely positive (CP-) indivisibility was noted in [22]. In
[23] it was shown that temporal correlations due to temporal non-separability,
temporal steering, and temporal non-locality form a hierarchy.

In [24], a novel approach to non-Markovianity was given based on breakdown of
conditional past-future correlations, in which the author proposes an
indicator of non-Markovianity that detects memory effects that are due even to
CP-divisible dynamics. In [25], the notion of operational divisibility (where
a non-Markovian process can be operationally divisible) has been associated
with the satisfaction of data processing inequality and monogamy of temporal
correlations.

This work is focused on quantifying non-Markovianity using a recently
introduced causality measure arising out of PDM formalism. Making use of
causality monotone, which is a measure of causal or temporal correlations,
introduced in [17, 26], we will show that breakdown of its monotonicity is
associated with non-Markovianity of a quantum channel. As we will show later,
the measure introduced in this work is with regards to the processes that are
CP-indivisible in the sense of [8], which will be explained shortly, and
positive (P-)indivisible in the sense of [27]. Studying open system dynamics
by considering spatial and temporal correlations on a equal footing may open
up an arena to study memory effects with indefinite causal order and help
understand a possible intimate connection between non-Markovianity and causal
order [28]. We shall be using the terms temporal correlations and causal
correlations (or relationships) interchangeably, since both mean the same in
PDM formalism.

##  II The pseudo-density matrix

A density matrix can be given a Pauli operator representation
$\rho=\frac{1}{2}\sum_{j=0}^{3}\alpha_{j}\sigma_{j},$ with
$\alpha_{j}=\langle\sigma_{j}\rangle=\text{Tr}(\rho\sigma_{j})$, where
$\sigma_{0}=I,\sigma_{1}=X,\sigma_{2}=Y,\sigma_{3}=Z$ are the Pauli operators.
Similarly, a two qubit joint state of two spatially separated systems $A$ and
$B$ can also be given a Pauli representation as:

$\displaystyle\rho^{\rm
AB}=\frac{1}{4}\sum_{i,j=0}^{3}\alpha_{ij}(\sigma_{i}\otimes\sigma_{j}),$ (1)

where
$\alpha_{ij}=\langle\sigma_{i}\otimes\sigma_{j}\rangle=\text{Tr}(\rho^{\rm
AB}\sigma_{i}\otimes\sigma_{j})$ are the joint expectation values of two Pauli
operators pertaining to symbols $i$ and $j$. Here, the system density
operators $\rho^{A}$ and $\rho^{B}$ act on different Hilbert spaces
$\mathcal{H}^{A}$ and $\mathcal{H}^{B}$ respectively, each of dimension 2.
When the tensor production structure in Eq. (1) fails, one talks of a non-
separable state; for example, entangled (Bell) states.

Now, we come to the framework to define correlations in time by associating a
tensor product structure to the qubit Hilbert spaces $\mathcal{H}_{t_{1}}$ and
$\mathcal{H}_{t_{2}}$ to the density matrix $\rho_{A}$ at time $t_{1}$ and
$\rho_{B}$ at time $t_{2}$ respectively. When such a tensor product structure
fails, one talks about the temporal correlations [23]. The time evolution is
generally obtained by using a qubit completely positive trace preserving map
$\mathcal{E}_{B\leftarrow A}$ such that $\rho_{B}=\mathcal{E}_{B\leftarrow
A}[\rho_{A}]$. The state $\rho_{A}$ may be thought of as an input to the
quantum channel $\mathcal{E}$. Now, this can be extended to $k$ sequence of
Pauli measurements representing $k$-qubit state as below:

$\displaystyle\mathcal{P}=\frac{1}{2^{k}}\sum_{i_{1}=0}^{3}\cdots\sum_{i_{k}=0}^{3}\langle\\{\sigma_{i_{j}}\\}^{k}_{j=1}\rangle\bigotimes^{k}_{j=1}\sigma_{i_{j}}$
(2)

where
$\langle\\{\sigma_{i_{j}}\\}^{k}_{j=1}\rangle=\text{Tr}\left[\left(\bigotimes\limits_{j=1}^{k}\sigma_{i_{j}}\right)\mathcal{P}\right]$
is the joint expectation value of $k$ Pauli observables where each measurement
event is denoted by $j$ for each $i$. In other words it is nothing but the
correlation function of a $k$ sequence of Pauli measurements $\sigma_{i_{j}}$
on the system [29]. Above state (2) shares all the features of a proper
density matrix except that it may not be positive semi-definite. However, the
PDM is Hermitian and has unit trace. Since PDM is defined through the weighted
sum over Pauli operators, the Hermiticity of Pauli operators giving rise to
all real valued weights guarantees the Hermiticity of PDM. Unit trace of the
PDM follows from the fact that Pauli operators are traceless and only trace of
the identity operators contribute:
$\text{Tr}[\mathcal{P}]=\text{Tr}[\frac{\langle I\cdots I\rangle}{2^{k}}I]=1$.
Even though, PDM is not a state globally (meaning correlated state with at
least one negative eigenvalue), under the partial trace it gives a physically
valid marginal state. For a simple case of two sequential measurements i.e.,
for $k=2$, the equation (2) reduces to

