# Generation of Entangled Photons via Cooper Pair Recombination in
Noncentrosymmetric Quantum Wells

Mehdi Biderang  Department of Physics, University of Toronto, 60 St. George
Street, Toronto, Ontario, M5S 1A7, Canada  DelQuanTech Inc., 1130-500 Doris
Ave., Toronto, Ontario, M2N 0C1, Canada  Asia Pacific Center for Theoretical
Physics (APCTP), Pohang, Gyeongbuk, 790-784, Korea  Erfan Hosseini  Institute
for Quantum Computing, University of Waterloo, 200 University Ave W, Waterloo,
Ontario, N2L 3G1, Canada  Alireza Akbari  Institut für Theoretische Physik
III, Ruhr-Universität Bochum, D-44801 Bochum, Germany  Asia Pacific Center for
Theoretical Physics (APCTP), Pohang, Gyeongbuk, 790-784, Korea  DelQuanTech
Inc., 1130-500 Doris Ave., Toronto, Ontario, M2N 0C1, Canada

###### Abstract

We theoretically explore the generation of entangled two-photon pairs through
Cooper pair recombination within a noncentrosymmetric [001]-quantum well
superconductor. Superconducting state is induced into the quantum well via
proximity effects, and featuring an admixture of Rashba and Dresselhaus
antisymmetric spin-orbit couplings. Our investigation highlights that the
highest achievable purity of entangled photon pairs emerges within scenarios
involving pure singlet Cooper pairs. Specifically, the conventional $s$-wave
gap function within the singlet pairings achieves the highest purity levels.
Furthermore, our findings underscore the significance of reducing spin-triplet
pairing amplitudes to attain entangled states of superior purity. This
reduction can be achieved by diminishing the amplitude of antisymmetric spin-
orbit couplings. In addition to purity considerations, our study delves into
the population of two-photon states. We observed that states featuring $s+p$\-
and $d_{x^{2}-y^{2}}+p$-wave Cooper pairings exhibit the highest population
values among the generated entangled two-photon states within a
noncentrosymmetric superconducting quantum well.

##  I Introduction

Entanglement stands as a cornerstone in modern quantum technologies due to its
pivotal role in a spectrum of applications, notably in quantum cryptography
[1, 2, 3, 4, 5, 6], computing [7, 8, 9, 10, 11], communication [12, 13, 14]
and metrology [15, 16, 17, 18]. Achieving reliable quantum entanglement
sources can make big advancement in classical sensing tools like radars and
lidars, which are using entangled photons to enhance the detection capability
of the system over the classical limit [19, 20, 21, 22]. The quest for robust
sources capable of producing entangled photon pairs has become paramount for
advancing these applications. In this pursuit, the convergence of
semiconductor and superconductor technologies has birthed an interdisciplinary
domain termed superconducting optoelectronics [23, 24, 25]. This field has
witnessed significant strides with the creation of hybrid devices, including
but not limited to, superconducting light-emitting diodes, quantum dots [26,
27, 28], and superconductor-based waveguide amplifiers [29, 30, 31].

A captivating avenue proposed for the generation of entangled photons involves
the construction of a P-N-S heterostructure, wherein the induction of
superconductivity in the n-type semiconductor through the proximity effect
holds promise [32]. Furthermore, the strategic integration of a quantum well
structure within the semiconductor layers not only facilitates the production
of entangled photon pairs but also enables the generation of pure
polarization-entangled photons [33]. These design intricacies exploit the
Rashba and Dresselhaus spin-orbit couplings (SOCs), which play a pivotal role
in manipulating the parity of the superconducting order parameter [34, 35].

Recent investigations have delved into the manifestations of Cooper-pair-based
two-photon gain in semiconductor-superconductor structures. These studies have
unveiled a broadband enhancement in ultrafast two-photon amplification,
delineating a comprehensive quantum-optical model encompassing both singly-
and fully-stimulated two-photon emission [30]. Moreover, the exploitation of
inherent angular momentum entanglement within the superconducting state has
showcased the generation of polarization-entangled photons through Cooper-pair
luminescence, circumventing the necessity for isolated emitters within
semiconductors [33].

Further explorations into the electroluminescence and photonic attributes of
forward-biased p-n junctions proximal to superconducting leads have unearthed
notable enhancements in electroluminescence within specific frequency ranges
in the presence of superconductivity [36]. The strategic coupling of
superconducting contacts with quantum dots has not only yielded intensified
luminescence at temperatures below $T_{c}$ but has also yielded higher purity
in photon entanglement, mitigating the adverse effects of excitonic energy
level splittings [37].

Moreover, investigations into the purity of generated photons have ventured
into triplet superconductors with fixed ${\bf d}_{{\bf k}}$, revealing
intriguing directional dependencies. Specifically, the induction of triplet
pairing via Rashba spin-orbit coupling has been identified as a mechanism for
generating pure entangled photons when the photon polarization axis aligns
parallelly with ${\bf d}_{{\bf k}}$. Conversely, induced triplet pairing
within a singlet superconductor has exhibited a degradation in state purity.
Strikingly, induced singlet pairing in a triplet superconductor has shown an
amplification in the production of entangled pairs [32]. These findings
underscore the intricate interplay between superconductivity, spin-orbit
interactions, and resultant entanglement properties within these
heterostructures.

The complex interdependencies between superconductivity, semiconductor
heterostructures, and associated quantum effects form a rich foundation for
further exploration. This confluence opens avenues for advancing both
fundamental understanding and practical applications in the realm of quantum
information processing [38, 39, 40]. The emergence of Rashba and Dresselhaus
SOCs in zinc-blende structures and quantum wells stands as a distinctive
feature that significantly shapes the electronic properties of these systems
[41, 42]. In noncentrosymmetric crystals like zinc-blende semiconductors, the
lack of inversion symmetry gives rise to the Rashba SOC, an effect resulting
from structural asymmetry. The Rashba term induces a momentum-dependent
splitting of electronic bands, impacting charge carriers’ spin dynamics. In
the presence of an asymmetric potential gradient, such as that encountered at
interfaces or within quantum wells, the Dresselhaus SOC becomes prominent.
Originating from structural asymmetry perpendicular to the interface, the
Dresselhaus term contributes an additional layer to the spin-orbit
interaction. Both Rashba and Dresselhaus SOCs play a crucial role in the
manipulation and control of spin states, paving the way for novel electronic
and optoelectronic phenomena in these materials [43, 44, 45, 46, 47].

Here, we study the effect of combination of Rashba and Dresselhaus SOCs and
varying the relative amplitude of singlet to triplet superconductivity on the
entanglement properties of the generated pairs of photons during the process
of Cooper pair recombination within a semiconductor-superconductor
heterostructure.

##  II Theoretical Model

The system under investigation is a P-N-S heterostructure, a configuration
expounded in the approach outlined in Ref. [32]. As depicted in Fig. 1, this
composite structure merges a superconductor with a semiconducting medium (p-n
junction). Within this arrangement, superconductivity manifests in the
semiconducting region through the infusion of Cooper pairs from the
superconducting contacts, facilitated by the proximity effect. The divergence
between the light-hole (LH) and heavy-hole (HH) bands assumes a pivotal role
in generating pure, entangled photons by means of the recombination of Cooper
pairs with the LH band [33].

###  II.1 Model Hamiltonian

The intrinsic physics of a p-n junction, encompassing the involvement of
photons, holes, and electrons, is described by the following Hamiltonian [32,
33]

$\displaystyle{\cal H}_{0}$ $\displaystyle={\cal H}_{\rm ph}+{\cal H}_{\rm
v}+{\cal H}_{\rm c}$ (1) $\displaystyle=\sum_{{\bf q},p}\omega_{{\bf
q}}a^{\dagger}_{{\bf q},p}a_{{\bf q},p}+\sum_{{\bf k},J}\varepsilon_{{\bf
k}}h^{\dagger}_{{\bf k},J}h_{{\bf k},J}+\sum_{{\bf k},j}\xi_{{\bf
k}}c^{\dagger}_{{\bf k},j}c_{{\bf k},j},$

where ${\cal H}_{\rm ph}$, ${\cal H}_{\rm v}$, and ${\cal H}_{\rm c}$
represent the Hamiltonian of photons, holes, and conduction electrons,
respectively. In this context, $a^{\dagger}_{{\bf q},p}$ represents the
creation operator for a phonon with momentum ${\bf q}$ and polarization $p=\pm
1$. Additionally, the operators $h^{\dagger}_{{\bf k},J}$ and
$c^{\dagger}_{{\bf k},j}$ generate a hole and an electron with momentum ${\bf
k}$, respectively, located in the heavy-hole and conduction bands. The angular
momentum of electrons and holes is denoted by $j=\pm 1/2$, and $J=\pm 3/2$,
respectively. The conservation of angular momentum during processes such as
electron-hole recombination or pair creation requires that $p=J+j$.

The process of electron-hole recombination and the generation of photons are
described by the light-matter interaction within the dipole approximation,
which can be expressed as:

${\cal H}_{\rm int}=\sum_{{\bf k},{\bf q},J,\sigma}{\cal B}_{{\bf k},{\bf
q}}a^{\dagger}_{{\bf q},\sigma}h_{{\bf q}-{\bf k},-J}c_{{\bf k},J+\sigma}+{\rm
H.c.},$ (2)

where, ${\cal B}_{{\bf k},{\bf q}}$ denotes the matrix elements governing the
scattering process between a state with momentum ${\bf k}$ and another state
with momentum ${\bf q}$, and vice versa.

Figure 1:  The schematic geometry of a semiconductor-superconductor
heterostructure (P-N-S), where superconductivity is induced within the n-type
semiconducting layer owing to the proximity of the s-wave superconductor.
Vectors $\hat{{\bf p}}$ and ${\bf d}_{{\bf k}}$ symbolize the photon’s
polarization axis, and the direction of spin-triplet pairing, respectively,
which have an angle $\varsigma_{{\bf k}}$.

