aainstitutetext: Institute of Physics, Jagiellonian University, Łojasiewicza
11, 30-348 Kraków, Polandbbinstitutetext: Centre for High Energy Physics,
Indian Institute of Science,
C.V. Raman Avenue, Bangalore 560012, India.ccinstitutetext: Institute for
Theoretical Physics and Astrophysics and Würzburg-Dresden Cluster of
Excellence ct.qmat Julius-Maximilians-Universität Würzburg
Am Hubland, 97074 Würzburg, Germanyddinstitutetext: Indian Institute of
Technology - Kanpur, Kanpur 208016, Uttar Pradesh, India

# Spread complexity for measurement-induced non-unitary dynamics and Zeno
effect

Aranya Bhattacharya c Rathindra Nath Das d Bidyut Dey c and
Johanna Erdmenger<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Using spread complexity and spread entropy, we study non-unitary quantum
dynamics. For non-hermitian Hamiltonians, we extend the bi-Lanczos
construction for the Krylov basis to the Schrödinger picture. Moreover, we
implement an algorithm adapted to complex symmetric Hamiltonians. This reduces
the computational memory requirements by half compared to the bi-Lanczos
construction. We apply this construction to the one-dimensional tight-binding
Hamiltonian subject to repeated measurements at fixed small time intervals,
resulting in effective non-unitary dynamics. We find that the spread
complexity initially grows with time, followed by an extended decay period and
saturation. The choice of initial state determines the saturation value of
complexity and entropy. In analogy to measurement-induced phase transitions,
we consider a quench between hermitian and non-hermitian Hamiltonian evolution
induced by turning on regular measurements at different frequencies. We find
that as a function of the measurement frequency, the time at which the spread
complexity starts growing increases. This time asymptotes to infinity when the
time gap between measurements is taken to zero, indicating the onset of the
quantum Zeno effect, according to which measurements impede time evolution.

##  1 Introduction

The application of information-theoretic tools to quantum systems is one of
the major current research directions in theoretical physics, ranging from
simple models in quantum mechanics to complicated quantum field theoretic and
holographic setups. One of the recent advances in this context is the study of
complexity, in order to characterise the evolution of a quantum state or
operator. Complexity is of particular interest in the context of holographic
conjectures Susskind:2014rva ; Brown_2016 . Motivated by holography, Nielsen’s
circuit complexity Nielsen_2006 , which is the first example relating
complexity to geometric concepts such as geodesics, was investigated for new
quantum mechanical and quantum field theoretical systems Jefferson:2017sdb ;
PhysRevLett.120.121602 ; PhysRevLett.122.231302 ; Flory:2020dja ;
Erdmenger:2020sup ; Bhattacharya:2022wlp 111For detailed review on these
developments, see the review Chapman:2021jbh  and references therein.. An
alternative proposal of complexity, free of ambiguities such as the choice of
gates or reference state, is Krylov complexity Parker:2018yvk . Krylov
complexity is defined for an orthonormal basis of operators (states) in the
Heisenberg (Schrödinger) picture using the Lanczos algorithm VM ;
Parker:2018yvk ; Balasubramanian:2022tpr  which quantifies the growth of an
operator (state) during time evolution. While most of the studies for Krylov
complexity to date have been for operators Barbon:2019wsy ; Dymarsky:2019elm ;
Dymarsky:2021bjq ; Yates:2021asz ; Caputa:2021sib ; Kim:2021okd ;
Caputa:2021ori ; Patramanis:2021lkx ; Trigueros:2021rwj ; Rabinovici:2021qqt ;
Fan:2022xaa ; Heveling:2022hth ; Bhattacharyya:2023dhp ; Muck:2022xfc ;
Kundu:2023hbk ; Banerjee:2022ime ; Du:2022ocp ; Guo:2022hui ;
Avdoshkin:2022xuw ; Bhattacharyya:2020qtd ; Iizuka:2023pov ; Iizuka:2023fba ;
Jeong:2023hrb ; Camargo:2022rnt , in Balasubramanian:2022tpr  an equivalent
definition for Schrödinger evolution of quantum states was proposed using the
Krylov basis. This is known as the Krylov spread complexity, which quantifies
how the information of a quantum state spreads under Hamiltonian evolution.
This spreading implies that the state becomes more complex. However, as the
physical Hamiltonian is usually hermitian, these studies considered unitary
evolution of quantum states Caputa:2022eye ; Afrasiar_2023 ;
Balasubramanian:2022dnj ; Caputa:2022yju ; Erdmenger:2023shk ; Pal:2023yik ;
Nandy:2023brt ; Chattopadhyay:2023fob ; Gautam:2023bcm ; Bhattacharjee:2022qjw
; Gill:2023umm ; Bento:2023bjn ; Aguilar-Gutierrez:2023nyk ; Craps:2023ivc .

In this paper, we address the question of how the Krylov spread complexity
behaves for non-unitary dynamics. In principle, the methods we develop are
general and can be applied to any non-unitary evolution of quantum state. For
definiteness, we focus on a specific case known as the quantum first passage
problem (QFPP), where the dynamics become non-unitary due to repeated
projective measurements Peres ; Dhar2013QuantumTO ; Dhar_2015 ;
Anastopoulos_2006 ; Erez_2008 ; PhysRevE.95.032141 . It is a well-known fact
from the postulates of quantum mechanics that these projective measurements
are non-unitary operations, and hence, the evolution of the system becomes
non-unitary. The effect of the measurement is captured perturbatively by an
effective non-hermitian Hamiltonian, where the time between two consecutive
measurements acts as the perturbation parameter.

Previously, Krylov complexity was investigated for unitary evolution of
isolated systems Parker:2018yvk ; Rabinovici:2021qqt , as well as for open
system non-unitary dynamics of operators Bhattacharya:2022gbz ; liu2022krylov
; Bhattacharjee:2022lzy ; Bhattacharya:2023zqt ; Bhattacharjee:2023uwx .
Moreover, an approach towards non-unitary dynamics for operator growth of open
quantum systems was given in Bhattacharjee:2022lzy ; Bhattacharya:2022gbz ;
Bhattacharya:2023zqt . These works are concerned with the operator Hilbert
space, where the evolution is generated by the Liouvillian superoperator
$\mathcal{L}=[H,.]$ instead of the Hamiltonian $H$.

The starting point of the present work is the tight-binding Hamiltonian
Dhar2013QuantumTO ; Dhar_2015 . Subject to the measurements of the QFPP222The
quantum first passage problem (QFPP) considers a quantum state evolving under
a lattice Hamiltonian subject to local projective measurements performed on
particular lattice sites Dhar_2015 ., when the perturbative treatment is
applied for modelling the non-unitary evolution, we obtain a non-hermitian
Hamiltonian that we use to evolve position eigenstates. We obtain the non-
hermitian total probability, the spread complexity, and the spread entropy to
identify the characteristics of non-unitarity by comparing them to a scenario
of unitary evolution without measurements.

We build upon the methodology of the bi-Lanczos algorithm previously utilized
in Bhattacharya:2023zqt  for the non-unitary evolution of operator complexity.
We extend this approach to investigating the state complexity for the non-
unitary evolution of a quantum system subjected to projective measurements at
regular intervals. Our analysis pursues a dual objective; first, to extend the
non-unitary bi-Lanczos approach to spread complexity, and second, to explore
the behaviour of spread complexity for measurement-induced quantum channels.
Additionally, we demonstrate how a specialized algorithm designed for complex
symmetric matrices can effectively halve the workload and memory requirements.
We refer to this new algorithm as a complex symmetric Lanczos algorithm.
Applying these techniques to the QFPP, we analyze the spread complexity and
entropy in the Krylov basis. Our investigation encompasses both open and
periodic boundary conditions, revealing that the behaviour of spread
complexity in this non-hermitian system is marked by an initial growth
followed by an extended decay region and saturation. Notably, the prolonged
decay region, as compared to unitary evolution, represents a novel complexity
behaviour induced by measurements in this non-hermitian context. We also
observe that different choices of the initial state yield distinct dynamics
and saturation values for complexity and entropy.

Finally, we explore a quench scenario, gradually changing the non-hermitian
perturbation parameter from zero to higher values, effectively increasing the
time gap between measurements in the system. By selecting eigenstates of the
tight-binding hermitian Hamiltonian as the initial states, we examine the
evolution of spread complexity under the effective non-hermitian quenched
Hamiltonian. In this scenario, for the single-particle ground state, the
complexity dynamics shifts from oscillatory behaviour to an initial phase of
rapid growth, followed by decay and saturation. For the single-particle first
excited state as the initial state, the growth of the spread complexity is
delayed when the time between two measurements is decreased. This indicates a
transition in the complexity behaviour based on the change in the measurement
frequency. This is reminiscent of a measurement-induced phase transition
measure1 ; measure2 . For a very high frequency of measurements, we find that
the initial state does not evolve in the Hilbert space. This is the
characteristic of the quantum Zeno effect Sudarshan_1977 ; FACCHI200012 ;
Facchi_2008 .

Our effective non-hermitian Hamiltonian is a complex symmetric matrix. In
addition to applying the bi-Lanczos algorithm of Gaaf ; 200SIAM ;
Gruning2011ImplementationAT , which builds upon the open system operator
complexity, we also implement our new complex symmetric Lanczos algorithm. We
find that for complex symmetric Hamiltonians, the complex symmetric Lanczos
works much more efficiently than the bi-Lanczos algorithm. This is due to the
fact that for the bi-Lanczos algorithm, the computational power needed is
larger due to the separate treatment of bra and ket states. Furthermore,
unlike bi-Lanczos, the complex symmetric algorithm requires to deal with a
single Krylov space only, and no conjugate Krylov space is needed. In all the
cases studied in this paper, we find exact numerical agreement between the
results generated by the bi-Lanczos and the complex symmetric Lanczos methods.

The paper is structured in the following way. We begin with a brief review of
the Krylov spread complexity in section 2. In Section 3, we discuss the
modified non-hermitian norm in Krylov basis that we use for the non-unitary
dynamics. We also review the bi-Lanczos algorithm and present the complex
symmetric Lanczos algorithm. In section 4, we begin by reviewing the first
passage problem of quantum mechanics in detail (section 4.1), followed by the
main results of this work. These are divided into three parts. Section 4.2.1
contains our results for the one-dimensional chain with open boundary
conditions for which we study the dependence of spread complexity and entropy
on the non-hermiticity parameter. We also obtain the dependence of complexity
and entropy on the spread of the initial state on the position eigenstate and
the distance of the initial spread from the position of the detector. In
section 4.2.2, we perform a similar study for the chain defined with periodic
boundary conditions. For the periodic boundary conditions, we find cases where
the evolution becomes similar to unitary evolution after some initial time
window. In section 4.2.3, we consider two different eigenstates of the
hermitian Hamiltonian before measurements as our initial states. We obtain the
effects of a non-hermitian quench on the spread complexity under the influence
of the measurement process. Finally, we provide a detailed summary of results
with future directions in section 5. In Appendix A, we find the time
dependence of the time-dependent total probability and an analytical form for
the return amplitude. In Appendix B.1, we summarize the behaviour of the
Lanczos coefficients, spread complexity and entropy for the unitary tight-
binding Hamiltonian. In Appendix B.2, we discuss the form of the Lanczos
coefficients for different non-hermitian cases.

##  2 A brief review of spread complexity in the Krylov basis

The spread complexity in the Krylov basis was proven to be the most optimized
measure of complexity that quantifies the spread of a quantum state as it
evolves Balasubramanian:2022tpr . The starting point is the Schrödinger
evolution of a quantum state,

$|\psi(t)\rangle=e^{-iHt}|\psi(0)\rangle.$ (1)

Below we briefly review the Schrödinger evolution in the Krylov basis
Balasubramanian:2022tpr ; Caputa:2022eye  and discuss the basic constituents
needed to define the spread complexity in this basis. This is particularly
useful for comparing with our new algorithms and normalisation for non-unitary
evolution.

We note that any eigenstate of the Hamiltonian is not a good starting point
for computing the spread complexity since the eigenstates of the Hamiltonian
do not change under the Hamiltonian evolution apart from a phase proportional
to the eigenvalue. According to Balasubramanian:2022tpr , an appropriate
starting point is to expand the exponential in Eq. (1),

$|\psi(t)\rangle=|\psi(0)\rangle-
itH|\psi(0)\rangle+\frac{(it)^{2}}{2}H^{2}|\psi(0)\rangle+\cdots+\frac{(it)^{n}}{n!}H^{n}|\psi(0)\rangle+\cdots.\,,$
(2)

where the initial state is identified as the first element of the Krylov
basis, $|K_{0}\rangle=|\psi(0)\rangle$. The further elements $|K_{n}\rangle$
are recursively constructed by implementing the generalized Lanczos algorithm
with the Hamiltonian acting on any general element $|K_{m}\rangle$ of the
Krylov space,

$H|K_{m}\rangle=a_{m}|K_{m}\rangle+b_{m}|K_{m-1}\rangle+b_{m+1}|K_{m+1}\rangle\,,$
(3)

where $a_{m}$ and $b_{m}$ are the Lanczos coefficients defined by

$\displaystyle a_{m}$ $\displaystyle=\langle
K_{m}|H|K_{m}\rangle,\,b_{m}=\langle A_{m}|A_{m}\rangle^{\frac{1}{2}},\,$
$\displaystyle\text{with}~{}~{}|A_{m}\rangle$
$\displaystyle=(H-a_{m-1})|K_{m-1}\rangle-b_{m-1}|K_{m-2}\rangle.$ (4)

For a hermitian Hamiltonian, the overall evolution is unitary, and the Lanczos
coefficients $a_{n}$ and $b_{n}$ are both real. In terms of the constructed
Krylov basis, the evolved state can be written in terms of the Lanczos basis
vectors as

$|\psi(t)\rangle=\sum_{n}\psi_{n}(t)|K_{n}\rangle.$ (5)

These $|\psi_{n}(t)|^{2}$’s correspond to the probabilities of the state being
in the $n$-th Krylov basis element $|K_{n}\rangle$ at time $t$ with the total
probability being $P=\sum_{n}|\psi_{n}(t)|^{2}=1$.333This notion of Krylov
probability shows decaying behaviour for non-hermitian cases, as shown in
Appendix A, unless renormalized to one at all times. The corresponding spread
complexity is defined as the average position of the state in the Krylov space
at time $t$. This, in the Schrödinger Krylov basis, is written as

$C_{S}(t)=\sum_{n}n|\psi_{n}(t)|^{2}.$ (6)

An alternative way to derive the Lanczos coefficients ($a_{n},\,b_{n}$) is to
start from the survival amplitude
$S(t)=\langle\psi(t)|\psi(0)\rangle=\psi_{0}(t)$ that measures the overlap
between the initial and final states. Then the moments of the survival
amplitude are calculated recursively method from the Hankel matrix formed out
of Parker:2018yvk ; Balasubramanian:2022tpr . The moments are defined by

$\mu_{n}=\frac{d^{n}}{dt^{n}}S(t)|_{t=0}.$ (7)

This moment recursion process can be in fact represented by an unnormalized
Markov chain where the transition weights in different levels of the chain are
simply the Lanczos coefficients ($a_{n}$ are the weights in the $n$th level
and $b_{n}$ are the weights connecting $n$th level to the $(n+1)$-st one)
Balasubramanian:2022tpr .

