# Nonvolatile Memory Cell Based on Memristor Emulator

Santosh Parajuli, Ram Kaji Budhathoki, Hyongsuk Kim Santosh Parajuli and Ram
Kaji Budhathoki are with the Department of Electrical and Electronics
Engineering, Kathmandu University, Nepal, e-mail<EMAIL_ADDRESS>ram.budhathoki@ku.edu.np). Hyongsuk Kim is with the Division of Electronics
and Information Engineering, Chonbuk National University, Jeonju 567-54896,
South Korea, e-mail:(hskim@jbnu.ac.kr).

###### Abstract

Memristor, one of the fundamental circuit elements, has promising applications
in non-volatile memory and storage technology as it can theoretically achieve
infinite states. Information can be stored independently in these states and
retrieved whenever required. In this paper, we have proposed a non volatile
memory cell based on memristor emulator. The circuit is able to perform read
and write operations. In this memristor based memroy cell, unipolar pulse is
used for writing and bipolar pulse is used for reading. Unlike other earlier
designs, the circuit does not need external read/write enable switches to
switch between read and write operations; the switching is achieved by the
zero average bipolar read pulse given after the completion of write cycle. In
our proposed memristor based memory cell, single bit can be read and any
voltages from 0 to 5 volts can be written. Mathematical analysis and the
simulation results of memristor emulator based read write circuit have been
presented to confirm its operation.

###### Index Terms:

Memristor, Memristor Emulator, Memristor Memory Cell, Memristor Read,
Memristor Write.

##  I Introduction

Memristor was first postulated by Leon Chua in 1971 as a fundamental circuit
element [1]. In a memristor when the current flows in one direction, it’s
resistance decreases and vice versa. When the current flow is stopped,
memristor retains its final state. This state retention property of memristor
makes it useful as a non-volatile memory element [2], [3].

Memristor is a natural application for resistive random access memory (ReRAM)
technology [4]; moreover, neuromorphic computation using ReRAM technology is
also becoming popular [2], [5], [6], [7]. By controlling programming signal
width and/or amplitude, memristor can be taken to any state; however, two
states: high resistance state (HRS) and low resistance state (LRS) are
commonly used and they represent bit 1 and bit 0 respectively [2], [8], [9],
[10].

Several research groups have proposed memristor emulators [11], [12], [13],
[14], [15]. Lopez et al. [12] and Yu et al. [15] have proposed a floating
memristor emulator circuit based on current conveyors. The emulator proposed
by Kim et al. [11] is built from off-the-shelf solid state components. In this
paper, [11] has been chosen for emulating a memristor because it has shown
promising results that provides an alternative solution of hp TiO2 memristor
model in real circuit.

Memristor memory cell architecture based on the memristor model has been
proposed by [9], [10], [16], [17]. Mohammad et al. [16] have explored various
aspects of memristor modeling and designed a memristor based memory. Likewise,
Sarwar et al. [17] has designed memristor based nonvolatile random access
memory, but the memristor in their work uses Spice model. The design analysis
of nonvolatile memristor memories, [9], has been used for designing a read
write circuit in this paper because their analysis have specifically targeted
key electrical memristor characteristics relevant to memory operations.

Even though many researchers have proposed memristor emulator circuit and
memory cell based on memristor model, _no memristor emulator based read write
circuit is available_ , which can correctly estimate the behavior of a real
physical memristor memory cell. In [9], memristor linear model is used for
realizing a memory element. In this paper memristor emulator is used for
realizing a memory cell. Furthermore, the need for external read write enable
switch in [9] and [10] is eliminated. The unipolar write pulse and bipolar
read pulse with redefined read circuit helps to eliminate the read write
enable switch. This not only completely removes switching loss in read write
enable switch but also reduces the access time. Due to read cycle distortion,
the memristor internal state may change and enter into the invalid state. This
read distortion issue of the memristor memory is also investigated. Read
access time duration in read pattern is proposed to preserve data integrity
during the read operation.

The memristor state in this work is assumed to be the voltage across capacitor
of the memristor emulator circuit; in fact, actual memristor state is obtained
by multiplying instantaneous capacitor voltage by a fixed resistor and adding
another fixed resistor as shown in [11]. Memristor state, memristor
resistance, memristance are used interchangeably throughout the paper. Rest of
the paper is organized as follows: section II gives brief introduction of
memristor linear model followed by the discussion of Kim’s emulator. Section
III discusses the proposed read write circuit and its operation. Simulation
results of our memristor emulator based read write circuit is presented in
section IV. Finally, a short conclusion is given in section V.

##  II Principle of Memristor and Memristor Emulator Circuit

From a circuit perspective, memristor is often modeled as a two terminal
device, with two electrodes and a sandwiched conductive channel which acts as
a switching layer [3], [18]. In general the memristive systems are defined as
[19]

$\frac{dx}{dt}=f(x,u,t)$ (1) $y=g(x,u,t)u$ (2)

where $u$ and $y$ denote the input and output of the system and $x$ denotes
the state of the system. From (1), it can be seen that, in a memristive system
the state evolves with time $t$. The equations can be modified by choosing
appropriate variables to define memristor for electronic circuit as

$\frac{dw}{dt}=f(w,v_{M},t)$ (3) $i_{M}=M(w,v_{M},t)v_{M}$ (4)

where $v_{M}$ and $i_{M}$ denote the current and voltage of the memristor and
$w$ denotes the internal state variable. $M$ denotes the memconductance
(memristance). As shown in (3) – the voltage input directly affects the state
variable $w$ – which is typical in first order memristive system [5].

Memristor linear model shows how memristance arises naturally in nanoscale
system due to coupling of electronic and ionic transport under an external
bias. Equivalent circuit for this model is shown in Fig. 1. Two state
dependent resistors are in series, and they represent ON and OFF channel
resistance respectively.

Figure 1: Linear Memristor Model.

The device dynamics and the i-v characteristics are described by (5) and (6)
respectively.

$\frac{dw}{dt}=\mu_{v}\frac{R_{ON}}{D}i_{M}$ (5)
$v_{M}=\left(R_{ON}\frac{w}{D}+R_{OFF}\left(1-\frac{w}{D}\right)\right)i_{M}$
(6)

where $\mu_{v}$ is average ion mobility, $R_{ON}$ is the resistance of doped
region, $R_{OFF}$ is the resistance of undoped region, and $w\in(0,D)$. This
model considers ohmic electronic conduction and linear ionic drift, and
assumes modulation of $w$ is due to drift of charged dopants [3].

The incremental configuration of memristor emulator [11] is used in this work.
One of the distinguished features of Kim’s emulator is that the memristor can
be programmed by giving pulse input and the programming information can be
stored in the capacitor for longer period of time. This feature of memristor
is important to use memristor as a computational memory element [20]. To
confirm the basic memristor properties, 1 volt sinusoidal waveform is given at
frequencies of 100 Hz and 400 Hz respectively. Current probe with voltage to
current ratio of 1 V/mA is used for plotting the current trace in
oscilloscope. Oscilloscope traces are shown in Fig. 2. Increased height with
increment in frequency in the pinched loop confirms incremental configuration
of the memristor. In the time domain, green trace is voltage variation across
the capacitor, and this voltage is considered as memristor state in our work.

