# Spread complexity and localization in $\mathcal{PT}$-symmetric systems

Aranya Bhattacharya  0000-0002-1882-4177<EMAIL_ADDRESS>Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków,
Poland  Rathindra Nath Das  0000-0002-4766-7705 das.rathindranath@uni-
wuerzburg.de Institute for Theoretical Physics and Astrophysics and Würzburg-
Dresden Cluster of Excellence ct.qmat, Julius-Maximilians-Universität
Würzburg, Am Hubland, 97074 Würzburg, Germany  Bidyut Dey  0000-0003-1527-7064
<EMAIL_ADDRESS>Dipartimento di Fisica, Università della Calabria
Arcavacata di Rende, I-87036 Cosenza, Italy,
I.N.F.N., Gruppo Collegato di Cosenza Arcavacata di Rende, I-87036 Cosenza,
Italy  Johanna Erdmenger  0000-0003-4776-4326<EMAIL_ADDRESS>wuerzburg.de Institute for Theoretical Physics and Astrophysics and Würzburg-
Dresden Cluster of Excellence ct.qmat, Julius-Maximilians-Universität
Würzburg, Am Hubland, 97074 Würzburg, Germany

###### Abstract

We present a framework for investigating wave function spreading in
$\mathcal{PT}$-symmetric quantum systems using spread complexity and spread
entropy. We consider a tight-binding chain with complex on-site potentials at
the boundary sites. In the $\mathcal{PT}$-unbroken phase, the wave function is
delocalized. We find that in the $\mathcal{PT}$-broken phase, it becomes
localized on one edge of the tight-binding lattice. This localization is a
realization of the non-Hermitian skin effect. Localization in the
$\mathcal{PT}$-broken phase is observed both in the lattice chain basis and
the Krylov basis. Spread entropy, entropic complexity, and a further measure
that we term the Krylov inverse participation ratio probe the dynamics of wave
function spreading and quantify the strength of localization probed in the
Krylov basis. The number of Krylov basis vectors required to store the
information of the state reduces with the strength of localization. Our
results demonstrate how measures in Krylov space can be used to characterize
the non-hermitian skin effect and its localization phase transition.

††preprint: APS123-QED

### Introduction.—

The study of complexity Nielsen  _et al._ (2006); Jefferson and Myers (2017);
Parker  _et al._ (2019); Chapman  _et al._ (2018); Balasubramanian  _et al._
(2022) in quantum systems has provided new insights into the dynamical
behaviour of chaotic systems and the structure of spacetime in general Brown
_et al._ (2016); Susskind (2016). It resulted in the development of new tools
and techniques in quantum many-body systems, quantum information theory, and
holographic theories. Complexity was initially defined to quantify the
difficulty of reaching one target state starting from an initial reference
state of a quantum system Nielsen  _et al._ (2006).

Here we use Krylov state complexity Balasubramanian  _et al._ (2022) to study
$\mathcal{PT}$-symmetric systems. The complexity in Krylov basis was first
introduced to study the operator growth in the Heisenberg picture under
unitary evolution and distinguish chaotic theories from integrable ones in
Parker  _et al._ (2019). A recent advancement to this study Balasubramanian
_et al._ (2022) was to extend the method to Schrödinger evolution of quantum
states and to show that the choice of Krylov basis minimizes the complexity.
This measure defined in the Schrödinger picture is dubbed as the spread
complexity Balasubramanian  _et al._ (2022), which quantifies the spread of an
initial state under the evolution of the corresponding system Hamiltonian. The
spread complexity shows a distinct characteristic behaviour in chaotic,
integrable and intermediate systems, providing new insights into the dynamics
of information scrambling and phase transition under time evolution Caputa and
Liu (2022); Afrasiar  _et al._ (2023); Balasubramanian  _et al._ (2023);
Caputa  _et al._ (2023); Erdmenger  _et al._ (2023); Pal  _et al._ (2023);
Nandy  _et al._ (2023); Chattopadhyay  _et al._ (2023); Gautam  _et al._
(2023); Bhattacharjee  _et al._ (2022); Gill  _et al._ (2023); Bento  _et al._
(2023); Aguilar-Gutierrez and Rolph (2023); Craps  _et al._ (2023);
Bhattacharya  _et al._ (2023a); Caputa  _et al._ (2024a, b); Huh  _et al._
(2023); Zhou and Chen (2024); Nandy  _et al._ (2024). These concepts were
extended to the non-unitary time evolution of open quantum systems
Bhattacharya  _et al._ (2022); Bhattacharjee  _et al._ (2023); Bhattacharya
_et al._ (2023b); Bhattacharjee  _et al._ (2024); Bhattacharya  _et al._
(2024). These studies exemplified how operator growth is modified in the
presence of decoherence, resulting in a prolonged decay of complexity before
reaching the saturation value. A further example of non-unitarity is provided
by systems subject to local projective measurements, for which it was found
that the spread complexity undergoes a phase transition as a function of the
measurement frequency, showcasing the quantum Zeno effect Bhattacharya  _et
al._ (2023a).

The probability distribution of a state in the Krylov basis gives rise to an
entropy function known as the Krylov entropy. An alternative complexity
measure is given by the exponential of this entropy. In classical mechanics,
the exponential of entropy is related to the accessible phase space volume.
Similarly, entropic complexity estimates the number of supports of the state
on the Krylov basis vectors, reflecting the system’s complexity. For unitary
dynamics, both the spread complexity and the entropic complexity give
equivalent estimates of complexity Balasubramanian  _et al._ (2022).

An integral component of Krylov space orthonormalization is provided by the
Lanczos coefficients. Recently, a connection between the fluctuations of the
Lanczos coefficients and Krylov localization was proposed in Rabinovici  _et
al._ (2022). The fluctuating Lanczos coefficients act as random hopping
potentials between different sites of the Krylov chain, and the Krylov
localization is then similar to Anderson localization. Wave function
localization in Krylov basis then corresponds to the suppression of
complexity.

In this paper, using Krylov state complexity to address the wave function
spreading dynamics in the presence of localization, we consider a
$\mathcal{PT}$-symmetric tight-binding model Jin and Song (2009).
$\mathcal{PT}$-symmetric systems provide a natural setup to study the emergent
dynamics which arise from a change of eigenspectrum from real to imaginary.
This change in the spectrum is realized by a Hamiltonian with complex
conjugate potentials at the first and last sites Jin and Song (2009),

$H=-J\sum_{l=1}^{N-1}\left(a_{l}^{\dagger}a_{l+1}+\text{h.c.}\right)+i\gamma\left(a_{1}^{\dagger}a_{1}-a_{N}^{\dagger}a_{N}\right).$
(1)

The single-particle solutions of this model exhibit two distinct phases,
depending on the relative values of the coupling strength, $J$ and the
strength of the on-site potential $\gamma$ of the Hamiltonian. The
$\mathcal{PT}$-unbroken phase has a real eigenspectrum, while the
spontaneously broken phase introduces two imaginary eigenvalues. These two
phases are connected by a critical point, at which the system has a two-fold
degeneracy with eigenvalue zero Jin and Song (2009). In the broken phase, one
pair of purely imaginary eigenvalues result in localization of the wave
function on the boundary of the tight-binding lattice, which is the known non-
hermitian skin effect Zhang  _et al._ (2023); Zeng (2022); Lin  _et al._
(2023); Kawabata  _et al._ (2023); Kong  _et al._ (2022); Yao and Wang (2018);
Yao  _et al._ (2018). The term skin effect refers to the wave function
localizing on one boundary (skin) of the lattice.

Here we observe a non-hermitian skin effect in the $\mathcal{PT}$-broken phase
in which, as time evolves, the original-basis state
$|\psi(t)\rangle=e^{-iHt}|\psi(t=0)\rangle$ becomes localized around the first
site at the left edge of the tight-binding lattice. This localization may also
be seen in the Krylov space in the following way. The dynamics of wave
function spreading in position basis may be mapped to the dynamics of a dual
tight-binding chain in Krylov space, constructed from a given generic initial
state $|\psi(t=0)\rangle$ Balasubramanian  _et al._ (2022), with the Lanczos
coefficients providing the couplings for the tight-binding chain in Krylov
space. Localization in the lattice site basis corresponds to localization in
the Krylov basis through a change of basis vectors.