###  II.2 Spin-orbit coupling and superconducting state

The system of interest exhibits a two-dimensional (2D) square lattice
structure lacking a center of inversion. More specifically, our focus lies on
a Zinc-Blend structure confined in the $[001]$ direction, forming a two-
dimensional configuration. The effective Hamiltonian describing the behaviour
of the conduction electrons is established through a spin-orbit coupled
single-band tight-binding model, accounting for nearest-neighbour hopping
interactions. Consequently, the expression for conduction electrons is defined
by [48, 49]

${\cal H}_{\rm c}(\vartheta)=\sum_{k,jj^{\prime}}\Big{[}\epsilon_{{\bf
k}}\hat{\sigma}_{0}+\lambda{\bf g}_{{\bf
k}}(\vartheta)\cdot\hat{\bm{\sigma}}\Big{]}_{jj^{\prime}}c^{\dagger}_{{\bf
k}j}c_{{\bf k}j^{\prime}}+{\rm H.c.},$ (3)

The operator $c^{\dagger}_{{\bf k}j}$ ($c_{{\bf k}j}$) creates (annihilates)
electrons with linear and angular momenta ${\bf k}$, and $j=\pm 1/2$ within
the conduction band. Here, $j=1/2$ ($j=-1/2$) refers to an electron state
$\mid\uparrow\rangle$ ($\mid\downarrow\rangle$). The term $\epsilon_{{\bf
k}}=-\mu-2t(\cos k_{x}+\cos k_{y})$ represents the kinetic energy, with $\mu$
as the chemical potential and $t$ as the amplitude for first-neighbor hopping.
Moreover, $\hat{\sigma}_{0}$ denotes the $2\times 2$ identity matrix, while
$\hat{\sigma}_{i}$ ($i=x,y,z$) represents the $i^{\rm th}$ component of the
Pauli matrices in the spin space. The antisymmetric vector

${\bf g}_{{\bf k}}(\vartheta)=\alpha(\vartheta){\bf g}^{\rm R}_{{\bf
k}}+\beta(\vartheta){\bf g}^{\rm D}_{{\bf k}}$ (4)

characterizes the strength $\lambda$ of the spin-orbit coupling, combining the
Rashba and Dresselhaus effects. The dimensionless coefficients
$\alpha(\vartheta)=\cos\vartheta$ and $\beta(\vartheta)=\sin\vartheta$
indicate the magnitudes of the Rashba and Dresselhaus SOCs, respectively. The
parameter $\vartheta\\!\in\\![0,\pi/2]$ modulates their relative
contributions. In our model, the Rashba and Dresselhaus ${\bf g}$-vectors are
given by,

${\bf g}^{\rm R}_{{\bf k}}=(\sin k_{y},-\sin k_{x});\quad{\bf g}^{\rm D}_{{\bf
k}}=(\sin k_{x},-\sin k_{y}).$ (5)

In our calculations, we perform a projection of the conduction band
Hamiltonian onto an effective pseudospin-1/2 model through the transformation
${\cal P}_{\pm}{\cal U}{\cal H}^{\prime}_{\rm c}{\cal U}{\cal P}_{\pm}$. Here,
${\cal P}_{\pm}$ projects the conduction band onto the helical bands with
$\xi=\pm 1$, while ${\cal U}$ diagonalizes the Hamiltonian of the conduction
electrons. The energy dispersion in the presence of SOC is defined by

$\varepsilon_{{\bf k},\xi}(\vartheta)=\epsilon_{{\bf
k}}+\xi\lambda\sqrt{\sin^{2}\\!k_{x}+\sin^{2}\\!k_{y}\\!+2\sin
2\vartheta\sin\\!k_{x}\sin\\!k_{y}},$ (6)

where $\xi={\bf k}\cdot{{\bf{j}}}/|{\bf k}|=\pm 1$ stands for the helicity of
the helical bands. When $\vartheta=0$ ($\pi/2$), corresponding to the pure
Rashba (Dresselhaus) case, the energy dispersion simplifies to

$\varepsilon_{{\bf k},\xi}(\vartheta=0,\pi/2)=\epsilon_{{\bf
k}}+\xi\lambda\sqrt{\sin^{2}\\!k_{x}+\sin^{2}\\!k_{y}},$ (7)

indicating a complete $C_{4v}$ point group symmetry of the crystal lattice. We
show the corresponding energy dispersion of these helical bands along the
$\Gamma{\rm KM}\Gamma$ path in the Fig. 2(a). However, the energy dispersion
is expressed as

$\varepsilon_{{\bf k},\xi}(\vartheta=\pi/4)=\epsilon_{{\bf k}}+\xi\lambda|\sin
k_{x}+\sin k_{y}|$ (8)

at $\vartheta=\pi/4$, where Rashba and Dresselhaus SOCs contribute equally.
Under this condition, the Hamiltonian symmetry reduces to $C_{2v}$, leading to
helical Fermi surfaces intersecting at $k_{y}\\!=\\!-k_{x}$.

Figure 2:  (a) Energy dispersion of helical bands along the $\Gamma$KM$\Gamma$
path for pure Rashba/Dresselhaus SOC. (b) Density of states (DOS) for
conduction electrons showcasing scenarios at $\vartheta=0$ (pure Rashba,
black) and ($\pi/2$) (pure Dresselhaus, black) as well as at $\vartheta=\pi/4$
(green), representing the equal contributions of Rashba and Dresselhaus SOCs.
For a clearer illustration, the black curve is shifted by 0.1. (c) and (d)
represent the Fermi surface topology at $\langle n\rangle=0.8$ for
$\vartheta=0$ and $\vartheta=\pi/2$, respectively. (e) and (f) Similar
visualizations for a filling of $\langle n\rangle=1.2$.

In Fig. 2(b), the density of states (DOS) of the noncentrosymmetric n-type
semiconductor is presented in its non-superconducting state for three
different scenarios: $\vartheta=0$ represents the pure Rashba case,
$\vartheta=\pi/4$ corresponds to equal contributions of Rashba-Dresselhaus,
and $\vartheta=\pi/2$ signifies the pure Dresselhaus spin-orbit coupling. The
Fermi surface (FS) topology is depicted in Figs. 2(c) and 2(e) for pure Rashba
or pure Dresselhaus SOC at filling fractions $\langle n\rangle=0.8$ (hole-
doping) and $\langle n\rangle=1.2$ (electron-doping), respectively. A
comparison with the DOS reveals the occurrence of van Hove singularities
(Lifshitz transitions) at half-filling, leading to changes in the FS topology.
Lastly, Figs. 2(d) and 2(f) exhibit the FS texture for the aforementioned
filling levels in the presence of equal contributions of Rashba and
Dresselhaus SOCs. The prominent consequence of the antisymmetric SOC is the
lifting of the two-fold spin degeneracy and the formation of helical bands
characterized by helicity $\xi=\pm 1$.

Our objective is to investigate the impact of different SOC contributions at
specified filling fractions and various types of superconducting gap
symmetries. This exploration aims to assess the purity of the generated
entangled photons resulting from the breakdown of Cooper pairs. The broken
inversion symmetry within the crystal structure results in the violation of
parity. Consequently, in the superconducting ground state, both spin-singlet
and spin-triplet pairings will simultaneously condense. It is widely
acknowledged that within the spin space, the superconducting gap function
$\hat{\Delta}_{{\bf k}}$ can be represented by a $2\times 2$ matrix,
formulated as follows:

$\hat{\Delta}_{k}=\hat{\Delta}^{\rm sing}_{k}+\hat{\Delta}^{\rm trip}_{k}={\rm
i}\Big{[}\psi_{{\bf k}}\hat{\sigma}_{0}+{\bf d}_{{\bf
k}}\cdot\hat{\bm{\sigma}}\Big{]}\hat{\sigma}_{y}.$ (9)

In this expression, $\psi_{{\bf k}}$ and ${\bf d}_{{\bf k}}$ denote the
momentum-dependent spatial components of the spin-singlet and spin-triplet
constituents of the superconducting order parameters, respectively [50]. These
components are characterized as even and odd functions with respect to the
momentum ${\bf k}$ [51]. The projection of the superconducting gap function
onto the helical bands gives rise to $\Delta_{{\bf k},\xi}=\psi_{{\bf
k}}+\xi|{\bf d}_{{\bf k}}|$. The orientation of ${\bf d}_{{\bf k}}$ concerning
the photon polarization axis $\hat{{\bf p}}$ is determined by the angle
$\varsigma_{{\bf k}}$. The permissible irreducible representations of the
$C_{4v}$ point group within a square lattice include $A_{1}$, $B_{1}$, and
$B_{2}$. The corresponding translationally invariant structure factors can be
found in Table 1.

Table 1: Structure factors ($f_{{\bf k}}$) of the singlet and triplet superconducting pairings for the allowed irreducible representations of the $C_{4v}$ point group in the presence of antisymmetric SOC. Irrep. | l | Symmetry | $f_{{\bf k}}$
---|---|---|---
$A_{1}$ | 0 |  $s$-wave | $1$
$A_{1}$ | 0 |  $s^{*}$-wave | $\cos k_{x}+\cos k_{y}$
$B_{1}$ | 2 |  $d_{xy}$-wave | $\sin k_{x}\sin k_{y}$
$B_{2}$ | 2 |  $d_{x^{2}-y^{2}}$-wave | $\cos k_{x}-\cos k_{y}$

Additionally, we assume that in the triplet channel, electrons are exclusively
condensed in p-wave pairing. Hence, the potential scenarios for the
superconducting ground state encompass $s+p$, $s^{*}+p$, $d_{x^{2}-y^{2}}+p$,
and $d_{xy}+p$ pairings. By introducing the dimensionless coefficient $0<r<1$
to quantify the contribution of even and odd parity pairings on the Fermi
surface, the superconducting gap functions in the singlet and triplet channels
are expressed as

$\psi_{{\bf k}}=r\Delta_{0}f_{{\bf k}};\quad{\bf d}_{{\bf
k}}=(1-r)\Delta_{0}{{\bf g}}_{{\bf k}}.$

It is widely recognized that the stability of triplet superconductivity is
contingent on ${\bf d}_{{\bf k}}||{\bf g}_{{\bf k}}$ [50]. Here, $\Delta_{0}$
represents the magnitude of the superconducting order parameter, and
${\hat{{\bf g}}_{{\bf k}}}\\!\\!=\\!{\bf g}_{{\bf k}}/|{\bf g}_{{\bf k}}|$
signifies the unit vector associated with the direction of the SOC ${\bf
g}$-vector. The structure factor $f_{{\bf k}}$ encapsulates the orbital
angular momentum of pairing within the superconducting gap, as elucidated in
Ref. [52].