The response of memristor with programming pulse input is shown in Fig. 3. In
all the four cases (Fig. 3(a), Fig. 3(b), Fig. 3(c), Fig. 3(d)), the increment
step size is smaller as more pulses are applied. This is because, in a
memristor present state is the cumulative effect of past applied pulses. This
observation leads memristor to be useful as a computational memory element
particularly for neuromorphic computing. Non volatility of the memristor state
during inter pulse period can also be seen.

The results in Fig. 2 and 3 are consistent with the results obtained in [11].

(a) Pinched hysteresis loop at 100 Hz and corresponding time domain waveform.

(b) Pinched hysteresis loop at 400 Hz and corresponding time domain waveform.

Figure 2: Response of Memristor Emulator in XY and YT Mode of an Oscilloscpoe.
In the time domain, yellow trace represents input voltage, green trace
represents voltage across capacitor, and blue trace represents waveform
generated by current through memristor. Time axis is at 5 mS/div. In YT mode,
yellow trace is at 1 V/div, blue trace is at 50 mV/div, and green trace is at
2 V/div.

(a) Change in memristor state with 1.5 volts amplitude programming pulse. The
programming pulse has equal ON and OFF time duration.

(b) Change in memristor state with 1.5 volts amplitude programming pulse. The
programming pulse has ON time less than OFF time.

(c) Change in memristor state with 1 volt amplitude programming pulse. The
programming pulse has equal ON and OFF time duration.

(d) Change in memristor state with 1 volt amplitude programming pulse. The
programming pulse has ON time less than OFF time.

Figure 3: Effect of Variation of Height and Width of the Programming Pulse on
the State of the Memristor.

##  III Proposed Read Write Circuit

Based on memristor response to pulse input, a read write circuit is proposed
in this paper. The block diagram of the read write circuit is shown in Fig. 4.
The comparator is used for detecting the state of the memristor (high or low)
depending on the nature of programming pulse. The maximum and minimum voltages
that can be written in the memristor are 5 volts and 0 volt respectively. The
read circuit should read high/low, after the read pulse is given. When the
memristor state voltage is above 2.5 volts, the read circuit should read high
(5 volts). Similarly, when it is below 2.5 volts, the read circuit should read
low (0 volt). The read circuit should perform two tasks: first, it should
detect read pulse and activate read circuit; second, it should compare the
output voltage of memristor with 2.5 volts to detect correct memristor state
(high or low). If there is no read pulse, the read circuit should not come
into action, and the read circuit should retain it’s original state. Any
bipolar pulse with zero average value could be chosen to read from memristor
without perturbing its internal state.

Flow chart in Fig. 5 depicts the general operation of read write circuit. The
detailed circuit diagram of proposed memristor emulator based read write
circuit is shown in Fig. 6. The circuit consists of a PMOS transistor, a
resistive attenuator, and two high gain operational amplifiers. The circuit is
developed based on Fig. 4 and Fig. 5. Q1 and first comparator serve the
purpose of detecting bipolar pulse and activating read circuit. Resistive
attenuator is formed by two 1 kilo ohms resistors. The final comparator
compares memristor current state with 2.5 volts and gives single bit output
(high or low). One of the unique features of this circuit is that it does not
need external read write enable switch to toggle between read write
operations, instead this is achieved by unipolar write pulse and bipolar read
pulse.

Figure 4: Block Diagram of the Proposed Read Write Circuit. Figure 5: General
Idea Depicting Operation of the Read Write Circuit.

### Analysis of the Proposed Read Write Circuit

The PMOS transistor in the circuit of Fig. 6, Q1, has same specifications as
the specifications of the transistors used for developing memristor emulator.
The threshold voltage ($V_{t}$) is -0.5 volt and device constant (K) is 1
mA/V2. The gate voltage ($V_{G}$) becomes positive with respect to ground
during entire write cycle due to positive unipolar nature of write pulses.
Since the read pulse is bipolar, the read circuit activates only during
negative interval of the read cycle.

Figure 6: Proposed Read Write Circuit.

When the input pulse is positive, for example +1 V, the transistor remains in
saturation region. The source potential ($V_{S}$) is computed as [21],

$(5-V_{S})=\frac{1}{2}K[V_{SG}-|v_{t}|]^{2}$ $(10-2\times
V_{S})=[V_{S}-1-0.5]^{2}$ $V_{S}=3.32V$

When the input pulse is negative, for example, -1 V volt, the transistor
enters into the triode region. Neglecting channel length modulation effect,
the source voltage is computed as,

$(5-V_{S})=K\left[(V_{SG}-|v_{t}|)\times V_{SD}-\frac{V_{SD}^{2}}{2}\right]$
$(5-V_{S})=[(V_{S}+1-0.5)\times V_{S}-0.5\times V_{S}^{2}]$ $V_{S}=2V$

If $V_{S}$ is above 2 volts, memristor is in write mode and vice versa.
Leaving 0.2 volts as safety margin, the source voltage is compared with 2.2
volts to detect read pulse and read duration.

${\text{Memristor Mode
}}=\left\\{\begin{array}[]{@{}ll@{}}\text{Write},&\text{if}\ V_{S}>2.2\text{
V}\\\ \text{Read},&\text{if}\ V_{S}<2.2\text{ V}\end{array}\right.$ (7)

The output of first comparator goes high only when the read pulse is detected;
this leaves negative terminal of second comparator at 2.5 volts. By comparing
memristor state with this 2.5 volts, final output is taken as the state of
memristor from the second comparator.

##  IV Experiments and Simulations

A read write circuit is designed using memristor emulator based on the linear
memristor model. Internal state of the memristor should not be disturbed by
the read cycle; it shall return to its original state after the completion of
read cycle. Simulation has been performed on the read write circuit by
applying zero average and non-zero average read pulse. From the simulation
results, it is seen that in zero average pulse the memristor state returns to
its original state, which does not happen with non-zero average pulse. In Fig.
7(a), memristor returns to it’s original state of 1.2 volts after the
application of zero average pulse; whereas, memristor fails to return to its
original state with non-zero average pulse as shown in Fig. 7(b). Therefore,
zero average bipolar pulse is chosen as a read pulse.

(a) Returning of memristor to its original state with zero average bipolar
read pulse.

(b) Memristor fails to return to its original state with non zero average
bipolar read pulse.

Figure 7: Effect of Zero Average and Non-zero Average Pulse on the State of
the Memristor.

If the state of the memristor is below 2.5 volts, the output of the read
circuit is zero for the read duration. Similarly, if it is above 2.5 volts,
the output of the read circuit is 5 volts. The complete response of the
memristor emulator based read write circuit is shown in Fig. 8. Originally,
the output of the read circuit is 5 volts. When the low state is detected, it
jumps to the low state (0 volt) as indicated in the Fig. 8(a). If high state
is detected during the read cycle, output of the read circuit does not change
its original state as indicated in the Fig. 8(b). Green trace represents
memristor state voltage. Blue trace represents programming and read pulse. Red
trace represents logic generated by the read write circuit.