$\displaystyle\mathcal{P}_{\rm
AB}=\frac{1}{4}\sum_{i,j=0}^{3}\langle\sigma_{i}\otimes\sigma_{j}\rangle(\sigma_{i}\otimes\sigma_{j}).$
(3)

It is important to note that PDM represents arbitrary mixtures of spatial and
temporal correlations. Whenever PDM is not positive semi-definite, i.e., when
it has at least one negative eigenvalue, then there exit causal (temporal)
correlations and PDM represents the time-like correlated system state. In
other words, the correlations are purely temporal as they are obtained for a
system between time $t_{1}$ and the same system at a later time $t_{2}$
evolved through the channel $\mathcal{E}_{B\leftarrow A}$. When PDM is
positive semi-definite, it is equivalent to the state of a space-like
correlated systems. Hence, PDM formalism accounts for spatial and temporal
correlations in unified way. Based on the above considerations, a two-point
PDM under any channel $\mathcal{E}$ can be given an alternative representation
[26]:

$\displaystyle\mathcal{P_{\rm
AB}}=\left(I\otimes\mathcal{E}\right)\left[\bigg{\\{}\rho\otimes\frac{I}{2},Q_{\mathsf{swap}}\bigg{\\}}\right]$
(4)

where
$Q_{\mathsf{swap}}:=\frac{1}{2}\sum_{i=0}^{3}\sigma_{i}\otimes\sigma_{i}$ and
$\\{\hat{a},\hat{b}\\}=\hat{a}\hat{b}+\hat{b}\hat{a}$ is the anti-commutator
of operators $\hat{a}$ and $\hat{b}$, and $\sigma_{i}$ are Pauli-X,Y and Z
operators with $\sigma_{0}=I$. Throughout the paper we consider one time use
of channel acting on the initial qubit state $\rho$ with measurements before
and after the use of the channel. This essentially creates two-point temporal
correlations [29]. E.g., for an identity channel $\mathcal{E}=I$ with an
initial state set to $\rho=\frac{I}{2}$, the PDM is obtained as:

$\displaystyle\mathcal{P}^{\rm\small
AB}=\left(\begin{array}[]{cccc}\frac{1}{2}&0&0&0\\\ 0&0&\frac{1}{2}&0\\\
0&\frac{1}{2}&0&0\\\ 0&0&0&\frac{1}{2}\\\ \end{array}\right)$ (9)

Note that it has eigenvalues
$\left\\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\\}$ which
implies that a PDM generally may not be positive semi-definite. In this work
we consider a simple case of two-point correlations obtained out of 2
sequential measurements before and after the single use of the channel.
Interestingly, through Choi-Jamiolkowski (CJ) isomorphism, Eq. (4) in fact can
be represented as

$\displaystyle\mathcal{P}_{\rm AB}=\\{\rho_{A}\otimes\frac{I}{2},\chi_{\rm
AB}\\},$ (10)

where

$\displaystyle\chi_{\rm AB}:=\sum_{i,j}(I_{A}\otimes\mathcal{E}_{B\leftarrow
A})(\ket{i}\bra{j}_{A}\otimes\ket{j}\bra{i}_{B}),$ (11)

is a state that is CJ isomorphic to the channel $\mathcal{E}_{B\leftarrow A}$.
The above Eq. (10) is known as the Jordan product representation. This feature
has been used to relate the causality measure with the quantum capacity of the
channel [26].

##  III Causality monotone and divisibility of dynamical map

It is known that under any local operations and classical communication (LOCC)
protocol, (spatial) entanglement shared between a two parties can not increase
- one of the properties on which the definition of an entanglement monotone
relies. Local operations on the other hand are generally completely positive
trace preserving (CPTP) or trace-non-increasing maps representing devices that
effect a transformation on the system state. They are given an operators-sum
representation: given a local operation $\mathcal{E}$, we have
$\mathcal{E}[\mathcal{\rho}]=\sum_{j}K_{j}\mathcal{\rho}K^{\dagger}_{j}$,
where are $K_{j}$ are the Kraus operators of the channel $\mathcal{E}$ acting
on the state $\mathcal{\rho}$. PDM can be used to define a causality monotone
which decreases under a CPTP Markovian channel. We will see shortly that this
assumption is violated for a non-Markovian (or CP-indivisible) map. However,
note that classical communication must be excluded since it induces causal
relationships, therefore not accounted for in order to define the causality
monotone.

