# Global Correlation and Local Information Flows
in Controllable Non-Markovian Open Quantum Dynamics

Xin-Yu Chen,1 Na-Na Zhang,2,3 Wan-Ting He,2
Xiang-Yu Kong,1 Ming-Jie Tao,4 Fu-Guo Deng,2 Qing<EMAIL_ADDRESS>Gui-Lu Long1<EMAIL_ADDRESS>1Department of Physics, Tsinghua
University, Beijing 100084, China
2Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing
Normal University, Beijing 100875, China
3School of Optoelectronics Engineering, Chongqing University of Posts and
Telecommunications, Chongqing 400065, China
4Faculty of Fundamental Science, Space Engineering University, Beijing 101400,
China

(27th August 2024)

###### Abstract

In a fully-controllable experiment platform for studying non-Markovian open
quantum dynamics, we show that the non-Markovianity could be investigated from
the global and local aspects. By mixing random unitary dynamics, we
demonstrate non-Markovian and Markovian open quantum dynamics. From the global
point of view, by tuning the base frequency we demonstrate the transition from
the Markovianity to the non-Markovianity as measured by the quantum mutual
information (QMI). In a Markovian open quantum process, the QMI decays
monotonically, while it may rise temporarily in a non-Markovian process.
However, under some circumstances, it is not sufficient to globally
investigate the non-Markovianity of the open quantum dynamics. As an essential
supplement, we further utilize the quantum Fisher information (QFI) flow to
locally characterize the non-Markovianity in different channels. We
demonstrate that the QMI in combination with the QFI flow are capable of
measuring the non-Markovianity for a multi-channel open quantum dynamics.

Keywords: Non-Markovian, Mixing-induced Quantum Non-Markovianity, Quantum
Simulation, Nuclear Magnetic Resonance, Quantum Mutual Information, Quantum
Fisher Information

## Introduction

Quantum coherence and entanglement lie at the heart of important resources for
quantum metrology and quantum information processing 1, 2. Due to interaction
with the environment, any quantum system inevitably suffers from decoherence
and quantum entanglement disappears at a finite time 3, 4. However,
environment does not always play a harmful role and some interesting phenomena
may rise in open quantum dynamics with structured baths. For example, in a
spin bath, multi-decoherence processes may be slower than single-decoherence
processes when applied with appropriate pulse sequences 5, 6. In radical-pair
mechanism for avian compass, the local magnetic environments compete against
the homogeneous geomagnetic field to affect the yield of chemical reaction and
thus effectively play the role of measuring instrument for the weak
geomagnetic field 7, 8. Interestingly, the environment can assist efficient
energy transfer towards the reaction center in natural photosynthesis 9, 10.
In a non-Markovian environment, the quantum Zeno effect can be utilized to
steer the state evolution and explore quantum entanglement to improve the
precision of metrology 11, 12, 13, 14.

Generally speaking, the dynamics of open quantum systems are described by the
quantum master equation, in which the well-known Markovian and Born
approximations have been applied 15. However, under some circumstances, the
quantum dynamics can significantly deviate from that predicted by the
Markovian quantum master equation, e.g. strong coupling to some vibrational
modes in the bath 16, or the system-bath couplings are comparable to the
intra-system couplings 17. Furthermore, the positivity may be violated in the
reproduced quantum dynamics by the non-Markovian quantum master equation when
some approximations are made in the deduction 18, 19, 21, 20. Therefore, it is
necessary to exactly describe the open quantum dynamics for the whole
parameter regime in a unified manner 22, e.g. the hierarchical equation of
motion (HEOM) 23.