In the $\mathcal{PT}$-unbroken phase, the time-evolved single-particle state
$|\psi(t)\rangle$ spans the entire system and oscillates across different
sites over time in both the original and the dual Krylov lattice. The spread
complexity, defined on the dual Krylov basis, is a distinctive indicator of
this phase transition. In the $\mathcal{PT}$-unbroken scenario, when all the
eigenvalues are real, there is an initial growth of complexity, followed by a
saturation phase with large oscillations. These oscillations can be attributed
to the unitary dynamics and finite-size effects of the tight-binding chain Pal
_et al._ (2023). In contrast, in the $\mathcal{PT}$-broken case, spread
complexity shows a sharp growth followed by stable saturation. Moreover,
compared to the $\mathcal{PT}$-unbroken phase, the saturation value of
complexity is significantly suppressed in the broken phase. This suppression
of the complexity saturation value is due to the localization of the state
along non-unitary time evolution. As the state is localized in the Krylov
basis, it has less number of supports on Krylov basis vectors. This results in
the suppression of complexity. These results reflect how complexity dynamics
of quantum state evolution, phrased using the Krylov space approach captures
the $\mathcal{PT}$-phase transition and non-hermitian skin effect.

To systematically quantify the strength of localization in the dual Krylov
basis, we use a dynamic measure, the Krylov inverse participation ratio
(KIPR), inspired by the inverse participation ratio (IPR) Edwards and Thouless
(1972); Misguich  _et al._ (2016) for Hermitian evolution. IPR in Hermitian
systems is known as a good measure to probe Anderson localization. Unlike the
conventional IPR, KIPR serves as a dynamic quantifier. A higher value of KIPR
indicates stronger localization in the Krylov space. The strength of
localization varies for different values of the critical parameter and the
spread of the initial state on the tight-binding lattice. In Bento  _et al._
(2023), KIPR is used to study localization in a Hermitian model. Our work
defines KIPR for non-Hermitian Hamiltonian dynamics. This definition is
different from KIPR in the Hermitian setup since, for a non-Hermitian
Hamiltonian, the eigenstates do not form an orthogonal set of basis vectors.
Nonetheless, using a version of the Lanczos algorithm adapted to complex
symmetric Hamiltonians Bhattacharya  _et al._ (2023a), we build a complex
orthogonal Krylov basis that is applicable to $\mathcal{PT}$-symmetric
systems. We note that in Beetar  _et al._ (2023) $\mathcal{PT}$-phase
transitions were studied using Krylov operator complexity.

### Main result.—

We find a localization-delocalization transition along with the
$\mathcal{PT}$-phase transition in the model described by the Hamiltonian Eq.
(1). In the $\mathcal{PT}$-broken phase, the time-evolved wave function
localizes at the left-most lattice site of the tight-binding chain,
manifesting the non-hermitian skin effect. We find that the saturation values
of spread entropy and entropic complexity experience a suppression
proportional to the localization strength quantified by the KIPR. In the
$\mathcal{PT}$-broken phase, we find that this suppression of the saturation
value of spread complexity is not correlated with the localization strength.
In this phase, spread complexity is no longer a reliable probe of the strength
of localization, unlike in the unitary phase Rabinovici  _et al._ (2022).
However, in this phase, entropic complexity still reflects the localization
strength: it measures the effective Hilbert space dimension required to store
the entropy distribution and is, therefore, highly sensitive to the
localization strength.

###  $\mathcal{PT}$ phase transition and localization.—

We discuss the phase transition properties of the system governed by the
$\mathcal{PT}$-symmetric tight-binding Hamiltonian of Eq. (1) and define the
spread complexity. This Hamiltonian has uniform nearest-neighbour hopping
interactions and features two complex conjugate imaginary on-site potentials,
denoted as $\pm i\gamma$. The tunnelling strength $J$ is set to one throughout
this paper. The action of the parity operator $\mathcal{P}$ and the time-
reversal operator $\mathcal{T}$ is given by $\mathcal{T}i\mathcal{T}=-i$ and
$\mathcal{P}a_{l}^{\dagger}\mathcal{P}=a_{N+1-l}^{\dagger}$. The Hamiltonian
exhibits $\mathcal{PT}$ symmetry, i.e, $\left[H,\mathcal{PT}\right]=0$. The
$\mathcal{PT}$-unbroken phase corresponds to the case when all eigenstates of
the Hamiltonian given in Eq. (1) are also eigenstates of the $\mathcal{PT}$
operator. In this case, all the eigenvalues are real, and the evolution is
unitary. In the $\mathcal{PT}$-broken phase, $N-2$ eigenvalues are real, and
two eigenvalues form a complex-conjugate pair Jin and Song (2009),

$\displaystyle E_{\kappa\pm}=\pm i2J\sinh(\kappa).$ (2)

Here $\kappa$ satisfies the transcendental equations

$\displaystyle\gamma^{2}=\begin{cases}J^{2}\frac{\sinh\left(\kappa\left(N+1\right)\right)}{\sinh\left(\kappa\left(N-1\right)\right)}~{}~{}~{}\text{for}~{}~{}~{}N=2n+1,\\\
J^{2}\frac{\cosh\left(\kappa\left(N+1\right)\right)}{\cosh\left(\kappa\left(N-1\right)\right)}~{}~{}~{}\text{for}~{}~{}~{}N=2n.\end{cases}$
(3)

We find that the eigenvectors corresponding to the complex energy eigenvalues
$E_{\kappa+}$ and $E_{\kappa-}$ are the left edge state $|E_{+}\rangle$ and
the right edge state $|E_{-}\rangle$, respectively. As shown in Fig. 1,
$|E_{+}\rangle$ and $|E_{-}\rangle$ are localized around the left and right
edge of the lattice, respectively. These states are not eigenstates of
$\mathcal{PT}$ operator, thus breaking the $\mathcal{PT}$-symmetry. The skin
effect refers to the time-evolved wave function localizing on one boundary or
the skin of the lattice.

As discussed in Jin and Song (2009); Graefe  _et al._ (2008), the transition
between the $\mathcal{PT}$-symmetric and broken phases occurs at the following
parameter values of the imaginary coupling $\gamma$ of the Hamiltonian Eq.
(1),

$\displaystyle\gamma=\begin{cases}J\sqrt{(n+1)/n},~{}~{}~{}~{}\text{for}~{}~{}~{}~{}~{}~{}N&=~{}2n+1,\\\
J,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for}~{}~{}~{}~{}~{}~{}N&=~{}2n.\end{cases}$
(4)

As shown in Jin and Song (2009), at the transition point Eq. (4), two energy
eigenvalues become zero, corresponding to exceptional points Jin and Song
(2009); Graefe  _et al._ (2008).

We now apply the Krylov state complexity Balasubramanian  _et al._ (2022) to
our model. Starting from an initial state $|\psi(0)\rangle=|K_{0}\rangle$, we
construct the Krylov basis $\\{|K_{n}\rangle\\}$ using an adaptation of
Lanczos algorithm to complex symmetric Hamiltonians Bai  _et al._ (2000);
Bhattacharya  _et al._ (2023a), such as Eq. (1). The Krylov basis vectors form
a matrix $Q$, which follows the complex orthogonality relation,
$Q^{T}Q=I_{n}$. In this basis, the Hamiltonian takes a tri-diagonal form as
$\tilde{T}=Q^{T}HQ$. The complex symmetric Lanczos algorithm is reviewed in
Appendix I. The time-evolved state $|\psi(t)\rangle$ may be expanded in the
Krylov basis $|K_{n}\rangle$ as
$|\psi(t)\rangle=\sum_{n}\tilde{\psi}_{n}(t)|K_{n}\rangle$ and the spread
complexity and entropy are defined by

$C(t)=\sum_{n}np_{n},~{}S(t)=-\sum_{n}p_{n}\ln p_{n}.$ (5)

Here $p_{n}=|\tilde{\psi}_{n}(t)|^{2}$ and we dynamically normalize the time-
evolved state by
$\mathcal{N}=\sqrt{\text{Tr}[e^{-iHt}|\psi(0)\rangle\langle\psi(0)|e^{iH^{\dagger}t}]}$
to always fix the total probability to one.