In analogy to the Krylov entropy for operator growth Barbon:2019wsy , the
spread entropy in the Schrödinger picture is given by

$S_{S}(t)=-\sum_{n}|\psi_{n}(t)|^{2}\ln|\psi_{n}(t)|^{2}.$ (8)

In fact, the spread complexity can be understood as an exponentiation of the
spread entropy Balasubramanian:2022tpr .

For chaotic evolution, the spread complexity shows linear growth up to
exponential times concerning the degrees of freedom of the system, followed by
saturation to a plateau after a brief decay phase Balasubramanian:2022tpr .
The corresponding Krylov spread entropy demonstrates logarithmic growth,
followed by a region of linear growth before reaching a plateau for chaotic
evolution. Furthermore, these Krylov spread measures, including probability,
complexity, and entropy, can be utilized to characterize different quantum
phases of matter Afrasiar_2023 , such as topological phases Caputa:2022eye .
Therefore, these measures in the Krylov spread basis are a good probe of
various physical phenomena in unitarily evolving quantum systems.

##  3 Non-unitary evolution and modified Lanczos algorithms

Here, we generalise concepts for the Krylov spread measures to the case of
non-unitary evolution. We begin by presenting a modified normalisation for the
density matrix in Krylov space. We continue by presenting two distinct
algorithms for evaluating the Lanczos coefficients in the non-unitary case.

The unitarity of quantum mechanical evolution arises from the hermiticity of
the Hamiltonian $H$. Consequently, the evolution operator $U=e^{-iHt}$ is
inherently unitary by definition. Nevertheless, instances exist where the
effective evolution of a system deviates from unitarity. Two notable scenarios
include i) open system dynamics: this pertains to the time dynamics of a
subsystem within the broader system, exhibiting non-unitarity due to
interactions with the larger universe and ii) measurements: non-unitarity
emerges when measuring the state at any given moment.

Examining the characteristics of non-unitary evolution, we first consider the
case of an effective non-hermitian Hamiltonian $H$ (with $H\neq H^{\dagger}$),
leading to a complex eigenspectrum expressed as $H=H_{1}+iH_{2}$, where both
$H_{1}$ and $H_{2}$ are hermitian Hamiltonians. With this effective
Hamiltonian, the evolution takes the form

$|\psi(t)\rangle_{nh}=e^{-iHt}|\psi(0)\rangle_{nh}=e^{H_{2}t-iH_{1}t}|\psi(0)\rangle_{nh}.$
(9)

The non-hermitian part of the Hamiltonian, $H_{2}$, introduces a change in the
normalisation of the state.

###  3.1 Revised normalisation and definitions of probability, complexity and
entropy

According to (9), it is critical to adjust the normalisation continuously. To
do so, we introduce a revised normalisation in the Krylov basis that
renormalizes the state to unity at all times. The dynamical evolution of a
non-hermitian system initialized in a density matrix
$\rho_{nh}(0)=|\psi(0)\rangle_{nh}\langle\psi(0)|_{nh}$ is governed by newnorm

$\rho_{nh}(t)=\frac{e^{-iHt}\rho_{nh}(0)e^{iH^{\dagger}t}}{\text{Tr}[e^{-iHt}\rho_{nh}(0)e^{iH^{\dagger}t}]}.$
(10)

This ensures that the normalisation of the states under non-unitary evolution
is one at any given moment in time. We implement this normalisation (13) in
the Krylov basis in the following way. We have the initial matrix wave
function in the Krylov basis as

$|\Phi_{nh}(0)\rangle_{K}=\begin{pmatrix}1\\\ 0\\\ 0\\\ 0\\\ \vdots\\\ 0\\\
\end{pmatrix}_{K},$ (11)

which is equivalent to $\psi_{n}(t=0)=\delta_{n,0}$ for $\psi_{n}$ as given in
(5), by identifying the first element of the column matrix as $\psi_{0}(0)$.
Now, accordingly, we can define a density matrix in Krylov space,

$(\rho_{nh})_{K}(0)=|\Phi_{nh}(0)\rangle_{K}\langle\Phi_{nh}(0)|_{K}=\begin{pmatrix}1&0&&&&0\\\
0&0&0&&&\\\ &0&0&\ddots&&\\\ &&\ddots&\ddots&0&\\\ &&&0&0&\ddots\\\
0&&&&\ddots&0\\\ \end{pmatrix}_{K\times K}.$ (12)

This Krylov space density matrix undergoes evolution with respect to the non-
hermitian tridiagonalized form ($L$) of the non-hermitian Hamiltonian ($H$)
that is derived using the appropriate Lanczos algorithm. This evolution is
exactly similar to Eq. (10), with $L$ replacing $H$. Keeping the probability
conserved at all times, we work with a revised time-dependent normalisation of
a state vector in the Krylov basis and define the new state vector
$|\widetilde{\Phi}(t)\rangle_{K}$ with probability $P(t)$ as follows,

$|\tilde{\Phi}(t)\rangle_{K}=\frac{e^{-iLt}|\Phi_{nh}(0)\rangle_{K}}{\sqrt{\text{Tr}[e^{-iLt}(\rho_{nh})_{K}(0)e^{iL^{\dagger}t}]}}=\begin{pmatrix}\tilde{\phi}_{0}(t)\\\
\tilde{\phi}_{1}(t)\\\ \tilde{\phi}_{2}(t)\\\ \tilde{\phi}_{3}(t)\\\ \vdots\\\
\tilde{\phi}_{K-1}(t)\\\
\end{pmatrix}_{K},\,\,P(t)=|_{K}\langle\tilde{\Phi}(t)|\tilde{\Phi}(t)\rangle_{K}|=1.$
(13)

We use this Krylov probability in our further definitions of spread entropy
and complexity444Note that the square root over the trace in the denominator
of Eq. (13) is unity when the revised normalisation is considered. Therefore
this normalisation becomes exactly similar to the denominator of Eq. (10).,

$C_{S}(t)=\frac{\sum_{n}n|\tilde{\phi}_{n}(t)|^{2}}{P(t)},\,\,S_{S}(t)=-\frac{\sum_{n}|\tilde{\phi}_{n}(t)|^{2}\ln|\tilde{\phi}_{n}(t)|^{2}}{P(t)},$
(14)

where $\tilde{\phi}_{n}(t)$ is the $(n+1)$-th element of the matrix
$|\tilde{\Phi}(t)\rangle_{K}$. In terms of the normalisation given in Eq.
(13), the dynamically normalised probability $P(t)$ remains constant in time.
Hence, the definitions in Eq. (14) become formally similar to the definitions
for hermitian quantum systems as the denominator becomes $1$.

###  3.2 Construction algorithms for the Krylov basis of non-hermitian systems

As mentioned in section 2, one of the main features of the Krylov basis for
hermitian Hamiltonians is that the Hamiltonian becomes tri-diagonal. We now
summarize the details of the modifications on the Lanczos algorithm needed for
the non-hermitian Hamiltonian. For non-hermitian matrices, the Arnoldi
recursive technique Bhattacharya:2022gbz  is the simplest extension of the
Lanczos algorithm. In this approach, the non-hermitian Hamiltonian becomes an
upper Hessenberg matrix and thus not of tri-diagonal form. These extra terms
impede the standard computation of Krylov complexity Bhattacharya:2023zqt .

####  3.2.1 Bi-Lanczos algorithm

This issue is resolved by the bi-Lanczos algorithm that we now implement for
state complexity. The standard bi-Lanczos algorithm allows to tri-diagonalize
a non-hermitian matrix $A$ Gaaf ; 200SIAM ; Gruning2011ImplementationAT .
Since a non-hermitian matrix $A$ acts differently on a ket vector $|v\rangle$
than on a bra vector $\langle w|$, finding an orthogonal basis set that could
transform $A$ into a tridiagonal form is impossible. This problem is
circumvented by the bi-Lanczos algorithm by evolving two initial vectors,
$|q_{1}\rangle$ and $\langle p_{1}|=|q_{1}\rangle^{\dagger}$ by $A$ and
$A^{\dagger}$, instead of only $|q_{1}\rangle$ as in the usual Lanczos
algorithm. The initial state $|\psi(t=0)\rangle$ is chosen as $|q_{1}\rangle$.
From these two initial vectors, by using the two-sided Gram-Schmidt procedure,
a pair of bi-orthogonal bases are built, $\\{\langle p_{j}|\\}$ and
$\\{|q_{j}\rangle\\}$, which span the Krylov subspaces
$\mathcal{K}^{j}(A,|q_{1}\rangle)\equiv\\{A|q_{1}\rangle,A^{2}|q_{1}\rangle,\dots\\}$
and $\mathcal{K}^{j}(\langle p_{1}|,A)\equiv\\{\langle p_{1}|A,\langle
p_{1}|A^{2},\dots\\}$ respectively.

The usual Lanczos algorithm is realized by the action of a non-hermitian
matrix $A$ on a ket vector $|q_{j}\rangle$ from the left. The action of the
non-hermitian matrix $A$ on a bra vector $\langle p_{j}|$ can be realized as
$A^{\dagger}$ acting on a ket vector $|p_{j}\rangle$. These bases, which span
the two Krylov subspaces are bi-orthogonal to each other, i.e.

$\langle p_{i}|q_{j}\rangle\ =\delta_{ij}.$ (15)

Since $|\psi(t=0)\rangle=|p_{1}\rangle=|q_{1}\rangle$ is a normalised vector,
$\langle p_{1}|q_{1}\rangle=\langle q_{1}|q_{1}\rangle=1$ by definition. The
action of the non-hermitian matrix $A$ or $A^{\dagger}$ on general basis
vectors, $|q_{j}\rangle$ or $|p_{j}\rangle$, makes them imaginary for $j>1$.
This makes $\\{\langle p_{j}|\\}$ and $\\{|q_{j}\rangle\\}$ in general not
orthonormal, i.e.

$\langle p_{j}|p_{j}\rangle\neq 1\neq\langle
q_{j}|q_{j}\rangle\,\text{for}\,j>1.$ (16)

Starting from the first two initial vectors $|p_{1}\rangle$ and
$|q_{1}\rangle$, the two sets of vectors are generated by implementing three
terms of recursive relations given by

$\displaystyle c_{j+1}\,|q_{j+1}\rangle$ $\displaystyle=A\,|q_{j}\rangle-
a_{j}\,|q_{j}\rangle-b_{j}\,|q_{j-1}\rangle$ (17) $\displaystyle
b^{*}_{j+1}\,|p_{j+1}\rangle$ $\displaystyle=A^{\dagger}\,|p_{j}\rangle-
a_{j}^{*}\,|p_{j}\rangle-c^{*}_{j}\,|p_{j-1}\rangle.$ (18)

It is necessary to verify that each vector of Ket basis is bi-orthogonal with
respect to the previously constructed vectors of the Bra basis after every
iteration of the process. In this bi-orthogonal basis the matrix $A$ takes the
tri-diagonal form $T_{j}$,

$T_{j}=\begin{pmatrix}a_{1}&b_{2}&0&\dots&0\\\ c_{2}&a_{2}&b_{3}&&\vdots\\\
0&\ddots&\ddots&\ddots&\\\ \vdots&&c_{j-1}&a_{j-1}&b_{j}\\\
0&\dots&0&c_{j}&a_{j}\end{pmatrix}.$ (19)

The recursion relations given in Eq. (17) and Eq. (18) make it obvious that
this approach requires both $A$ and $A^{\dagger}$ to act on the vectors. This
method is simple because it only uses two three-term recursive relations.

Adapting the procedure given in equations (15)-(19) to non-hermitian systems
gives rise to an algorithm for the spread complexity as follows: starting from
two initial vectors chosen as the initial state
$|\psi(t=0)\rangle=|p_{1}\rangle=|q_{1}\rangle$, through iterative application
of the Hamiltonian and a particular orthogonalization process, we
systematically construct a bi-orthogonal set of basis vectors spanning the bi-
orthogonal Krylov spaces by the following way:

  1. 1.

Choose two initial normalised vectors $|p_{1}\rangle$ =
$|q_{1}\rangle=|\psi(t=0)\rangle$ such that $\langle p_{1}|q_{1}\rangle=1$.
Here to initiate the algorithm, we also need an initial set of values for the
diagonal and off-diagonal bi-Lanczos coefficients.

$b_{1}=c_{1}=0,~{}~{}a_{1}=\langle p_{1}|A|q_{1}\rangle=\langle
q_{1}|A|q_{1}\rangle.$ (20)

  2. 2.

Starting from the initial vectors $|p_{1}\rangle$ and $|q_{1}\rangle$ which
correspond to $j=1$, we implement the steps (a) to (f) given below to generate
further basis vectors of the two Krylov spaces, $|p_{j}\rangle$ and
$|q_{j}\rangle$, for $j>1$.

    1. (a)

The action of $A$ on $|q_{j}\rangle$ and, similarly $A^{\dagger}$ on
$|p_{j}\rangle$ gives rise to a set of vectors denoted by
$|r^{\prime}_{j}\rangle$ and $|s^{\prime}_{j}\rangle$ respectively.

$|r^{\prime}_{j}\rangle=A|q_{j}\rangle,\,\,|s^{\prime}_{j}\rangle=A^{\dagger}|p_{j}\rangle.$
(21)

    2. (b)

The vectors $|r_{j}\rangle$ and $|s_{j}\rangle$ correspond to the right hand
side of equations (17) and (18). Now in order to construct orthogonal basis
vectors, we subtract contributions from two vectors of previous basis
$|q_{j}\rangle,\,|q_{j-1}\rangle$ and $|p_{j}\rangle,\,|p_{j-1}\rangle$. We
obtain

$|r_{j}\rangle=|r^{\prime}_{j}\rangle-a_{j}|q_{j}\rangle-
b_{j}|q_{j-1}\rangle,\,|s_{j}\rangle=|s^{\prime}_{j}\rangle-
a_{j}^{*}|p_{j}\rangle-c^{*}_{j}|p_{j-1}\rangle,$ (22)

with complex Lanczos coefficients $a_{j},\,b_{j},$ and $c_{j}$. Their values
are known up to the level $j$.