Apart from the theories for simulation, many useful measures have been put
forward to assess the non-Markovianity of the open quantum dynamics 24, 25,
26, 27, 28. Because the Markovian open quantum process will erase the memory
of the initial state, quantum systems which start from different initial
states will result in the thermal equilibrium. The temporary rise in the trace
distance between two initial states can be viewed as a signature of the non-
Markovianity 29, 30. Because local trace-conserving complete positive maps do
not raise the entanglement, in Markovian processes the monotonic decay will be
observed for the entanglement between the system and an ancillary system.
Thus, the entanglement was also proposed for measuring the degree of non-
Markovian behavior in open quantum dynamics 31, e.g. in photosynthetic exciton
energy transfer 32, 20. However, because the entanglement characterizes only
the quantum correlation 33, it may be more convincing to use the quantum
mutual information (QMI), which is the total correlation including both
quantum and classical correlation, to explore the non-Markovianity 34.
Nevertheless, all the above measures only quantify the non-Markovianity from a
global point of view. Since a many-body system can interact with the
environments individually, it is quite natural to ask can we investigate the
non-Markovianity from a different point of view, e.g. from different channels.
The flow of the quantum Fisher information (QFI), which has been used to
estimate the error bound of quantum metrology, was proposed to study the
information exchange between the system and bath 35. In Markovian processes,
the QFI flow is unidirectional from the system to the bath, while it is
bidirectional in non-Markovian processes. Especially, the QFI flows in
different channels can be used to study the local information flows between
the system and bath, which can not be explored with the global measures. On
the other hand, it was suggested that mixing Markovian semigroups or unitary
dynamics may result in the emergence of non-Markovianity by virtue of an
ancillary system 36. It might be interesting to experimentally engineer and
mix unitary quantum dynamics to simulate a non-Markovian open quantum
dynamics.

In order to experimentally investigate the non-Markovianity from both the
global and local points of view, an experiment platform with both systematic
and environmental parameters accurately tunable is required. Recently, we have
theoretically developed and experimentally demonstrated an efficient quantum
simulation approach, which can exactly simulate the open quantum dynamics for
an arbitrary Hamiltonian and various types of spectral densities 37, 38, 39,
40. It effectively makes use of the bath-engineering technique 41 and the
gradient ascent pulse engineering (GRAPE) algorithm 42, 43 to exponentially
accelerate the exact simulation.

## Results

### Two-Qubit System

In open quantum systems, their dynamics are governed by the total Hamiltonian
15

$\displaystyle{H_{\mathrm{OQD}}}={H}_{\rm{S}}+{H}_{\rm{B}}+{H}_{\rm{SB}},$ (1)

where the system Hamiltonian of multi-level $|i\rangle$’s with energy
$\varepsilon_{i}$ and couplings $J_{ij}$’s is
${H}_{\rm{S}}=\sum_{i}\varepsilon_{i}|i\rangle\langle i|+\sum_{i\neq
j}J_{ij}|i\rangle\langle j|$, and the Hamiltonian of the bath reads
${H}_{\rm{B}}=\sum_{i,k}\omega_{k}a_{ik}^{\dagger}a_{ik}$ with $\omega_{k}$
and $a_{ik}^{\dagger}$ ($a_{ik}$) being frequency and creation (annihilation)
operator of $|i\rangle$’s $k$th harmonic oscillator. Generally, the system-
bath Hamiltonian can be of arbitrary form 41 and without loss of generality we
assume a pure-dephasing form ${H}_{\rm{SB}}=\sum_{i,k}g_{ik}|i\rangle\langle
i|(a_{ik}^{\dagger}+a_{ik})$ with coupling $g_{ik}$. All the information about
the system-bath interaction is given by the spectral density
$G(\omega)=\sum_{i,k}g_{ik}^{2}\delta(\omega-\omega_{k})$ 15.

Here, we consider a two-qubit system, one of which is the system qubit and the
other is the auxiliary qubit. In Fig. 1(a), we show the 13C-labelled
chloroform for the quantum simulation, where 13C is the system qubit and H is
the auxiliary qubit. In the quantum simulation, the total Hamiltonian reads

$\displaystyle{H}={H}_{\rm{S}}+{H}_{\rm{N}},$ (2)

where ${H}_{\rm{S}}$ and ${H}_{\rm{N}}$ correspond to the system Hamiltonian
and the noise Hamiltonian, respectively. The system Hamiltonian is

$\displaystyle{H}_{\rm{S}}=\frac{\omega_{\rm{s}}}{2}\sigma_{\rm{s}}^{z}+\frac{\omega_{\rm{a}}}{2}\sigma_{\rm{a}}^{z},$
(3)

where the subscript ${\rm{s}}$ (${\rm{a}}$) labels the system (ancillary)
qubit with $\omega_{i}$ and $\sigma_{i}^{z}$ being Zeeman energy and Pauli
operator ($i={\rm{s,a}}$), respectively. Initially, the total system is
prepared in the maximum-entangled state

$\displaystyle\left|\psi_{\rm{sa}}\left(0\right)\right\rangle=\left(\left|00\right\rangle+\left|11\right\rangle\right)/\sqrt{2}.$
(4)