A causality measure, based on PDM $\mathcal{P}$ was defined in [17] as

$\displaystyle f_{\rm cm}=\|\mathcal{P}\|_{1}-1,$ (12)

where $\|\mathcal{P}\|_{1}$ is the trace norm of the PDM given by
$\|\mathcal{P}\|_{1}=\text{Tr}[\sqrt{\mathcal{P}^{\dagger}\mathcal{P}}]$.
Here, $\mathcal{P}^{\dagger}$ is the Hermitian conjugate of $\mathcal{P}$. The
function $f_{\rm cm}$ satisfies all the necessary conditions for it to be a
monotone. Most notably it is monotonically decreasing function for CPTP maps.
Note that using Stinespring dilation theorem, a local quantum operation
$\mathcal{E}$ can be represented as a unitary operation on an extended Hilbert
space and since trace norm is known to monotonically decrease under partial
trace, any CPTP map is contractive under trace norm [30]. Here we make use of
more generalized causality measure introduced in [26], given by

$\displaystyle F:=\log_{2}(f_{\rm cm}+1)=\log_{2}\|\mathcal{P}_{\rm
AB}\|_{1},$ (13)

It is shown that for a maximally mixed input state $\rho=\frac{I}{2}$ in Eq.
(4), the function $F$ reduces to [29]

$\displaystyle\mathcal{N}:=\log_{2}\|(\chi_{\rm\small AB})^{\rm PT}\|_{1},$
(14)

where $PT$ denotes positive-partial-transpose. Here, $\chi_{\rm AB}$ is the
Choi matrix given in Eq.(11). Put in words, Eq. (14) is the logarithmic
negativity [31] of the Choi state of the channel $\mathcal{E}$.

Certain properties of $F$ are worth mentioning. $F=0$ if $\mathcal{P}$ is
positive semidefinite and $F=1$ for $\mathcal{P}_{2}$ obtained from two
consecutive measurements on a single qubit closed system, which is when the
initial state $\rho$ is not interacting with the external environment, which
is equivalent to saying that $\mathcal{E}=U$, where $U$ is some unitary.

The map $\mathcal{E}$ is said to be CP-divisible in a given interval
$\\{t+\tau,t_{0}\\}$ when the condition

$\displaystyle\mathcal{E}(t+\tau,t)\mathcal{E}(t,t_{0})=\mathcal{E}(t+\tau,t_{0})$
(15)

is satisfied [8], where $t_{0}\leq t\leq t+\tau$. As said earlier, $F$ is non-
increasing for local quantum operations, for a Markovian channel
$\mathcal{E}$, the following condition holds:

$\displaystyle F_{\mathcal{E}[\rho]}(t)\geq F_{\mathcal{E}[\rho]}(t+\tau),$
(16)

with $t+\tau\geq t$. As noted in Section (II), PDM formalism has natural
operational interpretation in the sense that it requires two experimental runs
to determine the causality measure; i.e., the qubit is measured before and
after the evolution down a channel. Consider that the qubit evolves from time
$t_{0}$ to $t$ via the channel $\mathcal{E}(t,t_{0})$, then PDM is obtained by
making measurement at initial time $t_{0}$ and final time $t$ and the
corresponding causality measure at $t$ is obtained as in Eq.(13). Now the same
qubit evolves under the channel $\mathcal{E}(t+\tau,t)$ and the PDM is
obtained by just making a measurement at time $t+\tau$, because it has been
measured already at time $t$, which is evident from the continuity of
evolution in Eq.(15) giving rise to continuity of $F$ in Eq.(16). Hence, a
single time argument is used for representation. Therefore, determination of
the condition of monotonicity in Eq. (16) essentially requires three
experimental runs.

It is in fact straightforward to generlaize the PDM formalism to $k$-time
instances. Interestingly, it was shown in [15] that a $3$-time PDM can be
suitably mapped to $2$-switch which gives rise to processes with indefinite
causal order. In [28], it was noted that every causally unordered process,
such as those studied using process matrices [32], can be realized by a non-
Markovian dynamics as described in the process tensor framework [33, 34]. In
[15] it was also shown that a process matrix can be mapped to PDM in four
different ways. Here, our work is motivated to realize a measure, similar in
spirit to [5], that quantifies dynamical memory in a given physical process
and a two-time PDM is used for simplicity.