(a) Reading low by the read circuit. The state of the memristor is below 2.5
volts.

(b) Reading high by the read circuit.The state of the memristor is above 2.5
volts.

Figure 8: Read Circuit Reading Low and High Memristor State During Negative
Cycle of the Bipolar Read Pulse.

During read cycle, the state of memristor changes; however, at the end of read
cycle the memristor should return to its original state. The read duration is
deliberately chosen as negative state of the read pulse. Due to read cycle
distortion, the memristor may enter into incorrect state. Therefore, in order
to preserve the data integrity, state of the memristor is read at the negative
half of the read pulse. As shown in Fig. 9, due to the read cycle distortion,
the state jumps above 2.5 volts (falls on the false region), but the state is
correctly read as low during negative state of bipolar read pulse presserving
data integrity. Therefore, the designed read circuit is not only simple and
non volatile but also resilient to read cycle distortion.

Figure 9: Reading at the Negative State of Read Pulse to Prevent False Reading
Due to Read Cycle Distortion.

##  V Conclusion

The application of memristor as a memory element has been explored through a
memristor emulator based memory cell. In this paper, we have designed and
developed a read write circuit based on memristor emulator, which can read a
single bit. Similarly, the circuit can be taken to any states between 0 to 5
volts representing different memory states. With unipolar write pulse and
bipolar read pulse, the circuit can be switched between read and write
operations without using an external read/write enable switch.

The read write operations of the proposed circuit are analyzed to ensure the
non volatility of the memristor states. Furthermore, the possible read
distortion is analyzed and the read access time duration in read pattern is
proposed to preserve data integrity during the read cycle.

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Non-Markovianity in High-Dimensional Open Quantum Systems using Next-generation Multicore Optical Fibers
Santiago Rojas-Rojas 0000-0002-2284-0305
Departamento de Física, Universidad de Concepción, casilla 160-C, Concepción, Chile
Millennium Institute for Research in Optics, Universidad de Concepción, casilla 160-C, Concepción, Chile
Daniel Martínez 0000-0002-8154-7141
Departamento de Física, Universidad de Concepción, casilla 160-C, Concepción, Chile
Millennium Institute for Research in Optics, Universidad de Concepción, casilla 160-C, Concepción, Chile
Departamento de Física, Universidad del Bío-Bío, Collao 1202, casilla 5C, Concepción, Chile
Kei Sawada 0009-0003-2396-681X
Departamento de Física, Universidad de Concepción, casilla 160-C, Concepción, Chile
Millennium Institute for Research in Optics, Universidad de Concepción, casilla 160-C, Concepción, Chile
Luciano Pereira 0000-0003-1183-2382
Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, España
Stephen P. Walborn 0000-0002-3346-8625
Esteban S. Gómez 0000-0003-3227-9432
Departamento de Física, Universidad de Concepción, casilla 160-C, Concepción, Chile
Millennium Institute for Research in Optics, Universidad de Concepción, casilla 160-C, Concepción, Chile
Nadja K. Bernardes 0000-0001-6307-411X
Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife-PE, Brazil
Gustavo Lima
Departamento de Física, Universidad de Concepción, casilla 160-C, Concepción, Chile
Millennium Institute for Research in Optics, Universidad de Concepción, casilla 160-C, Concepción, Chile

With the advent of quantum technology, the interest in communication tasks assisted by quantum systems has increased both in academia and industry. Nonetheless, the transmission of a quantum state in real-world scenarios is bounded by environmental noise, so that the quantum channel is an open quantum system. In this work, we study a high dimensional non-Markovian open quantum system in a multi-core optical fiber by characterizing the environmental interaction as quantum operations, which here correspond to phase flips between pairs of computational basis states.  The experimental platform is currently state-of-the-art for space division multiplexing optical fiber communication, which can serve as a robust channel for high-dimensional quantum communication. To test the channel, we perform a quantum communication task in the prepare-and-measure scenario. The non-Markovian nature of the system is demonstrated by implementing a Quantum Vault protocol. A better understanding of the phase noise in multi-core fibers could improve the stability and quality of several real-world communication protocols since they are a prime candidate to increase the telecom data transmission rate worldwide.

§ INTRODUCTION

Currently, optical fiber-based communication is the fastest method for information transmission <cit.>, mainly due to the multiple alternatives it offers for multiplexing techniques
<cit.>. Indeed, one promising method for space division multiplexing is based on Multicore Fibers (MCFs), where several cores are contained in the same cladding material in a single fiber, without noticeable core crosstalk <cit.>.  MCFs have allowed transmission rates up to 305 Tb/s through a 19-core MCF <cit.>, setting a new benchmark for ultrahigh-capacity transmission capabilities that surpass the current capacity of conventional single-mode fibers <cit.>. Moreover, using MCFs extends to multiple applications in quantum information (QI). For instance, quantum cryptographic protocols have been performed using 4- and 7-core fibers using high-dimensional quantum states (hereafter qudits) of single photons encoded on the available cores in an MCF <cit.>. This encoding allows the implementation of QI protocols with high fidelities <cit.>, and to build high-dimensional entangled photons sources compatible with the already installed fiber-communication architectures <cit.>.

In real large-scale networks, optical fibers are exposed to perturbations induced by environmental noise. As a result of these perturbations, information loss could be introduced according to the Markov hypothesis behind noise processes, spoiling the information transmission <cit.>. The Markovian nature of noise has significant consequences for quantum communication tasks and can be witnessed by a monotonic decay of channel capacities along the propagation. At the same time, the interest in quantum dynamics deviating from the Markov hypothesis, namely non-Markovianity (NM), has increased in recent years due to its theoretical relevance and possible application in the protection and processing of quantum information <cit.>. Moreover, the behavior of non-Markovian processes in quantum systems has been observed in photonic experimental simulations of environmental effects through different schemes <cit.>. To date, the use of NM as a resource for QI has been mainly focused on entanglement-based protocols <cit.>, while a formal resource theory for NM is still in early development <cit.>. Thus, a relevant goal to achieve is the experimental observation of NM in a simple scenario without considering entanglement, envisaging its application for QI tasks in the prepare-and-measure scenario. For instance, measuring non-Markovian effects through an adequately defined quantum channel capacity allows linking NM with the efficiency of a specific QI protocol <cit.>. Thereby, Ref. <cit.> has introduced the Quantum Vault (QV), a protocol for storing and retrieving information encoded in a quantum system subject to non-Markovian evolution. The efficiency of the QV is directly related to the fact that NM allows for a non-monotonic variation of
channel capacities. Hence, it is difficult for an eavesdropper to retrieve information during the evolution.