We consider that only the system qubit evolves under the influence of the
noise, with the noise Hamiltonian
${H}_{\rm{N}}=\beta_{\rm{s}}(t)\sigma_{\rm{s}}^{z}$. We can obtain the density
matrix of the total system
$\rho_{\rm{{sa}}}\left(t\right):=\left(\Lambda_{t}\otimes
I\right)\rho_{\rm{{sa}}}$ as

$\displaystyle\rho_{\rm{sa}}\left(t\right)=[|00\rangle\langle
00|+|11\rangle\langle 11|+g(t)|00\rangle\langle
11|+g^{\ast}(t)|11\rangle\langle 00|]/2,$ (5)

where the off-diagonal term is

$\displaystyle g(t)=\exp[-i(\omega_{\rm{s}}+\omega_{\rm{a}})t]f(t)$ (6)

with the norm $f(t)=\exp[-2\chi(t)]$ and the decoherence factor
$\chi\left(t\right)=\frac{4}{2\pi}\smallint_{-\infty}^{+\infty}\frac{d\omega}{\omega^{2}}S_{ii}\left(\omega\right)\sin^{2}(\frac{\omega
t}{2})$. Here the noise power spectrum
$S_{ii}\left(\omega\right)=\smallint_{-\infty}^{+\infty}d\tau\left\langle\beta_{i}\left(t+\tau\right)\beta_{i}\left(t\right)\right\rangle
e^{-i\omega\tau}$ is the Fourier transform of the noise correlation function
$\left\langle\beta_{i}\left(t+\tau\right)\beta_{i}\left(t\right)\right\rangle=\underset{T\rightarrow\infty}{\lim}\frac{1}{2T}\smallint_{-T}^{T}dt\beta_{i}\left(t+\tau\right)\beta_{i}\left(t\right)=\left(\frac{\alpha}{2}\right)^{2}\sum_{j=1}^{J}\left[F\left(\omega_{j}\right)\omega_{j}\right]^{2}\left(e^{i\omega_{j}\tau}+e^{-i\omega_{j}\tau}\right)$,
where the first equation is valid in the large-ensemble limit 44. Here,
$\beta_{i}\left(t\right)$ represents the distribution of noise in time domain,
and it can be written as
$\beta_{i}(t)=\alpha\sum_{j=1}^{J}F_{i}(j)\sin(\omega_{j}t+\psi^{(i)}_{j})$,
where $\alpha$ is the noise amplitude, $\omega_{j}=j\omega_{0}$ with base
frequency $\omega_{0}$ and cut-off frequency $J\omega_{0}$, $\psi^{(i)}_{j}$’s
are random numbers. And the types of noise rely on the function
$F\left(\omega_{j}\right)$.

### Quantum Mutual Information

Because by varying $\omega_{0}$ we can effectively tune the temporal behavior
of $\chi(t)$ 41, in Fig. 2 we investigate the non-Markovianity vs the base
frequency $\omega_{0}$. At a small $\omega_{0}$, e.g. $\omega_{0}=0.05$ MHz in
Fig. 2(b), the time evolution of the norm of the off-diagonal term in the
density matrix $f(t)$ manifests a monotonic decay. The unidirectional
information loss from the system to the bath is a signature of a Markovian
process. By increasing $\omega_{0}$, $f(t)$ turns to be oscillatory as shown
in Fig. 2(c)-(d). To explore the underlying physical mechanism, we investigate
the time dependence of $\chi(t)$ with respect to the change of the base
frequency. When $\omega_{0}$ is sufficiently small, $\chi(t)$ is quadratic
with time. As $\omega_{0}$ increases, $\chi(t)$ becomes linearly dependent on
time. However, if we further enlarges $\omega_{0}$, $\chi(t)$ will not be
monotonically increasing with the time and thus results in temporal rise in
$f(t)$. Because the temporary rise in the off-diagonal term of the density
matrix shows the information backflow from the bath to the system, both these
two processes are not Markovian. Notice that the quantum dynamics in Fig. 2(c)
is Markovian until $t_{f}=5\leavevmode\nobreak\ \mu$s, as $f(t)$ decays
monotonically in this period. Hereafter, up to $t_{f}=5\leavevmode\nobreak\
\mu$s, we explore the non-Markovianity by increasing $\omega_{0}$ as measured
by the QMI, i.e.,
$\mathscr{\mathfrak{\mathcal{N}}}_{0}\left(\Lambda\right)=\smallint_{0,\frac{d}{dt}\mathcal{I}>0}^{t_{f}}\frac{d}{dt}\mathcal{I}\left(\rho_{\rm{sa}}\right)dt$
34, which accumulates the rising part of the QMI,
$\mathcal{I}=S\left(\rho_{\rm{s}}\right)+S\left(\rho_{\rm{a}}\right)-S\left(\rho_{\rm{sa}}\right)$
with $S(\rho)=-\textrm{Tr}(\rho\log_{2}\rho)$. For a Drude-Lorentz spectral
density with
$F\left(\omega_{j}\right)=[2\lambda\gamma\omega_{0}\coth(\frac{\beta\omega_{j}}{2})/\omega_{j}(\omega_{j}^{2}+\gamma^{2})]^{1/2}$,
$\alpha_{i}=\sqrt{\pi/2}$, reverse temperature $\beta=1/k_{B}T$,
reorganization energy $\lambda$ and relaxation rate $\gamma$, the decoherence
factor is