It is known that non-Markovian dynamics may give rise to information back-flow
incurring revivals of quantum resources back to the system from the
environment. Since $F$ is a monotone, violation of condition (16) hence of
(15) implies that monotonicity will be broken under a CP-indivisible channel
hence signal non-Markovianity through revivals of temporal quantum
correlations.

##  IV Non-Markovianity measure based on causality monotone

Based on the above considerations, one can define a measure of non-
Markovianity using causality monotone $F$ for channel $\mathcal{E}$ acting on
an initial state $\rho$ as below:

$\displaystyle\mathcal{M}:=\underset{\rho}{\rm
max}\int_{\sigma_{(\rho,\mathcal{E},t)}>0}dt\;\sigma_{(\rho,\mathcal{E},t)}$
(17)

where

$\sigma_{(\rho,\mathcal{E},t)}=\frac{dF}{dt}.$

Here the integration is done over positive slope of $F$. Following [8, 18],
one may make use of an equivalent definition given by

$\displaystyle\mathcal{M}:=\underset{\rho}{\rm max}\int_{t_{0}}^{t_{\rm
max}}\biggl{|}\frac{dF}{dt}\biggr{|}dt+(F_{t_{\rm max}}-F_{t_{0}}).$ (18)

The integral (18) is such that for a non-Markovian process, the derivative of
$F$ is positive and $\mathcal{M}>0$. And for a time-dependent Markovian
process, the derivative of $F$ is negative and hence $\mathcal{M}=0$. The
channel becomes CP-indivisible when $\sigma_{(\rho,\mathcal{E},t)}>0$, which
we will show later that it corresponds to negativity of decay rate in the
canonical master equation representing the underlying dynamics.

It is shown [17] that $f_{\rm cm}$ satisfies the convexity condition $f_{\rm
cm}(\sum_{j}p_{j}\mathcal{P}_{j})\leq\sum_{j}p_{j}f_{\rm
cm}(\mathcal{P}_{j})$. Since it is monotonic under CPTP maps, one may define a
measure of non-Markovianity based on $f_{\rm cm}$ also. But $f_{\rm cm}$ is
not additive in which case the relation with the quantum capacity may not be
possible for the non-Markovianity measure based on $f_{\rm cm}$. Whereas $F$
is additive and therefore, convexity of $F$ can be sacrificed in favor of
additivity which will in fact be beneficial to relate the resulting causality
measure (13) with quantum capacity through Choi-Jamiolkoswski isomorphism
[26]. Note the similarity of $F$ with logarithmic negativity of a bipartite
entangled state, which is a well known entanglement monotone used to define a
measure of non-Markovianity [8]. Interestingly, for a maximally mixed initial
state (i.e., $\rho=\frac{I}{2}$), Eq. (13) gives exactly the same results as
does Eq. (14) - the logarithmic negativity of the Choi matrix of the map
$\mathcal{E}$ \- in quantifying non-Markovianity of quantum dynamics, and it
has been shown [26] to upper bound the quantum capacity of the channel
$\mathcal{E}$.

The method may be extended to $n$ parallel use of non-Markovian channel
$\mathcal{E}$ so that one may relate the causality measure to quantum capacity
thereby allowing for a more general capacity based non-Markovianity measure.
The measure does not require optimization in such case (i.e., with a maximally
mixed initial state), else requires optimization over all initial input
states. We may omit the discussion in this work. But, it is pertinent to point
out that for the causality measure to be an upper bound on quantum capacity,
it was assumed [26] that the sequential use of channels is uncorrelated.
Because, correlated quantum channels bring in classical memory effects [19]
and hence one can not consider $n$ parallel use of channels as a tensor
product $\mathcal{E}^{\otimes n}$. Extending the work of [26] to memory
channels [35] is an interesting open problem.

##  V Example 1: The damped Jaynes-Cummings model

Let us consider damped Jaynes-Cummings (JC) model [2] of a two level system
interacting with a dissipative bosonic reservoir at zero temperature. The
total system-reservoir Hamiltonian is

$\displaystyle H_{\rm\small tot}$
$\displaystyle=\frac{\omega_{0}\sigma_{z}}{2}+\sum_{j}\omega_{j}a_{j}^{\dagger}a_{j}+\sum_{j}(g_{j}\sigma_{+}a_{j}+g^{\ast}_{j}\sigma_{-}a^{\dagger}_{j})$
(19)

where $a_{j}^{\dagger}$ and $a_{j}$ are creation and annihilation operators,
and $g_{j}$ are coupling parameters and $\sigma_{+}=|1\rangle\langle 0|$ and
$\sigma_{-}=|0\rangle\langle 1|$. The above model has a Lorentian spectral
density
$J(\omega)=\gamma_{0}b^{2}/2\pi[(\omega_{0}-\Delta-\omega)^{2}+b^{2}]$, where
$\Delta=\omega-\omega_{0}$ is the detuning parameter which governs the shift
in frequency $\omega$ of the qubit from the central frequency $\omega_{0}$ of
the bath. $\omega_{0}$ represents the energy gap between ground state
$\ket{0}$ and exited state $\ket{1}$. $\gamma_{0}$ quantifies the strength of
the system-environment coupling and $b$ the spectral bandwidth. The reduced
dynamics of the qubit interacting with such an environment can be modeled as
an amplitude damping (AD) channel: $\mathcal{E}^{\rm\small
AD}[\rho]=\sum_{i}A_{i}\rho A_{i}^{\dagger}$ with the Kraus operators