An entropic definition of complexity is given by Balasubramanian  _et al._
(2022)

$C_{S}(t)=e^{S(t)},$ (6)

with $S(t)$ given by Eq. (5). This entropic complexity is independent of the
specific choice of monotonic weighting used in defining the spread complexity
in Eq. (5). It quantifies the minimal Hilbert space dimension necessary to
accommodate the probability distribution of Krylov basis weights.

We proceed by discussing the localization-delocalization transition that the
system Eq. (1) undergoes along with the $\mathcal{PT}$-phase transition. To
dynamically assess the strength of wave function localization in the dual
Krylov basis, we introduce the Krylov inverse participation ratio (KIPR),

$\text{KIPR}(t)=\sum_{n=0}^{D-1}|\langle
K_{n}|\psi(t)\rangle|^{4}=\sum_{n=0}^{D-1}|\tilde{\psi}_{n}(t)|^{4}.$ (7)

This measure acts as a dynamic probe of the wave function localization in the
Krylov basis over time. The KIPR$(t)$ is one when the state is fully localized
as $|\psi(t)\rangle=|K_{n}\rangle$ for a specific $n$. In cases when the time-
evolved state has support on all sites,
$|\psi(t)\rangle=\frac{1}{\sqrt{D}}\sum_{j=0}^{D-1}|K_{j}\rangle$, the KIPR is
given by ${1}/{D}$ in $D$-dimensional Krylov space.

### Localization in $\mathcal{PT}$-broken phase.—

Figure 1: Right (black dashed) and left (purple) edge states in the
$\mathcal{PT}$-broken phase with $N=40$, $J=1$ and $\gamma=1.3$, corresponding
to energy eigenvalues $E_{\kappa-}=-0.530769i$ and $E_{\kappa+}=0.530769i$,
respectively. We see that the edge states $|E_{\pm}\rangle$ are localized on
the left and right edges of the tight-binding lattice.

We now show that in the $\mathcal{PT}$-broken phase, the wave functions
localize in the lattice basis, giving rise to the non-hermitian skin effect.
The amplitude of the energy eigenstate $|E_{+}\rangle$ corresponding to the
energy eigenvalue $E_{\kappa+}=i2J\sinh{\kappa}$ grows exponentially with
time,

$\displaystyle|E_{+}(t)\rangle=e^{-iHt}|E_{+}(0)\rangle=e^{2J\sinh(\kappa)t}|E_{+}(0)\rangle.$
(8)

At late times, the maximum contribution in a time-evolved wave function
$\psi(t)$ comes from the state $|E_{+}\rangle$. The left edge state
$|E_{+}\rangle$ has maximum support on the first site of the actual lattice,
as shown in Fig. 1. Hence, at late times, a generic state localizes at the
first site of the lattice. Localization in the tight-binding lattice basis is
shown in Fig. 7 in Appendix III. This localization is known in the literature
Yao and Wang (2018); Yao  _et al._ (2018); Zeng (2022) as the non-hermitian
skin effect. On the other hand, in the $\mathcal{PT}$-unbroken phase, where
all the eigenvalues are real, the wave function does not localize and shows
oscillatory behaviour spanning all the lattice.

This localization-delocalization transition can also be seen in Krylov space.
In Fig. 2, we plot the time evolution of spread complexity, entropic
complexity and KIPR defined in Eq. (5), Eq. (6) and Eq. (7) in different
phases for the initial state spread over $12^{th}$-$18^{th}$ sites of the
chain with $N=40$ sites. Different phases can clearly be distinguished. In the
$\mathcal{PT}$-symmetric phase $(\gamma<1)$, the spread complexity shows an
initial growth followed by large oscillations that are caused by the real
spectra and by finite-size effects. In contrast to the symmetric phase, we
find that in the $\mathcal{PT}$-broken phase, $\gamma>1$, the spread
complexity saturates rapidly following a smaller but faster initial rise than
in the $\mathcal{PT}$-symmetric case. At $\gamma=1$, there is a critical point
at which the spread complexity shows lower initial growth and higher
saturation value compared to the symmetric and broken phases, respectively.
Moreover, the fluctuations are smaller at the critical point than in the
$\mathcal{PT}$-symmetric phase due to the transition to the localized regime.

The spread complexity of a state measures how much the state spreads in the
Krylov basis under the time evolution. For finite-size systems, since the
number of steps in the Lanczos algorithm is bounded by the dimension of the
Hilbert space, the support of the time-evolved state in the Krylov basis stops
growing once the state has explored the full Krylov space. After the time
required to explore the full Krylov space, the spread complexity either
saturates or oscillates. The saturation of complexity implies that the support
of the time-evolved state has reached a steady number of Krylov basis vectors.
In the $\mathcal{PT}$-symmetric phase, the time-evolved state is delocalized
in the Krylov space and has larger support in the Krylov basis as compared to
the $\mathcal{PT}$-broken phase. In the broken phase, the state localizes in
the Krylov space and, therefore, has a smaller complexity saturation value.

Now we turn to entropic complexity defined in Eq. (6). This may also be used
to probe the $\mathcal{PT}$\- phase transition. During the phase transition,
its qualitative behaviour is similar to the spread complexity. The time
dependence of the entropic complexity is shown in the central panel of Fig. 2
for the same initial state as for the spread complexity. The entropic
complexity also shows a lower value in the broken phase as compared to the
symmetric case, which again indicates localization of the time-evolved state
in the $\mathcal{PT}$-broken phase. As we will discuss below, in the broken
phase, the time evolution is different for spread and entropic complexities.

Figure 2: Time evolution of spread complexity (top), entropic complexity
(middle), Krylov inverse participation ratio (bottom) and spread entropy
(inset) in different phases for the initial state spread over
$12^{th}$-$18^{th}$ sites of the chain with $N=40$ sites and $J=1$. Spread
complexity and entropy saturation values are suppressed in the
$\mathcal{PT}$\- broken phase compared to the symmetric phase. In the
$\mathcal{PT}$-broken phase higher value of localization is measured by KIPR
compared to the symmetric phase.

The KIPR is plotted in the lowest panel of Fig. 2 for the initial state spread
around the centre of the tight-binding chain. Since at $t=0$, the initial
state is fully localized at a specific vector as
$|\psi(0)\rangle=|K_{0}\rangle$, the KIPR is one at $t=0$. In the symmetric
phase, the KIPR decays and eventually oscillates around zero, implying that
localization is absent. On the other hand, in the broken phase, the KIPR shows
decay followed by a ramp and finally saturates at a constant value. A higher
saturation value in KIPR indicates stronger localization, as explained in the
paragraph following Eq. (7). In the inset of Fig.2, we plot the spread entropy
for the same initial state as for the other three measures. The late-time
value of the KIPR at the critical point lies between the ones for the
$\mathcal{PT}$-unbroken and broken phases. Since the KIPR measures
localization, the higher saturation value of the KIPR in the broken phase
compared to the symmetric phase implies that the $\mathcal{PT}$-symmetry
transition and the localization-delocalization transition (LDT) occur in
parallel in the system considered, giving rise to a non-hermitian skin effect.

The place of localization within the Krylov space can be precisely determined
based on the choice of the initial state’s spread. We plot the amplitude of
the Krylov wave function, $|\tilde{\psi}_{n}(t)|^{2}$, for different initial
states at initial and late times (inset) in Fig. 3. At time $t=0$, as
expected, we notice all the initial states completely localized in the first
Krylov vector as $|\psi(0)\rangle=|K_{0}\rangle$. However, when the initial
state is spread between sites $l_{1}$ and $l_{2}$ of the lattice chain, the
resulting time-evolved state becomes localized on the Krylov basis vectors
$|K_{n}\rangle$ with $l_{1}\leq n\leq l_{2}$ at late times. This phenomenon is
represented in the inset of Fig. 3. The amplitudes of the overlap of a time-
evolved state on the Krylov basis at late times are not unique for all initial
states. Instead, they depend on the position of the initial states. When the
spread of the initial state includes the central site, the amplitude of the
overlap is higher compared to when this is not the case, as shown in the inset
of Fig. 3. See Appendix II for further details about the dependence of the
saturation value of complexity on different initial spreads.