$\displaystyle\chi\left(t\right)=\lambda\gamma\omega_{0}\sum_{j=1}^{J}\frac{\coth\left(\frac{\beta\omega_{j}}{2}\right)}{\omega_{j}\left(\omega_{j}^{2}+\gamma^{2}\right)}\sin^{2}\frac{\omega_{j}t}{2}.$
(7)

The reduced density matrices of the system and ancillary qubit are both
$\rho_{\rm{s}}(t)=\rho_{\rm{a}}(t)=I/2$, while the nonvanishing eigen-values
of the total density matrix are $\lambda_{\pm}=[1\pm f(t)]/2$. The von Neumann
entropy of the reduced density matrix of the system, ancilla and the composite
system can be respectively given as

$\displaystyle S\left(\rho_{\rm{s}}\right)$ $\displaystyle=$ $\displaystyle
S\left(\rho_{\rm{a}}\right)=1,$ (8) $\displaystyle
S\left(\rho_{\rm{sa}}\right)$ $\displaystyle=$
$\displaystyle-\frac{1+f\left(t\right)}{2}\log_{2}\frac{1+f\left(t\right)}{2}-\frac{1-f\left(t\right)}{2}\log_{2}\frac{1-f\left(t\right)}{2}.$
(9)

The QMI and its derivative are explicitly given as

$\displaystyle\mathcal{I}(\rho_{\rm{sa}})$ $\displaystyle=$ $\displaystyle
2+\frac{1+f(t)}{2}\log_{2}\frac{1+f(t)}{2}+\frac{1-f(t)}{2}\log_{2}\frac{1-f(t)}{2},$
(10) $\displaystyle\frac{d}{dt}\mathcal{I}\left(\rho_{\rm{sa}}\right)$
$\displaystyle=$
$\displaystyle-\chi^{\prime}\left(t\right)f\left(t\right)\log_{2}\frac{1+f\left(t\right)}{1-f\left(t\right)}.$
(11)

Thus, the measure of non-Markovianity in terms of total correlation is

$\displaystyle\mathscr{\mathfrak{\mathcal{N}}}_{0}\left(\Lambda\right)=-\smallint_{0,\frac{d}{dt}\mathcal{I}>0}^{t_{f}}\chi^{\prime}\left(t\right)f\left(t\right)\log_{2}\frac{1+f\left(t\right)}{1-f\left(t\right)}dt.$
(12)

In Fig. 2(a), we show the transition from the Markovianity to the non-
Markovianity by tuning the base frequency. The transition occurring at
$\omega_{0}=0.1$ MHz is consistent with the quantum dynamics in Fig. 2(c), as
$f(t)$ decays monotonically until $t_{f}=5\leavevmode\nobreak\ \mu$s. In Fig.
2(a), we also compare the QMI with the BLP 29 and the RHP 31. Interestingly,
all three measures show a similar dependence on the $\omega_{0}$ and the phase
transition in three cases accidentally occurs at $\omega_{0}=0.1$ MHz. These
observations are consistent with the theoretical discoveries in Ref. 45 that
the three criteria for Markovianity may coincide in many cases, although they
are different in general.

