# Measure of quantum Fisher information flow in multi-parameter scenario

Haijun Xing<EMAIL_ADDRESS>Graduate School of China Academy of
Engineering Physics, Beijing, 100193, China  Libin Fu<EMAIL_ADDRESS>Graduate School of China Academy of Engineering Physics, Beijing, 100193,
China

###### Abstract

We generalize the quantum Fisher information flow proposed by Lu et al. [Phys.
Rev. A 82, 042103 (2010)] to the multi-parameter scenario from the information
geometry perspective. A measure named the intrinsic density flow (IDF) is
defined with the time-variation of the intrinsic density of quantum states
(IDQS). IDQS measures the local distinguishability of quantum states in state
manifolds. The validity of IDF is clarified with its vanishing under the
parameter-independent unitary evolution and outward-flow (negativity) under
the completely positive-divisible map. The temporary backflow (positivity) of
IDF is thus an essential signature of the non-Markovian dynamics. Specific for
the time-local master equation, the IDF decomposes according to the channels,
and the positive decay rate indicates the inwards-flow of the sub-IDF. As
time-dependent scalar fields equipped on the state space, the distribution of
IDQS and IDF comprehensively illustrates the distortion of state space induced
by its environment. As example, a typical qubit model is given.

##  I introduction

Information interchange between the open system and its environment is a
critical viewpoint in studying the dynamics of open quantum systems. Memory
effect, i.e., the temporary revival of previously leaked information, is one
of the fascinating topics in this fields Breuer2002 . It is considered a
signature of the non-Markovianity firstly Rivas2014 ; Breuer2016 ; Vega2017 ;
Li2019a ; Li2019b ; Breuer2009 ; Luo2012b ; Lu2010 ; Laine2010 , then became a
vital method for manipulating quantum resources Breuer2009 ; Laine2010  such
as entanglement Maniscalco2008 ; Bellomo2007 ; Mazzola2009 ; Huelga2012 ;
Mirkin2019a ; Mirkin2019b , quantum interferometric power Dhar2015 , temporal
steering Chen2016 , quantum coherence, correlations Luo2012b , and quantum
Fisher information (QFI) Lu2010 ; Song2015 . With the booming of technologies
of control and manipulation open quantum systems Haikka2011 ; Yuan2017 ;
Myatt2000 ; Liu2011 ; Gessner2014 ; Haase2018 ; Anderssen2019 ; Wu2020 ;
Lu2020 , applications in ultracold atomic gases Haikka2011 ; Yuan2017 ,
quantum speed limit Deffner2013 ; Mirkin2016 ; Cimmarusti2015 , algorithms
Dong2018 ; Roy2019 , and thermal machines Thomas2018 ; Abiuso2019  are under
intensive studies in recent years.

Specifically, in the conventional memory-free dynamics, the information leaks
outwards from the system to the environment continuously, then dissolved. If
the environment has nontrivial structures, the information may be memorized by
the environment, then partially sending back into the system subsequently.
Though the rigorous quantification of these memory effects highly depends on
the interpretation of “information,” the distinguishability of quantum states
is one of the primary choices. It can be captured by the trace distance
Breuer2009 ; Laine2010  that measures the distinguishability of a pair of
states, and quantum Fisher information (QFI) Lu2010 ; Song2015  that we focus
on in this manuscript.

The QFI is an intrinsic measure of the local distinguishability of quantum
states via estimating a given parameter. It has tight connections with the
“distance” measures between quantum states, such as the Fubini-Study metric
Gibbons1992 , quantum geometric tensor Berry1989 , Bures distance
Braunstein1994 , quantum fidelity Braunstein1994 ; Wootters1981 , and relative
entropy Bengtsson2006 . Lu et al. Lu2010  define the quantum Fisher
information flow (QFIF) as a measure of the memory effect with the time-
variation of QFI with respect to a parameter previously encoded into the probe
state. It decomposes to sub-QFIF according to the channels of the time local
master equation. The temporal appearance of inwards (positive) sub-QFIF is
identified with the positive decay rates, hence becoming a signature of non-
Markovian dynamics. It performs well in single parameter cases. Furthermore,
its applications in quantum metrology and quantum speed limits are fruitful:
lots of achievements, both theoretical and experimental, have been made.

However, the practical systems are intrinsically multiple dimensional: 1) its
states generally locate in multi-dimensional state space and thus is
characterized by more than one parameter; 2) its dynamical evolution typically
involves the variation of more than one parameter. The single-parameter scheme
is therefore inadequate in thoroughly characterizing the dynamics of open
systems. The generalization of QFIF to multi-parameter cases is essential.

In this article, we generalize the QFIF to the multi-parameter scenario from
the information geometry perspective. We propose a measure named the intrinsic
density flow (IDF) with the time-variation of the intrinsic density of quantum
states (IDQS), a fundamental information-theoretic state-distinguishability
measure. The validity of IDF will be shown with its vanishing under the
parameter-independent unitary evolution and negativity (outwards-flow) under
the CP-divisible map. It indicates the positive (inwards) IDF is an essential
signature of non-Markovian dynamics. Specific to the dynamics generated by the
time-local master equation, the IDF is decomposable according to the channels.
The direction (sign) of the sub-flow is determined by the decay rates: the
temporary appearance of the positive decay rates indicates the backward sub-
flow in the corresponding channel. It violates the CP-divisible condition and
serves as a sufficient signature of the non-Markovian dynamics.

Furthermore, as time-dependent scalar fields equipped on the state space, the
distribution of IDF and IDQS are potent tools to exhibit the detailed picture
of the state space’s distorsion under the open dynamics. We will exemplify it
with the typical model of a two-level system under the non-Markovian
dissipative channels.

This manuscript is arranged as follows: In Sec. II, we review the QFI and QFIF
from the information geometry perspective. In Sec. III, the IDF is introduced
together with the IDQS. Specifically, a form of IDQS in the time-dependent
coordinates is given in Sec. III.3, for its tight connections with the
equation of motion. In Sec. IV, the dynamics generated by the time local
master equation are studied with the IDF. In Sec. V, the time-variation of
IDQS and IDF are studied with a two-level system under the dissipative
channels. At last, we conclude in Sec. VI.

##  II Review of QFI and QFIF from information geometry perspective

In the formation geometry, a given state $\hat{\rho}(\boldsymbol{x}_{0})$ is
equivalent to a point
$\boldsymbol{x}_{0}=(x^{1}_{0},x^{2}_{0},\dots,x^{d}_{0})$ in the
$d$-dimensional parameter space $\Omega_{0}$ via the model
$\mathcal{S}_{0}=\\{\hat{\rho}(\boldsymbol{x}_{0})|\boldsymbol{x}_{0}\in\Omega_{0}\\}$.
A Riemannian metric $\boldsymbol{g}^{F}$ named as the quantum Fisher metric
(QFM) with $d\times d$ entries

$\displaystyle g^{F}_{\mu\nu}(\boldsymbol{x}_{0})=$
$\displaystyle\frac{1}{8}\mathrm{Tr}\left[\\{\hat{L}_{\mu},\hat{L}_{\nu}\\}\hat{\rho}\right],$
(1)

$1\leqslant\mu,\nu\leqslant d$, is equipped on the space $\Omega_{0}$, where
$\\{\cdot,\cdot\\}$ is the anti-commutator, and the symmetric logarithmic
derivative $\hat{L}_{\mu}$ is defined via
$\partial_{\mu}\hat{\rho}(\boldsymbol{x}_{0})\equiv\frac{1}{2}\\{\hat{\rho},\hat{L}_{\mu}\\}$
implicitly Helstrom1976 ; Holevo1982 , with
$\partial_{\mu}\equiv\partial/\partial x_{0}^{\mu}$. The QFI
$\boldsymbol{\mathcal{F}}$ is four times of the QFM $\boldsymbol{g}^{F}$.

In the single-parameter metrology, the probe state $\hat{\rho}(\theta)$
sketches a curve $\boldsymbol{x}_{0}(\theta)$ in the initial parameter space
$\Omega_{0}$ with the shift of parameter $\theta$. The estimation of $\theta$
is thus equivalent to identifying a point $\boldsymbol{x}_{0}(\theta)$ on the
curve. The number of states distinguishable in a segment of the curve
$\boldsymbol{x}_{0}(\theta)$ is measured by the segment’s length. The length
is acquired by integrating the line element $ds$ with Wootters1981

$ds^{2}=\sum\nolimits_{\mu\nu}g^{F}_{\mu\nu}\dot{x}_{0}^{\mu}\dot{x}_{0}^{\nu}d\theta^{2}=g^{F}(\theta)d\theta^{2},$
(2)

where $\dot{\boldsymbol{x}}_{0}\equiv d\boldsymbol{x}_{0}(\theta)/d\theta$ is
the derivative of the curve $\boldsymbol{x}_{0}(\theta)$. Hence
${g}^{F}(\theta)^{1/2}$ is an intrinsic measure of the local density of states
distinguishable on the curve $\boldsymbol{x}_{0}(\theta)$. As the square of
this density, the QFM itself is also a measure of the distinguishability of
$\hat{\rho}(\theta)$ from its neighboring quantum states. Furthermore, it is
directly applicable in the quantum metrology as the upper bound of the
estimator’s precision GLM2004 ; GLM2006 ; GLM2011 .

When the dynamics $\Phi_{t}$ is applied, the states corresponding to the
initial coordinates $\boldsymbol{x}_{0}$ is changed to
$\hat{\rho}(\boldsymbol{x}_{0};t)\equiv\Phi_{t}\hat{\rho}(\boldsymbol{x_{0}})$,
with $\hat{\rho}(\boldsymbol{x}_{0};0)\equiv\rho(\boldsymbol{x}_{0})$. It
forms a new state space
$\mathcal{S}_{t}=\\{\hat{\rho}(\boldsymbol{x}_{0};t)=\Phi_{t}\hat{\rho}(\boldsymbol{x}_{0})|\boldsymbol{x}_{0}\in\Omega_{0}\\}$
at the given time $t$. Actually, $(\boldsymbol{x}_{0};t)$ is a coordinates of
the $(d+1)$-dimensional manifolds $\bigcup_{t}\mathcal{S}_{t}$ with $t$ as an
additional dimension.