$\displaystyle A_{1}=\left(\begin{array}[]{cc}1&0\\\ 0&\sqrt{1-r(t)}\\\
\end{array}\right)\quad;\quad A_{2}=\left(\begin{array}[]{cc}0&\sqrt{r(t)}\\\
0&0\\\ \end{array}\right).$ (24)

In this work, we will consider the case of no detuning ($\Delta=0$) i.e., when
the system qubit is in resonance with the central frequency modes of the
environment i.e., $\omega_{0}=\omega$. Here then, for damped JC model, $r(t)$
takes the form $r(t)=1-|G(t)|^{2}$, with

$\displaystyle
G(t)=e^{-\frac{bt}{2}}\left(\frac{b}{d}\sinh\left[\frac{dt}{2}\right]+\cosh\left[\frac{dt}{2}\right]\right),$
(25)

where $d=\sqrt{b^{2}-2\gamma_{0}b}$. The map (24) satisfies the completeness
condition $\sum_{j=0}A_{j}^{\dagger}A_{j}=I$.

Before going further, note that the operational meaning of the CP-divisibility
condition (15) and the monotonicity condition (16) were presented in Section
(III) with 3 time instances and 3 measurements on the qubit. In the following
calculations, we make one use of quantum channel which takes the qubit from
time $t_{0}$ to $t$, which in fact constitutes a continous "family" of
channels, which suffices for our purpose here. This is done for the simplicity
of the presentation of the measure, however calculating it for higher
dimensional PDM with multiple time measurements would be more involved but
straightforward.

Now, consider a general state

$\rho=\left(\begin{array}[]{cc}\sin^{2}(\theta)&\frac{1}{2}\sin(2\theta)\\\
\frac{1}{2}\sin(2\theta)&\cos^{2}(\theta)\\\ \end{array}\right).$ (26)

From Eq. (4) for an initial pseudo-pure state $\rho=|0\rangle\langle 0|$ which
is obtained when $\theta=\frac{\pi}{2}$, one obtains PDM for the above channel
as below:

$\displaystyle\mathcal{P}^{\rm\small AD}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&0&\frac{\sqrt{1-r(t)}}{2}&0\\\ 0&\frac{\sqrt{1-r(t)}}{2}&0&0\\\ 0&0&0&0\\\
\end{array}\right)$ (31)

which has the eigenvalues
$e_{1}=-\frac{\sqrt{1-r(t)}}{2},\,e_{2}=\frac{\sqrt{1-r(t)}}{2},\,e_{3}=1,\,e_{4}=0$.
Note that $e_{1}$ is always negative indicating that $\mathcal{P}^{\rm\small
AD}$ is not positive semi-definite. The measure (13) now reads

$\displaystyle F^{\small\rm AD}=\log_{2}\left(1+\sqrt{1-r(t)}\right)$ (32)

which is plotted in Fig. (1) for the considered noise in its Markov and non-
Markov regimes. Note that, for the particular case of initial state
$\rho=\ket{0}\bra{0}$, when $G(t)=\sqrt{1-r(t)}=1$, then $F=1$. That is when
there is no noise acting on the system, the temporal correlations are maximum.
And when $r(t)=1$, i.e., when the system is maximally damped, both
$e_{1}=e_{2}=0$, hence the causality measure $F=0$, which implies that causal
correlations die out at that point. One may observe the revivals of causal
correlations which happens when $G(t)$ becomes negative; see Fig. (1). When
all the eigenvalues are positive, then the PDM is called acausal in the sense
that there are no causal relationships, and if at least one eigenvalue is
negative then $F>0$ hence there exist causal correlations and the PDM then is
called causal. Note that when the system interacts with the environment in
weak-coupling regime, the causal correlations die slower than that compared
with in the strong-coupling regime. The reason is that the stronger the
coupling, the rapid the process. Interestingly, strong-coupling regimes make
PDM to become acausal at times, but because of non-Markovian nature, the
correlations can revive making PDM causal again. The normalized measure $C$,
normalized as $C=\frac{\mathcal{M}}{1+\mathcal{M}}$ from Eq. (18) for AD, such
that $0\leq C\leq 1$, is plotted in Fig. (2).