###  II.3 Two-photon density matrix

Our goal is to learn more about the process of two-photon generation resulting
from pair recombination. To achieve this goal, we plan to calculate the two-
photon density matrix using second-order time-dependent perturbation theory.
This calculation will be performed considering the various symmetries
associated with the superconducting scenarios within the system. The elements
of the two-photon density matrix are expressed as:

$\rho^{p_{1}p_{2}}_{p_{3}p_{4}}({\bf q}_{1},{\bf
q}_{2};\vartheta)=\langle\Psi_{t}(\vartheta)|a^{\dagger}_{q_{1},p_{1}}a^{\dagger}_{q_{2},p_{2}}a_{q_{1},p_{3}}a_{q_{2},p_{4}}|\Psi_{t}(\vartheta)\rangle,$
(10)

in which ${\bf q}_{1}$ and ${\bf q}_{2}$ denote the momenta of two photons,
while $p_{i}\\!=\pm 1$ signifies the polarization state of the photons. Energy
conservation dictates that $\omega_{{\bf q}_{1}}\\!+\\!\omega_{{\bf
q}_{2}}\\!=\\!2E_{g}$, where $E_{g}$ represents the band gap between the
heavy-hole and conduction electron bands. By utilizing Wick’s theorem, the
expectation values of $\rho^{p_{1}p_{2}}_{p_{3}p_{4}}({\bf q}_{1},{\bf
q}_{2};\vartheta)$ provide insights into the contributions arising from the
recombination of Cooper pairs in the generation of two entangled photons. In
addition,

$|\Psi_{t}(\vartheta)\rangle=\int^{t}_{-\infty}\\!\\!\\!\\!dt_{1}\int^{t_{1}}_{-\infty}\\!\\!\\!\\!dt_{2}{\cal
H}_{\rm int}(t_{1}){\cal H}_{\rm int}(t_{2})|\Psi_{0}\rangle,$ (11)

depicts the state of the second-order system at time $t$ in the interaction
picture, in which ${\cal H}_{\rm int}(t)\\!=\\!e^{\rm i{\cal H}_{0}t}{\cal
H}_{\rm int}e^{-\rm i{\cal H}_{0}t}$, where ${\cal H}_{\rm int}(t)$ denotes
the interacting potential at time $t$. Furthermore,

$|\Psi_{0}\rangle=|0\rangle|{\rm FS}\rangle|{\rm BCS}\rangle$

represents the system’s state, with $|0\rangle$, $|{\rm FS}\rangle$, and
$|{\rm BCS}\rangle$ indicating the vacuum state of photons, the Fermi sea of
the heavy hole band, and the BCS wave-function of the superconducting
electrons in the conduction band, respectively.

Using Eqs. (10) and (11), performing the time integrals, and summing over the
internal degrees of freedom, we obtain the $\vartheta$-resolved emission rate,
given by:

$\bar{\varrho_{\vartheta}}({\bf q}_{1},{\bf q}_{2})=\frac{1}{2N}\sum_{{\bf
k},\xi}\zeta^{{\bf k},\xi,\vartheta}_{{\bf q}_{1},{\bf q}_{2}}(\psi_{{\bf
k}},{\bf d}_{{\bf k}}){\cal M}_{{\bf k},\xi}(\psi_{{\bf k}},{\bf d}_{{\bf
k}}),$ (12)

in which the weight factor $\zeta^{{\bf k},\xi,\vartheta}_{{\bf q}_{1},{\bf
q}_{2}}(\psi_{{\bf k}},{\bf d}_{{\bf k}})$ encapsulates the rate of photon
pair generation resulting from the recombination of Cooper pairs, accounting
for the specific contributions of Rashba and Dresselhaus SOCs $\vartheta$,
which is given by

$\displaystyle\begin{aligned} \zeta^{{\bf k},\xi,\vartheta}_{{\bf q}_{1},{\bf
q}_{2}}&(\psi_{{\bf k}},{\bf d}_{{\bf k}})=\\\ &4\pi|B_{{\bf k},{\bf
q}_{1}}|^{2}|B_{{\bf k},{\bf q}_{2}}|^{2}\Big{|}\frac{\Delta_{{\bf
k},\xi}}{2E_{{\bf k},\xi}}\Big{|}^{2}\Big{[}n_{F}(-E_{{\bf
k},\xi})\Big{]}^{2}\times\\\ &\Bigg{[}\frac{n_{F}(E_{{\bf
k},\xi})n_{F}(E_{{\bf k},\xi})}{(E_{{\bf k}}-\omega_{{\bf
q}_{1}}+\varepsilon_{{\bf k},\xi})^{2}}+\frac{(1-n_{F}(E_{{\bf
k},\xi}))}{(E_{{\bf k}}+\omega_{{\bf q}_{1}}-\varepsilon_{{\bf
k},\xi})^{2}}\\\ &+2\frac{n_{F}(E_{{\bf k},\xi})n_{F}(E_{{\bf
k},\xi})}{(E_{{\bf k}}+\omega_{{\bf q}_{1}}-\varepsilon_{{\bf k},\xi})(E_{{\bf
k}}+\omega_{{\bf q}_{1}}-\varepsilon_{{\bf k},\xi})}\\\ &+\frac{n_{F}(E_{{\bf
k},\xi})n_{F}(E_{{\bf k},\xi})}{(E_{{\bf k}}-\omega_{{\bf
q}_{1}}+\varepsilon_{{\bf k},\xi})(E_{{\bf k}}-\omega_{{\bf
q}_{2}}+\varepsilon_{{\bf k},\xi})}\\\ &+\frac{(1-n_{F}(E_{{\bf
k},\xi}))(1-n_{F}(E_{{\bf k},\xi}))}{(E_{{\bf k}}+\omega_{{\bf
q}_{1}}-\varepsilon_{{\bf k},\xi})(E_{{\bf k}}+\omega_{{\bf
q}_{2}}-\varepsilon_{{\bf k},\xi})}\\\ &+\frac{n_{F}(E_{{\bf
k},\xi})(1-n_{F}(E_{{\bf k},\xi}))}{(E_{{\bf k}}+\omega_{{\bf
q}_{1}}-\varepsilon_{{\bf k},\xi})(E_{{\bf k}}-\omega_{{\bf
q}_{2}}+\varepsilon_{{\bf k},\xi})}\\\ &+\frac{n_{F}(E_{{\bf
k},\xi})(1-n_{F}(E_{{\bf k},\xi}))}{(E_{{\bf k}}-\omega_{{\bf
q}_{1}}+\varepsilon_{{\bf k},\xi})(E_{{\bf k}}+\omega_{{\bf
q}_{2}}-\varepsilon_{{\bf k},\xi})}\\\ &+({\bf q}_{1}\longleftrightarrow{\bf
q}_{2})\Bigg{]}\delta(\omega_{{\bf q}_{1}}+\omega_{{\bf
q}_{2}}-2\varepsilon_{{\bf k},\xi}).\end{aligned}$ (13)

Here, $n_{F}(E)=[1+\exp(E/T)]^{-1}$ represents the Fermi-Dirac distribution
function of electrons at temperature $T$. In the limit of $\lambda\rightarrow
0$, Eq. (12) converges to the expression reported in Ref. [32]. The basis for
the two-photon density matrix is describes by the circular right- or left-
handed polarization and is given by
$(|\\!L\\!L\rangle,|\\!L\\!R\rangle,|\\!R\\!L\rangle,|\\!R\\!R\rangle)$. Under
this basis, the matrix ${\cal M}_{{\bf k},\xi}(\psi_{{\bf k}},{\bf d}_{{\bf
k}})$ is defined as:

$\displaystyle{\cal M}_{{\bf k},\xi}(\psi_{{\bf k}},{\bf d}_{{\bf
k}})=\begin{pmatrix}\Upsilon_{{\bf k},\xi}&0&0&0\\\ 0&\eta_{{\bf
k},\xi}&\eta_{{\bf k},\xi}&0\\\ 0&\eta_{{\bf k},\xi}&\eta_{{\bf k},\xi}&0\\\
0&0&0&\Upsilon_{{\bf k},\xi}\\\ \end{pmatrix},$ (14)

where

$\eta_{{\bf k},\xi}={\cal A}^{2}_{{\bf k},\xi}+{\cal B}^{2}_{{\bf
k}}\cos^{2}\\!\varsigma_{{\bf k}};\quad\Upsilon_{{\bf k},\xi}=2{\cal
B}^{2}_{{\bf k},\xi}\sin^{2}\\!\varsigma_{{\bf k}}$

with the fractions of pairings in singlet ${\cal A}_{{\bf k},\xi}=\psi_{{\bf
k}}/|\\!|\hat{\Delta}_{{\bf k}}|\\!|$, and triplet channels ${\cal B}_{{\bf
k},\xi}=\xi{\bf d}_{{\bf k}}/|\\!|\hat{\Delta}_{{\bf k}}|\\!|$. Here

$|\\!|\hat{\Delta}_{{\bf k}}|\\!|=\Big{[}\prod_{\xi}\Delta_{{\bf
k},\xi}\Big{]}^{\frac{1}{2}}=\sqrt{\psi^{2}_{{\bf k}}+|{\bf d}_{{\bf
k},\xi}|^{2}},$

and $\varsigma_{{\bf k}}\\!\\!=\\!\arccos(\hat{{\bf p}}\cdot{\bf d}_{{\bf
k}}/|{\bf d}_{{\bf k}}|)$ represents the angle between ${\bf d}_{{\bf k}}$ and
the polarization axis of photons $\hat{{\bf
p}}=(\hat{x}\cos\phi+\hat{y}\sin\phi)\sin\theta+\hat{z}\cos\theta$. The
fractions of pairings satisfy the property ${\cal A}^{2}_{{\bf k},\xi}+{\cal
B}^{2}_{{\bf k},\xi}=1$.