Under the dynamics $\Phi_{t}$, we also have
$\hat{\rho}(\theta)\rightarrow\hat{\rho}(\theta;t)$. The corresponding QFI
$\mathcal{F}(\theta;t)$ is lost (revival) with the variation of the state
space $\mathcal{S}_{t}$. It can be accounted as the effects of outflow
(inflow) of the information. Thus, Lu et al. define the QFIF as Lu2010

$\mathcal{I}_{\mathrm{LWS}}(\theta;t)\equiv\frac{d\mathcal{F}(\theta;t)}{dt}=4\frac{dg^{F}(\theta;t)}{dt},$
(3)

i.e., the time variation of the QFI. From the information geometry
perspective, it measures the time-variation of the square of the density of
distinguishable states along the given curve $\boldsymbol{x}_{0}(\theta)$.

##  III Intrinsic density of quantum states and Intrinsic density flow

###  III.1 Intrinsic density of quantum states

To identify a state $\hat{\rho}(\boldsymbol{x}_{0};t)\in\mathcal{S}_{t}$ in
general cases, one should acknowledge all of the components of
$\boldsymbol{x}_{0}$ at a given time $t$. It is equivalent to localizing the
point $\boldsymbol{x}_{0}$ in the $d$-dimensional initial parameter space
$\Omega_{0}$, where $\hat{\rho}(\boldsymbol{x}_{0};t)$ is surrounded by states
in all of the “directions.” Furthermore, the dynamical evolutions generally
affect the distinguishability of quantum states in the multi-direction of the
state space $\mathcal{S}_{t}$. A single element of the QFM, i.e.,
$g^{F}(\theta)$, is thus inadequate for characterizing the distinguishability
of $\hat{\rho}(\boldsymbol{x}_{0};t)$ out of its neighborhood.

Theoretically, the local statistical distinguishability of quantum states in
$\Omega_{0}$ can be measured with the intrinsic density of quantum states
(IDQS) Xing2020

$\mathcal{D}_{Q}(\boldsymbol{x}_{0};t)=\sqrt{|\boldsymbol{g}^{F}(\boldsymbol{x}_{0};t)|}=\frac{dV(\mathcal{S}_{t})}{d^{d}\boldsymbol{x}_{0}},$
(4)

with $|\boldsymbol{A}|$ denoting the determinant of matrix $\boldsymbol{A}$,
where the invariant volume element $dV(\mathcal{S}_{t})$ quantifies the number
of quantum states locating in the element $d^{d}\boldsymbol{x}_{0}$. We
mention that for the pure state, IDQS is the measure that defines the
completeness relationship of (sub-manifolds of) the projective Hilbert spaces
Xing2020 ; Bengtsson2006 .

###  III.2 Intrinsic density flow

Although the state $\hat{\rho}(\boldsymbol{x}_{0};t)$ is evolving under the
map $\Phi_{t}$, the corresponding point $\boldsymbol{x}_{0}$ is stationary in
the initial parameter space $\Omega_{0}$. On the contrary, the QFM, thereby
IDQS, is time-dependent and capable of characterizing the distortion of state
space under the map. Specifically, IDQS is a qualified “information” measure
that meets the essential criteria Ruskai1994 ; Breuer2009 ; Breuer2016
satisfied by the trace distance: IDQS is non-negative, invariant for
parameter-independent unitary dynamics, and contraction under the parameter-
independent completely positive and trace-preserving (CPTP) maps. In a concise
form, we have (for proof, see Appendix A, B, and C)

$\mathcal{D}_{Q}(\boldsymbol{x}_{0};t)\geqslant\mathcal{D}_{Q}(\boldsymbol{x}_{0};t^{\prime})\geqslant
0,$ (5)

for arbitrary given state $\hat{\rho}(\boldsymbol{x}_{0};t)$ and
$\hat{\rho}(\boldsymbol{x}_{0};t^{\prime})\equiv\Lambda_{t^{\prime},t}[\hat{\rho}(\boldsymbol{x}_{0};t)]$
with $\Lambda_{t^{\prime},t}$ denoting an arbitrary parameter-independent CPTP
map, where the first equality is reached by the unitary channel
$\Lambda_{t^{\prime},t}[\cdot]=\hat{U}(t,t^{\prime})\cdot\hat{U}^{\dagger}(t,t^{\prime})$
with $\partial_{\mu}\hat{U}(t,t^{\prime})=0$. The unitary invariance of
$\mathcal{D}(\boldsymbol{x}_{0};t)$ indicates IDQS measuring an information
which conservative in the composite of system and environment. The contraction
of IDQS indicates the revival information is always smaller than the previous
leaking information.

Based on the above discussions, we define the intrinsic density flow (IDF) as

$\mathcal{I}(\boldsymbol{x}_{0};t)\equiv\frac{d}{dt}\mathcal{D}_{Q}(\boldsymbol{x}_{0};t),$
(6)

i.e., the time-variation of the IDQS. Its negative value indicates the leaking
of information from system to environment. The positive value indicates the
backflow of the previous leaking information, which is a signature of the non-
Markovianity. Specifically, the IDF has the following properties:

a1

IDF is not positive under the parameter-independent CP-divisible dynamics and
vanishes under the parameter-independent unitary dynamics. It directly results
from Eq. (5) and makes $\mathcal{I}(\boldsymbol{x}_{0};t)>0$ a sufficient
condition for the non-CP divisible dynamics.

a2

IDF is a linear function of $d\hat{\rho}/dt$ and
$\mathcal{D}(\boldsymbol{x}_{0};t)$ as (for details, see Appendix C)

$\mathcal{I}(\boldsymbol{x}_{0};t)=\mathcal{I}^{R}(\boldsymbol{x}_{0};t)\mathcal{D}(\boldsymbol{x}_{0};t),$
(7)

with the _relative intrinsic density flow_ (RIDF)

$\mathcal{I}^{R}(\boldsymbol{x}_{0};t)\equiv\frac{1}{2}\mathrm{tr}\left[\frac{1}{\boldsymbol{g}^{F}}\mathrm{Tr}(\boldsymbol{\mathcal{L}}\frac{d\hat{\rho}}{dt})\right],$
(8)

where $\mathrm{Tr}$ (tr) denotes the trace operation in Hilbert space
(parameter space), $\boldsymbol{\mathcal{L}}$ is a $d\times d$-dimensional
matrix with the entry

$\hat{\mathcal{L}}_{\mu\nu}=\frac{1}{2}\left[\hat{L}_{\nu}(2\hat{\partial}_{\mu}-\hat{L}_{\mu})+\hat{L}_{\mu}(2\hat{\partial}_{\nu}-\hat{L}_{\nu})\right].$
(9)

IDF thus inherits the linear structures of the master equation, as shown in
Sec. IV.

a3

By choosing $\boldsymbol{x}_{0}$ as a complete basis, i.e., a coordinate
system of the state space concerned, the IDQS
$\mathcal{D}(\boldsymbol{x}_{0};0)$ captures all the local information of the
initial state $\rho(\boldsymbol{x}_{0};0)$. Then effects of the dynamical
evolution $\Phi_{t}$ on all of the components of $\boldsymbol{x}_{0}$ are
accounted in $\mathcal{I}(\boldsymbol{x}_{0};t)$.

a4

The RIDF is independent of the parameterization model. For two time-
independent coordinates $\boldsymbol{x}_{0}$ and $\boldsymbol{y}_{0}$ of the
initial parameter space $\Omega_{0}$, we have

$\mathcal{I}^{R}(\boldsymbol{x}_{0};t)=\mathcal{I}^{R}(\boldsymbol{y}_{0};t).$
(10)

The corresponding IDFs only differ in a time-independent constant as
$\mathcal{I}(\boldsymbol{x}_{0};t)=\left|\frac{\partial\boldsymbol{y}_{0}}{\partial\boldsymbol{x}_{0}}\right|\mathcal{I}(\boldsymbol{y}_{0};t)$.
Hence, the direction of IDF is also independent of the parameterization model.

These properties make the IDF be a qualified measure of the local information
flow. Before further studies, we will introduce another form of IDF.

###  III.3 State space $\mathcal{S}$ and IDF

In practical studies, researchers favor to identify the state space
$\mathcal{S}_{t}$ at different times with
$(\boldsymbol{x}_{0};t)\sim(\boldsymbol{x}^{\prime}_{0};t^{\prime})$ if
$\hat{\rho}(\boldsymbol{x}_{0};t)=\hat{\rho}(\boldsymbol{x}^{\prime}_{0},t^{\prime})$.
We denote it as the state space
$\mathcal{S}\equiv\\{\hat{\rho}(\boldsymbol{x})=\hat{\rho}(\boldsymbol{x}_{0};t)|\boldsymbol{x}_{0}\in\Omega_{0},t\\}$,
where $\boldsymbol{x}$ serves as the coordinates of the state space
$\mathcal{S}$. A dynamical evolution is thus depicted by the movement in
$\mathcal{S}$: the initial state $\hat{\rho}(\boldsymbol{x}_{0})$ sketches an
orbit illustrated by the equations of motion $\boldsymbol{x}(t)$ with
$\boldsymbol{x}(0)=\boldsymbol{x}_{0}$. One can define an alternative IDQS in
space $\mathcal{S}$ as
$\mathcal{D}_{Q}(\boldsymbol{x})=\sqrt{|\boldsymbol{g}^{F}(\boldsymbol{x})|}$
with respect to the coordinates $\boldsymbol{x}$.

Figure 1: (color online). Two coordinates (parameterization schemes) form the
quantum metrology perspective. (a) The initial parameter $\boldsymbol{x}_{0}$
are encoded into the probe $\rho$ firstly, then sent to the channel
$\Phi_{t}$. The precision of $\boldsymbol{x}_{0}$’s estimation is upper
bounded by the QFM $\boldsymbol{g}^{F}(\boldsymbol{x}_{0};t)$. (b) The
parameter $\boldsymbol{x}$ are encoded after the state through the channel
$\Phi_{t}$. The precision of $\boldsymbol{x}$’s estimation is upper bounded by
the QFM $\boldsymbol{g}^{F}(\boldsymbol{x})$.