In this work, we present the study and experimental implementation of an NM dynamical map acting on high-dimensional path-encoded qudits transmitted through newly developed MCFs-based devices <cit.>.  The dynamics are realized by tuning the occurrence probability of specific error operations aimed to generalize the bit- and phase-flip channels. This platform provides a reliable testing ground to study noise operations in high-dimensional open quantum systems, where criteria to detect NM through channel capacities do not always coincide <cit.>.  The device can be described as a completely positive map composed of quantum channels where certain stochastic error operations are made. Similar probabilistic noise models have proven to be successful in developing an open systems theory <cit.>, while the parameterized evolution by error probabilities has been helpful in experimental simulations of environmental effects <cit.>.  We observe a non-monotonic behavior of three different capacities, allowing a proof-of-principle realization of a QV. Since it is critically based on NM, observing a QV indicates that our map describes a non-Markovian process in an extended regime.  The observed effects of error operations on particular channel capacities (for instance, the coherent information) enable its use for quantum data hiding protocols and quantum private channels developments <cit.>, which can lead to NM-based QI processing in modern optical fiber platforms.

A quantum map $\Lambda_t$ can be defined as a curve in the space of quantum channels. We use $t=1-p_0$ as the dynamical variable, where $p_0$ is the no-change probability. Assuming this probability to be null at the end of the evolution, the ending point corresponds to $t=1$.

§ NOISY QUANTUM MAPS IN MCFS

A general quantum process is defined by a family of one parameter dynamical maps $\{\Phi_{t,0}\}$, where $\Phi_{t,0}$ is a completely positive and trace preserving (CPTP) map for any $t>0$ <cit.>, taking an input density operator $\rho_0$ into an output density operator $\rho_t$. As depicted in Fig. <ref>, a quantum process can be described by a continuous evolution curve $\Lambda_t$ in the space of CPTP maps, such that $\Lambda_0=\mathds{1}$. An evolution is called CP-divisible if $\Phi_{t,0}=\Phi_{t,s}\Phi_{s,0}$, where $\Phi_{t,s}$ is completely positive (CP) for all stages $s$ such that $0\leq s \leq t$. All processes given by CP-divisible maps are called Markovian evolutions <cit.>. Otherwise, they are called non-Markovian.

In the following, we consider a noisy quantum process where a quantum system in an unknown state may be affected by a set of unitary transformations $U_k$ operating with their respective probability $p_k$. The formalism of quantum processes allows us to define such a dynamical map following an intuitive posit: when the system is just starting to evolve from its initial state $\rho$, it does not undergo any change yet, so the probability $p_0$ of it remaining unchanged (or being transformed only by $U_0={\mathds{1}}$) must be equal to one, with all the remaining probabilities $p_{k\neq 0}$  being null; as it evolves, the no-change probability $p_0$ decreases while the remaining probabilities increase. (A similar approach has been used for the experimental simulation of environmental effects on single qubits <cit.>.) This way, we associate the dynamical variable $t$ with a decay of $p_0$. Since all the relevant behavior for our study is observed for a finite value of $p_0$, knowing the particular dependence of $t$ on it is not crucial. Then, we can take $t=1-p_0$ as the dynamical parameter to define the noisy process, which is now enclosed by the curve in $0\leq t\leq 1$ or $1\geq p_0\geq 0$.

The explicit definition of our map makes use of the operator-sum representation for a quantum channel $\Phi(\rho)=\rho'$ in the framework of quantum operations, where all the environmental effects are captured by a complete set of Kraus operators $E_k$ acting on the system's Hilbert space as $\rho'=\Phi(\rho)=\sum_k E_k\rho E_k^\dagger$. The noisy process $\Lambda_t$ under our consideration is composed of a family of channels whose Kraus operators $E_i=\sqrt{p_i}U_i$ describe the occurrence of a  unitary error operation $U_i$ [see the examples below in (<ref>)] affecting the system with probability $p_i$, all the $p_i$ being defined in function of the no-change probability $p_0$. Let $\ket{\psi}$ be the state of  a single photon propagating through a $N$-core optical fiber:

\begin{equation}\label{eq:state}
\ket{\psi}=\frac{1}{\sqrt{N}}\sum\limits_{\ell=1}^Ne^{i\phi_\ell}\ket{\ell}\,,
\end{equation}

where $\ket{\ell}$ denotes a one-photon state in the core $\ell$. We will consider a set of error operations $U_i$ that permute the phase values  $\phi_\ell$ of the state in Eq. (<ref>).  Each permutation on its own is a noiseless operation, so the presence of noise is due to the probabilistic combination of them and somewhat extends the concept of bit-flip or phase-flip channels found in open qubit systems to higher dimensions. We can address different scenarios by allowing permutations only between cores on certain subsets (this has some relation to the prescription given in <cit.> to implement non-Markovian phase patterns). Let $s$ be the number of cores in these subsets: $s=1$ prevents reordering, so the only possible unitary transformation is the identity, leading to trivial Markovian dynamics. The $s=2$ case (Fig. <ref>) involves just  four possible operations, namely,

\begin{equation}\label{eq:n4s2}
\begin{split}
\begin{blockarray}{c|c}
\begin{block}{(c|c)}
\, \mathds{1}_2 \quad & 0 \\ \BAhline \, 0 \quad & \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \\
\end{block}
\end{blockarray}\;\;,
\quad
U_2 =
\begin{blockarray}{c|c}
\begin{block}{(c|c)}
\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} & \,  0 \quad
\\ \BAhline
0 & \, \mathds{1}_2 \quad   \\
\end{block}
\end{blockarray} \;\;,
\\
U_3& =\;
\begin{blockarray}{c|c}
\begin{block}{(c|c)}
\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} & \, 0 \quad
\\ \BAhline
\, 0 \quad & \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \\
\end{block}
\end{blockarray}\,,
\end{split}
\end{equation}

while $s=3$ and $s=4$ enable 6 and 24 unitary transformations, respectively, corresponding to the number of available permutations. It is worth remarking that instead of the most common method used in an experimental simulation of non-Markovian processes — that is, to consider and manipulate the state of the environment explicitly — our approach reflects the environmental influence by the variation on the specific probability of an error (permutation) occurring, similar to the scheme used in Ref. <cit.> to implement a dephasing channel or the simulation of classical non-Markovianity in <cit.>. In this way, we intend to make our experimental simulation consistent with the standard premise that only the state of the system is accessible to the experimenter, so it can be used as a building block to describe a general noise process for qudits, just like in qubit systems the bit-flip and phase-flip errors are components of the general depolarizing channel. The inaccessibility of the environment is also relevant for using our system as a secure channel.

The four possible operations when $N=4$ and $s=2$, with their respective probabilities $p_i$. Greek symbols represent the phases of the logical state at each core.

§ NON-MARKOVIAN EFFECTS IN QUANTUM INFORMATION CAPACITIES

Evolution of coherence and quantum mutual information under maps $\Lambda_{p_0}$ for $s=2$. (a) Relative entropy of coherence as a function of $t$ from maximally coherent initial states. If the relative phases between the terms of the state are null, its REC is constant, as depicted by the red line. The other cases are contained in the orange region. (b) Quantum mutual information at each stage of the map $\Lambda_t$. The beige region contains the results obtained from 10$^6$ normally-distributed mixed states. Green solid curves correspond to a chaotic input state ${\mathds{1}}/4$. (c) Evolution of coherent information. In this case, we show the maximum at each point obtained from a normal distribution of 10$^6$ initial mixed states (dashed line). Green solid curves correspond to a chaotic input state.  The vertical dashed lines indicate where the minimum is reached.