Figure 3: Localization of time-evolved initial states with different spreads
on the Krylov basis at time $t=0$ (main plot) and $t=100$ (inset plot). We
choose the total number of sites $N=40$, $J=1$, and $\gamma=1.3$. The
localization of the wave function in the Krylov basis is determined by the
spread of the initial state in the tight-binding chain.

According to our findings shown in Fig. 4, the value at which spread
complexity, entropic complexity, and KIPR saturate significantly depends upon
the precise location of the initial state. For the initial state spread over
the central site of the chain as shown in Fig. 3, the corresponding saturation
value of the spread complexity as shown by the blue curve in the first panel
of Fig. 4 occupies an intermediate position, even if the localization strength
is the highest. In contrast, the suppression of entropic complexity and spread
entropy for different initial states always occurs in accordance with the
strength of localization, as can be seen from the middle and bottom panels of
Fig. 4. This observation supports the finding that for the
$\mathcal{PT}$-broken phase, stronger localization does not always imply
stronger suppression of spread complexity. However, spread entropy and
entropic complexity are always suppressed inversely proportional to the
strength of localization.

Figure 4: Saturation of spread complexity (top), entropic complexity (middle),
KIPR (bottom) and spread entropy (inset) for different initial states in
broken phase. We choose the total number of sites $N=40$, tunnelling strength
$J=1$, and $\gamma=1.3$. The hierarchy of saturation values of complexity does
not necessarily follow from the strength of localization; instead, the
saturation value of spread entropy is suppressed accordingly.

The entropic complexity given in Eq. (6) measures the dimension of the minimum
Hilbert space required to contain the information of the time-evolved state.
In the $\mathcal{PT}$-broken phase, the effective Hilbert space dimension is
reduced proportionally to the localization strength. Therefore, we conclude
that in the presence of localization, the entropic complexity is a better
probe to measure the spread of the state as compared to the spread complexity
in Eq. (5).

This different behaviour of spread and entropic complexity can be understood
as follows. Due to the skin effect in the $\mathcal{PT}$-broken phase, the
wave function in the position basis is found to localize always at the first
site in lattice site basis. However, depending on the choice of initial state,
this first position basis corresponds to different Krylov basis vectors
$|K_{m}\rangle$. The further the initial state is from the first lattice site,
the higher the site number $m$ where the wave function localizes in Krylov
space as shown in Fig. 3. So, for different initial states corresponding to
the same strength of localization, the spread complexity will be higher for
the state localized furthest from the initial Krylov vector. This is due to
the fact that in the definition of the spread complexity in Eq. (5), the
Krylov basis vectors with higher site numbers contribute with higher weight
factor $n$. The dependence on the position of localization in the Krylov basis
causes the spread complexity to lose its sensitivity to the strength of
localization in the $\mathcal{PT}$-broken case. Therefore, we infer that for
quantum state evolution in the Krylov basis, increasing localization does not
necessarily imply suppression of the spread complexity in the non-unitary
$\mathcal{PT}$-broken phase. However, spread entropy and the entropic
complexity as defined in Eq. (6), are always suppressed for stronger
localization.

Conclusion.— For a tight-binding Hamiltonian with complex on-site potentials
at the edge sites, we have shown that in the $\mathcal{PT}$-symmetric phase,
the wave function is delocalized and has oscillatory behaviour at late times.
On the other hand, in the $\mathcal{PT}$-broken phase, the wave function is
localized at the left edge site of the tight-binding lattice for the edge
state corresponding to the positive-imaginary eigenvalue. This is the main
feature of the non-Hermitian skin effect. The skin effect occurs due to the
complex energy eigenvalues. Their corresponding eigenstates are localized at
the edge of the tight-binding lattice space. The skin effect entails a
localization-delocalization transition that occurs simultaneously with the
$\mathcal{PT}$-phase transition. In this work, we studied the spread
complexity and spread entropy as a means of characterizing this dynamical
behaviour. All four Krylov space probes we use in this work, namely, spread
complexity, spread entropy, entropic complexity, and KIPR, detect the
$\mathcal{PT}$-phase transition. Due to localization, the saturation values of
spread complexity, spread entropy and entropic complexity are suppressed in
the $\mathcal{PT}$-broken phase. This result for the non-unitary evolution and
PT-symmetric systems adds to the general insight Rabinovici  _et al._ (2022)
that localized quantum states are less complex.

The Krylov inverse participation ratio (KIPR) defined in Eq. (7) acts as a
quantifier for the localization strength in Krylov space. In the
$\mathcal{PT}$-symmetric phase, when there is no localization, the KIPR value
oscillates around $1/D$ where $D$ is the dimension of the Krylov space. In the
$\mathcal{PT}$-broken phase, the localization strength depends upon the
strength of the on-site imaginary potential, as well as on the spread of the
initial state on the tight-binding chain. The spread entropy and entropic
complexity are suppressed monotonically with the localization strength. On the
other hand, the spread complexity is not correlated with the localization
strength. This insensitivity is due to the position dependence of the spread
complexity.

###### Acknowledgements.

The authors would like to thank Vijay Balasubramanian, Souvik Banerjee, Pablo
Basteiro, Lorenzo Crippa, Diptarka Das, Saskia Demulder, Giuseppe Di Giulio,
Mario Flory, Domenico Giuliano, René Meyer, Sara Murciano, Kunal Pal, Kuntal
Pal, Subir Sachdev, Giorgio Sangiovanni and Aninda Sinha for useful
discussions and comments. The work of A.B. is supported by the Polish National
Science Centre (NCN) grant 2021/42/E/ST2/00234. R.N.D. and J.E. are supported
by Germany’s Excellence Strategy through the Würzburg‐Dresden Cluster of
Excellence on Complexity and Topology in Quantum Matter ‐ ct.qmat (EXC 2147,
project‐id 390858490), and by the Deutsche Forschungsgemeinschaft (DFG)
through the Collaborative Research centre “ToCoTronics”, Project-ID
258499086—SFB 1170. R.N.D. further acknowledges the support by the Deutscher
Akademischer Austauschdienst (DAAD, German Academic Exchange Service) through
the funding programme, “Research Grants - Doctoral Programmes in Germany,
2021/22 (57552340)”. This research was also supported in part by the Perimeter
Institute for Theoretical Physics. Research at Perimeter Institute is
supported by the Government of Canada through the Department of Innovation,
Science and Economic Development and by the Province of Ontario through the
Ministry of Research, Innovation and Science. B.D. acknowledges MHRD, India
for Research Fellowship. B.D. would also like to acknowledge the support
provided by Max Planck Partner Group grant MAXPLA/PHY/2018577. B.D. would
further like to acknowledge the support provided by the MATRICS grant
SERB/PHY/2020334.

## Supplemental Material

##  I Lanczos algorithm adapted to complex symmetric operator

We review the algorithm for constructing the Krylov basis associated with a
complex symmetric operator $A$ Bai  _et al._ (2000); Bhattacharya  _et al._
(2023a). The complex symmetry refers to the property, $A^{T}=A$ but
$A^{{\dagger}}\neq A$. The utilization of complex symmetry leads to a
reduction in both computational workload and storage requirements when
compared to the bi-Lanczos method. Notably, the algorithm demonstrates that
for a non-hermitian but complex symmetric operator, only one set of Krylov
basis vectors is sufficient for tridiagonalizing the operator, in contrast to
the bi-Lanczos algorithm, which demands two sets of basis vectors. The
diagonalizability of a complex symmetric matrix A depends on the feasibility
of selecting its eigenvector matrix Z such that
$Z^{T}AZ=\text{diag}(\lambda_{1},\lambda_{2},...,\lambda_{n})$. The matrix of
eigenvectors $Z$, adheres to the condition $Z^{T}Z=I_{n}$, indicating complex
orthogonality. The importance of complex orthogonality plays a pivotal role in
facilitating an efficient construction of the Krylov basis. To ensure this
complex orthogonality, even in the presence of complex vectors, a modified
Lanczos algorithm is employed.