For state $\hat{\rho}(\boldsymbol{x})$ with the equation of motion
$\boldsymbol{x}(t)$, we have the IDQS
$\mathcal{D}_{Q}(\boldsymbol{x}_{0};t)=\left|\frac{\partial\boldsymbol{x}}{\partial\boldsymbol{x}_{0}}\right|\mathcal{D}_{Q}(\boldsymbol{x}),$
and the corresponding IDF decomposes as

$\displaystyle\mathcal{I}($
$\displaystyle\boldsymbol{x}_{0};t)=\mathcal{D}_{Q}(\boldsymbol{x}_{0};t)$
$\displaystyle\times\left\\{\frac{1}{2}\mathrm{tr}\left[\frac{d}{dt}\log\boldsymbol{g}^{F}(\boldsymbol{x})\right]+\mathrm{tr}\left[\frac{d}{dt}\log\begin{pmatrix}\frac{\partial\boldsymbol{x}}{\partial\boldsymbol{x}_{0}}\end{pmatrix}\right]\right\\},$
(11)

where the first term describes shift of the point $\boldsymbol{x}(t)$ along
its orbit in the state space $\mathcal{S}$. The second term is contributed by
the Jacobian which connect the state space $\mathcal{S}$ and initial parameter
space $\Omega_{0}$ . From the initial information perspective, it depicts the
variation of the frame $\boldsymbol{x}$, i.e., the _background geometry_ of
$\mathcal{S}$.

We mention that, from the quantum metrology perspective, the coordinates
$\boldsymbol{x}_{0}$ and $\boldsymbol{x}$ are parameters encoded into the
probe and awaiting estimation. As shown in Fig. 1a (b),
$\boldsymbol{g}^{F}(\boldsymbol{x}_{0};t)$
($\boldsymbol{g}^{F}(\boldsymbol{x})$) depicts the QFI acquired via
parameterizing states before (after) the dynamical evolution $\Phi_{t}$. In
case Fig. 1 b, the channel $\Phi_{t}$ is actually part of the state
preparation.

##  IV Intrinsic density flow with time-local master equation

In this section, we study the dynamics of open quantum systems with the IDF.
Specifically, we focus on the state $\hat{\rho}(\boldsymbol{x}_{0};t)$ whose
evolution is governed by the time-local master equation Gorini1976 ;
Lindblad1976 ; Breuer2004 ; Breuer2016

$\frac{d}{dt}\hat{\rho}(\boldsymbol{x}_{0};t)=\mathcal{K}(t)\hat{\rho}(\boldsymbol{x}_{0};t),$
(12)

with the generator $\mathcal{K}(t)$ acting on the state $\hat{\rho}$ as

$\mathcal{K}(t)\hat{\rho}=-\mathrm{i}[\hat{H},\hat{\rho}]+\sum_{i}\gamma_{i}\left[\hat{A}_{i}\hat{\rho}\hat{A}_{i}^{\dagger}-\frac{1}{2}\\{\hat{A}_{i}^{\dagger}\hat{A}_{i},\hat{\rho}\\}\right],$
(13)

where all of the $\hat{H}$, $\gamma_{i}$, and $\hat{A}_{i}$ are generally
time-dependent. It is a generalization of the conventional Lindblad master
equation that all $\hat{A}_{i}$ and $\gamma_{i}$ are time-independent, and
$\gamma_{i}$ are non-negative. Eq. (13) leads a CP-divisible dynamic, if and
only if $\gamma_{i}$ is non-negative for all channel $\hat{A}_{i}$ at all of
the time Gorini1976 ; Lindblad1976 ; Breuer2016 . Hence, the temporary
appearance of negative $\gamma_{i}$ is taken as the signature of the non-
Markovian (non CP-divisible) dynamics. It is also necessary for the memory
effects and backflow of the information Piilo2008 ; Breuer2009b . We further
assume $\partial_{\mu}\hat{H}=0$ and $\partial_{\mu}\hat{A}_{i}=0.$ It
indicates the linearity of the von Neumann equation and inconsistent with the
quantum no-cloning theorem Lu2010 ; Wootters1982 . It also makes state
$\hat{\rho}(\boldsymbol{x}_{0};t)$ a stationary point in the initial parameter
space.

Firstly, we focus on a special case with $\gamma_{i}=0$ $\forall i$, where Eq.
(13) reduces to the unitary evolution. The corresponding IDF vanishes as
$\mathcal{I}(\boldsymbol{x}_{0};t)_{0}=0$, with the additional footnote $0$
denoting the unitarity. It directly results from the time-invariance of the
metric under unitary dynamics with (for proof, see Appendix A)

$\displaystyle\frac{d}{dt}g^{F}_{\mu\nu}(\boldsymbol{x}_{0};t)_{0}=0.$ (14)

It indicates that the parameter space $\Omega_{0}$ is frozen. It is
tremendously different from the picture in state space $\mathcal{S}$. In the
coordinates $\boldsymbol{x}$, the system may demonstrate very complicated
dynamics.

In the general cases of Eq. (13), the system exchanges information with the
environment through each of the channel $\hat{A}_{i}$ with $\gamma_{i}\neq 0$.
Specifically, the IDF decomposes as
$\mathcal{I}(\boldsymbol{x}_{0};t)=\sum_{i}\mathcal{I}_{i}(\boldsymbol{x}_{0};t)$,
where

$\mathcal{I}_{i}(\boldsymbol{x}_{0};t)\equiv\frac{\gamma_{i}}{2}\mathrm{tr}\left[\frac{1}{\boldsymbol{g}^{F}}\left(\frac{d}{dt}\boldsymbol{g}^{F}\right)_{i}\right]\mathcal{D}_{Q}(\boldsymbol{x}_{0};t)$
(15)

denotes the sub-IDF through the channel $\hat{A}_{i}$ with the derivatives
(for details, see Appendix D)

$(\frac{d}{dt}g^{F}_{\mu\nu})_{i}=\frac{1}{2}\mathrm{tr}\left\\{([\hat{A}_{i}^{\dagger},\hat{L}_{\nu}][\hat{A}_{i},\hat{L}_{\mu}]+[\hat{A}_{i}^{\dagger},\hat{L}_{\mu}][\hat{A}_{i},\hat{L}_{\nu}])\hat{\rho}\right\\}.$
(16)

For the matrix $(d\boldsymbol{g}^{F}/dt)_{i}$ is negative semidefinite, we
have the direction (sign) of sub-flows

$\mathcal{I}_{i}(\boldsymbol{x}_{0};t)\begin{cases}\geqslant
0,&\gamma_{i}(t)<0\\\ =0,&\gamma_{i}(t)=0\\\ \leqslant
0,&\gamma_{i}(t)>0.\end{cases}$ (17)

These results are full of physical implications: (1) The decomposition of IDF
according to the channel results from the time-local master equation’s
linearity to $d\hat{\rho}/dt$. (2) The direction of the sub-flow
$\mathcal{I}_{i}$ is controlled by $\gamma_{i}$. If there exist a channel such
that $\gamma_{i}<0$ for some time $t$, the corresponding sub-flow will flow
back to the system. It is consistent with the CP-divisible condition given by
the Gorimi-Kossakowski-Sudarshan-Lindblad theorem Gorini1976 ; Lindblad1976 .
We mention that these results are the natural generalization of the
proposition Eq. (6) in Lu2010 . However, it is now valid in the multi-
parameter scenario and independent of the parameterization scheme.

##  V Two-level systems

For two-level system with basis $\\{|0\rangle,|1\rangle\\}$, we parameterize
the general mixed state $\hat{\rho}$ as

$\hat{\rho}(\boldsymbol{n})=\frac{1}{2}\left(\hat{\mathbb{I}}_{2}+\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\right),$
(18)

with the Pauli matrices
$\hat{\boldsymbol{\sigma}}=(\hat{\sigma}_{1},\hat{\sigma}_{2},\hat{\sigma}_{3})$,
where $\mathcal{S}$ is the Bloch sphere, and coordinates $\boldsymbol{x}$ is
the Bloch vector $\boldsymbol{n}=(n^{1},n^{2},n^{3})$ , with
$|\boldsymbol{n}|^{2}\equiv\sum_{\mu}(n^{\mu})^{2}\leqslant 1$. We have the
density

$\mathcal{D}_{Q}(\boldsymbol{n})\equiv\sqrt{|\boldsymbol{g}^{F}(\boldsymbol{n})|}=\frac{1}{8\sqrt{1-|\boldsymbol{n}|^{2}}}.$
(19)

This density only depends on the radius $|\boldsymbol{n}|$, i.e., the state’s
purity. It results from the unitary invariance of IDQS.
$\mathcal{D}_{Q}(\boldsymbol{n})$ is divergent if $|\boldsymbol{n}|=1$. It is
induced by the radial element
$g^{F}_{nn}(\boldsymbol{n})=1/[4(1-|\boldsymbol{n}|^{2})]$, which depicts the
distinct statistical difference between pure and mixed states. However, the
statistical distance between any pair of pure and mixed states are still
finite, so is the volume given by this density. The minimum of
$\mathcal{D}_{Q}(\boldsymbol{n})$ is reached by the completely mixed state
with $\boldsymbol{n}=0$. Surprisingly, this minimum is non-zero. It depicts
the statistical difference between $\hat{\rho}(\boldsymbol{0})$ and its
neighboring states, although $\hat{\rho}(\boldsymbol{0})$ itself is usually
termed as information-free.

Figure 2: (color online). Two-level systems in dissipative channel
($W=3\lambda$). (a) $\gamma$ as a function of time; (b) The IDQS
$\mathcal{D}_{Q}(\boldsymbol{n}_{0};t)$ as a function of time, with
$r_{0}=\sqrt{0.9}$; (c) the IDF $\mathcal{I}(\boldsymbol{n}_{0};t)$ as a
function of time; (d) the RIDF $\mathcal{I}^{R}(\boldsymbol{n}_{0};t)$ as a
function of time. (a1)-(a3) $\mathcal{D}_{Q}(\boldsymbol{n})$ in
$n^{1}$-$n^{3}$ plane with $n^{2}=0$ at $\lambda t=0.02$, $0.5$, and $1.0$;
(b1)-(b3) The IDQS $\mathcal{D}_{Q}(\boldsymbol{n}_{0};t)$ in
$n^{1}_{0}$-$n^{3}_{0}$ plane with $n_{2}^{0}=0$ at $\lambda t=0.02$, $0.5$,
and $1.0$. (c1)-(c3) $\mathcal{I}(\boldsymbol{n_{0}};t)$ in
$n^{1}_{0}$-$n^{3}_{0}$ plane with $n^{2}_{0}=0$ at $\lambda t=0.1$, $0.5$,
and $1.0$; (d1)-(d3) $\mathcal{I}^{R}(\boldsymbol{n_{0}};t)$ in
$n^{1}_{0}$-$n^{3}_{0}$ plane with $n_{2}^{0}=0$ at $\lambda t=0.1$, $0.5$,
and $1.0$.