### Quantum Fisher Information

In addition to the QMI, the QFI flow was proposed to observe the information
exchange between the system and bath in different channels 35. Recently, Lu et
al. engineered multiple dissipative channels of an open system in diamond and
used QFI to quantify its non-Markovian dynamics 46. Here, we also use the QFI
flow to analyze the local information exhange between the system and the bath
in different channels. However, by virtue of bath-engineering technique, we
can effectively engineer individual channels from Markovian to non-Markovian
in the simulated open quantum dynamics. We further show that by mixing unitary
dynamics, we could obtain non-Markovian open quantum dynamics 36. We consider
a total system with two qubits, as shown in Fig. 1(b), where the system qubit
13C is the first channel and the auxiliary qubit H is the second channel 35.
By applying noise to different qubits, we can realize the information flow of
different channels. When the noise is applied to the system and the auxiliary
qubit, the corresponding channel 1 and channel 2 are opened respectively. When
the noise is applied on both qubit, both channels are turned on. We also
assume that the initial state is the maximum-entangled state, i.e.,
$|\Phi_{\rm{sa}}(0)\rangle=(|00\rangle_{\rm{sa}}+|11\rangle_{\rm{sa}})/\sqrt{2}$.
The density matrix of the total system at time $t$ can be diagonalized as

$\displaystyle\rho_{\rm{sa}}(t)=\sum_{i}P_{i}{{|\psi_{i}\rangle\langle\psi_{i}|}},$
(13)

where $|\psi_{i}\rangle$ is $i$th eigen-vector with eigen-value $P_{i}$. For
such a state $\rho_{\rm{sa}}$ and a generator $\hat{O}$, the QFI of a state
$\rho_{\rm{sa}}(\theta)=\exp(-i\theta\hat{O})\rho_{\rm{sa}}\exp(i\theta\hat{O})$
with respect to a parameter $\theta$ can be given as 47

$\displaystyle Q=2\sum_{P_{i}+P_{j}\neq
0}\frac{(P_{i}-P_{j})^{2}}{P_{i}+P_{j}}|\langle\psi_{i}|\hat{O}|\psi_{j}\rangle|^{2}.$
(14)

When the noise is only applied on the system qubit, i.e.,
${H}=\frac{\omega_{\rm{s}}}{2}\sigma^{z}_{\rm{s}}+\frac{\omega_{\rm{a}}}{2}\sigma^{z}_{\rm{a}}+\beta_{\rm{s}}(t)\sigma^{z}_{\rm{s}}$,
the QFI and its flow with respect to the generator
$\hat{O}=(\sigma^{z}_{\rm{s}}+\sigma^{z}_{\rm{a}})/2$ can be calculated as

$\displaystyle Q_{\rm{s}}$ $\displaystyle=$ $\displaystyle
4\exp(-4\chi_{\rm{s}}),$ (15) $\displaystyle\mathcal{F}_{\rm{s}}$
$\displaystyle=$ $\displaystyle-16\dot{\chi}_{\rm{s}}\exp(-4\chi_{\rm{s}}),$
(16)

where the decoherence factor is
$\chi_{\rm{s}}(t)=\frac{4}{2\pi}\int_{-\infty}^{\infty}\frac{d\omega}{\omega^{2}}S_{\rm{ss}}(\omega)\sin^{2}(\frac{\omega
t}{2})$. When the noise is only applied on the auxiliary qubit, i.e.,
${H}=\frac{\omega_{\rm{s}}}{2}\sigma^{z}_{\rm{s}}+\frac{\omega_{\rm{a}}}{2}\sigma^{z}_{\rm{a}}+\beta_{\rm{a}}(t)\sigma^{z}_{\rm{a}}$,
the QFI and its flow are respectively

$\displaystyle Q_{\rm{a}}$ $\displaystyle=$ $\displaystyle
4\exp(-4\chi_{\rm{a}}),$ (17) $\displaystyle\mathcal{F}_{\rm{a}}$
$\displaystyle=$ $\displaystyle-16\dot{\chi}_{\rm{a}}\exp(-4\chi_{\rm{a}}),$
(18)

where the decoherence factor is
$\chi_{\rm{a}}(t)=\frac{4}{2\pi}\int_{-\infty}^{\infty}\frac{d\omega}{\omega^{2}}S_{\rm{aa}}(\omega)\sin^{2}(\frac{\omega
t}{2})$. When both channels are turned on, i.e.,
${H}=\frac{\omega_{\rm{s}}}{2}\sigma^{z}_{\rm{s}}+\frac{\omega_{\rm{a}}}{2}\sigma^{z}_{\rm{a}}+\beta_{\rm{s}}(t)\sigma^{z}_{\rm{s}}+\beta_{\rm{a}}(t)\sigma^{z}_{\rm{a}}$,
the QFI and its flow are respectively