Figure 1: (Color online.) Breakdown of monotonicity of causality measure (13)
for AD for a given range of time. For the values $\gamma_{0}=3$ and $b=0.6$,
the curves with $\theta=0$ (dotted, yellow), $\theta=\frac{\pi}{4}$ (dot-
dashed, blue), and $\theta=\frac{\pi}{2}$ (bold, orange) represent non-
Markovian (NM) processes with revivals of temporal (causal) correlations. And
for $\gamma_{0}=0.6$ and $b=3$ the dashed purple curve, for an initial state
with $\theta=\frac{\pi}{2}$, represents time-dependent Markovian (TDM)
process, where the correlations fall monotonically. Figure 2: (Color online.)
Normalized measure $C$ of non-Markovianity (18) for AD, in its non-Markovian
regime given by $\frac{\gamma_{0}}{b}>\frac{1}{2}$, for $\Delta=0$, against
the coupling strength $\gamma_{0}$, with a fixed $b=1.1$. The integration
interval in Eq. (18) is chosen to be $t=0$ to $t_{\rm max}=2$, and the initial
state was set to $\rho=|0\rangle\langle 0|$, i.e., $\theta=\frac{\pi}{2}$ for
which the measure attains a maximum, which can be verified from Fig.(1) where
for $\theta=\frac{\pi}{2}$ PDM attains a maximum recurrence and that the
measure the above Figure is defined over the positive slope of the bold,
orange curve in Fig.(1) with $\theta=\frac{\pi}{2}$.

##  VI Comparison with some of the existing measures of non-Markovianity

###  VI.1 Comparison with the decay rate measure

A measure of non-Markovianity based on canonical decay rate was given by Hall-
Cresser-Li-Anderson (HCLA) in [12]. The canonical master equation
$\dot{\rho}(t)=\mathcal{L}[\rho(t)]$ of the process considered in (24) is
given by

$\displaystyle\frac{d\rho_{s}(t)}{dt}=\gamma(t)[\sigma_{-}\rho_{s}(t)\sigma_{+}-\frac{1}{2}\\{\sigma_{+}\sigma_{-},\rho_{s}(t)\\}],$
(33)

where $\gamma(t)=-2\Re[\frac{\dot{G}(t)}{G(t)}]$ is the time-dependent decay
rate, and $G(t)$ is the decoherence function given Eq.(25). Given that, the
HCLA measure is given as $\mathcal{N}_{\rm\small{HCLA}}=\int_{0}^{t_{\rm
max}}-\gamma(t)dt$, where the integral is over only the negative parts of
$\gamma(t)$. Now, as was noted earlier, the measure is defined over positive
slope of $\frac{dF}{dt}$, which, from Eq. (32), is found to be

$\displaystyle\frac{dF}{dt}=\gamma(t).\frac{G(t)}{2(1+\sqrt{G^{2}(t)})}$ (34)

where
$\gamma(t)=2\Re\bigg{(}\frac{\gamma_{0}}{\sqrt{1-\frac{2\gamma_{0}}{b}}\coth\left(\frac{1}{2}bt\sqrt{1-\frac{2\gamma_{0}}{b}}\right)+1}\bigg{)}$
is the decay rate with $\Delta=0$. Note that the denominator of (34) is always
positive. The fraction $\frac{dF}{dt}>0$ when $G(t)<0$ i.e., when the revivals
occur, and that is when $\gamma(t)<0$. Therefore, negativity of decay rate
gives rise to CP-indivisibility of $\mathcal{E}$ which is when the
monotonicity condition (16) of causality measure $F$ will be broken. The
measure of non-Markovianity according to negativity of decay rate is simply
$\mathcal{N}_{\rm HCLA}=-\int_{0}^{t_{\rm max}}\gamma(t)dt$.

Observe that the time-local generator $\mathcal{L}$ has infinite
singularities. At every instance when the generator becomes singular, the
dynamical map $\mathcal{E}^{\rm\small
AD}=T\exp\\{-\int_{t_{0}}^{t}\mathcal{L}(s)ds\\}$ becomes momentarily non-
invertible. Here, $t\geq s\geq t_{0}$ and $T$ is the time-ordering operator.
Interestingly, at each such point in time the PDM becomes positive semi-
definite, i.e., particularly for the considered example with
$\rho=\ket{0}\bra{0}$, the eigenvalues become: $e_{1}(t)=e_{2}(t)=e_{4}(t)=0$
and $e_{3}(t)=1$. In other words, the causality measure $F=0$ at each such
points (see Fig. (1)). One observes revivals of causal correlations after each
singularity when the decay rate becomes negative. It would an interesting
future direction to look at the relationship between singularities in a
dynamical map [36] and correlations in space-time.