In order to assess the effect — and usefulness — of NM in quantum information and quantum communication protocols, different  NM measures have been proposed, which are based on the monotonic decay of certain quantifiers under the action of CPTP maps. An increase or revival of these quantities along evolution indicates a backflow of information from the environment to the system and is a witness of NM evolutions <cit.>. The first information resource we can consider is the quantum coherence. If a system evolves under a CPTP map, such as $\Lambda_t$, its coherence cannot increase. Conversely, coherence is expected to decay monotonically under a Markovian noisy process, so a suitable NM witness is given by an increase in the relative entropy of coherence (REC) <cit.>, which has a closed form for a $N$-dimensional quantum state $\rho$:

\begin{equation}
C_r=S(\rho_{\rm diag})-S(\rho)\,.
\end{equation}

Here, $\rho_{\rm diag}$ is the diagonal part of the density matrix $\rho$, and $S$ denotes the von-Neumann entropy <cit.>. This relation with the entropic characteristics of the system poses the maximum REC as a valid measure of the channel capacity, providing an interpretation of coherence in terms of the average amount of information conveyed by a state. We evaluate its behavior under our map $\Lambda_t$, taking as initial states the set of maximally coherent pure states with $C_r=\log 4$.  The orange region depicts the respective evolution of $C_r$ in Fig. <ref> (a) for $s=2$ when the error operations are equally probable, i.e., $p_{i\neq 0}=(1-p_0)/3$. We observe how $C_r$ decays to a minimum at a certain $t$ corresponding to a finite no-change probability $p_0$, after which it increases. The minimum is reached when all the permutations are equally probable, i.e., when $p_i=p_\text{min}$ for all $i$.
The revival of REC in the regime $1-p_\text{min}<t<1$ (or equivalently $p_0<p_\text{min}$) breaks the monotonic decrease imposed by the Markov approximation.

On the other hand, diagonal density matrices correspond to incoherent states giving a null value of distance-based measures of coherence such as REC [green lines in Fig. <ref> (a)]. However, these states have maximal information content that can be transmitted through a quantum channel, which can be measured by proper capacities capturing the non-unital part of the dynamics <cit.>.
In analogy to the classical Shannon capacity, different quantities are used to estimate a bound for the average amount of transmitted information, depending on the particular protocol. First, the quantum mutual information,

\begin{equation}\label{eq:qI}
\end{equation}

limits the amount of classical information that can be transmitted through the channel $\Phi$. In this definition, the last term
corresponds to the entropy exchange $S(\rho,\Phi)$, which quantifies the entropy change with the environment <cit.>. This significant parameter is also used to define the coherent information <cit.>

\begin{equation}\label{eq:Ic}
\end{equation}

This quantity is important for several reasons, which are discussed in more detail in the supplementary material. Its fundamental definition considers the entanglement between the system and an ancilla, this being the reason why it is regarded as a proper bound of the nonclassical information conveyed by the quantum carrier <cit.>.

Like quantum mutual information, coherent information cannot be increased by any post-processing of the channel output. This property, the data processing inequality <cit.>, is related to the monotonic decay of both capacities under a Markov noisy process. A break of this behavior is a signature of NM evolution <cit.>.
In Fig. <ref>, we present the evolution of the quantum mutual information (b) and the coherent information (c) under $\Lambda_t$ for different initial states. Dark green curves correspond to the evolution of the capacities for a mixed input state $\rho=\mathds{1}/d$. Noticeably, both capacities $I$ and $I_c$ attest to the non-Markovian nature of the map: for mixed states, a minimum is again reached at $p_0=p_\text{min}$
. According to its definition, the decrease of $I$ (of $I_c$) up to this stage of the evolution indicates a loss of classical (quantum) correlations between the input and the output, with a consequent variation of entropy on the environment, or equivalently, a leak of information from the system to it. In the region $1-p_\text{min}<t<1$, both capacities rise again, indicating an information back-flow from the environment to the system. Note that since all the maximally coherent states of Fig. <ref> (a) are pure, their von Neumann entropy is null, so the related quantum mutual information is also null [red lines in Figs. <ref> (b,c)].

§ USING NON-MARKOVIAN DYNAMICS FOR INFORMATION STORAGE

§.§ The quantum vault

As far as they account for the maximal average amount of information that can be transmitted, the channel capacities introduced in the previous section identify resources that can be used to store and retrieve information by state preparation and measurement of a quantum state. This is the concept of the quantum vault, originally introduced in Ref. <cit.>. Consider that Alice wants to store some information by encoding it in a qudit at the input of a MCF. The system evolves by the map $\Lambda_t$, with the resource at each stage being quantified by some channel capacity $K(t)$ (which can be any of the capacities of Sec. <ref>). After the process is finished, i.e., for $t_f=1$ ($p_0=0$), Alice tries to recover her information. The more information that is contained in the final state, the more successful the retrieval is. Now, let us consider that at some stage of the evolution in the interval  $0<t<1$, an eavesdropper Eve can attempt to measure the state of the system and thus steal Alice's information. If Eve cannot retrieve as much information as Alice, then the system constitutes a quantum vault <cit.>. The noisy process $\Lambda_t$ has a key feature that enables the implementation of the quantum vault in the MCF, namely, the non-monotonic behavior of the different capacities introduced in the previous section. In this way, there is a finite interval where $K(t)<K(t_f)$, and less information can be retrieved.

The extension $\Delta t$ of the interval where the capacities increase at the end of the evolution can then be used to quantify the suitability of our system to serve as a QV. As stated in Sec. <ref> the minimum in the capacities is reached when all the allowed permutations are equally probable, i.e. $p_i\equiv p_\text{min}$. In a general $N$-core fiber, if $m$ is the integer quotient of $N$ divided by the size of the subsets $s$, there are $(s!)^m\,(N-ms)!$ possible permutations. Since we consider $t_f=1$, the uniform probability $p_\text{min}$ is equal to $\Delta t$ and takes the value

\begin{equation}\label{eq:pmin}
\Delta t = p_\text{min}=\frac{1}{(s!)^m\,(N-ms)!}\,.
\end{equation}

Until now, we have taken all the error operations to occur with the same probability, i.e. $p_{i\neq 0}=(1-p_0)/3$ in the 4-core fiber with $s=2$, leading to $\Delta t=0.25$. From Eq. (<ref>) it is clear that the value of $\Delta t$ is inversely proportional to the number of available permutations affecting the evolution (we present different cases in the supplementary material). In consequence, for high-dimensional systems, our model approximates a Markovian process. To reach the maximum efficiency and privacy that can be attained with our system, we will consider here a  simplified scenario in the $s=2$ case, where phase permutations can only occur simultaneously on both subsets of cores, i.e., $p_1=p_2=0$ and $p_3=1-p_0$. This allows for a revival time $\Delta t=0.5$.