    3. (c)

In order to obtain the Lanczos coefficients of the next step, we evaluate the
inner product of vectors constructed in the previous one by
$\omega_{j}=\langle r_{j}|s_{j}\rangle$. We define the upper and lower
diagonal coefficients $b_{j+1}$ and $c_{j+1}$ for the $(j+1)$st basis vector
of the $T_{j}$ matrix given in (19),

$c_{j+1}=\sqrt{|\omega_{j}|},\,\,b_{j+1}=\frac{\omega^{*}_{j}}{c_{j+1}}.$ (23)

    4. (d)

We proceed to construct the unit basis vectors $|q_{j+1}\rangle$ and
$|p_{j+1}\rangle$ of the two Krylov bases,

$|q_{j+1}\rangle=\frac{|r_{j}\rangle}{c_{j+1}},\,\,|p_{j+1}\rangle=\frac{|s_{j}\rangle}{b^{*}_{j+1}}.$
(24)

    5. (e)

In any Lanczos basis construction, full orthonormality can be lost due to
numerical instability caused by the finite-precision arithmetic Parlett1979 ;
Simon1984 . This happens because the contributions of only the previous two
basis vectors, instead of all previous basis vectors, are subtracted while
constructing a new basis vector (see Eq. (22)). The resulting errors
accumulate with increasing steps causing a loss of orthonormality. This
problem can be avoided by the full Gram-Schmidt orthogonalization procedure
that subtracts the contributions of all previous basis vectors while
constructing the new basis vector. To avoid this problem for the bi-Lanczos
algorithm, we implement the full bi-orthogonalization to ensure that Eq. (15)
is valid for all $i,j$,

$|q_{j}\rangle=|q_{j}\rangle-\sum_{l=1}^{j-1}\langle
p_{l}|q_{l}\rangle|q_{l}\rangle,\,\,|p_{j}\rangle=|p_{j}\rangle-\sum_{l=1}^{j-1}\langle
q_{l}|p_{l}\rangle|p_{l}\rangle.$ (25)

    6. (f)

Finally, after full bi-orthogonalization, we compute the diagonal coefficients
$a_{j+1}$ of the tri-diagonal matrix for $(j+1)$st step. In usual Lanczos,
this is just the expectation value of the operator $A$ with respect to the
newly constructed basis vector $|q_{j+1}\rangle$. In bi-Lanczos algorithm, the
modified definition of $a_{j+1}$ is given by sandwiching $A$ between the
$\langle p_{j+1}|$ ‘bra’ vector and the $|q_{j+1}\rangle$ ‘ket’ vector,

$a_{j+1}=\langle p_{j+1}|A|q_{j+1}\rangle$ (26)

and by returning to step $2$ for level $j+2$.

  3. 3.

If $\omega_{j}=0$ at $j=\mathcal{K}$, we end the recursion and obtain the
$\mathcal{K}$-dimensional Krylov space $\mathcal{K}$ Krylov basis vectors.
Here, $\omega_{\mathcal{K}}=0$ implies that no further linearly independent
basis vectors are left to be formed. This happens when the Krylov basis
vectors completely explore the full Hilbert space. Then, for a $N\times N$
non-hermitian matrix A, at the $\mathcal{K}^{th}$ step, two
$N\times\mathcal{K}$ matrices $P_{\mathcal{K}}\equiv[p_{1},p_{2},\dots
p_{\mathcal{K}}]$ and $Q_{\mathcal{K}}\equiv[q_{1},q_{2},\dots
q_{\mathcal{K}}]$ are formed by bi-Lanczos method such a way that,
$P^{\dagger}_{\mathcal{K}}AQ_{\mathcal{K}}=T_{\mathcal{K}}$.

This algorithm provides all the basis vectors $|p_{j}\rangle$, $|q_{j}\rangle$
of the full Krylov bases. In addition, we obtain the list of Lanczos
coefficients $a_{j}$ Eq. (26), $b_{j}$, and $c_{j}$ Eq. (23), which are the
diagonal, upper diagonal, and lower diagonal elements of the tri-diagonal form
of $A$ mentioned in Eq. (19) respectively.

From this algorithm it is clear that all $c_{j}$’s are real since
$c_{j}=\sqrt{|\omega_{j-1}|}$, but $b_{j}$ can be complex Eq. (23). In general
$c_{n}\neq b_{j}$ while $|c_{j}|=|b_{j}|$ Bhattacharya:2023zqt . We may expand
the wave function $|\psi(t)\rangle$ in both of the bases,

$\sum_{j=1}^{\mathcal{K}}\Phi^{q}_{j}(t)|q_{j}\rangle=|\psi(t)\rangle=\sum_{j=1}^{\mathcal{K}}\Phi^{p}_{j}(t)|p_{j}\rangle,$
(27)

where $\mathcal{K}$ is the dimension of the bi-orthogonal Krylov basis. In the
bi-orthogonal basis, the corresponding amplitudes of the wave function
$\ket{\psi(t)}$ have two components, $\Phi^{p}_{j}(t)$ and $\Phi^{q}_{j}(t)$
for the $|p_{j}\rangle$ and $|q_{j}\rangle$ bases respectively Eq. (27). Using
the normalisation (12) for each of these components, the total probability in
the bi-orthogonal Krylov basis is defined by555Here we introduce the
superscript (bl) to specify that these definitions are specific to the bi-
Lanczos algorithm while the tildes mean that we have performed the
normalisation of the coefficients to make the total probability $1$ at all
times.

$P^{(bl)}(t)=\sum_{n}|\tilde{\Phi}^{p*}_{n}(t)\tilde{\Phi}^{q}_{n}(t)|.$ (28)

The corresponding definitions of spread complexity and spread entropy are

$\displaystyle C^{(bl)}_{S}(t)$
$\displaystyle=\sum_{n}n|\tilde{\Phi}^{p*}_{n}(t)\tilde{\Phi}^{q}_{n}(t)|,$
(29) $\displaystyle S^{(bl)}_{S}$
$\displaystyle=-\sum_{n}\left(|\tilde{\Phi}^{p*}_{n}\left(t\right)\tilde{\Phi}^{q}_{n}\left(t\right)|\right)\ln[|\tilde{\Phi}^{p*}_{n}\left(t\right)\tilde{\Phi}^{q}_{n}\left(t\right)|].$
(30)

The bi-Lanczos algorithm and the above definitions of probability, complexity
and entropy apply to any general non-hermitian operator acting on the state
vectors. However, for large Hilbert space dimensions, this method has high
computational memory requirements. For this reason, we propose an alternative
approach for the complex symmetric Hamiltonians below.

####  3.2.2 Complex symmetric Lanczos algorithm

Here implement an alternative cost-efficient algorithm for complex symmetric
matrices that we will study in section 4.1 when considering a complex
symmetric Hamiltonian $H_{\text{eff}}^{T}=H_{\text{eff}}$ but
$H_{\text{eff}}^{{\dagger}}\neq H_{\text{eff}}$.

The property of complex symmetry is purely algebraic, exerting no impact on
the matrix spectrum. In general, for any given set of $n$ numbers, there
exists a complex symmetric $n\times n$ matrix $A$, whose eigenvalues precisely
match the prescribed numbers. A complex symmetric matrix may not always be
diagonalizable. While the complex symmetry of matrix $A$ does not put any
constraints on its eigenvalues, this specific algebraic property can be used
to significantly reduce the computational workload and storage demands
associated with the general non-Hermitian bi-Lanczos method.

The diagonalizability of a complex symmetric matrix, $A$ depends upon the
possibility of choosing its eigenvector matrix, $Z$, in such a manner that,
$Z^{T}AZ=\text{diag}(\lambda_{1},\lambda_{2},...,\lambda_{n})$, where
$\\{\lambda_{1},\lambda_{2},...,\lambda_{n}\\}$ is the set of its eigenvalues.
The eigenvector matrix $Z$ additionally satisfies, $Z^{T}Z=I_{n}$ which means
the $Z$ matrix is complex orthogonal. The complex orthogonality arising from
the inherent complex symmetric nature of the matrix is pivotal. It allows the
construction of a Krylov basis for such matrices in an efficient manner. To
achieve this, we employ a modified algorithm based on the conventional Lanczos
method. Even when dealing with complex vectors, we use the fact that the new
Krylov basis should maintain complex orthogonality. Consequently, the Lanczos
vectors constructed exhibit complex orthogonality.

Utilising these properties of complex symmetry, the tri-diagonal form of a
complex symmetric matrix $A$ can be obtained using the complex symmetric
Lanczos algorithm 200SIAM . The orthogonal basis $\\{|q_{j}\rangle\\}$ that
spans the Krylov space
$\mathcal{K}^{j}(A,|q_{1}\rangle)\equiv\\{A|q_{1}\rangle,A^{2}|q_{1}\rangle,\dots\\}$
is constructed starting with a normalized vector
$|q_{1}\rangle=|\psi(0)\rangle$. The complex orthogonality of the vectors in
this situation means that the product of a vector with its transpose is $1$,
i.e., $\langle q_{j}|q_{j}\rangle=\delta_{i,j}$ where in our notation $\langle
q_{j}|=(|q_{j}\rangle)^{T}$. The construction of the required basis
$\\{|q_{j}\rangle\\}$ involves a three-term recursion relation,

$\displaystyle\beta_{j+1}|q_{j+1}\rangle=A|q_{j}\rangle-\alpha_{j}|q_{j}\rangle-\beta_{j}|q_{j-1}\rangle.$
(31)

In this basis $\\{|q_{j}\rangle\\}$, the complex symmetric matrix $A$ takes
the tri-diagonal form $\tilde{T}_{j}$,

$\tilde{T}_{j}=\begin{pmatrix}\alpha_{1}&\beta_{2}&0&\dots&0\\\
\beta_{2}&\alpha_{2}&\beta_{3}&&\vdots\\\ 0&\ddots&\ddots&\ddots&\\\
\vdots&&\beta_{j-1}&\alpha_{j-1}&\beta_{j}\\\
0&\dots&0&\beta_{j}&\alpha_{j}\end{pmatrix}.$ (32)

Using the usual Lanczos algorithm with complex orthogonality, we obtain the
set of Krylov basis matrix,
$Q=[\hat{q}_{1}~{}\hat{q}_{2}~{}...~{}\hat{q}_{n}]$ that satisfies
$Q^{T}Q=I_{n}$. This yields the tri-diagonal form of the matrix, $A$ in this
basis, $\tilde{T}_{j}=Q_{j}^{T}AQ_{j}$.

Now we compare between the tri-diagonal matrices $T_{j}$ and $\tilde{T}_{j}$
constructed using the bi-Lanczos and the complex symmetric Lanczos methods
given in equations (19) and (32) respectively. An obvious advantage of the
complex symmetric Lanczos algorithm is that we just deal with a single set of
Krylov basis vectors, $|q_{j}\rangle$ (Eq. (31)). In the complex symmetric
Lanczos method, upper and lower diagonal elements of the tri-diagonal matrix
$\tilde{T}_{j}$ are the same. This makes the $\tilde{T}_{j}$ symmetric. On the
other hand, $T_{j}$ obtained by the bi-Lanczos algorithm is not symmetric (Eq.
(19)). The diagonal elements of $T_{j}$ and $\tilde{T}_{j}$ are the same i.e.,
$a_{j}=\alpha_{j}$. The $j^{th}$ upper as well as the lower diagonal elements
of $T_{j}$ and $\tilde{T}_{j}$ are not the same i.e., $b_{j}\neq\beta_{j}$
also $c_{j}\neq\beta_{j}$. However, absolute values are the same i.e.,
$|\beta_{j}|=|b_{j}|=|c_{j}|$. Hence $|T_{j}|=|\tilde{T}_{j}|$.

This algorithm for complex symmetric matrices will be of use below in section
4 where we study Hamiltonians of a complex symmetric form, in particular for
the QFPP.

##  4 Complexity and entropy under projective measurements

To study the different features of non-unitary evolution, we focus on a
particular example known as the “quantum first passage problem”. The methods
developed in the previous section are generic and applicable to any non-
unitary evolution, whereas the results of this section are problem-specific.
The problem-specific results indicate how sensitive the Krylov basis measures
are to specific properties of the QFPP. We first review the first passage
problem in section 4.1. Then, we discuss the numerical findings of this non-
unitary evolution in section 4.2.

###  4.1 First passage problem

The quantum first passage problem (QFPP) in quantum mechanics consists of
determining the probability that a particle starting from a specific initial
position reaches a particular final position for the first time during
evolution within a given time muga2000 . The QFPP is relevant in different
contexts, even beyond physics, including the study of chemical reactions,
electron transport in materials, and the behaviour of particles in biological
systems.

In Dhar_2015 ; Dhar2013QuantumTO , it was shown that the time evolution of a
quantum mechanical system under periodic projective measurements may be viewed
as a non-hermitian system with a non-unitary time evolution. We use the setup
presented in Dhar_2015 ; Dhar2013QuantumTO  as our prime example here. This
setup involves a quantum particle moving on a lattice governed by a tight-
binding Hamiltonian. The position of the particle is periodically measured to
determine whether the particle is present at specific predefined sites. We
begin by considering the unitary time evolution of the wave function between
the points at which detection occurs. Projective operations represent the
detection procedures, giving rise to non-unitarity.

We consider a system undergoing unitary time evolution subjected to repeated
projective measurements, effectively giving rise to the non-unitary evolution
of the system. In the most general case, it is important to note that this
system has a complex eigenspectrum without a complex conjugate pair of
eigenvalues. In general, we start with a Hamiltonian of the form,

$H=\underbrace{\sum_{l,m}H_{l,m}|l\rangle\langle
m|}_{H_{S}}+\underbrace{\sum_{\alpha,\beta}H_{\alpha,\beta}|\alpha\rangle\langle\beta|}_{H_{M}}+\underbrace{\sum_{\alpha,l}\left(V_{\alpha,l}|\alpha\rangle\langle
l|+V_{l,\alpha}|l\rangle\langle\alpha|\right)}_{V},$ (33)

where each state $|l\rangle$ represents a specific location on the lattice.
The complete set of sites is divided into two categories: those associated
with the system, designated by Roman indices $l$ and $m$, and those comprising
the domain of sites where measurements are conducted, denoted by Greek indices
$\alpha$ and $\beta$. In accordance with the notation put forth in previous
works Dhar_2015 , the terms in Eq. (33) are consecutively referred to as
$H_{S}$, $H_{M}$, and $V$. The unitary time evolution is

$|\psi(t)\rangle=U_{t}|\psi(0)\rangle,~{}~{}\text{with}~{}~{}U_{t}=e^{\textendash
iHt}.$ (34)

In order to take into account the projective measurements, we define the
projection operator defined at detection sites as $A=\sum_{\alpha\in
D}|\alpha\rangle\langle\alpha|$. This operator corresponds to a measurement
for detecting the particle within domain $D$, which contains a specific number
of sites. The complementary operator $B=1-A$ corresponds to the projection
onto the sites of the system subspace. The probability $p$ of detecting the
particle on measuring the state $|\psi\rangle$ can be found by calculating the
expectation value of $A$ concerning that state. Subsequently, the probability
of that state not being detected or the so-called survival probability defined
in Dhar_2015  is given by $P=\langle\psi|B|\psi\rangle=\left(1-p\right)$. In
the case of a positive detection outcome, the state right after the
measurement is $A|\psi\rangle$. On the other hand, if the state is not
detected right after the measurement, the state must have collapsed to
$B|\psi\rangle$. We repeat this procedure indefinitely until the state is
finally detected.