The introduced complex symmetric Lanczos algorithm, as detailed in Bai  _et
al._ (2000), is adept at deriving the tri-diagonal form of a complex symmetric
matrix A Bhattacharya  _et al._ (2023a). This algorithm constructs an
orthogonal basis ${|q_{j}\rangle}$ spanning the Krylov space
$\mathcal{K}^{j}(A,|q_{1}\rangle)\equiv{A|q_{1}\rangle,A^{2}|q_{1}\rangle,\dots}$.
The basis is initiated with a normalized vector, chosen as the initial state
$|q_{1}\rangle=|\psi(0)\rangle$. Complex orthogonality of the basis vectors is
ensured, with $\langle q_{j}|q_{j}\rangle=\delta_{i,j}$, where $\langle
q_{j}|=(|q_{j}\rangle)^{T}$. The basis construction involves a three-term
recursion relation

$\displaystyle\beta_{j+1}|q_{j+1}\rangle=A|q_{j}\rangle-\alpha_{j}|q_{j}\rangle-\beta_{j}|q_{j-1}\rangle,$
(S.1)

where $\beta_{n}=\langle q_{n-1}|A|q_{n}\rangle$ and $\alpha_{n}=\langle
q_{n}|A|q_{n}\rangle$. In the resulting basis ${|q_{j}\rangle}$, the complex
symmetric matrix A adopts the tri-diagonal form denoted as $\tilde{T}_{j}$,

$\tilde{T}_{j}=\begin{pmatrix}\alpha_{1}&\beta_{2}&0&\dots&0\\\
\beta_{2}&\alpha_{2}&\beta_{3}&&\vdots\\\ 0&\ddots&\ddots&\ddots&\\\
\vdots&&\beta_{j-1}&\alpha_{j-1}&\beta_{j}\\\
0&\dots&0&\beta_{j}&\alpha_{j}\end{pmatrix}.$ (S.2)

Through the utilization of the Lanczos algorithm, which enforces complex
orthogonality, the Krylov basis matrix set is obtained as
$Q=[|q_{1}\rangle~{}|q_{2}\rangle~{}...~{}|q_{n}\rangle]$, satisfying
$Q^{T}Q=I_{n}$. This process yields the tri-diagonal form of matrix A in this
basis, expressed as $\tilde{T}_{j}=Q_{j}^{T}AQ_{j}$.

##  II Relation between localization and saturation of complexity and entropy
in the $\mathcal{PT}$-broken phase

Figure 5: Time dependence of the spread complexity (left), KIPR (right) and
spread entropy (inset) in broken phase for different $\gamma$. The chosen
initial state is spread over $12^{th}$ \- $19^{th}$ sites of the lattice chain
consists of $N=40$ sites.

In Fig. 5, we plot spread complexity, spread entropy, and KIPR for different
$\gamma$ in the $\mathcal{PT}$-broken phase $(\gamma>1)$ with the same initial
state spread over $12^{th}$ \- $19^{th}$ sites of the lattice chain, which
consists of $N=40$ sites. The goal is to understand if stronger localization
of the time-evolved state in the Krylov space always necessarily means
suppression of complexity and entropy saturation values. We notice that the
saturation value of the spread complexity decreases as $\gamma$ increases.
However, for spread entropy, the saturation value falls when $\gamma$ rises
up-to $\gamma=1.75$. After $\gamma>1.75$, the saturation value of entropy
begins to rise with $\gamma$. The figure of KIPR provides a valuable
comparison between the saturation value of spread complexity and entropy with
the strength of localization. We observe that there is also a crossover for
the saturation value of the KIPR at $\gamma=1.75$, which is perfectly
consistent with the spread entropy case. Therefore, spread entropy in the
Krylov basis for a quantum state is always more suppressed when localization
is stronger; this is not necessarily the case for spread complexity.

Figure 6: Comparison between the late-time average saturation values of spread
entropy (red), the inverse of KIPR (green), and spread complexity (blue) for
different choices for the spread of the initial state. Notice that the spread
entropy and KIPR follow the exact same pattern while the spread complexity
saturation value keeps growing as the initial spread is taken far from the
first position basis.

The saturation value of spread complexity, entropy, and KIPR depends on the
choice of the initial state, as depicted in Fig. 4. We further validate this
pattern by varying the initial state spread over four sites from the left to
the right of the chain and observing the dependence of the saturation value of
complexity and entropy on changes in the saturation value of KIPR. The
saturation value of KIPR and the saturation value of spread entropy should
exhibit an inverse relationship, demonstrating that a stronger saturation
indicates the suppression of spread entropy. Thus, higher saturation values of
KIPR lead to lower saturation values of spread entropy. For visualization, we
define $\mathcal{C}$ and $\mathcal{S}$ as the average saturation values of
spread complexity and spread entropy, respectively, as

$\mathcal{C}=\lim_{t^{*}\to\infty}\frac{1}{t^{*}-t_{\text{sat}}}\int_{t_{\text{sat}}}^{t^{*}}C(t)dt,~{}~{}\mathcal{S}=\lim_{t^{*}\to\infty}\frac{1}{t^{*}-t_{\text{sat}}}\int_{t_{\text{sat}}}^{t^{*}}S(t)dt.$
(S.3)

We additionally define $\mathcal{K}$ as the average of the inverse of the KIPR
saturation value as

$\displaystyle\mathcal{K}=\lim_{t^{*}\to\infty}\frac{1}{t^{*}-t_{\text{sat}}}\int_{t_{\text{sat}}}^{t^{*}}\frac{1}{\text{KIPR}(t)}dt.$
(S.4)

Here, $t_{\text{sat}}$ is the time when complexity, entropy and KIPR values
get saturated and for a particular initial state, this time is same for all
these three quantities. Following our modified proposal, $\mathcal{S}$ should
be proportional to $\mathcal{K}$. This behaviour aligns precisely with our
findings in Fig. 6. The blue, red and green dots represent the $\mathcal{C}$,
$\mathcal{S}$ and $\mathcal{K}$ respectively. We have rescaled $\mathcal{C}$
and $\mathcal{S}$ by 0.2 and 3 times, respectively, for bringing all the
values in the same plot. We find $\mathcal{S}$ is suppressed according to the
strength of localization as measured by $\mathcal{K}$ but $\mathcal{C}$ keeps
growing linearly due to the monotonically increasing weight factor.

##  III Support of the time-evolved states in the tight-binding lattice basis

Figure 7: Localization of time-evolved different initial states on lattice
sites at initial (left) and late times (right).

In Fig. 7, we illustrate the localization of the initial states on the lattice
basis in the $\mathcal{PT}$-broken phase. We consider three different spreads
of the initial state. Initially, the states have equal support on the
different lattice sites for $N=40$. The tri-diagonal Hamiltonian in the
$\mathcal{PT}$-broken phase has an unbalanced gain parameter in the first
diagonal. Due to this unbalanced gain, the state localizes at the first site
of the lattice at late times. Correspondingly, the localization of the time-
evolved states in the Krylov basis is shown in Fig. 3. Since all the initial
states are taken as the $|K_{0}\rangle=\delta_{n,0}$, the states are localized
in a specific vector in the Krylov basis. The Hamiltonian for an arbitrary
initial state changes after performing the Lanczos algorithm, and the new tri-
diagonal matrix $\tilde{T}$ that we get in this way would always have the
first non-hermiticity present around the sites where the chosen initial state
had the first nonzero support. As soon as the Krylov wave function hits these
sites, it localizes and stays so afterwards. Therefore, in the
$\mathcal{PT}$-broken phase, complexity analysis is dependent on the initial
state, and the localization of the time-evolved state on the Krylov basis,
always helps us to understand which site exactly the chosen initial state had
the first nonzero support on. Comparing Fig. 3 and Fig. 7, we can see a dual
mapping between the localization of the states in the two different bases.
However, the dual mapping is approximate since the amplitudes of the
localization of initial states on the lattice at early times are the same. In
contrast, the amplitudes of the localization of initial states in the Krylov
basis at late times are not the same, see Fig. 3. These different amplitudes
of localization in the Krylov basis are the main reason for the different
strengths of localizations.