In the following, we show how a classical message can be stored in an MCF if it undergoes the non-Markovian process $\Lambda_t$. Consider an image composed of several pixels whose color is defined in the four-color model CMYK (cyan, magenta, yellow, black). This pattern of colors can be encoded directly in the quantum state of a photon propagating in the four-core MCF by associating each color with a state $\ket{e_j}$ of a chosen basis for each pixel (see  Fig. <ref>a). Suitable bases are those composed of maximally coherent states with $C_r=\log 4$.  We chose the basis consisting of the four orthogonal states:

\begin{equation}
\begin{split}
&\ket{e_1}=\frac{1}{2}\begin{pmatrix}  1 \\
i \\
i \\
-1 \end{pmatrix}\,,\quad\ket{e_2}=\frac{1}{2}\begin{pmatrix}  1 \\
i \\
-i \\
1 \end{pmatrix}\,,\\
&\ket{e_3}=\frac{1}{2}\begin{pmatrix}  1 \\
-i \\
-i \\
-1 \end{pmatrix}\,,\quad
\ket{e_4}=\frac{1}{2}\begin{pmatrix}  1 \\
-i \\
i \\
1 \end{pmatrix}\,.
\end{split}
\label{eq:mub4}
\end{equation}

which are all equally weighted superpostions of  the states $\ket{\ell}$ ($\ell=0,1,2,3$) corresponding to a single photon propagating in the $\ell$-th core.  As depicted in Fig. <ref>b, associating the basis with the CMYK code provides a straightforward visual representation of mixed states as color mixtures.

Implementation of the quantum vault protocol to store an image using the noisy map $\Lambda_t$ with $p_1=p_2=0$. (a) Each qudit in the 4-dimensional canonical basis is associated with a specific color of the cyan-magenta-yellow-black (CMYK) model. (b) Mapping the CMYK color model into the basis allows a direct equivalence between color mixtures and mixed states: the diagonal of the density matrix in this basis is the CMYK code of the respective color. (c) At the input stage, Alice stores a message as an image whose pixels have one of the base colors. As the system evolves, the distinguishability between different state pairs in the basis decreases, which is reflected by a decrease in the REC. At $t=1-p_{\rm min}$, the capacity is minimal, rendering the message of the image unreadable. Non-Markovian dynamics in the final stage of the evolution leads to an increase in distinguishability, allowing Alice to retrieve her information even if the final state of each pixel is different from its initial state (colors are swapped in the retrieved image). In the insets, the top and bottom images correspond to the predicted and measured images.

Let us consider that Alice stores a four-color image whose pixels correspond exactly to one of the four CMYK colors, as shown in Fig. <ref>c. As the evolution proceeds, each of these pixels is independently affected by the noisy map $\Lambda_t$, so they turn into a mixed state $\rho_t$.
In Fig. <ref>c, we plot the entropy of coherence $C_r$ along the evolution under $\Lambda_t$, with the respective images retrieved at specific stages in the simplified case $p_1=p_2=0$. This scenario allows $C_r$ to rise back to its initial value at the end of the evolution, as shown by the thick curve in the figure, so the information backflow is complete. This is reflected by the output image, all of whose pixels are in a pure color of the CMYK base, just like the image stored by Alice. The minimum of the capacity $C_r$ is reached when $p_3=p_0=1/2$ ($t=0.5$) all the pixels at this point are in one of two mixtures, rendering the message in the image unreadable.

§.§ Experimental design

Four-arm Mach Zehnder interferometer based on MCF and PMs to simulate non-markovian dynamics. A CW-1546 nm laser, 55 dB neutral attenuators, and an intensity modulator generate a weak coherent state with a mean photon number of $\mu=0.15$ at a repetition rate of  2 MHz. The single photon state is transmitted through an MCF DMUX and an MCF BS, generating a 4-dimensional state encoded on the 4 possible cores inside the MCF. An MCF link connects a DMUX to spatially separate the paths to prepare relative phases via PMs. During a 100 ms time interval, an FPGA-based active control system adjusts the voltages on the PMs to generate the state $\ket{e_i}$. Immediately thereafter, FPGA1 randomly adds a phase to the existing one to create the dynamical map $\Lambda_t$ with probability $t=1-p_0$. In the characterization stage, a second FPGA performs measurements over 5 MUBs to reconstruct the resulting density matrix. This is done by closing the interferometer with a MCF BS and collecting the single photon counts with four APDs.

In order to demonstrate a NM open quantum system, we exploit the use of an MCF in a 4-arm Mach-Zehnder interferometer to prepare a $d=4$ qudit and statistically apply a set of unitary operations $U_i$ on the state. By doing this, we simulate the state transition maps, revealing non-Markovian dynamics. The experiment consists of three main parts: state preparation, map implementation, and quantum state tomography characterization, and is presented in Fig. <ref>.

Qudit Source. A CW-1546 nm laser attenuated by neutral filters and a fiber intensity modulator controlled by a Field Programmable Gate Array (FPGA1) prepares light pulses with a repetition rate of 2 MHz, which are propagated by Telecom SMF-28 fibers. The combination of these three devices allows the generation of weak coherent pulses with an average photon number of $\mu=0.15$, which provides a good approximation to a single photon source since $95.7\%$ of the non-vacuum pulses are single photons. Next, an MCF demultiplexer (DMUX) integrates the SMF into a single core of a four-core MCF, while the other three cores remain unconnected. A backbone device in the setup is the MCF Beam Splitter (MCF-BS), which enables the superposition state on the propagation paths of the MCF<cit.>. An MCF-BS is fabricated by heating and stretching a section of the MCF, creating a tapered subsection of the fiber. This geometric deformation of the MCF structure changes the distance between the cores, allowing crosstalk between them. The MCF-BS was recently characterized by Cariñe et al. <cit.>, who proved that the output distribution corresponds to a $d=4$ Haddamard matrix $H_4$, allowing the generation of superposition states. Thus, the qudit state after MCF-BS is $\ket{\psi_0} = ( 1/2 ) \sum_{\ell=1}^4 e^{i\phi_\ell} \ket{\ell}$, where $\phi_k$ are individual phase drifts that will be addressed in the next stage.

Map Implementation. To extinguish parasitic phase drifts in state preparation and to apply the probabilistic dynamical map, we include a phase stabilization stage in the procedure. A set of three LiNbO$_3$ phase modulators (PM) prepares different sets of phases at different time intervals. For the 100 ms stabilization interval, the FPGA1 implements a stabilization procedure <cit.> to compensate for phase drifts between cores. This is done by optimizing the measurement probability in the same state prepared by Alice. This procedure is repeated until a target minimum output probability is reached, and a set of compensation phases $\phi_{\ell,s}$ is identified for each PM. Once this is achieved, the dynamic map $\Lambda_t$ is implemented for a 100 ms interval. This is done by applying unitary $U_i$ transformations (Fig. <ref>) over the state. Since the $U_i$ are phase flips, the PMs implement a phase of the form $\phi=\phi_\ell+\phi_{\ell,s}+\phi_{U_i^\ell}$, with probability $1-p_0=t$.