In this setup, the effective time evolution between two consecutive
measurements is of the form

$\tilde{U}_{\tau}=B\cdot e^{-iH\tau}\cdot B=B\cdot U_{\tau}\cdot B.$ (35)

Here, $\tau$ is the time interval between two consecutive measurements. The
total time evolution of the system is given by

$\displaystyle\mathcal{U}_{M+1}=\left(\tilde{U}_{\tau}\right)^{M}=B\cdot
U_{\tau}\cdot B\cdot U_{\tau}\cdot\cdot\cdot\underbrace{B\cdot
U_{\tau}}_{M~{}\text{times}}\cdot B,$ (36)

where in the last step, we have used $B^{2}=B$. The operator
$\mathcal{U}_{M+1}$ acts as the time evolution operator for $M+1$ number of
projective measurements. $B$ being not unitary leads to an effective non-
hermitian Hamiltonian through second-order perturbative calculations for small
$\tau$. This effective Hamiltonian is given by,

$\displaystyle\tilde{U}_{\tau}$
$\displaystyle=e^{-iH_{\text{eff}}\tau}+\mathcal{O}(\tau^{3})\rightarrow\mathcal{U}_{M+1}\approx
e^{-iH_{\text{eff}}t}~{}~{}\text{with}~{}~{}t=M\tau,~{}~{}$ (37)
$\displaystyle\text{where}~{}~{}H_{\text{eff}}$
$\displaystyle=H_{S}-\frac{i\tau}{2}\sum_{l,m}\sum_{\alpha}V_{l,\alpha}V_{\alpha,m}|l\rangle\langle
m|.$ (38)

We notice that $H_{\text{eff}}$ is a linear combination of a hermitian part
$H_{S}$ and a non-hermitian part. This effective Hamiltonian acts on all the
sites except for the sites where the detectors are present.666From now
onwards, we refer to the system as the chain without the detector sites. The
second term in 38 determines the complex part of the eigenspectrum. The first
hermitian part always contributes to the real part of the eigenvalues. It is
important to note that the limit $\tau\rightarrow 0$ does not coincide with
setting $\tau=0$ in the effective non-hermitian Hamiltonian from the
beginning. This is because the effective Hamiltonian is derived from a
perturbative expansion. In the mathematical expression, $\tau=0$ gives back
the hermitian Hamiltonian, whereas the physical limit $\tau\rightarrow 0$
represents almost continuous measurements. Additionally, interpreting
$\tau\rightarrow\infty$ as the unitary limit, where the time gap between
measurements becomes very large, is not accurate. In this scenario, the
perturbative treatment breaks down.

We consider two one-dimensional models, i) one with open boundary conditions
and ii) one with periodic boundary conditions. We start with the initial
tight-binding Hamiltonian of $N$ sites,

$H_{\text{TB}}=-\sum_{l=1}^{N-1}\left(|l+1\rangle\langle l|+|l\rangle\langle
l+1|\right)$ (39)

and place the detector at the end, on the $N$th point. So, for this particular
model, the operator $A$ takes the form, $A=|N\rangle\langle N|$ and $B$
operator is given as $B=\sum_{l=1}^{N-1}|l\rangle\langle l|$. Using these, we
can get the $H_{\text{eff}}$ for this model up to first order in $\tau$ as,

$H_{\text{eff}}=-\sum_{l=1}^{N-2}\left(|l+1\rangle\langle l|+|l\rangle\langle
l+1|\right)-\frac{i\tau}{2}|N-1\rangle\langle N-1|.$ (40)

The second term on the right-hand side represents the $V_{\text{eff}}$ that is
non-hermitian. In this case, the endpoints of the chain are open. But we can
also use periodic boundary conditions as $|N+1\rangle=|1\rangle$. In this
case, the operator $A$ and $B$ are still the same, but the effective
Hamiltonian changes as the $V_{\text{eff}}$ is different. In this case, we
find

$H_{\text{eff}}=-\sum_{l=1}^{N-2}\left(|l+1\rangle\langle l|+|l\rangle\langle
l+1|\right)-\frac{i\tau}{2}\left(|N-1\rangle\langle N-1|+|1\rangle\langle
1|+|1\rangle\langle N-1|+|N-1\rangle\langle 1|\right).$ (41)

The periodic boundary conditions require some attention. This Hamiltonian has
$(N-2)/2$ eigenvalues with two-fold degeneracy. In this case, following the
form of the energy eigenstates discussed in Dhar_2015 , half of them remain
unaffected by these measurement processes. In contrast to open boundary
conditions, the survival probability does not decay to zero at late times
unless the initial state is localised at the $(N/2)^{th}$ site for an even
number of $N$ Dhar_2015 . However, the survival probability decays to zero at
late times only if the initial state is localised at the $(N/2)^{th}$ site.

###  4.2 Results for Krylov state complexity and entropy

The total Krylov probability, as defined in section 3.1, remains constant even
for non-unitary evolution. However, in Appendix A, we present an additional
definition of total probability more commonly used in the literature, which is
time-dependent for non-unitary evolution. Here, we follow the normalized
spread complexity and spread entropy as given in Eq. (14).

We now proceed to discuss the numerical results for the spread complexity and
entropy exhibited by the first passage problem. The presentation of our
findings is divided into three distinct parts. Initially, we examine the
spread complexity of the non-hermitian tight-binding chain, considering both
i) open (section 4.2.1) and ii) periodic boundary conditions (section 4.2.2).
This analysis encompasses different values for the non-Hermiticity
perturbation parameter $\tau$, as well as different positions and spreads of
the initial state. Subsequently, in iii) we shift our focus to the
investigation of Krylov spread complexity within a specific quench scenario
(section 4.2.3), where we initialise the system with the eigenstates of the
hermitian tight-binding Hamiltonian and evolve it using the effective non-
hermitian Hamiltonian. In addition to the Krylov complexity, we also explore
the behaviour of the Krylov entropy and total probability across all
aforementioned systems, leading us to identify the characteristics attributed
to the non-hermitian nature of the system.

####  4.2.1 Open boundary conditions

We start with a tight-binding chain with open boundary conditions, under the
repeated projective measurements whose effective time evolution is governed by
the effective non-hermitian Hamiltonian given in Eq. (40). In the following
subsections, we first discuss the time evolution of the actual wave function
and how it reaches a steady state. Then we discuss the time evolution of the
spread complexity and spread entropy.

#### Steady state

We consider the QFPP with open boundary conditions. Steady states are those
states that remain invariant up to small fluctuations under the action of the
Hamiltonian. These states for non-equilibrium systems are similar in nature to
the thermal states of unitary evolution. For the QFPP, as an initial state
evolves following Eq. (10), it explores the full Krylov basis and eventually
reaches a steady state. This results in the saturation of the spread
complexity and entropy. For open boundary condition in the QFPP, we find that
the average position of the steady state is located at the $l$-th site of the
tight-binding chain of length $N=2l$, as we see in Figure 1. The steady state
profile does not depend on the position of the initial state or the time
$\tau$ between two measurements. For the dependence of the spread complexity
on initial state chosen for constructing the Krylov basis, given in Eq. (11),
we find that for initial states which already have a similar spread as the
steady state reached later, the spread complexity has a very low saturation
value. Under repeated action of the Hamiltonian, the Krylov space wave
function explores the neighbouring sites around the initial state. This
results in a strong dependence of the complexity dynamics on the choice of the
initial state.

Figure 1: Spread of the steady state at late times $t=30000$ for different
spread of initial state, for open boundary conditions with total number of
sites $N=22$ and non-hermiticity parameter $\tau=0.1$. The spread of the
steady state is the same for different initial states.

#### Spread complexity

We plot the temporal behaviour of the Krylov spread complexity given by Eq.
(14) for the tight-binding chain in Figure 2, for different values of the non-
hermiticity parameter $\tau$ in Figure 2(a) and for different spreads of the
initial state in Figure 2(b).

(a) Time variation of spread complexity for different values of $\tau$.

(b) Time variation of spread complexity for different spreads of initial
state.

Figure 2:  (a) Time dependence of spread complexity for different non-
hermiticity parameters $\tau=0.1$ (green), $0.2$ (red), and $0.5$ (blue) with
fixed total sites $N=42$ and initial spread of $18\leq l\leq 22$. (b) Time
dependence of spread complexity for different spreads of the initial state
$9\leq l\leq 13$ (red), $10\leq l\leq 12$ (green), $2\leq l\leq 6$ (pink),
$2\leq l\leq 4$ (orange), $16\leq l\leq 20$ (dotted blue), and $18\leq l\leq
20$ (dotted cyan) with fixed values of $N=22$ and $\tau=0.1$. The insets
display the very early-time behaviour of the spread complexity. The decay
region is prolonged for smaller $\tau$ and larger initial spread.

The Krylov spread complexity initially experiences growth followed by
prolonged oscillations and eventually saturates to a constant value. The
amplitude of these oscillations diminishes over time, and the complexity
stabilizes at a saturation point much lower than the initial peak value. The
oscillatory behaviour has different origins. Firstly, it reflects the
integrability of the underlying system. Moreover, it is caused by the repeated
projective measurements as well as the finite length of the chain. As the
inset of Figure 2(a) shows, changes in the measurement frequency result in a
change in the oscillation period. The saturation of complexity at late times
indicates a steady state as explained the paragraph above the steady states.

For determining the dependence of spread complexity on the non-hermiticity
parameter, we plot the spread complexity for the initial state spreading over
sites $18\leq l\leq 22$ for different non-hermiticity parameters $\tau=0.1$
(green), $0.2$ (red), and $0.5$ (blue) in Figure 2(a). Note that $\tau$
corresponds to the time elapsing between two measurements. We notice that a
more rapid decay of the oscillation amplitude occurs for higher values of
$\tau$, resulting in saturation on a shorter time scale. This reveals that the
longer the system evolves between two consecutive measurements, the quicker it
reaches the steady state corresponding to the saturation. Larger time
intervals between two consecutive measurements allow for the effect of the
interaction to spread further into the entire system.

Moreover, it is essential to note that the saturation value of spread
complexity in Figure 2(a) remains unaffected by changes in $\tau$. It is in
agreement with previous results for the open system non-unitary operator
complexity reported in Bhattacharya:2023zqt , where changing the non-hermitian
couplings in the Lindbladian does not change the saturation value of the
complexity. This suggests that while a larger value of $\tau$ makes the system
reach a steady state sooner, the steady state complexity for a given initial
state is universal and independent of $\tau$.

We find that the value and the rate of the saturation rely on the initial
state’s position and spread in the chain of the tight-binding model (see
Figure 2(b)). In particular, a narrower spread of the initial state leads to a
higher saturation value for the spread complexity. It also reaches saturation
sooner as compared to states with broader initial spreads. As is evident from
Figure 2(b), the spread complexity corresponding to the state initially
spreading over $9\leq l\leq 13$ (red line) takes longer to saturate and
saturates at a lower value as compared to the state initially spreading over
$10\leq l\leq 12$ (green line). This observation indicates that for a larger
spread of the initial state, the system needs more time to reach the steady
state. This steady state (Figure 1) is spread around the central site and,
therefore, more complex compared to the initial states that spread away from
the central site.

If we choose the initial state spread around the centre, it can reach the
steady state in less time with less saturation value of complexity. For
example, in Figure 2(b), in comparison to the spread complexity for the
initial spread over $18\leq l\leq 20$ (cyan dotted line), the spread
complexity for the initial spread over $10\leq l\leq 12$ (green line)
saturates quicker in time and at a lower value. This indicates that the
farther the initial state is from the central site of the chain, the more
complex the universal steady state is compared to the initial state.

The spread complexity further demonstrates a symmetric nature with spread (for
example, $2\leq l\leq 4$ and $18\leq l\leq 20$) around the central site
($11$-th site) for the tight-biding chain of length $N=22$. If we begin the
evolution with two initial states at the same distance from the centre of the
chain and the same initial width, their spread complexities are identical. In
Figure 2(b), the overlap of the spread complexity for the initial spread
$2\leq l\leq 4$ ( orange line) and $18\leq l\leq 20$ ( cyan dotted line) (also
for the initial spread $2\leq l\leq 6$ ( pink line) and $16\leq l\leq 20$ (
blue dotted line) clearly indicates the above statement. This behaviour can be
traced back to the symmetrical nature of the survival probability
$S(t)=\langle\psi(0)|\psi(t)\rangle$ in this system Dhar_2015  (see Appendix A
for analytical expressions). The decay of the time-dependent Krylov
probability described in Appendix A also shows an identical symmetric nature
with respect to the central site for the position of the initial state.

#### Spread entropy

Next, we turn to the time evolution of spread entropy, defined in Eq. (14),
for the chain with open boundary conditions. The time dependence of the spread
entropy is given in Figure 3. We notice that spread entropy shows rapid
initial growth followed by an oscillation with decreasing amplitude and,
eventually, saturation. Figure 3(a), shows temporal behaviour of spread
entropy for different non-hermiticity parameters $\tau=0.1$ (green), $0.2$
(red), and $0.5$ (blue). The dependence of spread entropy on the non-
hermiticity parameter is qualitatively equivalent to that reported previously
for the spread complexity. Higher values of $\tau$ result in a more rapid
saturation because of the rapid decline of the oscillation amplitude of the
spread entropy. Also, the saturation value of the spread entropy is the same
for different $\tau$. This observation strengthens our previous conclusion
about the universality of the steady state for a given initial state.

(a) Time variation of spread entropy for different $\tau$.

(b) Time variation of spread entropy for different spreads of initial state.

Figure 3:  (a) Time dependence of spread entropy for different non-hermiticity
parameters $\tau=0.1$ (green), $0.2$ (red), and $0.5$ (blue) with fixed total
sites $N=42$ and initial spread of $18\leq l\leq 22$. (b) Time dependence of
spread entropy for different spreads of the initial state $9\leq l\leq 13$
(red), $10\leq l\leq 12$ (green), $2\leq l\leq 6$ (pink), $2\leq l\leq 4$
(orange), $16\leq l\leq 20$ (dotted blue), and $18\leq l\leq 20$ (dotted cyan)
with fixed value of $N=22$ and $\tau=0.1$. We observe that spread entropy
behaves in a qualitatively similar way to spread complexity (see Figure 2).

Next, we summarise our findings about the spread entropy for different choices
of initial states. The value and rate of saturation of the spread entropy,
similar to spread complexity, depend on the position and spread of the initial
state. Spread entropy initially experiences growth, then oscillates, and
eventually saturates at a constant value. We plot the spread entropy with time
for different spread and positions of the initial state in Figure 3(b).
Different colour plots indicate the same configuration of the initial state as
Figure 2(b). The spread entropy in Figure 3(b) corresponding to the states
initially spread around the central site ($9\leq l\leq 13$ and $10\leq l\leq
12$ for $N=22$ sites) of the chain saturate sooner to a lower value than other
choices ($2\leq l\leq 4$ and $18\leq l\leq 20$). For unitary evolution, the
similar behaviour of spread entropy and complexity motivated
Balasubramanian:2022tpr  to conjecture that the spread complexity can be
approximated by the exponential of spread entropy. Our results imply that this
conjecture also applies to non-hermitian Hamiltonian evolution.

####  4.2.2 Periodic boundary condition

Here, we look at the QFPP with periodic boundary conditions characterized by
the effective non-hermitian Hamiltonian given in Eq. (41). In this case, we
first discuss the time evolution of the wave function and characterise the
profile of the steady state in the position basis. Then, we discuss the time
dependence of spread complexity and spread entropy using the non-hermitian
normalisation in Eq. (13) and the definitions given in Eq. (14). We report the
results for varying initial states in this section as it has important
physical consequences for periodic boundary conditions, which are
significantly different from the open boundary conditions.777The behaviour for
varying $\tau$ is identical to the behaviour reported for open boundary
conditions. In the spread complexity and entropy profiles, the decay region is
less profound if the initial states are spread near the detector. In this
case, we also see more oscillations around the saturation value.

#### Steady states

Here we discuss the steady states for QFPP with periodic boundary conditions,
evolving under the effective non-hermitian Hamiltonian given by Eq. (41).
Unlike the open boundary conditions discussed in section 4.2.1, the steady
states are different for different choices of initial states. The spread of
the steady state for different initial states is plotted in Figure 4. A
localised steady state is found only when the position of the initial state is
around the central site ($18\leq l\leq 24$) of the chain of length $N=42$.