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Manuscript submitted to ACM Manuscript submitted to ACM

# Design Considerations for Data Daemons: Co-creating Design Futures to
Explore Ethical Personal Data Management

Wiebke Toussaint<EMAIL_ADDRESS>,  Olya Kudina  ,  Aaron Yi Ding
Technology, Policy & Management, TU DelftThe Netherlands ,  Alejandra Gomez
Ortega  ,  Jered Vroon  ,  Jacky Bourgeois  Industrial Design Engineering, TU
DelftThe Netherlands ,  Julian Harty  0000-0003-4052-0054 Commercetest Ltd.,
The Open UniversityUnited Kingdom ,  Gürkan Solmaz  NEC Laboratories
EuropeGermany and  Ella Peltonen  University of OuluFinland

###### Abstract.

Mobile applications and online service providers track our virtual and
physical behaviour more actively and with a broader scope than ever before.
This has given rise to growing concerns about ethical personal data
management. Even though regulation and awareness around data ethics are
increasing, end-users are seldom engaged when defining and designing what a
future with ethical personal data management should look like. We explore a
participatory process that uses design futures, the Future workshop method and
design fictions to envision ethical personal data management with end-users
and designers. To engage participants effectively, we needed to bridge their
differential expertise and make the abstract concepts of data and ethics
tangible. By concretely presenting personal data management and control as
fictitious entities called Data Daemons, we created a shared understanding of
these abstract concepts, and empowered non-expert end-users and designers to
become actively engaged in the design process.

personal data management, personal tracking data, data ethics, participatory
design, design futures

††ccs: Human-centered computing Participatory design††ccs: Human-centered
computing Ambient intelligence

##  1\. Introduction

Connected digital devices, including laptops, smartphones, wearables, and
diverse Internet of Things (IoT) devices, produce ongoing personal tracking
data streams that track what, when, where and with whom we move, interact, and
socialise. Personal tracking data have become important drivers of mobile and
digital services, with businesses even viewing them as the ”new gold” (Moon,
2019). Consequently, our virtual and physical behaviour is tracked more
actively and with a broader scope today than ever before. This is seldom in
the interest of individual end-users, even if they voluntarily accept the
risks (Phelan et al., 2016). Ethical concerns pertaining to the processing and
management of personal data have thus moved to the forefront of policy debate.
However, translating ethical guidelines and policies into regulatory
mechanisms and standards remains a challenge (Mittelstadt et al., 2016).
Similarly, ethical concerns do not translate readily into design
considerations for data-driven products and services. Despite increasing
awareness around data ethics, and regulation around the use and management of
personal data, many people have resigned themselves to the belief that their
personal data is more likely to be abused than to serve them (Phelan et al.,
2016; Colnago et al., 2020). Even though end-users are most likely to be
adversely impacted by extractive data practices, participatory approaches to
ethical personal data management remain scarce (Whitman et al., 2018). The
abstract nature of data (Vandenberghe and Slegers, 2016) and differential
expertise of participants (Baumer, 2017) are considered key barriers to
engaging end-users in the design process of data-driven applications. High
data and digital literacy requirements often exclude both application
designers and end-users from actively participating in the design of ethical
data-driven services (Roeck et al., 2011).

This paper presents a first exploration of how the perspectives of end-users
and designers can inform the future development of ethical personal data
management. Using design futures and a participatory approach, we overcome the
barriers presented by abstraction and the inherent data and digital literacy
requirements of data ethics with a fictitious concept that we call Data
Daemons. The Data Daemons enabled us to co-create design fictions and design
considerations that make ethical personal data management more concrete and
reflective of the values of a group of expert and non-expert participants. We
see this as a valuable contribution that builds on participatory design from
the HCI community to bring end-user and designer perspectives into ethical
personal data management.

##  2\. Background and Related Work

###  2.1. The Status Quo of Personal Data Management

Collecting and processing personal tracking data can be highly invasive and
seldom in the best interest of end-users (Goodman, 2014). Many online services
and mobile applications harvest and trade personal tracking data with a
multitude of data brokers and third-party service providers. As Helles et al.
(Helles et al., 2020) show, the online tracking business feeds an attention
economy in which end-users are more unsuspecting subjects than collaborating
partners.

Treating end-users as subjects is contested in the data science domain.
Metcalf and Crawford (Metcalf and Crawford, 2016) point out the growing
disconnect between the limited ethical considerations in data science research
on the one hand, and established tools and practices of research ethics in
disciplines that traditionally conduct human subject research on the other
hand. Ethical challenges arise, in part, from data science methods making data
”infinitely connectable, indefinitely repurposeable, continuously updatable
and … removed from … context” (Metcalf and Crawford, 2016). These same
conditions inherently dominate the nature of personal tracking data. Risks
arising from unethical data practices are not limited to traditional physical
harms, but include concerns like the rise of new digital divides, infringement
on information privacy, wrong conclusions due to decontextualisation, and
misleading claims of objectivity and accuracy (Boyd and Crawford, 2012).

###  2.2. Participatory Design Perspectives on Personal Data and Algorithms

Participatory Design is an established tradition in the design of information
and communication technologies (Robertson and Simonsen, 2013). It stipulates
that the users of technology must be able to influence its design, even if
they have no expertise or prior knowledge in technology design. Furthermore,
it encourages mutual learning between users and designers, acknowledging that
users who lack expertise in technology design may not know what they want
without knowing what is possible. Participatory Design has been used to engage
end-users on personal data. For example, Rosenbank et al. (Rosenbank and
Feckenstedt, 2016) present an art-based approach that involves participants in
co-designing their digital shadows with extracts of personal metadata. In
Metadating (Elsden et al., 2016), Elsden et al. take a speculative approach to
explore how people use personal data to communicate their identity in a
future-focused speed-dating event. Looking at design interventions,
Vandenberghe and Slegers (Vandenberghe and Slegers, 2016) use anthropomorphism
as a strategy to learn about the personal health data information needs of
end-users during the ideation phase of a future health data application.
Colnago et al. (Colnago et al., 2020) conduct semi-structured interviews to
elicit the opinions of potential users on different privacy-related design
configurations to inform design considerations for personalized privacy
assistants. Baumer (Baumer, 2017) highlights the need for Participatory Design
in algorithm design, but also points to the challenge that engaging with
algorithm design requires significant expertise, which participants typically
do not have.

While promising, participatory research on ethical personal data management
and algorithms is still an emerging field (Whitman et al., 2018). In
particular, Whitman et al. note that most participatory research, like the
research discussed above, focuses on facilitating participants’ engagement
with data, or centering end-users in the process of developing effective and
engaging tools. Participatory research that focuses on values and ethics of
personal data management and algorithm design is scarce.

###  2.3. Using Fictions and Futures to Explore What Could and Should Be

Design futures are a strategy for actively recruiting the future into design
practice (Reeves et al., 2016). Reeves et al. point out that to be of value,
design futures need to be considered legitimate by a community that extends
beyond the visionaries. They suggest that participatory approaches and in
particular design fictions can be used to enhance the legitimacy of design
futures by enabling broader participation in envisioning the future. The
Future workshop (Vidal, 2005) is a facilitated approach that guides
participants through a process of co-creating fantasy futures. It was
developed as a form of structured citizen engagement, and emphasises group
dynamics, collective work and collaborative learning, and facilitated,
creative problem solving. Design fictions invite the narrative and future-
oriented characteristics of science fiction into the practical and material
world of design (Bleecker, 2009). They are widely used in HCI to explore
future scenarios, to map out design possibilities, to engage users in
participatory design, or to offer critique through story-telling and character
development (Baumer et al., 2020). We initiated our participatory research
with the Future workshop to invite participants into a co-creative process.
Design fictions emerged as a participatory tool later on to continue engaging
participants in the concepts that were inspired by the workshop.

##  3\. Co-creating Data Daemons

We approached this study as an exploratory and emergent process, with a focus
on active participation and co-creation. Our starting point was the goal to
explore design futures for ethical personal data management in an online
Future workshop. The Future workshop brought together a group of 11
participants to scope the problem space, envision ideal fantasy futures and
identify barriers towards them. The participants were a demographically
heterogeneous group of researchers with 10 different nationalities from 4
continents, 6 male and 5 female researchers, and an age span of two decades.
Participants’ professional backgrounds were in both technical and non-
technical disciplines. While all participants had an interest in ethical
personal data management, expertise was clustered around specific aspects of
the topic – ethics, personal data, data management and data-driven products
and services. Some participants self-identified as non-experts, bringing
perspectives from design, industry, and their personal lives. The output of
the workshop can be viewed online111https://miro.com/app/board/o9J_kkfAKjE=/.