$\displaystyle Q_{\rm{sa}}$ $\displaystyle=$ $\displaystyle
4\exp[-4(\chi_{\rm{s}}+\chi_{\rm{a}})],$ (19)
$\displaystyle\mathcal{F}_{\rm{sa}}$ $\displaystyle=$
$\displaystyle-16(\dot{\chi}_{\rm{s}}+\dot{\chi}_{\rm{a}})\exp[-4(\chi_{\rm{s}}+\chi_{\rm{a}})].$
(20)

Notice that
$\mathcal{F}_{\rm{sa}}\neq\mathcal{F}_{\rm{s}}+\mathcal{F}_{\rm{a}}$ implies
that the overall QFI flow does not equal to the summation of the QFI flows
from individual channels 35, as will be illustrated in Fig. 3(i).

By using the quantum simulation approach 37, 38, we again simulate the time
evolution of density matrix $\rho(t)$ and compare it to that by the HEOM. The
HEOM has been widely used in studying open quantum systems 23, 48, 49, as it
faithfully reproduces accurate open quantum dynamics. Nevertheless, the
computational complexity of our quantum simulation method is significantly
reduced as compared to that of the HEOM 37, 38. By straightforward
calculation, we obtain the QFI $Q(t)$ and then numerically fit the
experimental data to attain the QFI flow $\mathcal{F}(t)=\partial_{t}Q(t)$.
First of all, we turn on the noise of the system qubit. The off-diagonal term
of the density matrix $f(t)$ decreases monotonically, cf. Fig. 3(a), and thus
this channel is Markovian as the QFI flow is negative all the time in Fig.
3(g). Then, if we only switch on the noise of the auxiliary qubit, there are
oscillations in $f(t)$, cf. Fig. 3(b), which suggests positive QFI flow in
Fig. 3(h) when $f(t)$ is rising during the corresponding period. Finally, we
examine the case when the noises are applied to both qubits. In Fig. 3(c,f),
both of $f(t)$ and the QFI decrease monotonically along with the time. The
process is Markovian since the QFI flow is always negative as shown in Fig.
3(i). However, the summation of the QFI flows over two individual channels can
be positive and thus
$\mathcal{F}_{\rm{sa}}\neq\mathcal{F}_{\rm{s}}+\mathcal{F}_{\rm{a}}$, because
the total noise can exert influence on all channels 35. Therefore, we may
safely arrive at the conclusion that it is inadequate to only use global
measures such as entanglement and QMI to characterize the non-Markovianity of
the open quantum dynamics. The local measures, e.g., the QFI flow, can subtly
survey the non-Markovianity inside the open quantum system.