Figure 4:  Spread of the steady state at late times ($t=50000$) for different
spread of initial state for periodic boundary conditions with $N=42$ total
sites and non-hermiticity parameter $\tau=0.1$. For periodic boundary
conditions, the spread of the steady state depends on the initial state,
unlike the open boundary condition case shown in Figure 1.

We find that the steady state is located at the central site of the chain only
when the position of the initial state is in the middle ($l$-th) site of the
lattice chain of length $2l$. However, when the initial state is spread close
to the detector ($2\leq l\leq 7$ and $35\leq l\leq 40$ for $N=42$), then non-
unitary time evolution does not lead to any localised steady state. The wave
function is distributed with varying probability at different sites of the
chain. At late times, the wave function for the initial state spread on one
side oscillates around this distribution.

Next we report the behaviour of the spread complexity and entropy, which are
consistent with these findings at late times.

#### Spread complexity

We plot the time dependence of spread complexity Eq. (14) in Figure 5(a) for
different spreading of initial state, keeping $N=42$ and $\tau=0.1$ fixed.
When the initial spread includes the central site ($l$-th site for $N=2l$),
the complexity shows similar growth, decay, and saturation behaviour found for
the open boundary conditions. For example, see the spread complexity of the
initial state with an initial spread of $18\leq l\leq 24$ (green plot in
Figure 5(a)). The saturation value for this case is highly suppressed as
compared to the other plots in the figure. This suppression of complexity
arises from the fact that, for an initial state distributed around the centre
of the chain, the dimension of the Krylov space reduces to half of the total
dimension. This follows since the steady state is localised at the centre of
the tight-binding chain when the initial state is spread around this centre
(Figure 4). Therefore, the steady state is reached sooner, and the saturation
value of complexity is less. However, if the initial state is spread away from
the centre of the chain, the time-evolved state never reaches a localised
steady state, as shown in Figure 4. It oscillates near a state which has
support on all sites. This is reflected by the oscillations shown in the red
and blue plots for complexity and entropy in Figures 5(a) and 5(b),
respectively.

(a) Time variation of spread complexity for different spreads of initial
state.

(b) Time variation of spread entropy for different spreads of initial state.

Figure 5:  (a) Time dependence of spread complexity and (b) Time dependence of
spread entropy for different spreads of the initial state $18\leq l\leq 24$
(green), $2\leq l\leq 8$ (red), $34\leq l\leq 40$ (dotted blue) with fixed
value of $N=42$ and $\tau=0.1$ with periodic boundary condition. For initial
state spread around the centre of the chain (green), the oscillations die out
quickly. For off-centre initial spread (red, dotted blue), we observe an
oscillatory saturation phase.

When the initial spread does not include the central site, the complexity
behaves as in an effectively unitary system at late times, as indicated by a
short decay period and prolonged oscillatory saturation region. As we notice,
in Figure 5(a), the spread complexity associated with the initial states of
spreading $2\leq l\leq 8$ (red line) and $34\leq l\leq 40$ (dotted blue line)
do not have the prolonged decay region. They oscillate with a small amplitude
after initial growth. However, the amplitude of the oscillation depends on the
size of the spread of the initial state. The oscillation amplitude is larger
for a larger spread.

The corresponding time-dependent Krylov probability, defined under Eq. 5, also
exhibits signs of effective unitarity. This is discussed in detail in Appendix
A and shown in Figure 9. The total time-dependent probability stops decaying
at a relatively early time and saturates at a constant value. For the off-
centre spread where the time-dependent probability does not decay completely
to zero, the spread complexity plots show a higher saturation value (resulting
from a less profound decay) than the cases with initial spread around the
centre of the chain.

#### Spread entropy

In Figure 5(b), we show the spread entropy for different initial states. The
spread entropy shown here is similar to the corresponding spread complexity
shown in Figure 5(a). For the states with the initial spread around the centre
of the chain, the spread entropy again shows large oscillations, which
decrease with time and saturate to a fixed value. For the states with an
initial spread near the detector, spread entropy grows and oscillates around a
saturation value without showing any decay. The red and blue plots of spread
entropy in Figures 5(b) and 5(a) also clearly show the symmetric nature around
the central site, as reported previously for steady state and spread
complexity. In the spread entropy, we also find the effective unitary
behaviour for initial states with off-centre spread.

####  4.2.3 Quenching eigenstates of the hermitian Hamiltonian

We now turn to the Krylov spread complexity for a quench from a hermitian to a
non-hermitian Hamiltonian at time $t=0$. We consider the system with open
boundary conditions through the same study as for the case of periodic
boundary conditions. We apply a typical quench protocol where the initial
state of the system is taken as one of the eigenstates of the hermitian tight-
binding Hamiltonian $H_{\text{TB}}$ given in Eq. (39). This state, being an
eigenstate of $H_{\text{TB}}$, evolves trivially under the hermitian
Hamiltonian as it can only change up to a phase. This trivial evolution for
all the times before $t=0$ (starting from $t\rightarrow-\infty$) does not
contribute anything to the spread complexity. We then assume that there is a
sudden quench at time $t=0$ that shifts $H_{\text{TB}}$ to non-hermitian
$H_{\text{eff}}$ given in Eq. (40) with open boundary condition. The quench
Hamiltonian $H_{q}$ as a function of time is noted below,

$H_{q}=\begin{cases}H_{\text{TB}}=H_{\text{TB}}^{\dagger},&-\infty\leq t<0\\\
H_{\text{eff}}\neq H_{\text{eff}}^{\dagger},&~{}~{}~{}~{}~{}t\geq
0.\end{cases}$ (42)

The action of the complex symmetric Hamiltonian constructs the corresponding
Hilbert space after $t=0$. Note that in sections 4.2.1 and 4.2.2, our initial
states were position eigenstates spread over a few sites of the lattice.
Therefore, they, not being eigenstates of the tight-binding Hamiltonian or the
effective non-hermitian Hamiltonian, would spread under both unitary and non-
unitary evolution. However, in this quenching case, the initial state being an
eigenstate of the hermitian Hamiltonian evolves non-trivially only after the
quench. Therefore, all the evolution that we observe for the state in this
case is just due to the measurement process after intervals of $\tau$.

Let us assume the system is prepared in its single-particle ground state or
the single-particle first excited state of the tight-binding hermitian
Hamiltonian and evolves nontrivially under the effective non-hermitian
Hamiltonian starting at time $t=0$. For such a sudden quenching, we analyse
the behaviour of the spread complexity with varying non-hermiticity parameters
$\tau$. First, we consider the i) the single-particle ground state, and, then
ii) the single-particle first excited state of the hermitian Hamiltonian.

### Quenching the single-particle ground state

In this case, we evolve the single-particle ground state of the hermitian
Hamiltonian, denoted by $|\psi_{\text{GS}}\rangle$, with non-hermitian
$H_{\text{eff}}$.

$|\psi(t=0)\rangle=|\psi_{\text{GS}}\rangle.$ (43)

Note that $|\psi_{\text{GS}}\rangle$ has maximum support on the central site
of the chain as shown in Figure 6(a) with the distribution similar to the
steady state of the open boundary conditions (Figure 1).

Figure 6(b) reveals that when the time gap ($\tau$) between two measurements
is large, then there is a notable initial growth in the complexity. The
subsequent decay and saturation structure mirrors previous findings in the
open boundary condition scenario. Conversely, for smaller values of $\tau$,
the initial growth is relatively insignificant and is comparable to the
oscillations in the saturation region. This observation indicates minimal
effects of the non-hermitian perturbation on the complexity.

(a)

(b)

Figure 6:  (a) Single-particle ground state of tight-binding hermitian
Hamiltonian for $N=22$ and (b) Spread complexity for $N=22$ and different
values of $\tau=0.05$ (blue), $0.1$ (green), and $0.2$ (red) for the initial
state as the single-particle ground state of unperturbed hermitian
Hamiltonian. For more frequent measurements, characterized by smaller $\tau$,
the initial growth of spread complexity is suppressed and it oscillates around
a constant value.

A transition in complexity behaviour is observed based on the $\tau$ parameter
of the Hamiltonian. Changing from very small values of $\tau$ of the order of
$10^{-2}$ to the values of the order $10^{-1}$ results in a sharp increase in
complexity at the initial time followed by a long decay period and a higher
value of saturation in Figure 6(b). Now, $\tau$ being the time gap between
measurements, this change indicates a shift in the measurement frequency. As
we decrease $\tau$ to very small values, it indicates a very high frequency of
measurements. In this limit, the complexity oscillates around a constant value
without proper growth or decay.

To understand this, we first recall the probability density of the ground
state (Figure 6(a)) of the hermitian Hamiltonian expressed in the lattice site
position basis, as given by Eq. (39). This has a similar profile as the steady
states in open boundary conditions (Figure 1). For very small gaps between the
measurements, the state does not get enough time to expand and explore the
whole Krylov space. The high frequency of the measurements keeps the initial
profile of the density matrix almost unchanged. This is the closest limit
possible to the quantum Zeno effect in this model Sudarshan_1977 ;
FACCHI200012 ; Facchi_2008 , since for $\tau=0$ there is no time evolution at
all. For larger gaps between two consecutive measurements, the initial state
spreads further in the Hilbert space before reaching the final steady state.
This transition in complexity is a purely measurement-induced phenomenon. It
is reminiscent of the measurement-induced phase transition in quantum systems.

Due to the fact that $|\psi_{\text{GS}}\rangle$ as shown in Figure 6(a) is
already similar to the steady state shown in Figure 1 in terms of support on
the sites, the overall value of spread complexity is suppressed. For an
enhanced quantum Zeno effect, it is thus necessary to consider instead the
evolution of the single-particle first excited eigenstate of the hermitian
Hamiltonian, denoted by $|\psi_{\text{FES}}\rangle$ and shown in Figure 7(a).
Since this state has a minimum at the central site, it differs significantly
from the final steady state. Thus the quantum Zeno becomes more prominent.

(a)

(b)

Figure 7:  (a) Single-particle first excited state of unperturbed hermitian
Hamiltonian for $N=22$ and (b) Spread complexity for $N=22$ and different
values of $\tau=0.05$ (blue), $0.1$ (green), and $0.2$ (red) for the initial
state as the single-particle first excited state eigenstate of unperturbed
hermitian Hamiltonian. For smaller $\tau$, the spread complexity starts
growing at a later time indicating the Zeno effect.

### Quenching the single-particle first excited state

When the first excited state $|\psi(t=0)\rangle=|\psi_{\text{FES}}\rangle$, is
taken as initial state, it has to explore a significant number of Krylov basis
vectors before reaching the steady state. We plot the spread complexity for
this choice of initial state in Figure 7. We notice a clear difference in
complexity profile from Figure 6(b). The larger saturation value of the
complexity indicates that the steady state is much more complex for the first
excited state as compared to the ground state. From the plots for different
values of $\tau$, we also clearly notice that the spread complexity starts
growing much earlier (red plot) for larger values of $\tau=0.2$. This clearly
indicates the measurement-induced phase transition in correlation with the
quantum Zeno effect Sudarshan_1977 ; FACCHI200012 ; Facchi_2008 . As we
decrease the interval between two consecutive measurements $\tau$, the state
takes longer to start evolving. Therefore, we find that when $\tau$ is
infinitesimally small, the spread complexity stays at zero for longer. This is
precisely equivalent to the statement of the quantum Zeno effect, where a
system can not undergo any evolution if projective measurements are made
continuously. It is also worth noticing that saturation values of the spread
complexity are the same for different values of $\tau$. It indicates that they
indeed reach the same steady state after non-unitary time evolution.

##  5 Discussion and applications

As we showed, the spread complexity may be used to characterize the non-
unitary evolution of states induced by measurements at regular intervals. We
exemplified this by considering the quantum first passage problem in which
regular projective measurements track whether the state has reached the
detector site, making the system effectively non-unitary. Our numerical
results as given in section 4.2 display the effects of non-unitarity based on
varying spreads of the initial state and the non-hermitian perturbation
parameter $\tau$ that corresponds to the time between two consecutive
measurements. We characterize this non-unitary behaviour using the total
probability in the Krylov basis, the spread complexity and the spread entropy.
In the following, we discuss the main conclusions from our findings.

### Non-unitarity

We implement a time-dependent normalisation for state vectors in the Krylov
basis. This normalisation keeps the total probability in the Krylov basis
constant at value one with time. As discussed in section 4.2, for the quantum
first passage problem (QFPP) the spread complexity and spread entropy behave
in a qualitatively similar way as the Krylov complexity for operators in open
quantum systems as considered in Bhattacharya:2023zqt . The spread complexity
in the QFPP grows initially and displays an elongated decay phase before
saturating to a constant value (see Figure 2(b)). Note that as discussed in
Appendix B.1, even for unitary evolution there is a very small decay of the
spread complexity after the peak, i.e. there is an overshooting of the
asymptotic value. However, in the case of non-unitary evolution, this decay
region persists for a longer period of time. Therefore, the saturation value
is significantly suppressed as compared to the peak value. This decay is
caused by the imaginary part of the diagonal coefficients $\alpha_{n}$ of the
tri-diagonal matrix $\tilde{T}_{j}$. The saturation value of the spread
complexity is determined by a steady state of the non-unitary quantum system.
This steady state in non-unitary is analogous to thermal states with
effectively unitary time evolution. For open boundary conditions, we find that
the steady state is universal and unique for all choices of initial states and
non-unitarity parameter $\tau$ (see Figure 1). On the other hand, for periodic
boundary conditions, there is a hierarchy of steady states for different
choices of initial state (see Figure 4).

### Non-hermiticity parameter $\tau$

According to Eq. (38), the parameter $\tau$ controls the amount of non-
hermiticity. It corresponds to the time between two successive measurements.
While the measurements introduce non-unitarity into the problem, states evolve
unitarily during the time gap $\tau$ between the two such measurements.
Frequent projective measurements impede this evolution. The system’s non-
unitarity actually spreads during the unitary evolution windows. Therefore,
increasing $\tau$ results in a faster decay of the spread complexity and makes
it saturate sooner (see Figure 2(a)). The same saturation value of both
complexity and entropy for different values of $\tau$ indicates that given an
initial choice of state, the steady state is universal and invariant of
$\tau$. A larger value of $\tau$ implies faster equilibration to the steady
state.

For the quench setup, we also find that the saturation value is independent of
the non-hermiticity parameter $\tau$, as shown in Figure 7. However, the time
when the complexity starts growing increases when the interval $\tau$ between
two measurements is decreased. This is reminiscent of the quantum Zeno effect
Sudarshan_1977 ; FACCHI200012 ; Facchi_2008 . This effect precisely amounts to
frequent repeated measurements impeding the state evolution. This occurs for
$\tau\rightarrow\,0$. In the Zeno regime, the spread complexity vanishes, as
the state does not evolve at all.