###  3.1. Emergence of Data Daemons

During the workshop two challenges became evident: firstly, because future
visions emerge from experienced reality, ideas were limited by existing data
management practices, technologies and paradigms, which at best incorporate
ethics as an afterthought. While we expected alternative futures to exist, we
needed a way to inspire a “what if” perspective to provoke them. Secondly, we
realised that building fantasy futures of abstract concepts like ethics and
data was difficult, as participants had different backgrounds and a vastly
different understanding of what these concepts mean. To overcome these
differences we needed a bridge that could connect different standpoints.

These challenges inspired us to conceptualise Data Daemons. Data Daemons would
be owned by individuals, and have the responsibility of managing and
controlling their owner’s personal data in their owner’s best interest. They
merge the functionality of daemons found in computer programmes, and daemons
invoked in the popular fantasy series ”His Dark Materials” by Sir Philip
Pullman. In computer science, daemons are programmes that run routine
background processes and service requests in multitasking operating systems.
They operate independently without user intervention, and can interact with
other processes. Pullman’s fantasy daemons are animal-shaped, physical
manifestations of our inner self. They are spiritual guides that are at the
same time external to their humans, yet a part of them. Combining these two
kinds of daemons led us to Data Daemons: artificial entities that know us
intimately and perform data management tasks on our behalf. Data Daemons
manage all aspects of our personal tracking data and provide useful services
like the daemons that already execute tasks in computing systems. In addition,
they also concretise relationships between people and their personal tracking
data to be like the relationships between Pullman’s human characters and their
daemons. In our ideal future, Data Daemons thus embody the relationship
between humans and the systems that collect and process our personal tracking
data. They are the virtual manifestation of our digital self.

###  3.2. Co-creative Approach

This shared conceptualisation of fictitious Data Daemons created a common
language for workshop participants, while also liberating them to be more
ambitious and creative in imagining their ideal ethical personal data futures.
The Future workshop inspired a series of follow-up meetings and writing
exercises to further develop the concept of Data Daemons to explore design
considerations for ethical personal data management. In the follow up meetings
we selected design fictions as a tool to co-create narrative prototypes of
Data Daemons that highlight design considerations for them. The design
fictions were refined through several iterations.

Figure 1. Co-creative approach to developing Data Daemons

The co-creative process that emerged is shown in Figure 1, which depicts the
interdependent relationship between the Data Daemon concept, design fictions
of Data Daemons, and design considerations for Data Daemons. We used the Data
Daemons concept to explore design considerations and create design fictions.
The design fictions highlight design considerations. Reflecting on the design
considerations helped to refine the design fictions over several iterations.
With each iteration the design fictions and design considerations became more
nuanced, expressing and clarifying the Data Daemon concept with greater
detail, and with it concretising participants’ expectations for ethical
personal data management of data-driven products and services. Next we present
the design fictions as refined in the final iteration.

##  4\. Design Fictions of Data Daemons

The design fiction scenarios presented here are fictional reflections on
personal Data Daemons by three personae with different values and concerns
that are influenced by their data and digital literacy. Proactive Pippa
(Section 4.1) is highly data-conscious and an Internet native. Mother Maria
(Section 4.3) feels overwhelmed by the rapid pace of digital change. Middle-
aged Regular Robert (Section 4.2) captures the middle ground between the two.
The design fictions demonstrate how people with varying degrees of data
literacy reflect on their interaction with, and experience of, their Data
Daemon.

###  4.1. Design Fiction 1: Pippa improves her performance and health with her
data

PIPPA: Data-conscious Internet native in her late twenties. Wary of fake news
and social media manipulation. Suffers from stress and migraines. Into
mindfulness.

”Pivo, my data concierge, knows me intimately. He knows what I want, and
guards my preferences. I request them from him on a monthly basis to help me
reflect on my behaviour and activities. Pivo can summarise whether I have
exercised enough, who I have seen the most, and where I have been. Based on
that, he recommends what I should change in the coming month to meet my
personal and career goals. Of course I don’t have to accept his
recommendations, and I sometimes reject them to ensure that I can still make
independent decisions. Since working with Pivo I have finally managed to
prioritise self-care. He infers my default data collection and processing
options for all my digital services based on my monthly reflections. Pivo
highlights data donation opportunities that fit my goals, for example,
donating data to inform the design of a new product that can help me cope with
migraines. Recently, the loud city noises and weather conditions have
intensified my migraines. When I got on a busy bus last Tuesday, I was
automatically guided to a quiet seat away from the harsh sunlight. Thanks to
Pivo, the data from my various health monitors are now combined with hyper-
local weather forecasts and the public transport systems to plan my route,
minimising adverse effects. Pivo has arranged that just enough data is
revealed on a temporary basis with the weather app and transit systems. The
shared data automatically expires at the end of my journey. I get the benefit
of personalised route planning, while the various systems can calibrate and
improve their accuracy based on real life conditions. My health insurance
actually contributes to the cost of Pivo and the services he negotiates,
because this is much more effective _and_ significantly cheaper than medical
care. No-one else needs to access my sensitive data, least of all my employer!
Work pays the balance of Pivo’s subscription because I’m able to do better
work as a result of the services he facilitates. Keeping me healthy is still
good business value.”

###  4.2. Design Fiction 2: Robert’s data empowers him to do more and do good

ROBERT: Mid-career professional in the EU. Discusses gadgets and tech trends
with colleagues. Reads Hacker News. Expects tech to serve him and likes to
feel good.

”In the early days, when Ruba, my Data Daemon, was not yet fully trained, it
would ask for my preferences almost daily. But now, I barely notice Ruba
anymore. Sometimes it checks in briefly to confirm we are still aligned, which
we usually are. Ruba’s services extend well beyond managing my personal data
ownership of course, as we leave traces all over the place. We really
established our relationship when I got my smart washing machine. Ruba guided
me through the setup and interpreted the options for sharing my data. It took
less time than setting up my sharing preferences on social media. When cookies
started to be introduced on my home devices, I rejected all of them at first,
just like I did with Internet browsers. I then realised that by accepting
certain cookies, Ruba was able to compile reviews and perspectives that
contrast my own to broaden my horizon. Ruba quickly adapted to accept useful
cookies on my behalf. And when cookies collect data for targeted advertising,
Ruba skillfully haggles good returns and ensures that the advertisements I am
shown do not make me feel like I’ve sold my soul. I even donate some of my ad-
view revenue towards planting trees. Ruba suggested that I share my anonymised
speech data, even for commercial purposes, because it knows that I like
openness and believe in win-win situations. Donating data helps improve speech
recognition for everyone – and I get a small compensation that I usually use
for a cup of coffee. Thanks to Ruba’s excellent negotiations I strike a fairly
even balance, paying only a little for some digital services each month. For
example, I pay to get unrestricted access to a selection of curated websites,
including two news platforms, without any advertisements. However, I give free
access to my anonymised home energy usage data to contribute towards a more
sustainable world.”

###  4.3. Design Fiction 3: Maria benefits from her smart services with peace
of mind

MARIA: Mother of two teenage daughters. Overwhelmed by the rapid pace of
digital change. Seeks stability and connection to the familiar. Chicken curry
champion (family secret).

”I have to confess, I did not want a Data Daemon. I was scared by the idea of
having to interact with something so vaporous. But being digitally
disconnected really isolated me from my family. So my daughter convinced me to
try Myra. Myra is a tangible Data Daemon, with a minimalist touchscreen
interface that allows me to effectively manage my interactions with her. Myra
does not bother me for tiny tasks, only important events and emergencies. She
is my anchor in the digital world, which is very foreign to me. Myra ensures
that things run smoothly, always. First, Myra helped me go through the privacy
settings on my smartwatch, the only smart device I own. I usually get anxious
and confused with PC software updates asking to share my data, so I had a lot
of questions about terminology and functionality. Myra gently guided me, gave
insightful tips and responded to all my questions, patiently. She even helps
me to digest complicated, long, consent forms! Nowadays I trust Myra and feel
relieved I no longer have to navigate the digital world alone. Myra keeps
track of the data from my smart watch and my daughter says I can double-check
her decisions. I could even ask Myra to step aside if I wanted to be in
control, but I would need to ask for help to learn how to do that. Myra scans
the data streams for suspicious activity and catches unusual behaviour. My
daughter explained that Myra takes preventative actions to protect my health
tracking data, rather than alarming me unnecessarily. Myra has a good
relationship with all the family’s Data Daemons. She regularly fetches family
updates, which makes me happy. Myra also collects news from neighbours and
insights from experts, which helps me stay in touch and stay informed.”