###  V.1 Dissipative channels

In this sub-section, we study a typical model where a two-level system is
immersed in a dissipative environment Breuer1999 ; Lu2010 ; Breuer2016 . In
the interaction picture, the system undergoes a dynamic generated by the
master equation

$\displaystyle\frac{d}{dt}\hat{\rho}(t)=$
$\displaystyle\gamma(t)\left[\hat{\sigma}_{-}\hat{\rho}\hat{\sigma}_{+}-\frac{1}{2}\\{\hat{\sigma}_{+}\hat{\sigma}_{-},\hat{\rho}\\}\right],$
(20)

with the raising (lowering) operator $\hat{\sigma}_{+}$ $(\hat{\sigma}_{-})$.
By assuming the environment has a Lorentzian spectral density with vanishing
detuning, we have $\gamma(t)=-2\dot{h}(t)/h(t)$ with the characteristic
function

$h(t)=\begin{cases}e^{-\lambda
t/2}\left[\cosh(\frac{dt}{2})+\frac{\lambda}{2}\sinh(\frac{dt}{2})\right],&W<\frac{\lambda}{2}\\\
e^{-\lambda
t/2}\left[\cos(\frac{dt}{2})+\frac{\lambda}{2}\sin(\frac{dt}{2})\right],&W\geqslant\frac{\lambda}{2},\end{cases}$
(21)

$d=\sqrt{|\lambda^{2}-4W^{2}|}$, where $W$ measures the system-environment
coupling strength, $\lambda$ defines the width of the spectral density. In the
weak coupling regime with $W<\lambda/2$, $\gamma(t)$ is non-negative. The
system undergoes a Markovian dynamics, where the information—measured by both
of the trace distance Laine2010  and QFI Lu2010 —is lost continuously. For
conciseness, we mainly focus on the strong coupling regime with
$W\geqslant\lambda/2$, where $h(t)$ displays an oscillatory behavior and the
non-Markovian dynamics emergent with the negative $\gamma(t)$.

We begin with the equations of motion

$\displaystyle n^{\mu}(t)$ $\displaystyle=h(t)n^{\mu}_{0},\quad\mu=1,2,$ (22a)
$\displaystyle n^{3}(t)$ $\displaystyle=h(t)^{2}(1+n^{3}_{0})-1.$ (22b)

Under this equation, the Bloch sphere contracts to ground state $|0\rangle$
firstly with the positive $\gamma(t)$, then partially swell back corresponding
to the negative $\gamma(t)$, as shown by Fig. 2 a1-a3. Together with Eq. (11)
and (19), we have the IDF

$\mathcal{I}(\boldsymbol{n}_{0};t)=-\frac{\gamma}{2}\left[3+\frac{(1+n^{3})^{2}}{1-|\boldsymbol{n}|^{2}}\right]\mathcal{D}_{Q}(\boldsymbol{n}_{0};t)$
(23)

with the IDQS

$\mathcal{D}_{Q}(\boldsymbol{n}_{0};t)=\frac{h(t)^{4}}{8\sqrt{1-|\boldsymbol{n}|^{2}}}.$
(24)

As shown by Fig. 2a, $\gamma(t)$ oscillates between the positive and negative
values. It induces the information flow $\mathcal{I}(\boldsymbol{n}_{0};t)$ to
propagate outwards and inwards, and the IDQS
$\mathcal{D}_{Q}(\boldsymbol{n}_{0};t)$ increases and decreases
correspondingly. The time-local extremum of the IDQS is given at the
transition times of inwards and outwards flow, when the IDQS vanishes with the
Bloch sphere shrinks to the point $|0\rangle$. In Fig. 2b (2c), we illustrate
the IDQS (IDF) of four initial states: $\hat{\rho}_{1}(0,0,r_{0})$,
$\hat{\rho}_{2}(r_{0},0,0)$, $\hat{\rho}_{3}(0,0,-r_{0})$, and
$\hat{\rho}_{4}(0,0,0)$, with blue, red, green, and purple lines,
respectively. At $t=0$, $\hat{\rho}_{1}$, $\hat{\rho}_{2}$, and
$\hat{\rho}_{3}$ locate in the same shell with radius $r_{0}=\sqrt{0.9}$, the
maximally mixed state $\hat{\rho}_{4}$ locates in the center of the space.
They show the same time-dependent non-Markovian behavior. The state
$\hat{\rho}_{3}$ near the stationary point $|0\rangle$ has the biggest IDQS
over the four states. We mention that the distinguishability of the
“information free” state $\hat{\rho}_{4}$ from its neighboring states are
still captured by IDQS and IDF, which exhibited the same dynamical
characteristics as other states.

For both the IQDS and IDF are scalar fields equipped on the parameter space
$\Omega_{0}$, the variation of their distributions provides us an interesting
viewpoint to study the dynamics of the open system. As shown by Fig. 2 c2 and
c3, the IDF takes its maximum at point $|1\rangle$ in the beginning of the
first contraction. It induces the tremendous decrease of the IDQS for states
in its neighborhood, as shown by Fig. 2 b2 and exemplified by state
$\hat{\rho}_{1}$. The IDQS always takes its maximum (minimum) in the point
$|0\rangle$ ($|1\rangle$) in the subsequent evolutions. It is induced by the
fact that the inward flow’s magnitude is always smaller than the previous
outwards flow, although the flow takes its maximum at state $|0\rangle$ in the
following stages.

Furthermore, the RIDF captures a parameterization-independent signature of the
dynamics. As a time-dependent scalar field equipped on the parameter space,
its distribution is an overall description of the relative strength of state
space’s distortion. Specific for this dissipative model, we have

$\mathcal{I}^{R}(\boldsymbol{n}_{0};t)=-\frac{\gamma}{2}\left[3+\frac{(1+n^{3})^{2}}{1-|\boldsymbol{n}|^{2}}\right].$
(25)

Its distribution is shown by Fig. 2 d1-d3. It indicates the IDQS near point
$|1\rangle$ is lost and acquired with a more significant relative strength,
not only in the first contraction but also in the full evolutions. It consists
of the insights that the excited states component $|1\rangle\langle 1|$ is
mostly influenced by the dissipative channel. Furthermore, the gradient and
range of $\mathcal{I}^{R}(\boldsymbol{n}_{0};t)$ indicate the uniformity of
the time-variation of IDQS fields. As exemplified by Fig. 2 d and d1, the IDQS
leaks with a larger gradient in the first contraction. In the following stage,
as shown by Fig. 2 d2 and d3, the gradient and range are tremendously
decreased. The IDQS at all of the points are lost and revival with roughly the
same RIDF. It results from that the Bloch sphere shrinks to an ellipsoid
highly localized around the point $|0\rangle$ after the first contraction.

##  VI Conclusions

In conclusion, we have generalized the quantum Fisher information flow (QFIF)
Lu2010  to the multi-parameter scenario from the information geometry
perspective. We propose a measure named the intrinsic density flow (IDF) with
the time-variation of quantum states’ local distinguishability, quantified by
the intrinsic density of quantum states. The validity of IDF has been shown
with its vanishing under the parameter-independent unitary evolution and
negativity (propagate outwards) under the completely positive and trace-
preserving map. It makes the positive (inwards) IDF be a vital signature of
non-Markovian dynamics.

Specific for dynamics generated by the time local master equation, we have
shown the IDF is decomposable according to the channels. The direction (sign)
of the sub-flow is determined by the decay rates: the temporary appearance of
the positive decay rates indicates the sub-flow in the corresponding channel
flowing backward. It violates the CP-divisible condition and serves as an
essential signature of the non-Markovian dynamics.

Not only having tight connections with the non-Markovian dynamics, but the
IDQS and IDF themselves are also significant for the studies of open quantum
systems. As time-dependent scalar fields equipped on the state space, the
distribution of IDF and IDQS are potent tools to exhibit the global picture of
the state space’s variation under the open dynamics. We have exemplified it
with the qubit system under the non-Markovian dissipative channel.

Via ten years of productive studies, the QFIF exhibits its values in both
theoretical and experimental aspects. Our research and the IDF provide a path
to generalize these studies to the multi-parameter cases. Furthermore, its
tight connections with the information geometry have been built, which gives
us the systematic methods to study the dynamics of the open quantum systems.
We mention that IDF is one of many measures provided by the quantum Fisher
metric, though possibly the most important one. Other explorations are in
processing. The potential value of this field is promising. We expect this
article can catalyze more studies of the open quantum system from the
information geometrical perspective.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(NSFC) (Grant No. 12088101, No. 11725417, and No. U1930403) and Science
Challenge Project (Grant No. TZ2018005).

##  Appendix A Proof of Eq. (14): The IDF vanishes under the parameter-
independent unitary channel.

We will prove the invariance of QFM metric

$\frac{d}{dt}{\boldsymbol{g}}^{F}(\boldsymbol{x}_{0};t)=0,$ (26)

under the unitary evolution $U$ generated by

$\frac{d}{dt}\rho(\boldsymbol{x}_{0};t)=-\mathrm{i}[H,\rho],$ (27)

where $H$ is assumed Hermitian and parameter-independent with $H=H^{\dagger}$
and $\partial_{\mu}H=0$.