### Initial state dependence

The spread complexity also depends on the choice of the initial state by
considering two further conditions, namely i) the spread of the initial state
in the position basis and ii) the distance of the initial spread from the
location of the detector. For both these cases, our results indicate that the
effect of non-hermiticity induced by the measurement occurs at a given
distance from the detector after the time necessary for the signal to reach
the detector. Therefore, i) increasing the initial spread and ii) decreasing
the distance of the initial spread from the detector both make the spread
complexity decay and saturate faster to different values (see Figure 2(b)).
These different saturation values represent how complicated the steady state
is compared to the initial state. For open boundary conditions, in the tight-
binding chain the average position of the steady state is located at the
centre. For periodic boundary conditions, however, when the initial state is
spread close to the detector and does not include the central site of the
chain, there is no localised steady state, as it keeps oscillating over the
entire tight-binding chain. Both spread complexity and entropy respect the
hierarchy of the saturation value in a coherent manner.

### Behaviour of Lanczos coefficients

We find evidence that the behaviour of Krylov complexity or entropy is not
entirely dictated by the Lanczos ascent and descent behaviours. Previous
papers Parker:2018yvk ; Balasubramanian:2022tpr ; Erdmenger:2023shk  have
stressed the different behaviour of Lanczos coefficients distinguishing the
integrable or chaotic nature of the theory. In particular, in the operator
growth hypothesis, the growth and saturation behaviour of complexity was
correlated to the growth of the Lanczos coefficients Parker:2018yvk ;
Alishahiha_2023 . Here we find for non-unitary systems that the spread
complexity and entropy still show the characteristic time evolution of growth-
peak-decay-saturation form, even if the coefficients are oscillatory. This is
discussed in Appendix B.2. As long as the operator generating the evolution is
written in a tridiagonal form, the complexity shows regimes of growth, decay
and saturation. We note that even for a tight-binding chain without
measurements, corresponding to taking $\tau=0$ and thus a limiting unitary
case, spread complexity and entropy grow and saturate though the Lanczos
coefficients are oscillatory (refer to Appendix B.1).

### Spread entropy

The spread entropy in the Krylov basis saturates to a plateau after initial
growth (as shown in Figures 3(b) and 5(b)). For non-unitary evolution with
open and periodic boundary conditions in QFPP, the approximate relation
between spread entropy ($S_{S}$) and complexity ($C_{S}$), conjectured
previously for unitary systems in Balasubramanian:2022tpr  ($C_{S}\approx
e^{S_{S}}$), persists. In the case of periodic boundary conditions, we observe
high oscillations in the saturation phase of spread entropy, showing effective
unitary behaviour when the initial states are spread near the detector and do
not include the central site (see Figure 5).

### Distance from detector

The distance of the spread of the initial state from the detector plays an
important role in the complexity dynamics. If the detector is placed at one
boundary ($N$th site) for a finite number of sites ($N$ sites), the nature of
the plots is symmetric for when i) the initial spread is between
$\left(n_{2}-n_{1}\right)$ sites and ii) the initial spread is between
$\left(N-(n_{2}-n_{1})\right)$ sites. This symmetry results from the
expression of the survival amplitude Dhar2013QuantumTO ; Dhar_2015  mentioned
in Eqs. (47) and (49). Since the Lanczos coefficients can be obtained from the
moments of survival amplitude VM  as shown in Appendices A.1.2 and A.2.2, our
work shows that this symmetry is reflected in the behaviour of spread
complexity and entropy as well.

### Periodic BC and effective unitarity

QFPP with periodic boundary condition requires additional attention as
discussed in section 4.2.2. This may also be seen by considering the survival
probability instead of building the Krylov basis using the Lanczos algorithm.
Previous studies in Dhar_2015  showed that in this case, the survival
probability, defined in appendix A.1.2 in Eq. (45)888Note that the definition
of survival probability is slightly different in the paper Dhar_2015 . There
it is denoted as $\langle\psi(t)|\psi(t)\rangle$. decays to zero only if the
initial spread of the state for a $N$ site Hamiltonian is symmetric with
respect to the $\frac{N}{2}$-th site. Else, it decays to a non-zero constant
value and remains saturated. This saturation value is $0.5$ in cases when the
initial spread does not contain the central site. We find the exact same
behaviour in the time-dependent spread probability (Appendix A). Therefore,
our results in this setup indicate a correspondence between the survival
probability and the time-dependent Krylov spread probability999This is to be
expected since we already know that the Lanczos elements can be equivalently
found directly from the moments of the survival probability.. We also find
that after the time-dependent probability becomes constant, the spread
complexity and entropy also behave similarly to unitary setups (Appendix B.1),
i.e., the complexity and entropy both saturate after decaying for a lesser
amount of time. Therefore, in this case, we observe an effective unitary
nature of the system after the survival probability saturates to non-zero
constant values. The Krylov basis can capture this effect very elegantly. Of
course, for other cases when the initial spread contains the central site, the
behaviour is like the usual non-unitary cases mentioned earlier, as the time-
dependent probability decays all the way to zero.

### Quench

In section 4.2.3, our initial state is the single-particle ground state of the
actual hermitian Hamiltonian, and therefore, it evolves under the unitary
evolution with a phase factor only. This phase factor does not contribute to
the spread complexity. This means that if we evolve this state unitarily for
time $t_{1}$, the complexity remains at zero. At time say, $t_{1}$, we
introduce the detector and start measurements at regular intervals $\tau$
given by Eq. 42. Now, the ground state of the hermitian Hamiltonian starts
evolving nontrivially since it is not an eigenstate of the effective non-
hermitian Hamiltonian. As a result, complexity starts showing non-unitary
evolution. This can be understood as a unitary-to-non-unitary quench taking
place at time $t_{1}$ that is time zero in our plots in section 4.2.3. Spread
complexity is a good probe of this transition since it changes behaviour at
the point of the transition time. By modulating the measurement frequency, we
discern a notable shift in the behaviour of spread complexity. For smaller
values of the time gap between measurements of the order $10^{-2}$, we observe
that the complexity oscillates around a constant value. Conversely, as we
increase the value of $\tau$ to the order $10^{-1}$, the complexity exhibits
an initial phase of rapid growth, succeeded by a prolonged decaying region and
eventual saturation — a characteristic signature of spread complexity for non-
unitary evolution. Consequently, we infer that more frequent measurements
restrict the system to evolve for only shorter duration post-measurement,
thereby constraining the growth of complexity. This is reminiscent of the
measurement-induced phase transition phenomena measure1 ; measure2  in the
context of complexity growth.

The transition in the complexity behaviour becomes more prominent for the
quench of the single-particle first excited state. Even though the initial
spread of this state on the lattice sites is much different from the steady
state (see Figure 7(a) and Figure 1), the state does not go through any
nontrivial evolution after the quench for a long time if the measurement
frequency is very high. This is because of the quantum Zeno effect taking
place that suggests frequent measurements impede the time evolution and
spreading of a quantum state. Finally, after a very long time that depends on
the value of the $\tau$ parameter, complexity starts growing rapidly, and it
saturates to a constant value when the system reaches a steady state. In this
case, contrary to all the previous cases, the evolution directly leads to a
steady state without much oscillation.

To conclude further, our results show that the spread complexity and entropy
in the Krylov basis can act as very good probes for different phenomena during
a non-unitary evolution ranging from the varying non-hermiticity to effective
unitarity in periodic boundary conditions and unitary-nonunitary transition
through quench. This article successfully generalizes the concept of spread in
the Krylov basis to non-unitary setups. Our work further strengthens the idea
of implementing the modified approach of bi-Lanczos and complex symmetric
Lanczos for non-unitary cases as advocated in Bhattacharya:2023zqt . We find
an agreement between the non-unitary Lanczos coefficients to the ones derived
directly from the non-unitary survival amplitude for the first time. It shows
that the survival amplitude is a more fundamental way to compute Lanczos
coefficients in the sense that the moment recursion method remains unchanged
for a non-unitary setup, whereas the construction through the actual Lanczos
algorithm needs to be modified.

### Applications of Tight-Binding model

Since we have studied the effect of measurement using a tight-binding
Hamiltonian, our results can be useful in multiple physical situations where a
tight-binding Hamiltonian is used in model building. A couple of examples of
very active areas of research both from theory and experiment points of view
are,

i) electron transport in one-dimensional systems Karamlou_2022 ,

ii) the behaviour of electrons confined to one-dimensional structures, such as
quantum wells and quantum dots Sanderson2009QuantumDG .

For these cases, the diagonal and off-diagonal entries of the Hamiltonian can
be understood as the orbital energies of an atomic orbital and the hopping
amplitudes of an electron to jump to a different overlapping orbital of
another atom, respectively. These help to determine the electronic transport
properties of nanostructures. Our results indicate the simple fact that
measurements of these diagonal and hopping amplitudes are expected to give
imaginary results in general, i.e. to have a complex phase. This imaginary
nature will reflect the non-hermiticity of the Hamiltonian due to the effect
of measurements. Therefore, the effective non-hermitian Hamiltonian
($H_{\text{eff}}$), under which the system evolves when measurements are made,
can be written from those observed imaginary hopping amplitudes. The methods
introduced in this paper can be used to determine the exact time dynamics by
performing the Krylov basis analysis and studying the spread complexity for
different electronic configurations. We hope that this may be useful in
further characterization of these nanostructures.

### Future directions

The techniques developed in this work are expected to also be useful in the
studies of open system dynamics in the Schrödinger picture, where a mixed-
state density matrix evolves with a non-hermitian Lindbladian, similar in
spirit to the study of Bhattacharya:2022gbz ; Bhattacharya:2023zqt  for
operators. These studies of open system dynamics are also interesting from the
perspective of black hole evolution in a thermal environment. This can be
extended to the Schwinger-Keldysh path integrals study open quantum field
theories building on Baidya:2017eho ; Geracie:2017uku . It will also be
interesting to perform a similar study for the two-dimensional tight-binding
Hamiltonian, which is expected to provide information about the time dynamics
of electronic transport in 2D graphene-like materials. Another direction that
we are pursuing presently, and hope to report soon, is to study PT-symmetric
Hamiltonians and how spread complexity and entropy probe the PT-symmetry
breaking in such systems spread2upcoming .

A final future direction is to perform a similar study in holography. There
are recent studies concerning measurements in the SYK model as well as
holographic measurements in JT gravity Jian:2021hve ; Antonini:2022lmg . It
will be very interesting to check whether there is an exact match of Krylov
spread complexities under holographic measurements, extending the work on
holographic Krylov complexity of Rabinovici:2023yex .

## Acknowledgements

The authors would like to thank Vijay Balasubramanian, Souvik Banerjee, Pablo
Basteiro, Pawel Caputa, Mario Flory, Giuseppe Di Giulio, Taishi Kawamoto, René
Meyer, Sara Murciano, Pratik Nandy, Pingal Pratyush Nath, Subir Sachdev,
Himanshu Sahu, Aninda Sinha, Zhuo-Yu Xian, Kunal Pal, Kuntal Pal, and Diptarka
Das for useful discussions and comments on related topics at various stages.
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 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.

##  Appendix A  Time-dependent probability for non-unitary evolution

Here we discuss details of the time-dependent probability in the Krylov basis
before implementing the new normalisation described in section 3.1. This
probability captures the rate of decay of the state through the detector site
due to repeated measurements.

###  A.1  Open boundary conditions

We begin with the results for open boundary conditions with $N$ sites, with
the detector being present at the $N^{th}$ site only. This is characterized by
the effective Hamiltonian given in Eq. (40). We first examine the decay of the
total probability of the state in the Krylov space. After we spread the time-
evolved state on the Krylov basis (using the complex symmetric Lanczos
construction), the final state expansion looks like the following,

$|\psi(t)\rangle=\sum_{n=1}^{D}\psi_{n}(t)|K_{n}\rangle.$ (44)

First, we investigate the temporal behaviour of the time-dependent probability
in the Krylov space $\sum_{i=1}^{D}|\psi_{n}(t)|^{2}$. Note that this is
different from the notion of probability we discussed in Eq. (10), where it
was renormalised to $1$ for all times. The time-dependent probability provides
us with an understanding of how the information is lost slowly from the system
to the environment due to repeated measurements.

####  A.1.1 Probability decay

Due to the non-unitary nature of the time evolution, the preservation of the
time-dependent probability is not possible. The decay of time-dependent
probability to zero exemplifies the dissipative characteristics induced by the
measurements in the system. Notably, in Figure 8(a), as the value of $\tau$
increases from $0.1$ (green) to $0.5$ (blue) through $0.2$ (red) for an
initial spread in the sites $18\leq l\leq 22$ for a chain of length $40$, the
decay rate becomes faster, indicating a faster dissipative process or more
rapid detection of the state at the detector. This indicates that increasing
the non-hermiticity parameter results in a stronger interaction between the
system and the rest of the universe through the detection measurement.

(a) Decay of time-dependent probability with time for $N=40$ and the initial
spread of $18\leq l\leq 22$ for different values of non-hermitian perturbation
parameter $\tau=0.1$ (green), $0.2$ (red), and $0.5$ (blue).

(b) Decay of time-dependent probability with time for $N=20$ and $\tau=0.1$
for different spreads of the initial state $8\leq l\leq 12$ (green), $9\leq
l\leq 11$ (orange), $2\leq l\leq 6$ (red), $2\leq l\leq 4$ (pink), $15\leq
l\leq 19$ (dotted blue), and $17\leq l\leq 19$ (dotted cyan).

Figure 8: Decay of time-dependent probability with open boundary conditions
for different values of the non-hermiticity parameter and initial spread. The
probability decays faster for larger $\tau$ and off-centre spread of initial
state.

The decay rate of time-dependent probability also depends on the initial state
configuration. In Figure 8(b), we show the time dependence of this probability
for the different spreading of the initial state over a chain with total sites
$N=20$, keeping the non-hermiticity parameter fixed. Rates of decay for the
total probability depend on how wide the initial state is spread over the
sites. In particular, a wider spread of the initial state corresponds to a
slower decay rate. For example, in Figure 8(b), the total probability
corresponding to the initial spread $8\leq l\leq 12$ (green line) decays
slower than the total probability corresponding to the initial spread $9\leq
l\leq 11$ (orange line).

We also observe that the decay rates of total probability depend on the
location of the initial state. Total probability decays slowly with time for
the state initially spread around the midpoint of the system. In Figure 8(b),
for instance, the total probability for initial spreads of $2\leq l\leq 6$
(red) decays more quickly than the total probability for initial spreads of
$8\leq l\leq 12$ (green). Consequently, as the initial states deviate from the
centre of the system, decay rates of total probability increase.

Furthermore, despite the detector being situated solely at the end of the
chain, the decay rates exhibit symmetry with respect to the distance from the
centre rather than the distance from the detector. As from Figure 8(b), the
initial state corresponding to the red line $(2\leq l\leq 6)$ and blue dotted
line $(15\leq l\leq 19)$. Also, the pink line $(2\leq l\leq 4)$ and cyan
dotted line $(17\leq l\leq 19)$ spread over symmetrically from the centre of
the chain. The total probability corresponding to all the initial states with
a symmetric spread around the centre decays at the same rate. This behaviour
aligns well with the symmetrical nature of the survival probability in this
system Dhar_2015 .

In the next subsection, we discuss an alternative approach to obtaining the
same set of complex Lanczos coefficients.