Figure 3:  The purity of photon pairs, $\Gamma$, for various combinations of
superconducting gap functions, considering different amplitudes of singlet and
triplet channels for $\theta=0$. Notably, $r=0$ and $r=1$ represent instances
of pure triplet and singlet pairings, respectively.  Figure 4:  The purity of
generated entangled photon pairs, $\Gamma$, with respect to the amplitudes of
singlet and triplet pairings for the various gap functions. Each curve reports
at different values $\theta$, by setting $\phi=0$.

Excluding the contribution of the weight factor $\zeta^{{\bf
k},\xi,\vartheta}_{{\bf q}_{1},{\bf q}_{2}}(\psi_{{\bf k}},{\bf d}_{{\bf
k}})$, the purity of the two-photon states is obtained by

$\displaystyle\Gamma$ $\displaystyle=\frac{1}{2N}\sum_{{\bf k},\xi}{\rm
Tr}[{\cal M}_{{\bf k},\xi}(\psi_{{\bf k}},{\bf d}_{{\bf k}})]^{2}.$ (15)
$\displaystyle=\frac{1}{2N}\sum_{{\bf k},\xi}\Big{[}{\cal B}^{4}_{{\bf
k},\xi}\sin^{4}\\!\varsigma_{{\bf k}}+2\Big{(}{\cal A}^{2}_{{\bf k},\xi}+{\cal
B}^{2}_{{\bf k},\xi}\cos^{2}\\!\varsigma_{{\bf k}}\Big{)}^{2}\Big{]}.$

As we haven’t considered the influence of Rashba and Dresselhaus contributions
on the structure of the superconducting gap function, the matrix ${\cal
M}_{{\bf k},\xi}(\psi_{{\bf k}},{\bf d}_{{\bf k}})$ remains
$\vartheta$-independent. However, the emission rate matrix
$\bar{\varrho_{\vartheta}}$ is obviously $\vartheta$-dependent.

Figure 5:  Variation in the purity of entangled photons concerning the
parameter $r$ across different values of the azimuthal angle $\phi$, at
$\theta=\pi/2$. This scenario corresponds to the polarization axis of photons
lying in the $xy$-plane. Note that the curves corresponding to $\phi=0$ and
$\phi=\pi/2$ exhibit identical patterns, as do those for $\phi=\pi/6$ and
$\phi=\pi/3$.  Figure 6:  The impact of variations in the polar angle $\theta$
on the purity of entangled photon pairs. The figure presents diverse
contributions of singlet and triplet pairings, specifically at $\phi=0$,
displaying various ratios of singlet and triplet amplitudes  Figure 7:
Demonstrating the impact of variations in the azimuthal angle $\phi$ on the
purity of entangled photon pairs. The figure showcases diverse contributions
of singlet and triplet pairings, specifically at $\theta=\pi/2$, where the
$\hat{{\bf p}}$-vector resides in the $xy$-plane, presenting various ratios of
singlet and triplet amplitudes.  Figure 8:  Density matrix of polarized two-
photon states generated through the recombination of Cooper pairs for
$\theta=\pi/2$, and $\phi=0$. The left, middle, and right panels illustrate
the outcomes for pure singlet ($r=1$), an equal mix of singlet and triplet
($r=0.5$), and pure triplet ($r=0$) pairings, respectively. Within each
column, the rows correspond to different gap functions: $s+p$-, $s^{*}+p$-,
$d_{x^{2}-y^{2}}+p$-, and $d_{xy}+p$-wave.

##  III Results and Discussion

Our investigation now focuses on assessing the purity of the polarization
state of the two photons, considering various scenarios dictated by the
interplay of distinct physical parameters involved in the process. For a
specific scenario where the photon’s polarization axis is $z$-direction
($\theta=0$), the purity of the two-photon state, $\Gamma$, is depicted in
Fig. 3, concerning the diverse admixture of singlet and triplet pairings
across various superconducting gap functions. When $r=1$, representing pure
triplet $p$-wave pairing, the purity reaches its minimum value. Intriguingly,
the addition of singlet pairing alongside dominant triplet Cooper pairs leads
to a reduction in purity. However, around $r=0.5$, where singlet and triplet
components of the gap function are of comparable magnitudes, purity begins to
increase. At $r=1$, where the superconducting gap function demonstrates a pure
singlet texture, the purities achieve their maximum values. Notably,
conventional singlet $s$-wave pairing exhibits the highest achievable purity.
Consequently, conventional $s$-wave superconductors emerge as the most
suitable candidates for generating entangled photons via Cooper pair
recombination. Fig. 4 showcases the purity of polarized photons concerning the
parameter $r$ for various $\theta$ by considering $\phi=0$. For the case of
$\theta=\pi/2$, where the polarization vector lies in the $xy$-plane,
entangled photon purity maximizes across different superconducting pairing
symmetries. Specifically, for pure triplet $p$-wave pairing ($r=0$), maximum
purity is attained when both $\hat{{\bf p}}$ and ${\bf d}$ vectors lie in the
same plane. Furthermore, Fig. 5 exhibits the purity versus the singlet/triplet
pairing ratio for various values of the azimuthal angle $\phi$ at
$\theta=\pi/2$. Notably, for $\phi=\pi/4$, the purity of generated entangled
photons reaches its peak. Additionally, owing to the intrinsic $C_{4v}$ point
group symmetry, complementary angles yield identical results.

To assess the impact of the angle $\varsigma_{{\bf k}}$ on the purity
behaviour of generated entangled photons, Figs. 6 and 7 depict the behaviour
of entangled photon purity concerning changes of $\theta$ and $\phi$ angles
across various combinations of singlet/triplet pairings. It’s noticeable that
the purity for pure singlet pairing remains constant regardless of these
angles. These graphs affirm that the highest purity is achievable only in the
case of $s$-wave pairing. Moreover, within the singlet channel, the $s$\- and
$d_{xy}$-wave pairings exhibit the maximum and minimum values of purity,
respectively. The $C_{4v}$ symmetry of the crystalline lattice dictates that
the purity pattern remains unchanged under a $\phi=\pi/4$ rotation around the
$z$ axis.

So far, we have only explored the impact of the superconducting gap texture on
the purity of the generated entangled photons. Given that the direct influence
of the Rashba/Dresselhaus contributions hasn’t been accounted for in the
symmetry of the superconducting gap function, the parameter $\vartheta$
doesn’t influence the magnitude of the polarization purity of the photons. To
investigate the effect of the band structure on the purity of the photons, we
conducted calculations involving the matrix elements of the
$\vartheta$-resolved emission rate. We present the emission rate
$\bar{\varrho}_{\vartheta}({\bf q}_{1},{\bf q}_{2})$, in Fig. 8, for various
superconducting gap function textures across different singlet-triplet
admixture ratios $r$. The resulting two-photon states exhibit correlations
that could lead to the generation of entangled states such as $|\Psi_{\rm
ph}\rangle=(|LR\rangle+|RL\rangle)/\sqrt{2}$. Therefore, for the production of
entangled two-photon states, the focus should be on attaining higher purity in
generating $|LR\rangle$ and $|RL\rangle$ states. Fig. 8 reveals that in the
case of pure singlet pairing $(r=1)$, only $|LR\rangle$ and $|RL\rangle$ two-
photon states are produced. However, inducing triplet pairing due to the
presence of spin-orbit coupling results in the generation of $|LL\rangle$ and
$|RR\rangle$ states, which do not lead to entangled pairs. Given that Rashba
and/or Dresselhaus SOCs are intrinsic to zinc-blend compounds and
heterostructures, triplet Cooper pairs exist alongside singlet pairs, leading
to reduced purity in entangled two-photon states. To achieve higher purity,
it’s crucial to control the strength of antisymmetric SOC to decrease the
amplitude of odd parity spin-triplet superconductivity. Furthermore, we find
that within the parity-mixed superconducting states, the scenarios involving
$s+p$\- and $d_{x^{2}-y^{2}}+p$-wave pairings exhibit the most significant
occurrences of $|LR\rangle$ and $|RL\rangle$ states. These configurations are
more predisposed to generate entangled two-photon pairs. Additionally, the
emission rate remains unaffected by both $\vartheta$ and filling $\langle
n\rangle$. To examine the influence of Rashba/Dresselhaus mixture and charge
carrier filling, determining the symmetry of the superconducting ground state
concerning these parameters is necessary, as investigated in Refs. [53, 54].

##  IV Summary

We have conducted an exploration into the generation of entangled two-photon
states through Cooper pair recombination within a P-N-S heterostructure. The
semiconductor components utilized in our investigation adopted a zinc-blend
crystal structure, embodying a combination of Rashba and Dresselhaus
antisymmetric SOCs. Our research encompassed an analysis of various types of
superconducting gap functions, considering diverse ratios between singlet and
triplet pairings. Additionally, we scrutinized the influence of alterations in
the contribution of Rashba/Dresselhaus SOCs alongside electron filling levels
on both the purity and prevalence of distinct pairs of polarized two-photon
states. Given the inevitable presence of spin-triplet Cooper pairs in
structures and compounds featuring antisymmetric spin-orbit coupling, we
delved into examining the ramifications of altering the angle between the
spin-triplet ${\bf d}$-vector and the polarization axis of photons $\hat{{\bf
p}}$.