###  VI.2 Comparison with trace distance.

One may consider comparing the measure (13 and 17) with trace distance between
a pair of initial states $\\{\rho_{1},\rho_{2}\\}$ under a channel
$\mathcal{E}$, given by
$\mathcal{D}(\rho_{1},\rho_{2})=\frac{1}{2}\text{tr}|\mathcal{E}[\rho_{1}]-\mathcal{E}[\rho_{2}]|$.
The distinguishability of a pair of initial states undergoing a CPTP map
decreases over time. It can be shown [5] that $\ket{+}$ and $\ket{-}$ are an
optimal pair for an AD process given in Eq. (24). However, for a qubit
undergoing AD, if the temporal correlations are set up, meaning that the
future qubit under AD gets correlated with itself in the past such that both
qubit states (in time) can not be written in a tensor product form, then it is
seen in this work that the initial state $\ket{0}$ maximizes the measure (18)
for AD (See, caption in Fig. (1)). Interestingly, TD under AD for initial pair
of states $\ket{+}$ and $\ket{-}$ reads
$\mathcal{D}(\rho_{+},\rho_{-})=\sqrt{1-r(t)}$, which is exactly equal to the
causality measure (12) for an input state $\ket{0}$ under AD i.e.,
$f_{cm}=\sqrt{1-r(t)}$. Hence we have

$\displaystyle F_{\rho=|0\rangle\langle
0|}=\log_{2}(1+\mathcal{D}(\rho_{+},\rho_{-})),$ (35)

where $\rho_{\pm}=|\pm\rangle\langle\pm|$ are the initial pair of orthogonal
states and $\rho=|0\rangle\langle 0|$ is the initial input of the PDM. For
different input states to PDM, the relationship might be more involved.

One may say that revival of temporal correlations can be interpreted as
equivalent to information back-flow via recurrence of trace-distance, given
that the measures under comparison are optimized over initial input states.
Therefore, one must keep in mind the optimization over input states while
comparing the causality based non-Markovianity measure with that of others.

##  VII Example 2: non-Markovian generalized amplitude damping channel

As a second example, let us consider non-Markovian generalized amplitude
damping channel (GADC) introduced in [6]. This example is given in order to
exemplify a feature of the causality measure that it can detect non-
Markovianity of the channel while trace distance doesn’t when memory effects
are coming solely from non-unital part.

In [6], it was pointed out that trace distance may fail to detect non-
Markovianity for non-unital channels. Here, we show that causality measure
detects memory originating from both unital and non-unital parts. We consider
a generalized amplitude damping (GAD) channel given by the Kraus operators
representation [37, 6]

$\displaystyle E_{1}$
$\displaystyle=\sqrt{1-p(t)}\left[\begin{array}[]{cc}1&0\\\
0&\sqrt{1-\lambda(t)}\end{array}\right];$ (38) $\displaystyle E_{2}$
$\displaystyle=\sqrt{1-p(t)}\left[\begin{array}[]{cc}0&\sqrt{\lambda(t)}\\\
0&0\end{array}\right];$ (41) $\displaystyle E_{3}$
$\displaystyle=\sqrt{p(t)}\left[\begin{array}[]{cc}\sqrt{1-\lambda(t)}&0\\\
0&1\end{array}\right];$ (44) $\displaystyle E_{4}$
$\displaystyle=\sqrt{p(t)}\left[\begin{array}[]{cc}0&0\\\
\sqrt{\lambda(t)}&0\end{array}\right].$ (47)

with $p(t)=\sin^{2}(\omega t)$ and $\lambda(t)=1-e^{-t}$.

From Eq. (4), for an initial state $\rho=\ket{0}\bra{0}$, one obtains PDM for
GAD as below

$\displaystyle\mathcal{P}^{\small\rm
GAD}=\left(\begin{array}[]{cccc}1-p\lambda&0&0&0\\\
0&p\lambda&\frac{\sqrt{1-\lambda}}{2}&0\\\ 0&\frac{\sqrt{1-\lambda}}{2}&0&0\\\
0&0&0&0\\\ \end{array}\right)$ (52)

which has the eigenvalues $h_{1}=0,\,h_{2}=(1-\lambda p),$ and
$h_{3,4}=\frac{1}{2}\left(\lambda
p\pm\sqrt{1-\lambda+\lambda^{2}p^{2}}\right)$. Now the measure (13) for GAD
reads

$\displaystyle F^{\small\rm
GAD}=\log_{2}\left(\sqrt{1+2x+y}+\sqrt{1-2x+y}+2z\right)-1.$ (53)

where $x:=\sqrt{\lambda^{2}p^{2}\left(\lambda\left(\lambda
p^{2}-1\right)+1\right)}$ and $y:=\lambda\left(2\lambda p^{2}-1\right)$ and
$z:=\sqrt{(\lambda p-1)^{2}}$. In the above expressions we have used the
notation $\lambda(t)=\lambda$ and $p(t)=p$, for simplicity.