## Discussion

In the above investigations, a specific model in which the noise Hamiltonian
commutes with the system Hamiltonian is utilized in order to obtain the
analytical results. However, in practice the system-bath interaction generally
do not commutate with the system Hamiltonian. To show the broad applicability
of our proposal, we use a generic model in Ref. 10, where the environment
assists efficient energy transfer in photosynthetic four-chromophore system.
Since it has been shown in Refs. 37, 38 that the NMR experiments faithfully
reproduce the exact results of the HEOM simulation, here we compare the time
evolution of QFI flows $\mathcal{F}_{ij}(t)$ and QMI $\mathcal{N}_{0}$ by the
quantum simulation and HEOM in Fig. 4. In Fig. 4(a), the QMI discontinuously
rises and almost reaches the steady state around $t=18$ ms. The total QFI flow
can be divided into the flows in different channels as
$\mathcal{F}=\sum_{i,j}\mathcal{F}_{ij}=\sum_{i,j}\gamma_{ij}\mathcal{J}_{ij}$
35, where $\mathcal{J}_{ij}=-\textrm{Tr}(\rho[L,A_{ij}]^{\dagger}[L,A_{ij}])$
with $\rho$ and $L$ being the density matrix and the symmetric logarithmic
derivative respectively, $\gamma_{ij}$ is the decoherence rate associated with
the jump operator $A_{ij}=|\varepsilon_{i}\rangle\langle\varepsilon_{j}|$ in
the Lindblad-form quantum master equation with $|\varepsilon_{i}\rangle$ being
$i$th eigen state of ${{H}_{\rm{S}}}$, i.e.,
${{H}_{\rm{S}}}=\sum_{i}\varepsilon_{i}|\varepsilon_{i}\rangle\langle\varepsilon_{i}|$.
Here, $\gamma_{ij}$ is approximated by the Redfield theory without the
Markovian approximation 17. The matrix elements of $L$ are given as
$L_{ij}=2\langle\psi_{i}|\partial_{\theta}\rho|\psi_{j}\rangle/(p_{i}+p_{j})$,
where $\rho(t)=\sum_{j}p_{j}(t)|\psi_{j}(t)\rangle\langle\psi_{j}(t)|$. In
order to evaluate the QFI, we perform a unitary transformation
$U(\theta)=\exp{[(\sigma^{z}_{\rm{s}}+\sigma^{z}_{\rm{a}})\theta]}$ on the
initial state $(|00\rangle+|11\rangle)/\sqrt{2}$ before the open quantum
dynamics 35, i.e., we adopt the same generator as in the second experiment. By
varying $\theta$ with a small quantity $d\theta$, we could repeat the above
procedure and obtain $\partial_{\theta}\rho$. Since
$[{{H}_{\rm{S}},{H}_{\rm{SB}}}]\neq 0$, there are generally two kinds of
channels for the QFI, i.e., the dephasing and dissipative channels. In Fig.
4(b), we only show the three dissipative channels responsible for the energy
transfer between the nearest neighbours in the eigen basis, i.e.,
$\mathcal{F}_{j,j+1}$. Obviously, the QFI flows in the three channels are not
equal. Because the coupling between $|1\rangle$ ($|3\rangle$) and $|2\rangle$
($|4\rangle$) is larger than their site-energy difference, the energy explores
coherent relaxation to transfer between them 10, and thus the
$\mathcal{F}_{34}$ ($\mathcal{F}_{12}$) can be positive. Since
$|00\rangle\equiv|1\rangle$ ($\sim|\varepsilon_{4}\rangle$) is the state
(eigen state) with the highest site energy (eigen energy), $\mathcal{F}_{34}$
oscillates with the largest amplitude. On the contrary, since
$|11\rangle\equiv|4\rangle$ ($\sim|\varepsilon_{1}\rangle$) is the state
(eigen state) with the lowest site energy (eigen energy), $\mathcal{F}_{12}$
oscillates with a smaller amplitude. However, due to the smaller coupling
between $|2\rangle$ and $|3\rangle$ with respect to their large energy gap,
the energy transfer between them is incoherent and thus results in a lower
rate. Therefore, $\mathcal{F}_{23}$ is negative and its oscillation is
negligibly small. Here, we show that although the QMI $\mathcal{N}_{0}$
indicates that the open quantum dynamics in the photosynthetic light-
harvesting is not Markovian, the QFI flows subtly reveal the local
characteristics for different channels.

In this paper, we apply a recently-developed experimental platform for the
investigation on the non-Markovian open quantum dynamics. By tuning the base
frequency, we can effectively change the open quantum dynamics from
Markovianity to non-Markovianity. Furthermore, by turning on/off the noise on
either qubit, we can observe the QFI flows from different channels. As a
generalization, we apply these two measures on the investigations of the open
quantum dynamics of photosynthetic energy transfer. In addition to the global
measures such as the QMI, the local measure, i.e., the QFI flows, can show the
local information backflow, which clearly demonstrate non-Markovianity from a
local point of view. Notably, we have experimentally realized mixing-induced
non-Markovianity with unitary dynamics. Here, the random fields $\beta_{i}(t)$
essentially play the role as the ancillary system, which exchange information
with the system, and thus the mixing-induced non-Markovianity emerges 36. To
conclude, we demonstrate that the combination of the QMI and QFI flow are
fully capable of characterizing a multi-channel open quantum dynamics.

## Methods

### Quantum Simulation

In Refs. 41, 38, both the pure-dephasing and energy relaxation form of system-
bath interaction have been experimentally realized for various of spectral
densities.