In accordance with the previous results, our calculations reinforce the notion
that pure singlet pairings yield the highest purity levels compared to both
mixed and pure triplet pairings. Additionally, within the singlet gap
functions, we observed that the $s$\- and $d_{xy}$-wave pairings exhibit the
highest and lowest purity levels, respectively. In noncentrosymmetric crystals
featuring antisymmetric spin-orbit coupling, the induced spin-triplet pairing
tends to diminish the population of entangled two-photon states. Our findings
further revealed that in superconducting ground states with mixed-parity
pairings, the purity amplitude of entangled two-photon states reaches its peak
at $(\theta,\phi)=(\pi/2,\pi/4)$. Finally, it is important to note that our
analysis does not account for the impact of changes in the ratio of
Rashba/Dresselhaus SOCs or electron filling on the symmetry of the
superconducting gap function. Therefore, the resulting purity and population
of generated entangled two-photon pairs remain unaffected in our current
study.

## Acknowledgments

M. B. and A. A. highly appreciate Hae-Young Kee for the inspiring discussions.
A. A. acknowledges the financial support from the German Research Foundation
within the bilateral NSFC-DFG Project No. ER 463/14-1.

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# Quantum Reservoir Probing of Quantum Information Scrambling

Kaito Kobayashi  Yukitoshi Motome  Department of Applied Physics, the
University of Tokyo, Tokyo 113-8656, Japan

###### Abstract

Quantum information scrambling is the dynamical process where initially
localized quantum information spreads throughout an entire system. Here, by
extending the idea of quantum reservoir computing, we propose the quantum
reservoir probing (QRP) to diagnose quantum information scrambling. As a
paradigmatic example, we investigate scrambling in a one-dimensional quantum
Ising chain. We show that the QRP successfully distinguishes between
quasiparticle-mediated information propagation in a free fermion system and
correlation-mediated scrambling in a quantum chaotic system. Furthermore, via
the scanning of read-out operators, the QRP can elucidate the dynamical
distribution of quantum information in the Hilbert space, a distinct advantage
over the conventional methods with out-of-time-order correlators and
tripartite mutual information.

The dynamical phenomenon of locally encoded quantum information spreading over
a system, referred to as quantum information scrambling [1, 2], has been of
fundamental importance in modern physics, offering valuable insights in a wide
range of fields. For instance, in high-energy physics, it provides profound
elucidation of the information paradox of black holes [1, 3] and the
holographic duality between quantum and gravitational systems [4, 5, 6].
Similarly, in condensed matter physics, it is a potent tool for scrutinizing
nonequilibrium quantum many-body phenomena, such as quantum chaos [7, 8, 9,
10, 11, 12, 13], thermalization dynamics [14, 15, 16, 17, 18, 19], and
dynamical quantum phase transitions [20, 21, 22, 23].

Quantification of scrambling has been achieved primarily through the
application of the out-of-time-order correlator (OTOC), which probes
spatiotemporal correlations via the degree of noncommutativity between two
initially commuting local operators at different times [24, 25, 26, 27, 28].
This operator overlap suggests information spreading, and the decay rate of
OTOC is conjectured to be strongly related to the Lyapunov exponent in the
semiclassical regime [29, 30, 31]. Alternatively, the tripartite mutual
information (TMI) serves as an operator-independent metric for scrambling [9,
32, 33, 34]; it quantifies the amount of information initially localized in
one part of the system that can be retrieved from the correlations present
between the two remaining partitions. Both OTOC and TMI are suggested to
encapsulate essentially the same feature of scrambling [9], and advancements
in experimental methodologies have empowered the measurements of these
quantities in trapped ions [35, 36, 37], nuclear magnetic resonance systems
[38, 39, 40], and superconducting qubits [41, 42, 43, 44].

Meanwhile, these quantities are insufficient in capturing the distribution of
information in the Hilbert space and time. Indeed, the scrambling nature
becomes evident in the initial growth [45, 31, 28, 46] or the long-time limit
[47, 48, 49, 23] of OTOC for restricted pairs of specific operators, rather
than its explicit values at individual points in time for arbitrary operator
pairs. Besides, TMI quantifies the information stored in all types of
correlations [32], lacking the capability to pinpoint specific operators of
interest. Due to this absence of suitable witnesses, a comprehensive analysis
of how information scrambles, extending beyond the conventional question of
whether scrambling occurs or not, remains largely unexplored.

In this Letter, we propose a framework to directly trace information
propagation in quantum many-body systems, which we call quantum reservoir
probing (QRP). This is built upon the principles of quantum reservoir
computing (QRC) [50], a brain-inspired information processing algorithm [51]
leveraging strong nonlinearity and high dimensionality inherent in quantum
reservoirs [52, 53, 54, 55, 56, 57, 58, 59]. While QRC focuses on exploiting
the system for computation, our QRP is specifically dedicated to investigating
the system itself. Through the lens of QRP, we study a quantum Ising model
with transverse and longitudinal magnetic fields, and successfully capture the
dynamics of quantum information, elucidating the inherent dynamical
characteristics of the system. In addition, by scanning the read-out operators
employed in the QRP, we can estimate the dynamical distribution of information
across multiple degrees of freedom, which probes relevant information
propagation channels. In conjunction with the experimental simplicity, our QRP
substantially advances the understanding of quantum information scrambling by
revealing the underlying physical mechanisms of information propagation.

Figure 1:  (a) Schematic of our 1D quantum spin chain. Both qubit $0$ and
qubit $1$ are simultaneously employed for input. The qubits $1,2,\cdots$
evolve with the Hamiltonian in Eq. (1), while the qubit $0$ retains its
initial state at each input. (b) Quantum dynamics of the expectation value
$\langle O(t)\rangle$ with input $\\{s_{k}\\}$ given at time interval
$t_{\mathrm{in}}$. (c) Concept of virtual time $\tau$. $\langle
O(\tau+kt_{\mathrm{in}})\rangle$ is used in the calculation of output
${y}_{k}(\tau)$. (d) Virtual time dependence of the output ${y}_{k}(\tau)$.
The gray dotted line represents the target value $\bar{y}_{k}$. The
performance at $\tau$ is evaluated based on the determination coefficient
between $\bm{y}(\tau)$ and $\bar{\bm{y}}$.

Let us define the QRP framework, taking a spin-$1/2$ Ising chain as an
example. The Hamiltonian is given by

$\mathcal{H}=-J\sum_{i=1}^{N-1}\sigma^{x}_{i}\sigma^{x}_{i+1}+h_{x}\sum_{i=1}^{N}\sigma^{x}_{i}+h_{z}\sum_{i=1}^{N}\sigma^{z}_{i}$
(1)

with $h_{z}$ and $h_{x}$ representing the transverse and longitudinal magnetic
fields, respectively; ${\sigma}_{i}^{x}$ and ${\sigma}_{i}^{z}$ are the $x$
and $z$ Pauli matrices at site $i$, and $J>0$ is the strength of the nearest-
neighbor Ising interaction, which we set $J=1$ as our energy scale. $N$
denotes the number of sites in the system, exclusive of qubit $0$ which is not
involved in the time evolution [Fig. 1(a)]; we take $N=7$ in the following
calculations. This model is known to be mapped to a free fermion system via
the Jordan-Wigner transformation in the case of $h_{x}=0$, whereas it shows
chaotic spectral statistics at $(h_{x},h_{z})=(-0.5,1.05)$ [60].

In the QRP protocol, we employ sequential quantum quenches in response to
stochastic inputs. Suppose $\\{s^{k}\\}$ is an input sequence where each
$s^{k}$ is randomly sampled from a uniform distribution: $s^{k}\in[0,1]$. At
each time interval $t_{\mathrm{in}}$, the density matrix
$\rho(kt_{\mathrm{in}})$ is updated as

$\rho(kt_{\mathrm{in}})\rightarrow|\psi_{\mathrm{in}}\rangle\langle\psi_{\mathrm{in}}|\otimes\mathrm{Tr}_{\\{01\\}}\left[\rho(kt_{\mathrm{in}})\right],$
(2)

where
$|\psi_{\mathrm{in}}\rangle=\sqrt{s^{k}}|00\rangle_{\\{01\\}}+\sqrt{1-s^{k}}|11\rangle_{\\{01\\}}$
and $\mathrm{Tr}_{\\{01\\}}$ denotes the partial trace performed over the
qubits $0$ and $1$. The dynamics until the next input is simulated through the
exact diagonalization with the operator of unitary dynamics
$e^{-i\mathcal{H}t_{\mathrm{in}}}$, and the expectation value of an operator
$O$ is calculated by $\langle O(t)\rangle\equiv\mathrm{Tr}[\rho(t)O]$ [Fig.
1(b)]. We here introduce virtual time $\tau$, which is reset to $0$ for every
$k$-th input, and is defined as $\tau\equiv t-kt_{\mathrm{in}}$ for
$kt_{\mathrm{in}}\leq t<(k+1)t_{\mathrm{in}}$ [Fig. 1(c)]. In the following,
we denote $\langle O(kt_{\mathrm{in}}+\tau)\rangle$ for general $k$ by
$\langle O(\tau)\rangle$.

Following the QRC framework [50], we consider short-term memory (STM) task.
The target output is given by $\bar{y}^{k}=s^{k-d}$ where $d$ denotes the
delay steps after input. The output from the reservoir at $k$-th step is
calculated as $y^{k}(\tau)=w_{o}(\tau)\langle
O(kt_{\mathrm{in}}+\tau)\rangle+w_{c}(\tau)$, where $w_{o}(\tau)$ and
$w_{c}(\tau)$ are the $k$-independent coefficients for the linear
transformation, trained through the least squared method to match
$\bm{\bar{y}}\equiv\\{\bar{y}^{k}\\}$ and
$\bm{{y}}(\tau)\equiv\\{{y}^{k}(\tau)\\}$ [61]. The target $\bar{y}^{k}$
remains invariant for all $\tau$ in $k$-th step, whereas the output $y(\tau)$
exhibits a temporal variation [Fig. 1(d)]. The performance is measured by the
determination coefficient
$R^{2}=\mathrm{cov}^{2}(\bm{y}(\tau),\bar{\bm{y}})/[\sigma^{2}(\bm{y}(\tau))\sigma^{2}(\bar{\bm{y}})]$
where $\mathrm{cov}$ and $\sigma^{2}$ denote covariance and variance,
respectively. $R^{2}$ takes the value between $0$ and $1$, and becomes close
to $1$ when $\bm{y}(\tau)$ and $\bar{\bm{y}}$ match closely. Since only linear
transformation is applied to the raw expectation value, the resultant $R^{2}$
should accurately estimate the amount of information stored in $\langle
O(\tau)\rangle$ without over- or under-estimation. Hereafter, $R^{2}$ for the
STM task with delay $d$ is denoted as $R^{2}_{d}$, and $t_{\mathrm{in}}$ is
set to $5$ [61]. As shown below, our QRP protocol primarily utilizes
$R^{2}_{d=0}$ employing various read-out operators to probe the quantum
information scrambling.