Figure 3: (Color online.) Breakdown of monotonicity of causality measure (13)
for $\omega=3$, (bold, red curve) and the behavior of trace distance (TD)
under non-Markovian GAD [6].

Note the revivals of temporal correlations for $\omega=3$ in Figure (3), which
is evident from the observation that $p(t)$ appears in the measure. Whereas,
the trace distance (TD) between a pair of initial states $\ket{+}$ and
$\ket{-}$ for the channel (47) evaluates to $\sqrt{1-\lambda(t)}$ (plotted in
Fig. (3)) which is independent of $p(t)$. It can be shown that TD is
independent of $p(t)$ for all initial pairs of orthogonal states under the
action of GAD.

Further, note that quantum (spatial) correlation, such as entanglement, based
measures such as given in [8, 38, 6, 39, 40], do detect non-Markovianity of
the above channel (47). Although, it is important to note the difference
between these measures and the current one. Our measure is based on causal
correlations between the same system measured at two different times $t_{1}$
and $t_{2}$ where the state of the system at $t_{2}$ is fixed by the evolution
effected by the channel $\mathcal{E}$. It is known that RHP measure is
equivalent to decay rate (or HCLA) measure by a factor of $\frac{2}{d}$, where
$d$ is the dimension of the system, i.e., $\mathcal{N}_{\rm
RHP}=\frac{2}{d}\mathcal{N}_{\rm HCLA}$ [12, 4]. Since the comparison of the
measure introduced in this paper has been done with the decay rate measure,
the comparison between entanglement based measures and the causality based
measure is straightforward, hence we do not propose to do it here.

One may obtain a measure of non-Markovianity using Eq. (18), which is not done
here, since the purpose of this example was to show that the causality measure
can detect non-Markovianity of non-unital channels.

##  VIII Conclusions

A measure of non-Markovianity is introduced based on causality monotone given
in Eq. (13) arising out of pseudo-density-matrix (PDM) formalism [17]. A more
generalized monotone [26], which is just logarithm of the trace norm of PDM,
is used to define the measure. The function $F$ in Eq. (14) is non-increasing
for local operations which are CPTP maps in general. Breakdown of monotonicity
of $F$ is an indication that the corresponding map is CP-indivisible. A
suitable quantifier of non-Markovianity is given in Eq. (18).

Its comparison with the decay rate measure for the case of damped JC model is
also done showing that negativity of decay rate entails breakdown of
monotonicity of causality measure hence of CP-divisibility of a channel. Since
causality measure, for certain input states of PDM, is shown to upper bound
the quantum capacity of a channel, it can be said that it upper bounds the
capacity based measures introduced [11].

Although the PDM formalism, as proposed, is for qubits, it is important to
note that it can be generalized to the systems of arbitrary dimensions across
time [17, 41]. Therefore, the measure presented in this work should also apply
to systems of arbitrary finite dimension, though it is exemplified with a
system of dimension 2 in the present paper. The framework, as it stands, is
also for multiple qubits in space-time, hence use of, say $n$ parallel use of
qubit channels is possible, which in fact enables one to quantify quantum
capacity of a channel using causality measure [26].

Interesting future direction would be to investigate its applicability to
various models of open system dynamics and also compare it with the recently
introduced non-Markovianity measure based on temporal steering [18], and with
that of other spatial correlation based measures such as entanglement [8],
accessible information [39], mutual information [38]. It is important to note
that causal correlations due to PDM contain the strongest form of quantum
direct cause and it was shown [23] that temporal steering can be a weaker form
of quantum direct cause. In this light, non-Markovianity measure proposed in
this work can be conjectured to be stronger than that proposed using temporal
steerable weight [18]. However, the proof is left for a future work.

It is worth noting that treating quantum mechanics beyond a definite causal
structure is under intense investigation [15]. As is known, PDM contains
correlations without definite causal order, it is evident from this work that
revival of causal correlations due to (non-Markovianity) CP-indivisibility
brings back indefiniteness of causal order.

## Acknowledgments

The author thanks R. Srikanth for insightful discussions and suggestions and
also S. Aravinda and Ranjith V. for helpful comments on the draft. This work
was partially supported by the Department of Science and Technology (DST),
Govt. of India, through the project EMR/2016/004019.

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