###### Proof.

We denote the symmetric logarithmic derivative of $\rho(\boldsymbol{x}_{0};0)$
as $L_{\mu}(\boldsymbol{x}_{0};0)$, which satisfying
$\partial_{\mu}\rho(\boldsymbol{x_{0}};0)=\frac{1}{2}\\{\rho(\boldsymbol{x}_{0};0),L(\boldsymbol{x}_{0};0)\\}$,
and $L_{\mu}^{\dagger}(\boldsymbol{x}_{0};0)=L_{\mu}(\boldsymbol{x}_{0};0)$.
Then

$L_{\mu}(\boldsymbol{x}_{0};t)=U(t,0)L(\boldsymbol{x}_{0};0)U^{\dagger}(t,0)$
(28)

is a valid symmetric logarithmic derivative of the state
$\rho(\boldsymbol{x}_{0};t)=U(t,0)\rho(\boldsymbol{x}_{0};0)U^{\dagger}(t,0)$,
which satisfies
$\partial_{\mu}\rho(\boldsymbol{x}_{0};t)=\frac{1}{2}\\{L_{\mu}(\boldsymbol{x}_{0};t),\rho(\boldsymbol{x}_{0};t)\\}$
and $L^{\dagger}_{\mu}(\boldsymbol{x}_{0};t)=L_{\mu}(\boldsymbol{x_{0}};t)$.
Insert it into the definition of QFM Eq. (1), we have

$\displaystyle g^{F}_{\mu\nu}(\boldsymbol{x}_{0};t)$ $\displaystyle=$
$\displaystyle\frac{1}{8}\mathrm{tr}\left[\left\\{L_{\mu}(\boldsymbol{x}_{0};t),L_{\nu}(\boldsymbol{x}_{0};t)\right\\}\rho(\boldsymbol{x}_{0};t)\right]$
(29) $\displaystyle=$ $\displaystyle g^{F}_{\mu\nu}(\boldsymbol{x}_{0};0).$

Eq. (26), i.e., Eq. (14) is thus proved. ∎

##  Appendix B Proof of Eq. (5): The IDQS is not increased under parameter-
independent CPTP map.

Firstly, we prove that the IDQS is not increased under parameter-independent
CPTP map $\Lambda_{t^{\prime},t}$ which reads

$\mathcal{D}_{Q}(\boldsymbol{x}_{0};t)\geqslant\mathcal{D}_{Q}(\boldsymbol{x}_{0};t^{\prime}),$
(30)

with
$\rho(\boldsymbol{x}_{0};t^{\prime})=\Lambda_{t^{\prime},t}[\rho(\boldsymbol{x}_{0};t)]$
and $t^{\prime}\geqslant t$.

###### Proof.

We begin with the coordinates $\boldsymbol{y}_{0}$ in which
$\boldsymbol{g}^{F}(\boldsymbol{y}_{0};t)$ is diagonal, i.e.,
$\boldsymbol{g}^{F}_{\mu\nu}(\boldsymbol{y}_{0};t)=0$ for $\mu\neq\nu$. For
the QFI is not increased under CPTP map $\Lambda_{t^{\prime},t}$ in the single
parameter cases, we have

$\boldsymbol{g}^{F}_{\mu\mu}(\boldsymbol{y}_{0};t)\geqslant\boldsymbol{g}^{F}_{\mu\mu}(\boldsymbol{y}_{0};t^{\prime}),$
(31)

for all $\mu$. For $g^{F}(\boldsymbol{y}_{0};t^{\prime})$ is positive
semidefinite, we have

$\prod_{\mu}\boldsymbol{g}^{F}_{\mu\mu}(\boldsymbol{y}_{0};t^{\prime})\geqslant\left|\boldsymbol{g}^{F}(\boldsymbol{y}_{0};t^{\prime})\right|$
(32)

with the Hadamard’s inequality. Together with Eq. (31), we have

$\left|\boldsymbol{g}^{F}(\boldsymbol{y}_{0};t)\right|\geqslant\left|\boldsymbol{g}^{F}(\boldsymbol{y}_{0};t^{\prime})\right|.$
(33)

Multiplying $|\partial\boldsymbol{y}_{0}/\partial\boldsymbol{x}_{0}|$ on both
sides of Eq. (33), we have thus proved Eq. (30). ∎

The equality in Eq. (30) is reached by the unitary channel. It is the direct
results of Eq. (26). Furthermore, we have
$\mathcal{D}_{Q}(\boldsymbol{x}_{0};t)\geqslant 0$ with the positive semi-
definiteness of QFM. Eq. (5) is thus proved.

##  Appendix C IDF with the symmetric logarithmic derivative

Firstly, we expand the IDF as

$\mathcal{I}(\boldsymbol{x}_{0};t)\equiv\frac{d}{dt}\left|\boldsymbol{g}^{F}\right|^{\frac{1}{2}}=\frac{1}{2}|\boldsymbol{g}^{F}|^{\frac{1}{2}}\mathrm{tr}\left[\frac{d}{dt}\log\boldsymbol{g}^{F}\right],$
(34)

with
$\mathrm{tr}[\frac{d}{dt}\log\boldsymbol{g}^{F}]=\mathrm{tr}[\frac{d}{dt}\boldsymbol{g}^{F}/\boldsymbol{g}^{F}]$,
$\boldsymbol{g}^{F}\equiv\boldsymbol{g}^{F}(\boldsymbol{x}_{0};t)$ for
simplicity, and the element

$\frac{d}{dt}g^{F}_{\mu\nu}=\frac{1}{2}\mathrm{tr}[L_{\nu}\\{\rho,\dot{L}_{\mu}\\}+L_{\mu}\\{\rho,\dot{L}_{\nu}\\}+\\{L_{\mu},L_{\nu}\\}\dot{\rho}],$
(35)

with $\dot{A}\equiv\frac{d}{dt}A$ for sccinctness. For the derivative

$\frac{d}{dt}\partial_{\mu}\rho=\frac{1}{2}\mathrm{tr}\left[\\{\dot{\rho},L_{\mu}\\}+\\{\rho,\dot{L}_{\mu}\\}\right],$
(36)

we have the anti-commutaor

$\displaystyle\\{\rho,\dot{L}_{\mu}\\}$
$\displaystyle=2\partial_{\mu}\dot{\rho}-\\{\dot{\rho},L_{\mu}\\}=(2\partial_{\mu}-L_{\mu})\dot{\rho}-\dot{\rho}L_{\mu}.$
(37)

Insert it into Eq. (36), we have

$\displaystyle\frac{d}{dt}g^{F}_{\mu\nu}(\boldsymbol{x}_{0};t)$
$\displaystyle=\mathrm{Tr}\left[\mathcal{L}_{\mu\nu}(\dot{\rho})\right],$ (38)

with the operator

$\displaystyle\mathcal{L}_{\mu\nu}=\frac{1}{2}\left[L_{\nu}(2\partial_{\mu}-L_{\mu})+L_{\mu}(2\partial_{\nu}-L_{\nu})\right].$
(39)

##  Appendix D Fisher information flow with time-local master equation

In this appendix, we give the form of IDF under the time-local master equation
Eq. (13). We begin with the trace

$\displaystyle\mathrm{tr}\left[L_{\nu}\left(2\partial_{\mu}-L_{\mu}\right)\frac{d}{dt}\rho\right]_{i}$
$\displaystyle=$
$\displaystyle\mathrm{tr}\left[L_{\nu}\left(2\partial_{\mu}-L_{\mu}\right)\left(A_{i}\rho
A_{i}^{\dagger}-\frac{1}{2}\\{A_{i}^{\dagger}A_{i},\rho\\}\right)\right]$
$\displaystyle=$ $\displaystyle\mathrm{tr}\left[L_{\nu}A_{i}(L_{\mu}\rho+\rho
L_{\mu})A_{i}^{\dagger}-L_{\nu}L_{\mu}A_{i}\rho
A_{i}^{\dagger}-\frac{1}{2}L_{\nu}\\{A_{i}^{\dagger}A_{i},L_{\mu}\rho+\rho
L_{\mu}\\}+\frac{1}{2}L_{\nu}L_{\mu}\\{A_{i}^{\dagger}A_{i},\rho\\}\right]$
$\displaystyle=$
$\displaystyle\mathrm{tr}\left[A_{i}^{\dagger}L_{\nu}A_{i}L_{\mu}\rho+L_{\mu}A_{i}^{\dagger}L_{\nu}A_{i}\rho-
A_{i}^{\dagger}L_{\nu}L_{\mu}A_{i}\rho\right]$
$\displaystyle\;-\frac{1}{2}\mathrm{tr}\left[L_{\nu}A_{i}^{\dagger}A_{i}L_{\mu}\rho+L_{\mu}L_{\nu}A_{i}^{\dagger}A_{i}\rho+A_{i}^{\dagger}A_{i}L_{\nu}L_{\mu}\rho+L_{\mu}A_{i}^{\dagger}A_{i}L_{\nu}\rho-
L_{\nu}L_{\mu}A_{i}^{\dagger}A_{i}\rho-
A_{i}^{\dagger}A_{i}L_{\nu}L_{\mu}\rho\right]$ $\displaystyle=$
$\displaystyle\mathrm{tr}\left[A_{i}^{\dagger}L_{\nu}A_{i}L_{\mu}\rho+L_{\mu}A_{i}^{\dagger}L_{\nu}A_{i}\rho-
A_{i}^{\dagger}L_{\nu}L_{\mu}A_{i}\rho\right]$
$\displaystyle\;-\frac{1}{2}\mathrm{tr}\left[L_{\nu}A_{i}^{\dagger}A_{i}L_{\mu}\rho+L_{\mu}A_{i}^{\dagger}A_{i}L_{\nu}\rho+L_{\mu}L_{\nu}A_{i}^{\dagger}A_{i}\rho-
L_{\nu}L_{\mu}A_{i}^{\dagger}A_{i}\rho\right].$ (40)

Insert it into Eq. (38), we have the derivatives

$\displaystyle\frac{d}{dt}g^{F}_{\mu\nu}(\boldsymbol{x}_{0};t)_{i}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{tr}\left\\{\left[L_{\nu}\left(2\partial_{\mu}-L_{\mu}\right)+L_{\mu}\left(2\partial_{\nu}-L_{\nu}\right)\right]\frac{d}{dt}\rho\right\\}_{i}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{tr}\left\\{A_{i}^{\dagger}L_{\nu}A_{i}L_{\mu}\rho+L_{\mu}A_{i}^{\dagger}L_{\nu}A_{i}\rho+A_{i}^{\dagger}L_{\mu}A_{i}L_{\nu}\rho+L_{\nu}A_{i}^{\dagger}L_{\mu}A_{i}\rho\right.$
$\displaystyle\qquad\left.-L_{\nu}A_{i}^{\dagger}A_{i}L_{\mu}\rho-
L_{\mu}A_{i}^{\dagger}A_{i}L_{\nu}\rho-A_{i}^{\dagger}L_{\nu}L_{\mu}A_{i}\rho-
A_{i}^{\dagger}L_{\mu}L_{\nu}A_{i}\rho\right\\}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\mathrm{tr}\left\\{\left([A_{i},L_{\nu}]^{\dagger}[A_{i},L_{\mu}]+[A_{i},L_{\mu}]^{\dagger}[A_{i},L_{\nu}]\right)\rho\right\\}.$
(41)