##  5\. Design Considerations for Data Daemons

Robert, Pippa, and Maria have diverse values, specific needs and unique ways
of taking advantage of their personal tracking data. Nonetheless the design
fictions have commonalities that highlight design considerations for ethical
personal data management across the personae. Data Daemons clearly hold a
position of power. They connect end users to opportunities that are useful for
them (e.g. helping Pippa handle her migraines better) and, where possible,
benefit society at large (e.g. motivating Robert to share his speech data).
Across the design fictions, the Data Daemons are deeply personal, accountable,
and adaptive. They interpret, discover, access, negotiate and trade personal
tracking data, and facilitate interaction and collaboration. These
capabilities emerged from reflections on the design considerations entailed in
the design fictions. By finding a common framing for expressing our
collective, desired design considerations for Data Daemons and thus clarifying
our shared understanding of the Data Daemons concept, we agreed that ethical
personal data management should include, as a minimum, aspects of:

  1. (1)

Supported personal choice and control

  2. (2)

Interaction and collaboration through negotiation

  3. (3)

Accountability and adaptation to value change

Below we discuss how the design fictions highlight these considerations, and
what this implies for ethical personal data management.

###  5.1. Supported personal choice and control

For Pippa and Maria, it is important that their Data Daemons present them with
choice and control. Pippa can choose to ignore recommendations and has control
over how often her Data Daemon presents feedback. She wants to avoid
overreliance and retain a degree of independence. Maria chooses an embodied
Data Daemon that makes interaction easier for her. As is apparent from the
design fictions, exerting choice and control over personal tracking data can
require considerable effort, background knowledge, insights into data usage,
and familiarity with your preferences and values. The lower a person’s data
literacy, and the more data is generated, the more burdensome its management
becomes. Consequently, choice and control cannot be realised without offering
support. The Data Daemons thus automate repetitive, dynamic, and complex data
management tasks and perform activities humans cannot do: encoding and tracing
data usage permissions and purposes, managing temporary data sharing and
expiration, and detecting, validating, and mitigating digital threats.

###  5.2. Interaction and collaboration through negotiation

The Data Daemons are integrated into their personae’s existing worlds, and
rely on interaction and collaboration through negotiation to help them achieve
advantage in it. In the design fictions, personal tracking data has no
intrinsic economic value. Rather, its value arises out of the affordances it
enables for the humans that generate it, and the benefits it creates for
society. This means that data is valuable in so far as it supports the
development of relations between humans and their environment. Ruba offers
Robert new perspectives and negotiates compensation for the usage of his data.
Pivo negotiates privacy sensitive usage agreements for Pippa’s data streams
and integrates with the health insurance system. Myra maintains relationships
with the Data Daemons of Maria’s family and enables her to stay connected to
the people she cares about. Discourse is a precondition for negotiation. Our
design fictions propose three types of discourse for the Data Daemons:
discourse with their persona, discourse with other Data Daemons, and discourse
with external systems. Discourse between a persona and their Data Daemon
assumes that data usage concepts can be interpreted and aligned with personal
values and preferences. Discourse with external systems and between Data
Daemons points to the need for a common language that can express, capture and
interpret relevant data usage concepts and concerns, as well as the purposes
for which data is being used.

###  5.3. Accountability and adaptation to value change

The Data Daemons are accountable to their personae. They accompany their
personae in their digital worlds, interpreting what their data means and
represents in an individualised manner that is digestible and relatable. Their
interactions are transparent, they offer explanations and adapt with the
individual needs of their personae: Ruba asks, checks in and confirms, Pivo
requests, summarises and recommends, and Myra guides gently and responds
patiently. In turn, the personae have gotten accustomed to their Data Daemons
offering services tailored to their unique values, needs and preferences. They
trust their Data Daemons and are willing to reveal intimate information; like
Pippa who has authorised Pivo to help her uncover blindspots through
personalised reflection and goal-setting. Additionally, Ruba exhibits the
ability to adapt to Robert over time, while Ruba and Maya offer decision-
making and prioritisation support to help Robert and Maria navigate the
digital world. By checking in, the Data Daemons acknowledge they may get out
of sync with their persona’s needs and values. This highlights that personal
data management is an ongoing conversation that needs to adapt over time,
rather than a once-off decision.

##  6\. Reflection on Co-creating Data Daemons

We set out to explore ethical personal data management for data-driven
products and services from the perspective of users and designers. Here we
reflect on the overall process and the specific contributions of the Future
workshop, Data Daemons, and design fictions. The main value of the work lies
in exploring a process for engaging a group of heterogeneous participants in
the abstract and technically challenging topic of ethical personal data
management. The process resulted in participants reaching consensus on design
considerations they deemed essential for realising a future with ethical
personal data management. The Future workshop set the scene and encouraged a
participatory atmosphere. The fantasy futures phase allowed us to creatively
engage with the topic and opened up a space for new ideas. However, the Future
workshop was limited in supporting an abstract problem space in which the
expertise of participants and their alignment on the underlying concepts
varied. Moreover, it was difficult to imagine radical alternatives to current
technologies and data management practices.

Introducing Data Daemons turned out to be crucial. They immediately made the
problem space tangible and participants took on the concept enthusiastically.
While views on Data Daemons differed, the concept enabled participants to
express their differences and align their perspectives on how they might
interact with data ecosystems in mediated ways. Over time, perspectives
converged and the Data Daemons became a tool for interrogating how end-users
would like to relate to their personal tracking data. The group saw Data
Daemons as a promising concept for considering ethical personal data
management more comprehensively and with greater imagination. The Data Daemon
concept itself merits further exploration. For example, we assumed that Data
Daemons operate in the best interest of their owners. In real life this may of
course not be the case and potential failures of Data Daemons should also be
considered.

Design fictions proved to be a useful tool for highlighting design
considerations and expressing the Data Daemons concept. The narrative approach
they offer made it possible for several participants to work collectively on a
single design fiction, and to augment, merge, and refine the design fictions
through iteration. Participants who did not actively create or refine a design
fiction in a particular iteration were nonetheless able to critique it and
offer additional insights and ideas based on the highlighted design
considerations. The participatory process through which the design fictions
and design considerations were developed gave legitimacy to the Data Daemon
concept and the aspects of ethical personal data management that were elevated
through it. We found this approach of using co-created narratives highly
effective for gathering insights on how data processing and data protection
guidelines and regulations may be put to practice ethically, in alignment with
user values.

In this study participants were self-selected, participated voluntarily, and
had affiliations with research institutions or academia. In future work it
will be important to assess the process’s effectiveness for participants with
limited knowledge of data or technology and citizens more broadly. A
participatory process such as we present here also does not exist
independently. Moving forward, we aim to embed it in established design
practices, in particular in the conceptualisation phase of value-sensitive
design where it has the potential to expand and contextualise conceptual and
empirical investigations within the value-sensitive design process.

##  7\. Conclusion

In this study we have shown how a participatory approach involving design
futures, a Future workshop, design fictions and the fictitious concept of Data
Daemons, can empower non-expert end-users and designers to become active
participants in the design process, capable of co-creating design
considerations for ethical personal data management. The outcome is evident:
it is not only possible, but necessary and valuable to include multiple
perspectives and levels of expertise in considering this technically
challenging and abstract topic. We hope this study inspires more research on
participatory approaches to engaging end-users in the design of ethical
personal data management.

###### Acknowledgements.

This paper evolved from discussions at the First International Workshop on
Ethical Data-Centric Design of Intelligent Behaviour at MobileHCI 2020.

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