Figure 2 represents the dynamics of the STM performance $R^{2}_{d}$ for
$d=0,1,2$ in a free fermion system with $(h_{x},h_{z})=(0.0,1.0)$ and a
quantum chaotic system with $(h_{x},h_{z})=(-0.5,1.05)$ (see Supplemental
Material [61] for an intermediante case). For the calculation of the output
$\bm{y}(\tau)$, the expectation values $\langle\sigma^{z}_{i}(\tau)\rangle$ at
each qubit $i$ are employed independently. Right after the input where
$\tau\ll 1$, the information of $s_{k}$, fed into the qubits $0$ and $1$,
remains predominantly within the qubit $1$, as evidenced by the almost unity
$R^{2}_{d=0}$ for qubit $1$, while being vanishingly small for the remaining
qubits. Subsequently, the information propagates through qubits $2,3,...$, and
as a result, $R^{2}_{d=0}$ gradually begins to manifest nonzero values from
the qubits close to qubit $1$.

Figure 2:  Performance for the STM task with delay $d=0,1,2$ for (a) the free
fermion system with $h_{x}=0.0$ and $h_{z}=1.0$ and (b) the quantum chaotic
system with $h_{x}=-0.5$ and $h_{z}=1.05$. $\langle\sigma^{z}_{i}\rangle$ is
utilized in the calculation of the output $y^{k}(\tau)$. The colors represent
each qubit, and the marker styles indicate different delays.

At $\tau=t_{\mathrm{in}}$ ($t=(k+1)t_{\mathrm{in}}$), the new information of
$s_{k+1}$ is provided to input qubits, and the information of $s_{k}$
remaining within the system is evaluated via the STM task with $d=1$. Upon
this input phase, the quantum state of the input qubits undergoes substantial
alteration, while the states of the other qubits remain intact. Indeed,
$R^{2}_{d-1}$ at $\tau\rightarrow t_{\mathrm{in}}$ and $R^{2}_{d}$ at $\tau=0$
are continuously connected, except for qubit 1 (Fig. 2). In other words,
$R^{2}_{d}$ at virtual time $\tau$ effectively corresponds to $R^{2}_{0}$ at
virtual time $\tau+t_{\mathrm{in}}d$ in the successively quenched condition.
Hence, the STM task with high delay $d$ evaluates the impact of $d$-times
repetition of the quench on the preservation of information $s_{k}$, which is
difficult to quantify in general, including by OTOC and TMI, due to the non-
simultaneous accessibility of the pre- and post-quenching quantum states.

The nature of information propagation is closely linked to quantum dynamics of
the system. In the case of free fermion system, information propagates
ballistically as illustrated in Fig. 2(a); $R^{2}_{d=0}$ for each qubit
manifests a unimodal dynamical behavior, demonstrating sequential peaks.
Indeed, this ballistic behavior suggests quasiparticle-mediated information
propagation. On the input quenching phase, a quasiparticle containing the
input information is excited at the input qubits. As the quasiparticle moves
along the chain, the peak of $R^{2}_{d=0}$ moves to the qubit where the
quasiparticle exists, resulting in the unimodal shape in Fig. 2(a). Proceeding
to the next input, a new quasiparticle is created and interferes with the
existing ones. Consequently, $R^{2}_{d=1}$ and $R^{2}_{d=2}$ evolve with
complicated dynamics, albeit the ballistic nature is similar to the case of
$d=0$.

In contrast, Fig. 2(b) supports diffusive information propagation in the
quantum chaotic system. The timeline for $R^{2}_{d=0}$ commencing its
ascension at each qubit is similar to that in Fig. 2(a); however, the
information accumulation process toward the maximum of $R^{2}_{d=0}$ proves to
be prolonged. Besides, after reaching its maximum, $R^{2}_{d=0}$ exhibits a
gradual decay, which is an opposite behavior to the abrupt decrease observed
in the unimodal dynamics. $R^{2}_{d=1}$ and $R^{2}_{d=2}$ also show a smooth
and gradual dynamics without any pronounced local behaviors that lead to a
significant surge in the performance of a single qubit. Notably, $R^{2}_{d=2}$
at $\tau\rightarrow t_{\mathrm{in}}$ approaches a uniform value across all
qubits, implying a nearly equitable distribution of information throughout all
the qubits.

Figure 3:  Virtual time dependence of the dynamical spin correlation
$\langle\sigma^{z}_{1}(0)\sigma_{i}^{z}(\tau)\rangle$: (a) free fermion system
and (b) quantum chaotic system.

For further elucidation of the physical origin of information propagation, the
dynamical spin correlation between the individual qubits and the input qubit
$1$, $\langle\sigma^{z}_{1}(0)\sigma_{i}^{z}(\tau)\rangle$, is examined in
Fig. 3. In the free fermion system, the dynamical spin correlation with $i\geq
2$ initiates an ascension, achieving its maximum, and thereafter undergoes an
attenuation, in turn with distance from the qubit $1$ [Fig. 3(a)]. Conversely,
in the quantum chaotic system, the dynamical spin correlation accumulates
progressively over time and remains moderate for long periods.

Figure 4 plots the relationship between the dynamical spin correlation
$|\langle\sigma^{z}_{1}(0)\sigma_{i}^{z}(\tau)\rangle|$ and the STM
performance $R^{2}_{d=0}$ when individual $\langle\sigma_{i}^{z}(\tau)\rangle$
is employed for the read-out. As suggested by previous studies on classical
magnetic physical reservoirs [62, 63, 64], the reservoir achieves higher
$R^{2}_{d=0}$ when the spin variable harnessed for the read-out is in strong
correlation with $\sigma^{z}_{1}(0)$. In particular, notwithstanding the
intricate dynamics exhibited by both variables in the quantum chaotic system
[Figs. 2(b) and 3(b)], the data collapse onto a single curve in Fig. 4(b)
suggests an almost one-to-one correspondence between
$|\langle\sigma^{z}_{1}(0)\sigma_{i}^{z}(\tau)\rangle|$ and $R^{2}_{d=0}$,
irrespective of the qubit position $i$ and virtual time $\tau$. This is
accentuated by comparison with the more dispersed plot for the free fermion
system in Fig. 4(a) (see Supplemental Material [61] for more quantitative
analysis). This observation strongly indicates that, in the quantum chaotic
system, information spreads diffusively to the entire system via the dynamical
spin correlation, which is qualitatively different from the ballistic
propagation via quasiparticles in the free fermion system.

We here emphasize that the QRP can estimate the amount of information
encompassed within an arbitrary operator $O(\tau)$ through $R^{2}_{d}$,
utilizing the resultant output $\bm{y}(\tau)$ derived from the linear
transformation of $\langle O(\tau)\rangle$. Through exhaustive scanning in
operators and virtual time, the QRP explores the dynamical distribution of
information across multiple degrees of freedom, thereby suggesting the
information propagation channels. Figures 5(a) and 5(c) represent
$R^{2}_{d=0}$ employing observables of $\langle\sigma_{2}^{z}(\tau)\rangle$,
$\langle\sigma_{3}^{z}(\tau)\rangle$,
$\langle\sigma_{2}^{x}(\tau)\sigma_{3}^{x}(\tau)\rangle$, and
$\langle\sigma_{2}^{z}(\tau)\sigma_{3}^{z}(\tau)\rangle$. Remarkably, in the
free fermion system, $R^{2}_{d=0}$ when using
$\langle\sigma_{2}^{z}(\tau)\sigma_{3}^{z}(\tau)\rangle$ remains negligibly
small, while $R^{2}_{d=0}$ utilizing
$\langle\sigma_{2}^{x}(\tau)\sigma_{3}^{x}(\tau)\rangle$ shows performance
commensurate with those derived from $\langle\sigma_{2}^{z}(\tau)\rangle$ and
$\langle\sigma_{3}^{z}(\tau)\rangle$ [Fig. 5(a)].

Figure 4:  Relations between the dynamical spin correlation
$|\langle\sigma_{1}(0)\sigma_{i}(\tau)\rangle|$ and the STM performance
$R^{2}_{d=0}$: (a) free fermion system and (b) quantum chaotic system.

We present a detailed performance utilizing other operators in the
Supplemental Material [61], yet it is pertinent to note that $R^{2}_{d=0}$
manifests nonzero value only when the $z$ spin on a single site
$\langle\sigma^{z}_{i}(\tau)\rangle$ or the $x$ component of the nearest-
neighbor spin correlation
$\langle\sigma_{i}^{x}(\tau)\sigma_{i+1}^{x}(\tau)\rangle$ is employed. The
latter also confirms the information propagation by quasiparticles, since the
interaction $\sigma_{i}^{x}\sigma_{i+1}^{x}$ is the very origin of
quasiparticle picture in this free fermion system.

Conversely, in the quantum chaotic system, $R^{2}_{d=0}$ exhibit nonzero
values when employing
$\langle\sigma_{2}^{x}(\tau)\sigma_{3}^{x}(\tau)\rangle$,
$\langle\sigma_{2}^{z}(\tau)\sigma_{3}^{z}(\tau)\rangle$ [Fig. 5(c)], as well
as other correlation operators even among distant qubits [61]. This is a clear
manifestation of quantum information scrambling into numerous degrees of
freedom [9]. In particular, $R^{2}_{d=0}$ using
$\langle\sigma_{2}^{x}(\tau)\sigma_{3}^{x}(\tau)\rangle$ and
$\langle\sigma_{2}^{z}(\tau)\sigma_{3}^{z}(\tau)\rangle$ show similar behavior
in the early time; however, they diverge at middle $\tau$ region and exhibit
qualitatively different long-time behavior. Thus, each type of the spin
correlation serves as an individual channel for information propagation.