It indicates the derivative matrix
$(d\boldsymbol{g}^{F}(\boldsymbol{x}_{0},t)/dt)_{i}$ is negative semi-
definite. We can diagonalize it with a real orthonormal matrix
$\boldsymbol{O}$ as

$\boldsymbol{O}\left(\frac{d}{dt}\boldsymbol{g}^{F}\right)_{i}\boldsymbol{O}^{T}=\mathrm{diag}\left[\lambda_{1}^{(i)},\lambda_{2}^{(i)},\dots,\lambda_{d}^{(i)}\right],$
(42)

with the element

$\displaystyle\lambda_{k}^{(i)}$ $\displaystyle=$
$\displaystyle\sum\nolimits_{\mu,\nu}O_{k\mu}\left(\frac{d}{dt}\boldsymbol{g}^{F}_{\mu\nu}\right)_{i}O_{\nu
k}^{T}$ (43) $\displaystyle=$
$\displaystyle-\mathrm{tr}\left\\{\left[A_{i},\sum\nolimits_{\nu}O_{k\nu}L_{\nu}\right]^{\dagger}\left[A_{i},\sum\nolimits_{\mu}O_{k\mu}L_{\mu}\right]\rho\right\\}$
$\displaystyle\leqslant$ $\displaystyle 0.$

Furthermore, $1/\boldsymbol{g}^{F}(\boldsymbol{x}_{0};t)$ is positive
definite, it indicates

$\alpha_{\mu}\equiv\left(\boldsymbol{O}\frac{1}{\boldsymbol{g}^{F}}\boldsymbol{O}^{T}\right)_{\mu\mu}>0.$
(44)

Hence, we have the trace

$\mathrm{tr}\left[\frac{1}{\boldsymbol{g}^{F}}\left(\frac{d}{dt}\boldsymbol{g}^{F}\right)_{i}\right]=\sum_{\mu}\lambda_{\mu}^{(i)}\alpha_{\mu}\leqslant
0.$ (45)

Insert it into Eq. (15), we have thus proved Eq. (17) together with
$\mathcal{D}(\boldsymbol{x}_{0};t)\geqslant 0$.

## References

  * (1) H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University press, Oxford) (2002).
  * (2) Á. Rivas, S. F. Huelga, and M. B. Plenio, Quantum Non-Markovianity: Characterization, Quantification and Detection, Rep. Prog. Phys. 77 094001(2014).
  * (3) H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Colloquium: Non-Markovian Dynamics in Open Quantum Systems, Rev. Mod. Phys. 88, 021002 (2016) and references therein.
  * (4) I. de Vega, and D. Alonso, Dynamics of Non-Markovian Open Quantum Systems, Rev. Mod. Phys. 89, 015001 (2017) and references therein.
  * (5) C.-F. Li, G.-C. Guo, and J. Piilo, Non-Markovian Quantum Dynamics: What Does It Mean? EuroPhys Lett. 127, 50001 (2019).
  * (6) C.-F. Li, G.-C. Guo, and J. Piilo, Non-Markovian Quantum Dynamics: What Is It Good For? EuroPhys. Lett. 128, 30001 (2019).
  * (7) H.-P. Breuer, E.-M. Laine, and J. Piilo, Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems, Phys. Rev. Lett. 103, 210401 (2009).
  * (8) E.-M. Laine, J. Piilo, and H.-P. Breuer, Measure for the Non-Markovianity of Quantum Processes, Phys. Rev. A 81, 062115 (2010).
  * (9) X.-M. Lu, X. Wang, and C. P. Sun, Quantum Fisher Information Flow and Non-Markovian Processes of Open Systems, Phys. Rev. A 82, 042103 (2010).
  * (10) S. Luo, S. Fu, and H. Song, Quantifying Non-Markovianity via Correlations, Phys. Rev. A 86, 044101 (2012).
  * (11) S. Maniscalco, F. Francica, R. L. Zaffino, N. Lo Gullo, and F. Plastina, Protecting Entanglement via the Quantum Zeno Effect, Phys. Rev. Lett. 100, 090503 (2008).
  * (12) B. Bellomo, R. Lo Franco, and G. Compagno, Non-Markovian Effects on the Dynamics of Entanglement, Phys. Rev. Lett. 99, 160502 (2007).
  * (13) L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, and B. M. Garraway, Sudden death and sudden birth of entanglement in common structured reservoir, Phys. Rev. A 79, 042302 (2009).
  * (14) S. F. Huelga, Á. Rivas, and M. B. Plenio, Non-Markovianity-Assisted Steady State Entanglement, Phys. Rev. Lett. 108, 160402 (2012).
  * (15) N. Mirkin, P. Poggi, and D. Wisniacki, Entangling Protocols Due to Non-Markovian Dynamics, Phys. Rev. A 99, 020301(R) (2019).
  * (16) N. Mirkin, P. Poggi, and D. Wisniacki, Information Backflow as a Resource for Entanglement, Phys. Rev. A 99, 062327 (2019).
  * (17) H. S. Dhar, M. N. Bera, and G. Adesso, Characterizing Non-Markovianity via Quantum Interferometric Power, Phys. Rev. A 91, 032115 (2015).
  * (18) S.-L. Chen, N. Lambert, C.-M. Li, A. Miranowicz, Y.-N. Chen, and F. Nori, Quantifying Non-Markovianity with Temporal Steering, Phys. Rev. Lett. 116, 020503 (2016).
  * (19) H. Song, S. Luo, Y. Hong, Quantum Non-Markovianity Based on the Fisher-Information Matrix, Phys. Rev. A 91, 042110 (2015).
  * (20) C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe, and D. J. Wineland, Decoherence of Quantum Superpositions through Coupling to Engineered Reservoirs, Nature (London) 403, 269 (2000).
  * (21) B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, Experimental Control of the Transition from Markovian to Non-Markovian Dynamics of Open Quantum Systems, Nat. Phys. 7, 931 (2011).
  * (22) Y.-N. Lu, Y.-R. Zhang, G.-Q. Liu, F. Nori, H. Fan, and X.-Y. Pan, Observing Information Backflow from Controllable Non-Markovian Multichannels in Diamond, Phys. Rev. Lett. 124, 210502 (2020).
  * (23) M. Gessner, M. Ramm, T. Pruttivarasin, A. Buchleitner, H.-P. Breuer, and H. Häffner, Local Detection of Quantum Correlations with a Single Trapped Ion, Nat. Phys. 10, 105 (2014).
  * (24) P. Haikka, S. McEndoo, G. De Chiara, G. M. Palma, and S. Maniscalco, Quantifying, Characterizing, and Controlling Information Flow in Ultracold atomic gases, Phys. Rev. A 84, 031602(R) (2011).
  * (25) J.-B. Yuan, H.-J. Xing, L.-M. Kuang, and S. Yi, Quantum Non-Markovian Reservoirs of Atomic Condensates Engineered via Dipolar Interactions, Phys. Rev. A, 95, 033610 (2017).
  * (26) J. F. Haase, P. J. Vetter, T. Unden, A. Smirne, J. Rosskopf, B. Naydenov, A. Stacey, F. Jelezko, M. B. Plenio, and S. F. Huelga, Controllable Non-Markovianity for a Spin Qubit in Diamond, Phys. Rev. Lett. 121, 060401 (2018).
  * (27) G. Andersson, B. Suri, L. Guo, T. Aref, and P. Delsing, Non-exponential decay of a giant artificial atom, Nat. Phys. 15, 1123 (2019).
  * (28) K.-D. Wu, Z. Hou, G.-Y. Xiang, C.-F. Li, G.-C. Guo, D. Dong, and F. Nori, Detecting Non-Markovianity via Quantified Coherence: Theory and Experiments, npj Quantum Inf. 6, 55 (2020).
  * (29) S. Deffner and E. Lutz, Quantum Speed Limit for Non-Markovian Dynamics, Phys. Rev. Lett. 111, 010402 (2013).
  * (30) A. D. Cimmarusti, Z. Yan, B. D. Patterson, L. P. Corcos, L. A. Orozco, and S. Deffner, Environment-Assisted Speed-up of the Field Evolution in Cavity Quantum Electrodynamics, Phys. Rev. Lett. 114, 233602 (2015).
  * (31) N. Mirkin, F. Toscano, and D. A. Wisniacki, Quantum-speed-limit Bounds in an Open Quantum Evolution, Phys. Rev. A 94, 052125 (2016).
  * (32) Y. Dong, Y. Zheng, S. Li, C.-C. Li, X.-D. Chen, G.-C. Guo, and F.-W. Sun, Non-Markovianity-assisted High-Fidelity Deutsch–Jozsa Algorithm in Diamond, npj Quantum Inf. 4, 3 (2018).
  * (33) S. Singha Roy and J. Bae, Information-theoretic Meaning of Quantum Information Flow and Its Applications to Amplitude Amplification Algorithms, Phys. Rev. A 100, 032303 (2019).
  * (34) G. Thomas, N. Siddharth, S. Banerjee, and S. Ghosh, Thermodynamics of Non-Markovian Reservoirs and Heat Engines, Phys. Rev. E 97, 062108 (2018).
  * (35) P. Abiuso and V. Giovannetti, Non-Markov Enhancement of Maximum Power for Quantum Thermal Machines, Phys. Rev. A 99, 052106 (2019).
  * (36) G. W. Gibbons, Typical States and Density Matrics, J. Geom. Phys. 8, 147 (1992).
  * (37) M. V. Berry, “The Quantum Phase, Five Years After” in Geometric Phases in Physics (A. Shapere and F. Wilczek (eds.), World Scientific, Singapore, 1989).
  * (38) S. L. Braunstein, and C. M. Caves, Statistical Distance and the Geometry of Quantum States, Phys. Rev. Lett. 72, 3439(1994).
  * (39) W. K. Wootters, Statistical Distance and Hilbert Space, Phys. Rev. D 23, 357 (1981).
  * (40) I. Bengtsson and K. Zyczkowski, Geometry of Quantum States (Cambridge university press, New York, 2006).
  * (41) C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976).
  * (42) A. S. Holevo, Probabilistic and Statistical Aspect of Quantum Theory (North-Holland, Amsterdam, 1982).
  * (43) V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-Enhanced Measurements: Beating the Standard Quantum Limit, Science 306, 1330 (2004).
  * (44) V. Giovannetti, S. Lloyd, and L. Maccone, Quanutm Metrology, Phys. Rev. Lett. 96, 010401 (2006),
  * (45) V. Giovannetti, S. Lloyd, and L. Maccone, Advances in Quantum Metrology, Nat. Photon. 5, 222 (2011).
  * (46) H. Xing and L. Fu, Measure of the Density of Quantum States in Information Geometry and Quantum Multi-parameter Estimation Phys. Rev. A 102, 062613 (2020).
  * (47) M. B. Ruskai, Beyond Strong Subadditivity? Improved Bounds on the Contraction of Generalized Relative Entropy, Rev. Math. Phys. 06, 1147(1994).
  * (48) V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely Positive Dynamical Semigroups of N‐level Systems, J. Math. Phys. (N.Y.) 17, 821 (1976).
  * (49) G. Lindblad, On the Generators of Quantum Dynamical Semigroups, Commun. Math. Phys. 48, 119 (1976).
  * (50) H.-P. Breuer, Genuine Quantum Trajectories for Non-Markovian Processes, Phys. Rev. A 70, 012106 (2004).
  * (51) J. Piilo, S. Maniscalco, K. Härkönen, and K.-A. Suominen, Non-Markovian Quantum Jumps, Phys. Rev. Lett. 100, 180402 (2008).
  * (52) H.-P. Breuer, and J Piilo, Stochastic Jump Processes for Non-Markovian Quantum Dynamics, Europhys. Lett. 85, 50004 (2009).
  * (53) W. K. Wootters and W. H. Zurek, A Single Quantum Cannot be Cloned, Nature (London) 299, 802 (1982).
  * (54) H.-P. Breuer, B. Kappler, and F. Petruccione, Stochastic Wave-function Method for Non-Markovian Quantum Master Equations, Phys. Rev. A 59, 1633 (1999).