In order to simulate the dynamics of open quantum systems in NMR, we apply a
time-dependent local magnetic field
$\beta_{i}=\alpha_{i}\sum_{j=1}^{J}F_{i}(j)\sin(\omega_{j}t+\psi^{(i)}_{j})$
to the state $|i\rangle$, i.e.,

$\displaystyle{{H_{\mathrm{QS}}}}={{H}}_{\mathrm{S}}+{{H}}_{\mathrm{N}},$ (21)

where ${{H}_{\rm{S}}}$ and ${{H}_{\rm{N}}}=\sum_{i}\beta_{i}\sigma_{i}^{z}$
correspond to the system Hamiltonian and the noise Hamiltonian, respectively.
In the simulation, we initially prepare a large number of ensemble in the same
initial state, which afterwards are subject to different Hamiltonians
${{{H_{\mathrm{QS}}}}}$’s determined by $\\{\psi_{j}^{(i)},j=1,\cdots,J\\}$.
By averaging over the ensemble, we obtain the exact time evolution of the
density matrix of the open quantum system predicted by the hierarchical
equation of motion (HEOM), if we assume the system Hamiltonians in the open
quantum dynamics and quantum simulation are the same, and the spectral density
and power spectral density are equal 37, 38. Physical speaking, the coherent
and incoherent dynamics are determined by the system Hamiltonian and spectral
density, respectively. As long as these two quantities are the same, the open
quantum dynamics can be faithfully reproduced by the quantum simulation.
Finally, the theoretical Hamiltonian ${{H_{\mathrm{QS}}}}$ is decomposed into
a series of experimentally feasible pulse sequences by the GRAPE algorithm. In
this way, we can apply our quantum simulation approach for efficient
simulating the exact open quantum dynamics.

DATA AVAILABILITY
The data that support the findings of this study are available from the
corresponding author QA upon reasonable request.

ACKNOWLEDGEMENTS
We thank stimulating discussions with X.-M. Lu, B.-X. Wang and J.-W. Wen. This
work was supported by the National Basic Research Program of China under Grant
No. 2017YFA0303704, the National Natural Science Foundation of China under
Grant Nos. 61727801, 11774197, 11474181, 11674033, 11505007, 11474026, and
Beijing Natural Science Foundation under Grant No. 1202017. N.-N. Zhang was
partially supported by Chongqing University of Posts and Telecommunications
under Grant No. A2021-263.

COMPETING INTERESTS
The authors declare no competing interests.

AUTHOR CONTRIBUTIONS
X.Y.C., N.N.Z. and W.T.H. contributed equally to this work. All work was
carried out under the supervision of G.L.L., Q.A., and F.G.D. X.Y.C. and
X.Y.K. performed the experiments. N.N.Z., W.T.H., M.J.T., X.Y.C. and Q.A.
analysed the experimental data. N.N.Z., W.T.H., M.J.T. and X.Y.C. wrote the
HEOM simulation code and performed the numerical simulation. All authors
contributed to writing the manuscript.

Supplementary information See Supplementary Information for the details about
theoretical deduction, numerical simulation, and complete experimental data.

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Figure 1: Schematic for quantum simulation of non-Markovian open quantum
dynamics. (a) 13C-labelled chloroform for the quantum simulation in NMR. (b)
Carbon atom acts at as the system qubit, while the hydrogen atom is the
ancillary qubit.  Figure 2: Non-Markovianity measured by global correlation.
(a) Phase diagram of QMI $\mathcal{N}_{0}$ vs base frequency $\omega_{0}$.
Quantum dynamics of off-diagonal term in the density matrix $f(t)$ for (b)
$\omega_{0}=0.05$ MHz; (c) $\omega_{0}=0.1$ MHz; (d) $\omega_{0}=0.19$ MHz.
Other parameters are $\gamma_{\textrm{NMR}}=0.9$ MHz,
$\lambda_{\textrm{NMR}}=0.2$ kHz, $\omega_{J}=5$ GHz,
$T_{\textrm{OQD}}=3\times 10^{4}$ K, $N=150$. The blue, green, and black solid
lines are theoretically obtained for the QMI, RHP, and BLP respectively, while
the red and green dots are the respect experimental data for the QMI and the
RHP. Figure 3: The QFI flows in different channels. Propagation of (a) off-
diagonal term of the density matrix $f(t)$, (d) the QFI $Q(t)$, (g) the QFI
flow $\mathcal{F}(t)$, when only CH1 is open. (b), (e), (h) are the
corresponding results when only CH2 is open. (c), (f), (i) are the
corresponding results when both of the two channels are open.  Figure 4: Non-
Markovianity in the photosynthetic energy transfer. (a) The QMI
$\mathcal{N}_{0}$ and (b) the QFI flows from three dissipative channels
$\mathcal{F}_{j,j+1}$ ($j=1,2,3$) in a generic two-qubit system by the quantum
(symbol) and HEOM simulations (curve).