To lucidly demonstrate the advantages of QRP, we show the dynamics of OTOC and
TMI in Figs. 5(b) and 5(d). Here, we calculate OTOC for the qubit $i$ as
$F\equiv\langle\sigma^{z}_{i}(\tau)\sigma^{z}_{1}(0)\sigma^{z}_{i}(\tau)\sigma^{z}_{1}(0)\rangle$,
and TMI between qubits $0$, $2$, and $3$ as $I_{3}(0:2:3)\equiv
S_{\\{0\\}}+S_{\\{2\\}}+S_{\\{3\\}}-S_{\\{0\\}\cup\\{2\\}}-S_{\\{0\\}\cup\\{3\\}}-S_{\\{2\\}\cup\\{3\\}}+S_{\\{0\\}\cup\\{2\\}\cup\\{3\\}}$
where $S_{X}$ is the von Neumann entropy. We note that $I_{3}(0:2:3)$
evaluates the amount of information of $s_{k}$ that spreads into the
correlations between the qubits $2$ and $3$, using the qubit $0$ as a
reference for $s_{k}$.

Figure 5:  (a, c) Virtual time dependence of STM performance $R^{2}_{d=0}$.
The skyblue (green) line represents $R^{2}_{d=0}$ using single spin
$\langle\sigma_{2}^{z}(\tau)\rangle$ ($\langle\sigma_{3}^{z}(\tau)\rangle$)
for calculation, whereas the blue (pink) line illustrates $R^{2}_{d=0}$ when
the spin correlation $\langle\sigma_{2}^{x}(\tau)\sigma_{2}^{x}(\tau)\rangle$
($\langle\sigma_{2}^{z}(\tau)\sigma_{2}^{z}(\tau)\rangle$) is utilized. (b, d)
Dynamics of TMI and OTOC. The black line displays the TMI among qubits $0$,
$2$, and $3$. The skyblue (green) line plots $F(\tau)$ for qubit 2 (qubit 3).
(a, b) Free fermion system, (c, d) quantum chaotic system.

At the initial stage, $F(\tau)$ for the qubits $2$ and $3$ begin to decrease,
slightly before the ascension of $R^{2}_{d=0}$ that utilize
$\langle\sigma^{z}_{2}(\tau)\rangle$ and $\langle\sigma^{z}_{3}(\tau)\rangle$,
respectively. In the intermediate stage, information of $s_{k}$ delocalizes
across qubits $2$ and $3$, as evidenced by both the negative $I_{3}(0:2:3)$
and the nonzero $R^{2}_{d=0}$ calculated from the spin correlations. As $\tau$
approaches to $t_{\mathrm{in}}$, $F(\tau)$ converges to $1$ in the free
fermion system and $0$ in the quantum chaotic system corresponding to the
nature of scrambling [9, 38], the distinction of which has already been
discussed from the dynamics of $R^{2}_{d}$. Through these comparative
observations, we confirm that our QRP protocol endorses the identical
conclusion concerning scrambling as deduced by OTOC and TMI. Consequently, the
QRP can be harnessed for the evaluation of quantum scrambling, which offers
greater experimental feasibility in comparison to OTOC or TMI, as it solely
requires expectation values of pertinent operators, such as spins and spin
correlations.

It is worth emphasizing again that the OTOC and TMI are not defined with
respect to information propagation channels. In fact, although a significant
difference is observed in $R^{2}_{d=0}$ utilizing
$\langle\sigma_{2}^{z}(\tau)\sigma_{3}^{z}(\tau)\rangle$ in Figs. 5(a) and
5(c), we cannot deduce this difference solely from the behavior of OTOC or TMI
illustrated in Figs. 5(b) and 5(d). Therefore, the capability of QRP to
evaluate information stored in specific correlations, and thus information
distribution over the Hilbert space, represents significant advantages over
OTOC and TMI, enhancing our understanding of quantum information scrambling.

To summarize, we have proposed QRP as a means to diligently investigate
quantum information scrambling. In the quantum Ising chain with transverse and
longitudinal magnetic fields, the QRP has deduced an identical characteristic
of scrambling as observed by conventional methodologies, notwithstanding its
experimental practicability. Moreover, it enables us to elucidate the
dynamical information distribution in the Hilbert space by scanning the read-
out operators. Our study provides an efficient approach for further
exploration of the dynamics of quantum information in a plethora of exotic
quantum many-body systems, specifically within the context of quantum chaos,
quantum gravitation, and quantum computing.

###### Acknowledgements.

We thank Y. Kato for fruitful discussions. This research was supported by a
Grant- in-Aid for Scientific Research on Innovative Areas “Quantum Liquid
Crystals” (KAKENHI Grant No. JP19H05825) from JSPS of Japan and JST CREST
(Nos. JP-MJCR18T2 and JP-MJCR19T3). K. K. was supported by the Program for
Leading Graduate Schools (MERIT-WINGS).

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Supplemental Material for “Quantum Reservoir Probing of Quantum Information
Scrambling”
Kaito Kobayashi and Yukitoshi Motome

Department of Applied Physics, the University of Tokyo, Tokyo 113-8656, Japan

(Dated: )

##  I I. DETAILS OF THE QRP SCHEME

In this section, we present a comprehensive scheme of the QRP. We consider an
input sequence $\\{s^{k}\\}$ drawn from a uniform distribution
$s^{k}\in[0,1]$. The information of each input value $s_{k}$ is imparted to
the quantum system by manipulating the quantum state of qubits $0$ and $1$, as
described in Eq. (2) in the main text. Starting from the ground state of the
Hamiltonian [Eq. (1) in the main text], totally 5000 inputs are provided.
Among these, $l^{\mathrm{w}}=1000$ inputs are used for washing out the initial
conditions, $l^{\mathrm{tr}}=2000$ inputs are employed for training, and
$l^{\mathrm{ts}}=2000$ inputs are utilized for testing.

The input time interval $t_{\mathrm{in}}=5$ is chosen to be sufficiently long,
allowing information to propagate and reach the end of the system,
specifically qubit $7$. If $t_{\mathrm{in}}$ were too short, the analysis of
information propagation far from the input qubits would be inadequate. On the
other hand, if $t_{\mathrm{in}}$ were too long, it would complicate the
analysis of scrambling due to reflections at the edges of the system. Indeed,
in Fig. 5 of the main text, the dynamics of OTOC and TMI appear to saturate at
around $\tau\sim 5$, which validates the appropriateness of our choice for
$t_{\mathrm{in}}=5$ in this specific system.

The system undergoes time evolution by the Hamiltonian [Eq. (1)], and the
density matrix at virtual time $\tau$ at $k$-th step is given by

$\rho(\tau)=e^{-i\mathcal{H}\tau}\\{|\psi_{\mathrm{in}}\rangle\langle\psi_{\mathrm{in}}|\otimes\mathrm{Tr}_{\\{01\\}}\left[\rho(kt_{\mathrm{in}})\right]\\}e^{i\mathcal{H}\tau}.$
(S1)

When utilizing $O$ as the read-out operator, we observe $\langle
O(kt_{\mathrm{in}}+\tau)\rangle$ at each $k$-th step and construct a two-
dimensional internal state vector $\bm{X}_{k}(\tau)$, which includes an
additional component with the constant value $1$, as

$\bm{X}_{k}(\tau)=(\langle O(kt_{\mathrm{in}}+\tau)\rangle,1).$ (S2)

From internal state vectors during the training and testing phases, we
construct an $(l^{\mathrm{tr}}\times 2)$-dimensional matrix
$\bm{X}^{\mathrm{tr}}(\tau)=\\{\bm{X}_{k}(\tau)\\}_{k=l^{\mathrm{w}}+1}^{l^{\mathrm{w}}+l^{\mathrm{tr}}}$
and an $(l^{\mathrm{ts}}\times 2)$-dimensional matrix
$\bm{X}^{\mathrm{ts}}(\tau)=\\{\bm{X}_{k}(\tau)\\}_{k=l^{\mathrm{w}}+l^{\mathrm{tr}}+1}^{l^{\mathrm{w}}+l^{\mathrm{tr}}+l^{\mathrm{ts}}}$,
respectively. The sequence of target outputs for the STM task with delay $d$
are correspondingly given by
$\bm{\bar{y}}^{tr}\equiv\\{s_{k-d}\\}_{k=l^{\mathrm{w}}+1}^{l^{\mathrm{w}}+l^{\mathrm{tr}}}$
and
$\bm{\bar{y}}^{\mathrm{ts}}\equiv\\{s_{k-d}\\}_{k=l^{\mathrm{w}}+l^{\mathrm{tr}}+1}^{l^{\mathrm{w}}+l^{\mathrm{tr}}+l^{\mathrm{tw}}}$
in a $\tau$-independent way. The weight vector for the linear transformation,
$\bm{w}(\tau)=(w_{o}(\tau),w_{c}(\tau))^{\top}$, is trained to satisfy
$\bm{\bar{y}}^{\mathrm{tr}}=\bm{X}^{\mathrm{tr}}(\tau)\bm{w}(\tau)$, and the
optimal solution that minimizes the least squared error is obtained as

$\bm{w}(\tau)={\bm{X}^{\mathrm{tr}}}^{+}(\tau)\bm{\bar{y}}^{\mathrm{tr}}(\tau)$
(S3)

where ${\bm{X}^{\mathrm{tr}}}^{+}(\tau)$ is the Moore-Penrose pseudoinverse-
matrix of $\bm{X}^{\mathrm{tr}}(\tau)$. For the evaluation of the reservoir
performance, the similarity between the testing output, calculated as
$\bm{y}^{\mathrm{ts}}(\tau)=\bm{X}^{\mathrm{ts}}(\tau)\bm{w}(\tau)$, and the
desired output $\bm{\bar{y}}^{\mathrm{ts}}$ is assessed using the
determination coefficient

$R^{2}(\tau)=\frac{\mathrm{cov}^{2}(\bm{y}^{\mathrm{ts}}(\tau),\bar{\bm{y}}^{\mathrm{ts}})}{\sigma^{2}(\bm{y}^{\mathrm{ts}}(\tau))\sigma^{2}(\bar{\bm{y}}^{\mathrm{ts}})}.$
(S4)

where $\mathrm{cov}$ and $\sigma^{2}$ represent covariance and variance.
$R^{2}(\tau)$ approaches $1$ when output $\bm{y}^{\mathrm{ts}}(\tau)$ and
target $\bar{\bm{y}}^{\mathrm{ts}}(\tau)$ match well; otherwise it becomes
close to $0$.