matters, no adjectives should be included. They should always be a noun.
Actions & poses are word-level or phrase-level actions that the character or
the object in the illustration performs. Example actions & poses include
riding a bus, standing still, or traveling.
Themes & moods are words not directly present in the illustration, but those
that can potentially convey the overall theme or mood of the illustration.
Example themes & moods include imaginative, eco-friendly, or sad. They should
be adverbs, preferably one word.
Your task is to generate three descriptions of the illustration that the user
can draw based on the given keywords. The three descriptions should be
significantly different from each other. Each description should include three
things: "Caption" and "Objects".
"Caption" is a simple description of the overall image of the description.
This should include some objects in the "Objects" list. Keep it concise. Do
not make it long and do not include unnecessary adjectives.
"Objects" is a list of the objects depicted in the illustrations, and a short
description of them. The objects should be one of the given "Subject matters",
or something related to them. You don’t have to include all given subject
matters. The given action and concept should also be considered for generating
the object’s detail.

Examples:
User:
Subject matter: ball, cat, dog
Action & pose: jumping
Theme & mood: playful, peaceful
Assistant:
1\.
Scene: In the backyard, cats and dogs play happily with a ball which is their
favorite toy.
Objects: [(ball, a brightly colored ball), (dog, a dog jumping with its tongue
sticking out), (cat, an agile cat in mid-jump)]
2\.
Scene: A beach where a dog is united in its pursuit of a bouncing ball.
Objects: [(ball, a sandy beach ball), (dog, a spotted dog leaping for the
ball)]
3\.
Scene: In a residential living room, the cat and dog both defy gravity in
their jumps to catch the toy.
Objects: [(toy, a small rubber toy with vibrant stripes), (dog, a bouncy
terrier flying mid-air), (cat, a Siamese cat also in mid-jump)]
User:
Subject matter: dinosaur, drink, hat, chair
Action & pose: relaxing
Theme & mood: sunny, relaxed
Assistant:
1\.
Scene: A dinosaur wearing a hat, lounging with a drink on a chair under the
sun.
Objects: [(dinosaur, a large but friendly looking dinosaur), (drink, a fruity
cocktail with a tiny umbrella), (hat, a wide-brimmed straw hat), (chair, a
comfortable looking lounge chair)]
2\.
Scene: Dinosaur relaxes in a sophisticated environment with a drink and plops
a hat on the chair next to it.
Objects: [(dinosaur, a dinosaur in a suit), (drink, a fancy drink in a crystal
glass), (hat, a stylish trilby), (chair, a plush velvet chair)]
3\.
Scene: A kid with a hat is playing on a picnic mat with a toy dinosaur.
Objects: [(kid, a kid with pink shirts), (dinosaur, a small green toy
dinosaur)]
User:
Subject matter: dog, teeth
Action & pose:
Theme & mood: care
Assistant:
1\.
Caption: A caring veterinarian examining a dog’s teeth.
Objects: [(dog, a dog sitting calmly), (teeth, a pair of dental tools), (vet,
a veterinarian wearing a lab coat)]
2\.
Caption: A child diligently brushing their dog’s teeth.
Objects: [(dog, a dog lying on its back), (teeth, a toothbrush and dog
toothpaste), (child, a child brushing the dog’s teeth)]
3\.
Caption: A dental hygienist showing a dog owner the correct brushing technique
for their pet’s teeth.
Objects: [(dog, a dog standing on a dental examination table), (teeth, a
dental mirror and toothbrush), (hygienist, a dental hygienist demonstrating
the brushing technique)]

###  A.4. Prompt: Matching Layout with Objects

System Prompt:
You are an intelligent bounding box matcher. I will provide you with a caption
that describes an illustration, a list of the objects that are included in the
illustration, and a list of bounding boxes. Your task is to match bounding
boxes to each object to make the illustration most balanced and realistic.
Each bounding box is in the format of (object name, [top-left x coordinate,
top-left y coordinate, box width, box height]). The bounding boxes are
represented as a proportion. The top-left corner has coordinates [0, 0]. The
bottom-right corner has coordinates [1, 1]. The bounding boxes should not go
beyond the image boundaries.

Examples:
User:
A realistic image of landscape scene depicting a green car parking on the left
of a blue truck, with a red air balloon and a bird in the sky
[air balloon, car, bird, truck]
[0.041, 0.783, 0.442, 0.179], [0.525, 0.699, 0.408, 0.263], [0.261, 0.458,
0.222, 0.264], [0.525, 0.458, 0.279, 0.195]
Assistant:
[(’car’, [0.041, 0.783, 0.442, 0.179]), (’truck’, [0.525, 0.699, 0.408,
0.263]), (’balloon’, [0.261, 0.458, 0.222, 0.264]), (’bird’, [0.525, 0.458,
0.279, 0.195])]
User:
A realistic top-down view of a wooden table with two apples on it
[apple, apple, wooden table]
[0.219, 0, 0.562, 1], [0.402, 0.138, 0.195, 0.195], [0.402, 0.667, 0.195,
0.195]
Assistant:
[(’wooden table’, [0.219, 0, 0.562, 1]), (’apple’, [0.402, 0.138, 0.195,
0.195]), (’apple’, [0.402, 0.667, 0.195, 0.195])]
User:
A realistic scene of three skiers standing in a line on the snow near a palm
tree
[skier, skier, skier, palm tree]
[0.487, 0.131, 0.142, 0.441], [0.661, 0.131, 0.143, 0.441], [0.836, 0.131,
0.142, 0.441], [0.795, 0.613, 0.183, 0.387]
Assistant:
[(’skier’, [0.487, 0.131, 0.142, 0.441]), (’skier’, [0.661, 0.131, 0.143,
0.441]), (’skier’, [0.836, 0.131, 0.142, 0.441]), (’palm tree’, [0.795, 0.613,
0.183, 0.387])]
User:
An oil painting of a pink dolphin jumping on the left of a steam boat on the
sea
[dolphin, steam boat]
[0.273, 0, 0.245, 1], [0.032, 0.455, 0.135, 0.420]
Assistant:
[(’steam boat’, [0.273, 0, 0.245, 1]), (’dolphin’, [0.032, 0.455, 0.135,
0.420])]
User:
Immersed in his imagination, a boy is indoors enacting a prehistoric tale
using four toy dinosaurs.
[dino toys, dino toys, dino toys, boy, dino toys]
[0.250, 0.218, 0.566, 0.563], [0.074, 0.556, 0.137, 0.284], [0.074, 0.76,
0.137, 0.284], [0.659, 0.041, 0.254, 0.134], [0.464, 0.840, 0.195, 0.120]
Assistant:
[(’boy’, [0.250, 0.218, 0.566, 0.563]), (’dino toys’, [0.074, 0.556, 0.137,
0.284]), (’dino toys’, [0.074, 0.76, 0.137, 0.284]), (’dino toys’, [0.659,
0.041, 0.254, 0.134]), (’dino toys’, [0.464, 0.840, 0.195, 0.120])]
User:
Two pandas in a forest without flowers
[panda, panda]
[0.114, 0.399, 0.183, 0.441], [0.733, 0.106, 0.150, 0.441]
Assistant:
[(’panda’, [0.114, 0.399, 0.183, 0.441]), (’panda’, [0.733, 0.106, 0.150,
0.441])]

###  A.5. Prompt: Generating Layout based on Image Caption

System Prompt:
You are an intelligent bounding box generator. I will provide you with a
caption for an illustration and a list of the objects. Your task is to
generate the bounding boxes for the objects based on the caption. The images
are of size 512x512. The top-left corner has coordinates [0, 0]. The bottom-
right corner has coordinates [512, 512]. The bounding boxes should not go
beyond the image boundaries. Each bounding box should be in the format of
(object name, [top-left x coordinate, top-left y coordinate, box width, box
height]). If an object must appear several times by the provided caption,
multiple bounding boxes may be added for the object.