####  A.1.2 Lanczos coefficients from survival amplitude

Like the usual case of hermitian Hamiltonians, the Lanczos coefficients can
also be calculated by using the moments obtained from the survival amplitude
VM ,

$S(t)=\langle\psi(t)|\psi(0)\rangle.$ (45)

The initial state spread in the position basis can be expressed as,

$|\psi(t=0)\rangle=\frac{1}{\sqrt{|l_{2}-l_{1}+1|}}\sum_{l=l_{1}}^{l_{2}}|l\rangle.$
(46)

Following the technique described in Dhar_2015 , using the $V_{\text{eff}}$ as
perturbation and calculating till the first order in the perturbation, we
obtain the general analytical form of the survival amplitude for the open
boundary condition as,

$S(t)=\frac{1}{l_{2}-l_{1}+1}\sum_{s=1}^{N}\sum_{l=l_{1}}^{l_{2}}\sum_{k=l_{1}}^{l_{2}}\frac{2}{N}\sin\left(\frac{\pi
ks}{N}\right)\sin\left(\frac{\pi ls}{N}\right)\exp\left[-t\gamma_{s}\right]$
(47)

where $\gamma_{s}=\left(\frac{\tau}{N}\sin^{2}\left(\frac{\pi
s}{N}\right)+2i\cos\left(\frac{\pi s}{N}\right)\right)$. From this, it is
straightforward to get any the higher moment as,

$\mu_{n}=\frac{d^{n}}{dt^{n}}S(t)|_{t=0}=\frac{1}{l_{2}-l_{1}+1}\sum_{s=1}^{N}\sum_{l=l_{1}}^{l_{2}}\sum_{k=l_{1}}^{l_{2}}\frac{2(-\gamma_{s})^{n}}{N}\sin\left(\frac{\pi
ks}{N}\right)\sin\left(\frac{\pi ls}{N}\right).$ (48)

The Lanczos coefficients can be obtained from these moments recursively VM .
The exact match between the Lanczos coefficients obtained from these moments
and those from the complex symmetric Lanczos algorithm can be easily verified
numerically. We have checked this all along and found agreement in each of the
cases we studied. Therefore, it is evident that while the Lanczos construction
needs to be modified for non-hermitian Hamiltonians in order to obtain
meaningful Lanczos coefficients, the moment recursion method from survival
amplitude is unaffected.

Figure 9: Time variation of total probability with periodic boundary
conditions for different spreads of the initial state keeping fixed $N=40$ and
$\tau=0.1$. The black straight line marks the $0.5$ constant line. The time-
dependent probability for off-centre initial spread stops decaying earlier and
saturates at a constant value characterizing an effective unitarity.

###  A.2 Periodic boundary conditions

Next, we look at the system with periodic boundary conditions, when the
evolution is governed by the effective Hamiltonian given as Eq. (41). In this
case, we discuss two main features. Firstly, the Krylov probabilities have an
interesting feature with respect to the spread of the initial states, namely
an effective unitarity. Secondly, we discuss the implications of this
effective unitarity at late times on the complexity and entropy. We only
report the results for varying initial states as this is the case where it is
significantly different from the open boundary conditions.

####  A.2.1 Time-dependent probability

We discussed earlier in section 4.1 that the decay of the survival probability
saturates at $0.5$ when the initial spread of the state does not include the
middle site $(N/2)$ at all, i.e. the spread is in one half of the chain. We
also find similar behaviour of the total Krylov probability. Figure 9 shows
the time dependence of total probability for different spreading of the
initial state with periodic boundary conditions for a fixed number of total
sites $N=42$ and non-hermiticity parameter $\tau=0.1$. The total probability
associated with the initial state of spreading $18\leq l\leq 24$ (green line)
and $20\leq l\leq 22$ (orange line) decays to zero. On the other hand, we
notice the saturation at $0.5$ of the total probability corresponding to the
initial state spread over $2\leq l\leq 4$ (Magenta line), $38\leq l\leq 40$
(dotted cyan line), $34\leq l\leq 40$ (dotted blue line), and $2\leq l\leq 7$
(pink line). This is due to a symmetry that decreases the Krylov space
dimension by half when the central site of the chain is not included in the
initial state in a symmetric fashion. If the initial state is spread
symmetrically around the central site, it has equal access to both halves;
hence, the symmetry does not manifest in the plots. Thus the probability
decays completely to zero at late times. However, suppose the initial state is
distributed non-symmetrically around the centre of the chain. In that case,
the access is divided non-symmetrically into the two halves of the symmetry,
and we observe the probability to saturate still but to values lower than
$0.5$. This observation is in line with the survival probability property
mentioned in the previous works Dhar_2015 ; Dhar2013QuantumTO .

On the other hand, decreasing the non-hermiticity parameter and increasing the
spread of the initial state, in this case, results in a slower rate of decay,
as we observed in the case of the open boundary conditions.

(a) Total probability for unitary evolution remains unchanged with time.

(b) Time evolution of spread complexity for $N=201$ (red), and $N=101$ (blue).

(c) Time evolution of spread entropy for $N=201$ (red), and $N=101$ (blue).

Figure 10: Total probability, spread complexity, and spread entropy for
unitary evolution for different numbers of total sites, starting with an
initial state spreading over the centre of the chain. The complexity and
entropy show growth and saturation at a higher value for larger system size
$N$.

####  A.2.2 Lanczos coefficients from survival amplitude

Finally, to complete this section, we again ensure that all of these results
can be reproduced, starting from the survival amplitude and computing its
moments. If we take an initial state like Eq. (46) into consideration, then
the expression for the survival amplitude for periodic boundary conditions has
the following analytic form as,

$S(t)=\frac{2}{N(l_{2}-l_{1}+1)}\sum_{l=l_{1}}^{l_{2}}\sum_{k=l_{1}}^{l_{2}}\left(\sum_{s=0}^{\frac{N}{2}-1}\Phi_{2s+1}(l)\Phi_{2s+1}(l)\exp\left(-\gamma_{2s+1}t\right)+\sum_{s=1}^{\frac{N}{2}-1}\Phi_{2s}(k)\Phi_{2s}(l)\exp\left(-2it\cos\left(\frac{2\pi
s}{N}\right)\right)\right).$ (49)

Using this, all the higher moments can be written in general as,

$\mu_{n}=\frac{2}{N(l_{2}-l_{1}+1)}\sum_{l=l_{1}}^{l_{2}}\sum_{k=l_{1}}^{l_{2}}\left(\sum_{s=0}^{\frac{N}{2}-1}\Phi_{2s+1}(l)\Phi_{2s+1}(l)\left(-\gamma_{2s+1}\right)^{n}+\sum_{s=1}^{\frac{N}{2}-1}\Phi_{2s}(k)\Phi_{2s}(l)\left(-2i\cos\left(\frac{2\pi
s}{N}\right)\right)^{n}\right),$ (50)

where we denote $\sin\left(\frac{\pi ks}{N}\right)$ as $\Phi_{s}(l)$ to make
the equation more compact. From here, the Lanczos coefficients can be obtained
in an iterative manner VM .

##  Appendix B Unitary spread complexity and non-unitary Lanczos coefficients

To illustrate the crucial differences in the behaviour of spread complexity
and entropy for non-unitary evolution and unitary evolution, we here provide a
numerical analysis for unitary evolution.

###  B.1 Unitary spread in Krylov basis

In this section, we provide plots of spread complexity for unitary evolution
under the tight-binding Hamiltonian given in Eq. (39). The Hamiltonian is very
simple, and of the following form,

$H=\begin{pmatrix}0&-1&&&&0\\\ -1&0&-1&&&\\\ &-1&0&\ddots&&\\\
&&\ddots&\ddots&-1&\\\ &&&-1&0&\ddots\\\ 0&&&&\ddots&\ddots\\\
\end{pmatrix}_{N\times N},$ (51)

where $N$ is the number of sites. We take $N=201$, $98\leq l\leq 102$ (red)
and $N=101$, $48\leq l\leq 52$ (blue) and plot the total probability, spread
complexity and spread entropy in Figure 10.

####  B.1.1 Probability and complexity

As expected, Figure 10(a) shows that the probability is conserved throughout.
This ensures that the evolution is indeed unitary. The spread complexity
saturates to a plateau after initial linear growth (see Figure 10(b)).
Therefore the oscillating decay part of spread complexity in the cases studied
in the main text is attributed to the non-hermiticity of the effective
Hamiltonian, due to which the evolution becomes non-unitary. We also observe
that as we increase the system size, the initial growth and the saturation
value increases, which is in agreement with previous literature on the subject
Balasubramanian:2022dnj ; Parker:2018yvk .

(a) $\alpha_{n}$ for unitary evolution.

(b) $\beta_{n}$ for unitary evolution.

Figure 11: Behaviour of $\alpha_{n}$ and $\beta_{n}$ for unitary evolution. We
choose $N=201$ and spreading of the initial state in $98\leq l\leq 102$. There
is no Lanczos ascent or descent behaviour in the Lanczos coefficients,
implying that this is not required for the typical time dependence of
complexity and entropy.

####  B.1.2 Krylov spread entropy

The spread entropy also attains a plateau after the initial logarithmic growth
in the unitary case, as shown in Figure 10(c) and the saturation value
increases with increasing Hilbert space dimensions. In a similar spirit to
spread complexity, we are therefore able to identify the indefinite growth of
the Krylov spread entropy discussed in the main text to be a consequence of
the non-unitary evolution.

(a) Real parts of $\alpha_{n}$ for $\tau=0.1$(dotted red) and
$\tau=0.01$(dashed blue).

(b) Imaginary parts $\alpha_{n}$ for $\tau=0.1$(dotted red) and
$\tau=0.01$(dashed blue).

(c) Real parts of $\beta_{n}$ for $\tau=0.1$(dotted red) and
$\tau=0.01$(dashed blue).

(d) Imaginary parts of $\beta_{n}$ for $\tau=0.1$(dotted red) and
$\tau=0.01$(dashed blue).

Figure 12: Real and imaginary parts of $\alpha_{n}$ and $\beta_{n}$ by
considering open boundary conditions for different non-hermiticity parameters
$\tau$. In these plots, we choose total sites $N=101$ with a spreading of the
initial state between $48\leq l\leq 52$. The real parts of the coefficients
$\alpha_{n}$ and $\beta_{n}$ remain invariant under change of $\tau$, while
the imaginary parts change showing the effect of the non-hermitian Hamiltonian
evolution.

(a) Real parts of $\alpha_{n}$ for $\tau=0.1$ (dotted red) and $\tau=0.01$
(dashed blue).

(b) Imaginary parts of $\alpha_{n}$ for $\tau=0.1$ (dotted red) and
$\tau=0.01$ (dashed blue).

(c) Real parts of $\beta_{n}$ for $\tau=0.1$ (dotted red) and $\tau=0.01$
(dashed blue).

(d) Imaginary parts of $\beta_{n}$ for $\tau=0.1$ (dotted red) and $\tau=0.01$
(dashed blue).

Figure 13: Real and imaginary parts of $\alpha_{n}$ and $\beta_{n}$ with open
boundary conditions for different values of non-hermiticity parameter $\tau$.
Here, the initial state spreading between $95\leq l\leq 99$ of the total sites
$N=101$. For an off-centre spread, the non-hermiticity parameter $\tau$ only
substantially changes the imaginary parts of the coefficients.

####  B.1.3 Lanczos coefficients for unitary evolution

In Figure 11, we plot the Lanczos coefficients for unitary evolution of the
tight-binding Hamiltonian with $N=201$ and initial state to be spread in
$98\leq l\leq 102$. It is instructive to observe that the coefficients do not
show any distinct ascent or descent, apart from some peaks, and decay very
near the boundary and for most of the time, the behaviour is oscillatory. This
implies that although the integrable or chaotic nature of a system can be a
result of particular scaling of the ascent and descent of $\beta_{n}$ and
$\alpha_{n}$ coefficients in some cases, this ascending or descending
behaviour is not necessary to give rise to the expected typical spread
complexity and entropy profiles.

(a) Real parts of $\alpha_{n}$.

(b) Imaginary parts of $\alpha_{n}$.

(c) Real parts of $\beta_{n}$.

(d) Imaginary parts of $\beta_{n}$.

Figure 14: Real and imaginary parts of the $\alpha_{n}$ and $\beta_{n}$ for
different non-hermiticity parameter $\tau$ values. Here, we choose the number
of total sites $N=102$ with periodic boundary conditions, and initial state
localise over $2\leq l\leq 6$. For periodic boundary conditions, the
fluctuations in the real part of $\beta_{n}$ coefficients increase for larger
$\tau$ along with the change in the imaginary part for both sets of
coefficients.

(a) Real $\alpha_{n}$

(b) Imaginary $\alpha_{n}$

(c) Real $\beta_{n}$

(d) Imaginary $\beta_{n}$

Figure 15: Real and imaginary parts of the $\alpha_{n}$ and $\beta_{n}$ for
different non-hermiticity parameter $\tau$ values. Here, we choose the number
of total sites $N=102$ with periodic boundary conditions, and the spreading of
the initial state is $96\leq l\leq 100$. This shows that all real and
imaginary Lanczos coefficients are exactly the same as Figure 14.

###  B.2 Lanczos coefficients for the non-hermitian Hamiltonian

In this subsection, we discuss the real and imaginary parts of the Lanczos
coefficients $\alpha_{n}$ and $\beta_{n}$ for the non-hermitian effective
Hamiltonian derived from the complex symmetric Lanczos algorithm.

We consider open boundary conditions with $N=101$ total sites while the
spreading of initial state is over $48\leq l\leq 52$ (Figure 12) and $95\leq
l\leq 99$ (Figure 13). We notice that the real parts of the $\beta_{n}$ and
$\alpha_{n}$ remain nearly unchanged for different degrees of non-hermiticity
$(\tau)$. However, the imaginary parts show small deviations. Note that the
non-hermitian Hamiltonian is perturbatively generated with a perturbation
parameter. However, from the plots of the probability, spread complexity and
entropy in the main text, it is clear that these small deviations give rise to
distinguishable features in the above-mentioned quantities.

We plot the Lanczos coefficients for periodic boundary conditions with total
$N=102$ sites, and the initial state spread over $2\leq l\leq 6$ in Figure 14.
For the same system size, we also plot the Lanczos coefficients with initial
state spread over $96\leq l\leq 100$ sites in Figure 15. From the plots of
Lanczos coefficients for the periodic boundary conditions (Figures 14 and 15),
we notice that in terms of the distance from the centre of the chain, the
symmetry persists, resulting in exactly identical plots. This symmetry
therefore shows up in the spread complexity (Figure 5(a)) and entropy (Figure
5(b)) plots as well. The origin of this symmetry is the survival amplitude
given in Eq. (49), which is a function of the system size $N$, location of the
initial state $l_{1}\leq l\leq l_{2}$ and time $t$. The dependence on system
size $N$ and the location of the initial spread $l_{1},\,l_{2}$ are such that
if two initial states are spread over $l_{11}\leq l\leq l_{12}$, and
$l_{21}\leq l\leq l_{22}$ respectively, the survival probability remains same
when

$\displaystyle(l_{22}-l_{21})$ $\displaystyle=(l_{12}-l_{11}),$ (52)
$\displaystyle|\frac{N}{2}-\frac{l_{22}+l_{21}}{2}|$
$\displaystyle=|\frac{N}{2}-\frac{l_{12}+l_{11}}{2}|.$ (53)

The modulus in Eq. (53) is due to the reason that the spread can be on either
side of the centre of the chain. Therefore, the symmetry exists for two
initial states, which are of the same width, and the centre of the width has
an equal distance from the centre of the chain.