##  II II. QUANTITATIVE ANALYSIS OF THE RELATIONSHIP BETWEEN THE DYNAMICAL
SPIN CORRELATIONS AND STM PERFORMANCE

In Fig. 4 in the main text, a significant relationship between the dynamical
spin correlation and the performance of the STM task is suggested,
particularly in the case of the quantum chaotic system. To quantify the
deviation from a perfect one-to-one correspondence between these two
quantities, we introduce a measure of data deviation $\Delta$. For a given
integer $0\leq m\leq M-1$, we define $\Lambda_{m}$ as a set of
$\\{(i,\tau)\\}$ that satisfies

$\frac{m}{M}\leq|\langle\sigma_{1}(0)\sigma_{i}(\tau)\rangle|<\frac{m+1}{M},$
(S5)

where $M$ denotes the number of windows; we here set $M=4000$. Assuming the
one-to-one correspondence, we adopt the average of $R^{2}_{d=0}$ over
$\Lambda_{m}$, denoted by $\overline{\left(R^{2}_{d=0}\right)}_{m}$, as the
anticipated value corresponding to
$|\langle\sigma_{1}(0)\sigma_{i}(\tau)\rangle|$ within that window. The data
deviation $\Delta$ is then defined as the summation of the squared deviation
from $\overline{\left(R^{2}_{d=0}\right)}_{m}$, which is given by

$\Delta\equiv\sum_{m=0}^{M-1}\sum_{(i,\tau)\in\Lambda_{m}}\left[\left(R^{2}_{d=0}\right)_{(i,\tau)}-\overline{\left(R^{2}_{d=0}\right)}_{m}\right]^{2}.$
(S6)

In the quantum chaotic system, the value of $\Delta$ is evaluated to be
$\Delta\simeq 0.0299$, which is approximately one order of magnitude smaller
compared to the value of $\Delta\simeq 0.2866$ for the free fermion system.
This quantitative analysis provides strong support for the correlation-
mediated information propagation in the quantum chaotic system.

##  III III. QUANTIFICATION OF INFORMATION STORED IN VARIOUS DEGREES OF
FREEDOM

In this section, we investigate the amount of information stored in various
types of operators using the QRP protocol. We specifically focus on the
information encompassed within qubits $2,3$, and $4$, and calculate
$R^{2}_{d=0}$ employing the spin $\sigma_{i}^{(x,z)}(\tau)$ and spin
correlations $\sigma_{i}^{(x,z)}(\tau)\sigma_{j}^{(x,z)}(\tau)$ over these
qubits. Figure S1 illustrates $R^{2}_{d=0}$ for three different systems: the
free fermion system with $(h_{x},h_{z})=(0.0,1.0)$, the quantum chaotic system
with $(h_{x},h_{z})=(-0.5,1.05)$, and the intermediate system with
$(h_{x},h_{z})=(-0.02,1.002)$. The parameters for the intermediate system is
very close to the free fermion system, whereas it is no longer integrable and
the quasiparticle picture does not hold.

In the free fermion system, $R^{2}_{d=0}$ exhibits a nonzero value when either
$\langle\sigma^{z}_{i}(\tau)\rangle$ [Fig. S1(d)] or
$\langle\sigma_{i}^{x}(\tau)\sigma_{i+1}^{x}(\tau)\rangle$ [Figs. S1(g) and
S1(m)] is utilized as the read-out operator. Otherwise $R^{2}_{d=0}$ becomes
negligibly small, indicating information is not stored in operators such as
$\langle\sigma^{x}_{i}(\tau)\rangle$ [Fig. S1(a)],
$\langle\sigma_{i}^{z}(\tau)\sigma_{j}^{z}(\tau)\rangle$ [Figs. S1(g), S1(m),
and S1(s)], $\langle\sigma_{i}^{x}(\tau)\sigma_{j}^{z}(\tau)\rangle$,
$\langle\sigma_{i}^{z}(\tau)\sigma_{j}^{x}(\tau)\rangle$ [Figs. S1(j), S1(p),
and S1(v)], and $\langle\sigma_{i}^{x}(\tau)\sigma_{j\neq
i+1}^{x}(\tau)\rangle$ [Fig. S1(s)]. In contrast, in the quantum chaotic case,
$R^{2}_{d=0}$ manifests nonzero values for all the read-out operators shown in
Fig. S1, including correlations between qubits $2$ and $4$ despite the
distance between the qubits.

Remarkably, in the intermediate system, $R^{2}_{d=0}$ employing
$\langle\sigma_{i}^{z}(\tau)\rangle$ and
$\langle\sigma_{i}^{x}(\tau)\sigma_{i+1}^{x}(\tau)\rangle$ are
semiquantitatively the same as those in the free fermion system, as evidenced
the almost identical pairs of figures, namely, Figs. S1(d) and S1(f), Figs.
S1(g) and S1(i), and Figs. S1(m) and S1(o). However, due to the breakdown of
the quasiparticle picture, or more precisely, due to the symmetry breaking of
$\sigma_{i}\leftrightarrow-\sigma_{i}$, different types of information
propagation emerge in addition to the mediation of quasiparticles. This leads
to qualitatively different distributions of information compared to the free
fermion system. Indeed, the $x$ component of each spin
$\langle\sigma_{i}^{x}(\tau)\rangle$ manifests nonzero $R^{2}_{d=0}$ [Fig.
S1(c)], and the spin correlations of
$\langle\sigma_{i}^{x}(\tau)\sigma_{j}^{z}(\tau)\rangle$ and
$\langle\sigma_{i}^{z}(\tau)\sigma_{j}^{x}(\tau)\rangle$ [Figs. S1(l), S1(r),
and S1(x)] also exhibit nonzero $R^{2}_{d=0}$. The latter suggests that the
correlations $\langle\sigma_{i}^{x}(\tau)\sigma_{j}^{z}(\tau)\rangle$ and
$\langle\sigma_{i}^{z}(\tau)\sigma_{j}^{x}(\tau)\rangle$ act as information
propagation channels between qubits $i$ and $j$.

As emphasized in the main text, OTOC is not sensitive to information
propagation channels. Figure S2 illustrates OTOCs with different spin
operators, such as
$F^{xx}_{i}\equiv\langle\sigma^{x}_{i}(\tau)\sigma^{x}_{1}(0)\sigma^{x}_{i}(\tau)\sigma^{x}_{1}(0)\rangle$,
$F^{zz}_{i}\equiv\langle\sigma^{z}_{i}(\tau)\sigma^{z}_{1}(0)\sigma^{z}_{i}(\tau)\sigma^{z}_{1}(0)\rangle$,
$F^{xz}_{i}\equiv\langle\sigma^{x}_{i}(\tau)\sigma^{z}_{1}(0)\sigma^{x}_{i}(\tau)\sigma^{z}_{1}(0)\rangle$,
and
$F^{zx}_{i}\equiv\langle\sigma^{z}_{i}(\tau)\sigma^{x}_{1}(0)\sigma^{z}_{i}(\tau)\sigma^{x}_{1}(0)\rangle$,
for the free fermion system and the intermediate system. Despite the
qualitative difference of $R^{2}_{d=0}$ observed in Fig. S1, OTOCs in the free
fermion model and the intermediate model manifest almost the identical values,
as evidenced the almost overlapping figures in Fig. S2. Therefore, the
distinction between these two models can only be discerned by scrutinizing the
dynamic distribution of information via the QRP. This sensitivity of QRP would
enables the revelation of the unknown nature of quantum information
scrambling, which remains hidden from OTOC and TMI analyses.

Figure S1:  Virtual time dependence of reservoir performance $R^{2}_{d=0}$ for
the STM task employing various operators of qubits $2,3,4$ for the read-out.
The legends on the rightmost side show the correspondence between markers and
operators: (a, d, g, j, m, p, s, v) free fermion system, (b, e, h, k, n, q, t,
w) quantum chaotic system, and (c, f, i, l, o, r, u, x) intermediate system
with $(h_{x},h_{z})=(-0.02,1.002)$.  Figure S2:  Virtual time dependence of
OTOC: (a)
$F^{xx}_{i}=\langle\sigma^{x}_{i}(\tau)\sigma^{x}_{1}(0)\sigma^{x}_{i}(\tau)\sigma^{x}_{1}(0)\rangle$,
(b)
$F^{zz}_{i}=\langle\sigma^{z}_{i}(\tau)\sigma^{z}_{1}(0)\sigma^{z}_{i}(\tau)\sigma^{z}_{1}(0)\rangle$,
(c)
$F^{xz}_{i}=\langle\sigma^{x}_{i}(\tau)\sigma^{z}_{1}(0)\sigma^{x}_{i}(\tau)\sigma^{z}_{1}(0)\rangle$,
and (d)
$F^{zx}_{i}=\langle\sigma^{z}_{i}(\tau)\sigma^{x}_{1}(0)\sigma^{z}_{i}(\tau)\sigma^{x}_{1}(0)\rangle$
for $i=2$ and $3$. The skyblue and green lines represent OTOCs in the free
fermion model, and the pink and orange lines shows OTOCs in the intermediate
model.