Examples:
User:
A realistic image of landscape scene depicting a green car parking on the left
of a blue truck, with a red air balloon and a bird in the sky
[car, truck, air balloon, bird]
Assistant:
[(’car’, [0.041, 0.549, 0.412, 0.31]), (’truck’, [0.525, 0.553, 0.408,
0.313]), (’balloon’, [0.129, 0.016, 0.283, 0.264]), (’bird’, [0.578, 0.082,
0.279, 0.195])]
User:
A realistic top-down view of a wooden table with two apples on it
[wooden table, apple]
Assistant:
[(’wooden table’, [0.039, 0.289, 0.922, 0.422]), (’apple’, [0.293, 0.441,
0.195, 0.195]), (’apple’, [0.547, 0.441, 0.195, 0.195])]
User:
A realistic scene of three skiers standing in a line on the snow near a palm
tree
[skier, palm tree]
Assistant:
[(’skier’, [0.01, 0.297, 0.271, 0.328]), (’skier’, [0.543, 0.375, 0.236,
0.308]), (’skier’, [0.289, 0.338, 0.242, 0.303]), (’palm tree’, [0.789, 0.205,
0.201, 0.49])]
User:
An oil painting of a pink dolphin jumping on the left of a steam boat on the
sea
[steam boat, dolphin]
Assistant:
[(’steam boat’, [0.453, 0.439, 0.502, 0.291]), (’dolphin’, [0.041, 0.486,
0.369, 0.24])]
User:
A cute cat and an angry dog without birds
[cat, dog]
Assistant:
[(’cat’, [0.1, 0.131, 0.529, 0.632]), (’dog’, [0.589, 0.232, 0.412, 0.445])]
User:
Two pandas in a forest without flowers
[panda]
Assistant:
[(’panda’, [0.059, 0.335, 0.414, 0.441]), (’panda’, [0.516, 0.338, 0.434,
0.432])]
User:
Immersed in his imagination, a boy is indoors enacting a prehistoric tale
using two toy dinosaurs.
[boy, dino toys]
Assistant:
[(’boy’, [0.367, 0.076, 0.301, 0.486]), (’dino toys’, [0.15, 0.469, 0.201,
0.254]), (’dino toys’, [0.543, 0.475, 0.188, 0.248])]

###  A.6. Example Outputs from the Technical Pipeline

Figure 9 shows some examples of inputs and outputs for three technical
pipelines of — (1) element extraction pipeline, (2) keyword recommendation
pipeline, and (3) recombination generation pipeline. Figure 10 shows some more
examples from the keyword extraction pipeline.

Figure 9. Examples of inputs and outputs given to ’s pipelines. From the input
image, (a) captions describing a given image and (c) arrangement of an image
are acquired. Using captions as input, (b) keywords that are found in this
image are extracted and categorized in subject matter, action & pose, and
theme & mood. From the (c) extracted arrangement, a layout variator generates
two different recommended layouts similar to the original one, containing 2
and 3 bounding boxes respectively. The keyword recommendation pipeline is used
for (e) recommending keywords that are relevant to the input keywords. Based
on the input keywords, (f) descriptions are generated with image captions and
the details of objects. The final (g) generated image is created from the
descriptions and converted into a sketch style.

Examples of inputs and outputs given to ’s pipelines. In this example, an
illustration of a witch and octopus having dinner together is given as an
input image. From the input image, (a) nine captions describing a given image
(e.g., ”a cartoon illustration of a woman in a nun costume and an octopus”)
and (c) arrangement of an image (shown as a bounding box) are acquired. Using
captions as input, (b) keywords that are found in this image are extracted and
categorized in subject matter (e.g., woman, table), action & pose (e.g.,
sitting at a table, eating sushi), and theme & mood (e.g., humorous, surreal).
From the (c) extracted arrangement, a layout variator generates two different
recommended layouts similar to the original one, containing 2 and 3 bounding
boxes respectively. The keyword recommendation pipeline is used for (e)
recommending keywords (e.g., seahorse, performing magic show, festive) that
are relevant to the input keywords (e.g., woman, giving a present, humorous).
Based on the input keywords, (f) descriptions are generated with image
captions (”In a surreal underwater Christmas scene, a woman in scuba gear
surprises an octopus with a gift”) and the details of objects (e.g., Christmas
tree: a miniature Christmas tree underwater). The final (g) generated image is
created from the descriptions and converted into a sketch style.

Figure 10. Examples of the reference image with corresponding human-labeled
ground truth labels and predicted keywords from the keyword extraction
pipeline of . Keywords predicted by our system can often be more descriptive
and innovative than the ground-truths, which are highlighted in bold.

Two examples of the reference images with corresponding human-labeled ground
truth labels and predicted keywords from the keyword extraction pipeline of .
Keywords predicted by our system (right) can often be more descriptive and
innovative than the ground truths. For example, there is an image of a couple
standing in a purple desert and the big moon in the background. The ground-
truth labels only contain the keyword of ”sand”, but our pipeline predicted
more precisely with the keyword ”desert”. Also, it generates some innovative
mood keywords for this image such as ”surreal”. Another example image is an
illustration of a group of dogs celebrating a birthday with a cake. Our
pipeline predicted a keyword of ”party” or ”standing around a birthday cake”,
which was not shown in the ground-truth data.

##  Appendix B User Study

###  B.1. Baseline System Interface

Figure 11 shows the interface of the baseline system used for the user study.
The baseline system looks similar to the . There is no keyword extraction
feature in the left panel, but it allows participants to manually add keyword
notes. In the center, there is the same interactive mood board with the , but
there is no keyword suggestion panel. The right panel enables users to
manually configure the layout and prompts for image generation, instead of
selecting the keywords to combine. Other features such as mood board
interactions (zoom, add/delete images) and saving favorite sketches were
provided the same as the . Participants could employ ChatGPT for various
purposes other than this interface.

Figure 11. Screenshot of baseline system. (a) Keyword Note Panel: Users can
add keywords manually on each image. (b) Interactive Mood Board: Users can
organize the reference images on the mood board, along with the added keyword
notes. (c) Sketch Generation Panel: Users can configure the overall layout of
the generated image by manipulating boxes in the layout controller (beige) on
the top of the panel. Additionally, users can provide prompts for the entire
image and specific parts. They can specify more image details by clicking the
”Add object” button. To get the generated sketches, users can click the
”Generate sketch” button. (d) ChatGPT: Users were also provided with ChatGPT
on a separate screen.

Screenshot of baseline system. (a) Keyword Note Panel: Users can add keywords
manually on each image. (b) Interactive Mood Board: Users can organize the
reference images on the mood board, along with the added keyword notes. (c)
Sketch Generation Panel: Users can configure the overall layout of the
generated image by manipulating boxes in the layout controller (beige) on the
top of the panel. Additionally, users can provide prompts for the entire image
and specific parts. They can specify more image details by clicking the ”Add
object” button. To get the generated sketches, users can click the ”Generate
sketch” button. (d) ChatGPT: Users were also provided with ChatGPT on a
separate screen.

###  B.2. Interview questions

These are the questions used for the semi-structured interview after the two
idea generation sessions with baseline and tools.

  1. (1)

Can you share the idea sketch that you think most creative in each topic, and
what was the main source of inspiration for those ideas?

  2. (2)

Comparing the baseline and , what were the main differences you noticed in for
the idea generation process?

  3. (3)

In each three main stages of idea generation—finding reference elements,
exploring ideas, and generating sketches—did you find one tool to be more
helpful than the other, and why?

  4. (4)

Were there any differences in your typical approach to idea generation when
using these tools? If so, how was it different from the typical work process?

  5. (5)

Which functionalities did you find most beneficial in both tools and in what
scenarios were they particularly useful?

  6. (6)

Were there any situations or specific sketches where the tools were especially
useful or not useful?

  7. (7)

In terms of image generation methods, what were the main differences between
baseline and , and when did you feel each method was more helpful?

  8. (8)

How did you feel about the output in sketch format, and do you think the
tool’s effectiveness would differ if outputs were presented as a completed
image rather than a sketch?

  9. (9)

How did you incorporate the generated images into your final idea sketch?

###  B.3. Additional User Study Results: Raw Usage Log

Figure 12 shows the full usage log for all 16 participants of the user study,
showing the timestamps of 3 types of user actions (adding keyword notes,
generating images, and completing a design idea sketch).

Figure 12. Usage log for all participants in both (Top) and baseline (Bottom)
condition. The red triangle indicates the timestamp when the participants
complete each sketch. The pink dot is the point when the participant added new
keywords for the reference image. The sky blue dot is the point when the
participant gave input into the image generation model. As shown in the
figure, P15 first conducted multiple keyword-adding and image-generation
actions, came up with all the design ideas, and then sketched all of them
later in the session all at once. During the interview, P15 explained that
they intended to focus exclusively on the sketching process, so they decided
to jot down half-baked design ideas as memos in a text and draw them
collectively. Unfortunately, the collected usage log only records the point
when the overall sketch is completed and does not capture the individual
instances of writing each memo. Therefore, we were unable to analyze which
actions affected each design idea, so the usage data of P15 was excluded from
the analysis of the relationship between action types and each sketching turn
(Figure 6).

Usage log for all participants in both (Top) and baseline (Bottom) condition.
The red triangle indicates the timestamp when the participants complete each
sketch. The pink dot is the point when the participant added new keywords for
the reference image. The sky blue dot is the point when the participant gave
input into the image generation model. As shown in the figure, P15 first
conducted multiple keyword-adding and image-generation actions, came up with
all the design ideas, and then sketched all of them later in the session all
at once. During the interview, P15 explained that they intended to focus
exclusively on the sketching process, so they decided to jot down half-baked
design ideas as memos in a text and draw them collectively. Unfortunately, the
collected usage log only records the point when the overall sketch is
completed and does not capture the individual instances of writing each memo.
Therefore, we were unable to analyze which actions affected each design idea,
so the usage data of P15 was excluded from the analysis of the relationship
between action types and each sketching turn (Figure 6).