Characterization. A quantum state tomography (QST) with five mutually unbiased bases (MUBs) in $d=4$ estimates the final qudit density matrix <cit.>, allowing the calculation of the required channel capacity. FPGA2 prepares the corresponding phases of each of the four phase-modulated MUBs in the characterization stage, each consisting of four orthogonal states of the form

\begin{equation}\label{eq:psio}
\ket{\psi_j}=\frac{1}{2}\left(e^{i\alpha_j}\ket{0}+e^{i\beta_j}\ket{1}+e^{i\gamma_j}\ket{1}+e^{i\delta_j}\ket{3}\right).
\end{equation}

One of the bases chosen was that of Eq. (<ref>). The fifth MUB (logical basis) was measured before each round of the experiment. The results were used to compute the experimental density matrix $\rho_{exp}$ using a maximum likelihood estimation routine <cit.>, and then the different channel capacities, such as the relative entropy of coherence.
To verify the channel dynamics, the evolution of the input state $\ket{e_2}$ was studied by applying a series  $U_i$'s probabilistically. This was repeated 200,000 times, so that the output state could be characterized using QST as described above. The red dots in Fig. <ref>c correspond to the evolution of the relative entropy of coherence  $C_r$, calculated from the output state $\rho_{exp}$. The experimental dynamics follow the theoretical prediction very closely, serving as a demonstration of the revival behavior of this channel capacity. To compute the error bars for the experimental data set, we performed a Monte Carlo routine by varying a Poisson distribution centered on the experimental counts and computed 1000 simulated density matrices. Error bars correspond to one standard deviation of the ensemble.
To demonstrate the open system as a QV, we sequentially prepared a set of 1024 qudits, whose state represents the CYMK color code corresponding to the 1024 pixels in the upper left inset of Fig. <ref>c. The output image was recovered by measurement in the encoding basis (<ref>).  In Fig. <ref>, we show a comparison between the theoretical and experimental images at three crucial steps of the evolution. The average fidelity of the reconstructed qudits is $98.5\%$, so the difference between the two images is almost negligible. It is clear that at $t = 0.5$ the encoded information cannot be retrieved, but as $t$ increases, the message becomes readable again, resulting in an effective backflow of information.
From one point of view, the experiment above is a simulation of open system dynamics.  However, the fact that the randomly applied unitaries applied by the “environment" are actually performed by the user allows the NM to be exploited in a controlled way. Let us illustrate this by outlining how the QV procedure applied to our MCF setup can be used to hide information efficiently. Suppose the photons are sent through an MCF fiber that serves as a quantum memory. Let us apply phases using the relevant probability distribution corresponding to $p_{min}$. Then, the  information that an eavesdropper can recover is minimum, as shown in Fig. <ref>c.  Using a classical register, we can make a record of the particular unitary $U_i$ that was applied to each photon sent.   When the photons exit the fiber, if we have access to the classical register, we can look up which unitary was applied to each photon and apply a second unitary chosen so that the complete evolution corresponds to the non-Markovian channel with $t=1$, where information recovery is maximum. A user who does not have access to the classical register does not know which unitaries have been applied to which photons and thus cannot generate a non-Markovian channel.  In this way,  the information hidden in the vault can only be recovered by users with access to the classical memory register.

§ CONCLUSIONS

Applying quantum technology in practical scenarios is bound to decoherence and dissipation, which are intrinsically related to environmental noise and outweigh the advantages that quantum systems could provide. In this context, studying open quantum systems and non-Markovianity is a relevant topic, as it could provide bright insights for fault-tolerant quantum communication tasks, real-world system evolution, or even practical quantum memories <cit.>.

We have shown that multicore optical fibers provide a suitable platform to implement NM maps through the probabilistic application of simple unitary operations. The set of errors used in our analysis serves as a reliable proving ground that can be extended to include more noisy operations affecting open systems in actual communication networks. In particular, the maps in our model are formed by unital channels, which in the qudit case have different properties than their qubit couterpart <cit.>. We have shown that the evolution of several channel characteristics deviates from the Markov hypothesis, leading to an interval where an information backflow from the environment to the system is observable as a revival of different channel capacities. We have shown that such information backflow enables the use of the system as a quantum vault, where information is harder to retrieve along the evolution than at the end of it. The quantum vault is a versatile protocol not only due to its simplicity but also because it can be extended to other operational capacities besides the ones studied here. For instance, the effect of NM on state distinguishability can find a direct application in bipartite secret sharing or secure key distribution.

The full revival of the entropy of coherence in the simplified scenario ($p_1=p_2=0$) is also observed for the other capacities. Regarding the coherent information, this result is especially relevant since it allows for protocols related to quantum data hiding. First, the channel at $t=1$ fulfills the equality $I_c(\Phi_{p_0=1},\rho)=S(\rho)$, i.e., the coherent information at the output reaches its maximal value, namely, the entropy of the input state. The full map $\Lambda_t$ then becomes a private quantum channel <cit.> (this result is also interpreted as a necessary condition for the channel to achieve perfect error correction <cit.>). Indeed, coherent information provides a direct link between the QV protocol and optimal hiding of quantum information <cit.>. From its definition, a decay in $I_c$ is equivalent to a loss of quantum correlations with a reference system not affected by the evolution. As far as the environment is inaccessible to Alice and Eve, correlations are hidden. The utilization of the environment for this purpose is the pivotal element of the QV. In our optimal case with $p_1=p_2=0$, the backflow is complete, so there is no information leak to the environment after the evolution.

A further study of transition map models in multicore fibers could be useful for both classical and quantum communication tasks since a deep understanding of the noise contribution could lead to the implementation of noise passive filters, thus reducing the complexity and cost of communication infrastructure networks.

§ ACKNOWLEDGMENTS

This work was supported by Fondo Nacional de Desarrollo Científico y Tecnológico (ANID) (Grants No. 3200779, 1200266, 1231940, 1200859) and ANID – Millennium Science Initiative Program – ICN17_012. K.S. acknowledges financial support from project UCO 1866. LP was supported by ANID-PFCHA/DOCTORADO-BECAS-CHILE/2019-772200275, the CSIC Interdisciplinary Thematic Platform (PTI+) on Quantum Technologies (PTI-QTEP+), and the Proyecto Sinérgico CAM 2020 Y2020/TCS-6545 (NanoQuCo-CM). N.K.B. acknowledges financial support from CAPES, CNPq Brazil (Universal Grant No. 406499/2021-7), and FAPESP (Grant 2021/06035-0). N.K.B. is part of the Brazilian National Institute for Quantum Information (INCT Grant 465469/2014-0).

§ SUPPLEMENTARY MATERIAL

§.§ Coherent information and channel privacy

We will now discuss in more detail the original definition of coherent information and its relation to the privacy of a quantum channel. Consider that in addition to the original system $A$, we have a replica of it (either physical or just a mathematical device) that serves as a reference or ancilla system $R$.  Consider a quantum channel $\Phi$ which acts only in the system $A$ and describes its interaction with the environment $E$. This channel transforms the initial state $\rho_A$ of $A$ into a final state $\rho'_A$. Generally, $\rho$ is a mixed state but can be purified in the bipartite Hilbert space ${\mathcal H}_A\otimes{\mathcal H}_R$. The purification of $\rho$ is given by the entangled state $\ket{\Psi_{AR}}$. We can express the evolution of the extended system $AR$ by means of the superoperator $\Phi\otimes {\mathds{1}}_R$ to obtain

\begin{equation}
\rho_{AR}'=\left\lbrace \Phi\otimes {\mathds{1}}_R \right\rbrace (\ket{\Psi_{AR}}\bra{\Psi_{AR}})\,.
\end{equation}

This translates into the following scheme:

[thick] (0,0) node(rhoa)[circle,draw,fill=white]$\rho_A$ –++(3,0) node(rhoap)[circle,draw,fill=white,inner sep=2.5pt]$\rho_A'$;
[thick,latex-latex] (1.5,0) –(1.5,-1) node[below,draw]$E$ node[midway,right]$\Phi$;
[thick,dotted](rhoa) –+(0,1) node[draw,solid,fill=white]$R$;

Now, consider the entropy $S(\rho_{AR}')$ of the bipartite system composed of the output and the ancilla. Since this subsystem is not affected by the evolution, $S(\rho_{AR}')$ can still be considered as an intrinsic property of the system $A$, depending only on its initial state $\rho_A$ and the channel $\Phi$ <cit.>. Still, if the environment is initially in a pure state, we can identify $S(\rho_{AR}')$ with its entropy after evolution. The previous statements allow us to define the entropy exchange,

\begin{equation}
\end{equation}

as the amount of information exchanged between $A$ and the environment. The last identity comes from the fact that the tripartite state in the extended system $ARE$ remains pure along the evolution. Upon the entropy exchange, we define the two intrinsic quantities introduced in Sec. <ref>, namely, the quantum mutual information and the coherent quantum information. This last quantity, given by

\begin{equation}
\end{equation}

measures the degree of quantum correlation retained by $R$ and $A$ during the evolution, so it is considered as a proper measure of the nonclassicality of the final state $\rho_{AR}'$. The previous definitions provide a formal connection between our map and the task  of quantum data hiding <cit.>. In this context, removing correlations between $A$ and $R$ allows to hide quantum information by transferring them to the environment (cf. <cit.>). This is quantified by the entropy exchange and the associated reduction of the coherent information.

Consider the evolution of the coherent information $I_c$ under the map $\Lambda_t$, shown in Fig. <ref> (e,f). This is obtained by taking the chaotic initial state $\rho_A={\mathds{1}}/4$, whose purification is given by a maximally entangled state. As $t$ increases, the corresponding quantum channel $\Phi_{p_0}$ reduces its coherent information due to the entropy exchange: the information is hidden in the environment. This highlights the key role of the inaccessible environment in our model for implementing the quantum vault. As the system interacts with the environment, neither Alice nor Eve can obtain the information stored in its initial state.

Conveniently enough, further evolution of the system in the non-Markovian regime allows for partial o complete information recovery. In the optimal case [Fig <ref> (a)] the information is fully restored, so the coherent information reaches its maximal value at the output:

\begin{equation}\label{eq:Icmax}
I_c(\rho_A')=I_c(\rho_A) =s(\rho_A)\,.
\end{equation}

This way, it is shown that in the optimal scenario, $\Lambda_t$ allows for a quantum private channel in the limit $p_0\longrightarrow 0$, after performing quantum data hiding during the evolution, maximally at $p_0=p_{\rm min}$. The previous results can be expressed by means of another information quantity obtained from a different partition of the extended system $ARE$, namely, the loss given by

\begin{equation}
\begin{split}
L(\rho,\Phi)&=S(\rho_R) + S(\rho_E') - S(\rho_{RE}')\\
&= S(\rho_A) + S(\rho,\Phi) -S(\rho_A')\,.
\end{split}
\end{equation}

Implementation of the quantum vault protocol to store an image, using the noisy map $\Lambda_t$ in three different scenarios. Each qudit in the 4-dimensional canonical basis is associated with a specific color of the cyan-magenta-yellow-black (CMYK) model. At the input stage, Alice stores a message as an image whose pixels have one of the base colors. As the system evolves, distinguishability between different pairs of states in the basis is reduced, which is reflected by a decrease in the quantum mutual information (which quantifies the capacity of a quantum channel to convey a classical message). At $t=1-p_{\rm min}$, the capacity is minimal, rendering the message of the image unreadable. Non-Markovian dynamics in the final stage of the evolution lead to an increase in the quantum mutual information with the respective revival of distinguishability, allowing Alice to retrieve her information. In the simplified scenario where $p_1=p_2=0$ (thick line), a full recovery is possible, even if the final state of each pixel is different from their initial state (colors are swapped in the retrieved image). In the uniform scenario $p_1=p_2=p_3=(1-p_0)/3$ (thin line), there is a partial revival of distinguishability, allowing one to read the message. Finally, the unrestricted scenario with $s=4$ (dashed line), where permutations between all the cores can occur, enables complete encryption of the image at the minimum, at the expense of a low recovery in the output.

Notice that it corresponds to the quantum mutual information between the input and the environment. The loss is non-negative, and it is null only if the state $\rho_{RE}'$ is separable, i.e. in the absence of correlations between the $R$ and $E$, meaning that no information has leaked into the environment. Indeed, it has been shown that $L(\rho,\Phi)=0$ is equivalent to Eq. (<ref>), allowing for a completely private channel <cit.>. This is attained in our optimal case, where all the information has flown back from the environment  to the system at the end of the evolution.

§.§ QV protocol in different scenarios

The implementation of the QV in Sec. <ref> considers the optimal scenario in the $s=2$ case, where phase permutations can only occur simultaneously on both subsets of cores ($p_1=p_2=0$ and $p_3=1-po$). With this restriction, the relative entropy of coherence increases back to its initial value, allowing a complete information backflow at the end of the evolution. In Fig. <ref>, we compare how the QV performs in this optimal case and more unrestricted scenarios. The information backflow in the former case is reflected by the output image, all of whose pixels are in a pure color of the CMYK base, just like the image initially stored by Alice. In the uniform case $p_{i\neq 0}=(1-p_0)/3$ (thin filled curve) evolution until $p_0=1/4$ put the pixels in the same two mixtures as before. However, their state is not pure at the end of the evolution so Alice can recover the information only partially. Finally, the more realistic case $s=4$, where permutations can occur between all the four cores (dashed curve), enables full encryption of Alice's message, with all the base states evolving to the same mixed state, or equivalently, all the pixels in the image becoming gray. This comes at the cost of a very poor recovery at the end of the evolution, reflected by a barely readable message in the output image. This is related to the fact that the value of $p_\text{min}$ in (<ref>) is inversely proportional to the number of available permutations affecting the evolution.