## References

  * (1) L. Susskind, _Computational Complexity and Black Hole Horizons_ , _Fortsch. Phys._ 64 (2016) 24–43, [1403.5695].
  * (2) A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, _Holographic Complexity Equals Bulk Action?_ , _Phys. Rev. Lett._ 116 (2016) 191301, [1509.07876].
  * (3) M. A. Nielsen, M. R. Dowling, M. Gu and A. C. Doherty, _Quantum Computation as Geometry_ , _Science_ 311 (2006) 1133–1135, [quant-ph/0603161].
  * (4) R. Jefferson and R. C. Myers, _Circuit complexity in quantum field theory_ , _JHEP_ 10 (2017) 107, [1707.08570].
  * (5) S. Chapman, M. P. Heller, H. Marrochio and F. Pastawski, _Toward a Definition of Complexity for Quantum Field Theory States_ , _Phys. Rev. Lett._ 120 (2018) 121602, [1707.08582].
  * (6) P. Caputa and J. M. Magan, _Quantum Computation as Gravity_ , _Phys. Rev. Lett._ 122 (2019) 231302, [1807.04422].
  * (7) M. Flory and M. P. Heller, _Conformal field theory complexity from Euler-Arnold equations_ , _JHEP_ 12 (2020) 091, [2007.11555].
  * (8) J. Erdmenger, M. Gerbershagen and A.-L. Weigel, _Complexity measures from geometric actions on Virasoro and Kac-Moody orbits_ , _JHEP_ 11 (2020) 003, [2004.03619].
  * (9) A. Bhattacharya, A. Bhattacharyya and S. Maulik, _Pseudocomplexity of purification for free scalar field theories_ , _Phys. Rev. D_ 106 (2022) 086010, [2209.00049].
  * (10) S. Chapman and G. Policastro, _Quantum computational complexity from quantum information to black holes and back_ , _Eur. Phys. J. C_ 82 (2022) 128, [2110.14672].
  * (11) D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi and E. Altman, _A Universal Operator Growth Hypothesis_ , _Phys. Rev. X_ 9 (2019) 041017, [1812.08657].
  * (12) V. Viswanath and G. Müller, _The Recursion Method: Application to Many-Body Dynamics, Springer Berlin, Heidelberg 1994_ , .
  * (13) V. Balasubramanian, P. Caputa, J. M. Magan and Q. Wu, _Quantum chaos and the complexity of spread of states_ , _Phys. Rev. D_ 106 (2022) 046007, [2202.06957].
  * (14) J. L. F. Barbón, E. Rabinovici, R. Shir and R. Sinha, _On The Evolution Of Operator Complexity Beyond Scrambling_ , _JHEP_ 10 (2019) 264, [1907.05393].
  * (15) A. Dymarsky and A. Gorsky, _Quantum chaos as delocalization in Krylov space_ , _Phys. Rev. B_ 102 (2020) 085137, [1912.12227].
  * (16) A. Dymarsky and M. Smolkin, _Krylov complexity in conformal field theory_ , _Phys. Rev. D_ 104 (2021) L081702, [2104.09514].
  * (17) D. J. Yates and A. Mitra, _Strong and almost strong modes of Floquet spin chains in Krylov subspaces_ , _Phys. Rev. B_ 104 (2021) 195121, [2105.13246].
  * (18) P. Caputa, J. M. Magan and D. Patramanis, _Geometry of Krylov complexity_ , _Phys. Rev. Res._ 4 (2022) 013041, [2109.03824].
  * (19) J. Kim, J. Murugan, J. Olle and D. Rosa, _Operator delocalization in quantum networks_ , _Phys. Rev. A_ 105 (2022) L010201, [2109.05301].
  * (20) P. Caputa and S. Datta, _Operator growth in 2d CFT_ , _JHEP_ 12 (2021) 188, [2110.10519].
  * (21) D. Patramanis, _Probing the entanglement of operator growth_ , _PTEP_ 2022 (2022) 063A01, [2111.03424].
  * (22) F. B. Trigueros and C.-J. Lin, _Krylov complexity of many-body localization: Operator localization in Krylov basis_ , _SciPost Phys._ 13 (2022) 037, [2112.04722].
  * (23) E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, _Krylov localization and suppression of complexity_ , _JHEP_ 03 (2022) 211, [2112.12128].
  * (24) Z.-Y. Fan, _Universal relation for operator complexity_ , _Phys. Rev. A_ 105 (2022) 062210, [2202.07220].
  * (25) R. Heveling, J. Wang and J. Gemmer, _Numerically probing the universal operator growth hypothesis_ , _Phys. Rev. E_ 106 (2022) 014152, [2203.00533].
  * (26) A. Bhattacharyya, D. Ghosh and P. Nandi, _Operator growth and Krylov Complexity in Bose-Hubbard Model_ , 2306.05542.
  * (27) W. Mück and Y. Yang, _Krylov complexity and orthogonal polynomials_ , _Nucl. Phys. B_ 984 (2022) 115948, [2205.12815].
  * (28) A. Kundu, V. Malvimat and R. Sinha, _State dependence of Krylov complexity in 2d CFTs_ , _JHEP_ 09 (2023) 011, [2303.03426].
  * (29) A. Banerjee, A. Bhattacharyya, P. Drashni and S. Pawar, _From CFTs to theories with Bondi-Metzner-Sachs symmetries: Complexity and out-of-time-ordered correlators_ , _Phys. Rev. D_ 106 (2022) 126022, [2205.15338].
  * (30) B.-n. Du and M.-x. Huang, _Krylov Complexity in Calabi-Yau Quantum Mechanics_ , 2212.02926.
  * (31) S. Guo, _Operator growth in SU(2) Yang-Mills theory_ , 2208.13362.
  * (32) A. Avdoshkin, A. Dymarsky and M. Smolkin, _Krylov complexity in quantum field theory, and beyond_ , 2212.14429.
  * (33) A. Bhattacharyya, S. S. Haque, G. Jafari, J. Murugan and D. Rapotu, _Krylov complexity and spectral form factor for noisy random matrix models_ , _JHEP_ 23 (2020) 157, [2307.15495].
  * (34) N. Iizuka and M. Nishida, _Krylov complexity in the IP matrix model_ , 2306.04805.
  * (35) N. Iizuka and M. Nishida, _Krylov complexity in the IP matrix model II_ , 2308.07567.
  * (36) H.-S. Jeong, K.-Y. Kim and Y.-W. Sun, _Island in dyonic black holes: doubly holographic theory_ , 2305.18122.
  * (37) H. A. Camargo, V. Jahnke, K.-Y. Kim and M. Nishida, _Krylov complexity in free and interacting scalar field theories with bounded power spectrum_ , _JHEP_ 05 (2023) 226, [2212.14702].
  * (38) P. Caputa and S. Liu, _Quantum complexity and topological phases of matter_ , _Phys. Rev. B_ 106 (2022) 195125, [2205.05688].
  * (39) M. Afrasiar, J. Kumar Basak, B. Dey, K. Pal and K. Pal, _Time evolution of spread complexity in quenched Lipkin-Meshkov-Glick model_ , 2208.10520.
  * (40) V. Balasubramanian, J. M. Magan and Q. Wu, _Tridiagonalizing random matrices_ , _Phys. Rev. D_ 107 (2023) 126001, [2208.08452].
  * (41) P. Caputa, N. Gupta, S. S. Haque, S. Liu, J. Murugan and H. J. R. Van Zyl, _Spread complexity and topological transitions in the Kitaev chain_ , _JHEP_ 01 (2023) 120, [2208.06311].
  * (42) J. Erdmenger, S.-K. Jian and Z.-Y. Xian, _Universal chaotic dynamics from Krylov space_ , _JHEP_ 08 (2023) 176, [2303.12151].
  * (43) K. Pal, K. Pal, A. Gill and T. Sarkar, _Time evolution of spread complexity and statistics of work done in quantum quenches_ , _Phys. Rev. B_ 108 (2023) 104311, [2304.09636].
  * (44) S. Nandy, B. Mukherjee, A. Bhattacharyya and A. Banerjee, _Quantum state complexity meets many-body scars_ , 2305.13322.
  * (45) A. Chattopadhyay, A. Mitra and H. J. R. van Zyl, _Spread complexity as classical dilaton solutions_ , _Phys. Rev. D_ 108 (2023) 025013, [2302.10489].
  * (46) M. Gautam, K. Pal, K. Pal, A. Gill, N. Jaiswal and T. Sarkar, _Spread complexity evolution in quenched interacting quantum systems_ , 2308.00636.
  * (47) B. Bhattacharjee, S. Sur and P. Nandy, _Probing quantum scars and weak ergodicity breaking through quantum complexity_ , _Phys. Rev. B_ 106 (2022) 205150, [2208.05503].
  * (48) A. Gill, K. Pal, K. Pal and T. Sarkar, _Complexity in two-point measurement schemes_ , 2311.07892.
  * (49) P. H. S. Bento, A. del Campo and L. C. Céleri, _Krylov Complexity and Dynamical Phase Transition in the quenched LMG model_ , 2312.05321.
  * (50) S. E. Aguilar-Gutierrez and A. Rolph, _Krylov complexity is not a measure of distance between states or operators_ , 2311.04093.
  * (51) B. Craps, O. Evnin and G. Pascuzzi, _A relation between Krylov and Nielsen complexity_ , 2311.18401.
  * (52) A. Peres, _Quantum theory: Concepts and methods_ , _Springer Dordrecht_ (1993) .
  * (53) S. Dhar, S. Dasgupta and A. Dhar, _Quantum time of arrival distribution in a simple lattice model_ , _Journal of Physics A: Mathematical and Theoretical_ 48 (2013) , [1312.5923].
  * (54) S. Dhar, S. Dasgupta, A. Dhar and D. Sen, _Detection of a quantum particle on a lattice under repeated projective measurements_ , _Physical Review A_ 91 (2015) , [1410.8701].
  * (55) C. Anastopoulos and N. Savvidou, _Time-of-arrival probabilities and quantum measurements_ , _Journal of Mathematical Physics_ 47 (Dec., 2006) 122106–122106, [quant-ph/0509020].
  * (56) N. Erez, G. Gordon, M. Nest and G. Kurizki, _Thermodynamic control by frequent quantum measurements_ , _Nature_ 452 (Apr., 2008) 724–727, [0804.2178].
  * (57) H. Friedman, D. A. Kessler and E. Barkai, _Quantum walks: The first detected passage time problem_ , _Phys. Rev. E_ 95 (Mar., 2017) 032141, [1611.05676].
  * (58) A. Bhattacharya, P. Nandy, P. P. Nath and H. Sahu, _Operator growth and Krylov construction in dissipative open quantum systems_ , _JHEP_ 12 (2022) 081, [2207.05347].
  * (59) C. Liu, H. Tang and H. Zhai, _Krylov complexity in open quantum systems_ , _Phys. Rev. Res._ 5 (2023) 033085, [2207.13603].
  * (60) B. Bhattacharjee, X. Cao, P. Nandy and T. Pathak, _Operator growth in open quantum systems: lessons from the dissipative SYK_ , _JHEP_ 03 (2023) 054, [2212.06180].
  * (61) A. Bhattacharya, P. Nandy, P. P. Nath and H. Sahu, _On Krylov complexity in open systems: an approach via bi-Lanczos algorithm_ , _JHEP_ 12 (2023) 066, [2303.04175].
  * (62) B. Bhattacharjee, P. Nandy and T. Pathak, _Operator dynamics in Lindbladian SYK: a Krylov complexity perspective_ , 2311.00753.
  * (63) B. Skinner, J. Ruhman and A. Nahum, _Measurement-Induced Phase Transitions in the Dynamics of Entanglement_ , _Phys. Rev. X_ 9 (2019) 031009, [1808.05953].
  * (64) R. Suzuki, J. Haferkamp, J. Eisert and P. Faist, _Quantum complexity phase transitions in monitored random circuits_ , 2305.15475.
  * (65) E. C. G. Sudarshan and B. Misra, _The zeno’s paradox in quantum theory_ , _Journal of Mathematical Physics_ 18 (april, 1977) 756–763.
  * (66) P. Facchi, V. Gorini, G. Marmo, S. Pascazio and E. Sudarshan, _Quantum zeno dynamics_ , _Physics Letters A_ 275 (2000) 12–19.
  * (67) P. Facchi and S. Pascazio, _Quantum zeno dynamics: mathematical and physical aspects_ , _Journal of Physics A: Mathematical and Theoretical_ 41 (oct, 2008) 493001, [0903.3297].
  * (68) S. W. Gaaf and E. Jarlebring, _The infinite bi-lanczos method for nonlinear eigenvalue problems_ , _SIAM Journal on Scientific Computing_ 39 (2017) S898–S919, [https://doi.org/10.1137/16M1084195].
  * (69) Z. Bai, J. Demmel, J. Dongarra, A. Ruhe and H. van der Vorst, _Templates for the solution of algebraic eigenvalue problems: A practical guide_ , _SIAM, Philadelphia_ (2000) .
  * (70) M. Gruning, A. Marini and X. Gonze, _Implementation and testing of lanczos-based algorithms for random-phase approximation eigenproblems_ , _Computational Materials Science_ 50 (2011) 2148–2156.
  * (71) C. Zheng and D. Li, _Distinguish between typical non-hermitian quantum systems by entropy dynamics_ , _Scientific Reports_ 12 (2022) .
  * (72) B. N. Parlett and D. S. Scott, _The lanczos algorithm with selective orthogonalization_ , _Mathematics of Computation_ 33 (1979) 217–238.
  * (73) H. D. Simon, _The lanczos algorithm with partial reorthogonalization_ , _Mathematics of Computation_ 42 (1984) 115–142.
  * (74) V. Delgado and J. G. Muga, _Arrival time in quantum mechanics_ , _Phys. Rev. A_ 56 (Nov., 1997) 3425–3435, [quant-ph/9704010].
  * (75) M. Alishahiha and S. Banerjee, _A universal approach to Krylov state and operator complexities_ , _SciPost Phys._ 15 (2023) 080, [2212.10583].
  * (76) A. H. Karamlou et al., _Quantum transport and localization in 1d and 2d tight-binding lattices_ , _npj Quantum Inf._ 8 (2022) 35, [2107.05035].
  * (77) K. Sanderson, _Quantum dots go large_ , _Nature_ 459 (2009) 760–761.
  * (78) A. Baidya, C. Jana, R. Loganayagam and A. Rudra, _Renormalization in open quantum field theory. Part I. Scalar field theory_ , _JHEP_ 11 (2017) 204, [1704.08335].
  * (79) M. Geracie, F. M. Haehl, R. Loganayagam, P. Narayan, D. M. Ramirez and M. Rangamani, _Schwinger-Keldysh superspace in quantum mechanics_ , _Phys. Rev. D_ 97 (2018) 105023, [1712.04459].
  * (80) A. Bhattacharya, R. N. Das, B. Dey and J. Erdmenger, _Spread complexity and localisation as probes of PT breaking (to appear)_ , .
  * (81) S.-K. Jian, C. Liu, X. Chen, B. Swingle and P. Zhang, _Measurement-Induced Phase Transition in the Monitored Sachdev-Ye-Kitaev Model_ , _Phys. Rev. Lett._ 127 (2021) 140601, [2104.08270].
  * (82) S. Antonini, B. Grado-White, S.-K. Jian and B. Swingle, _Holographic measurement and quantum teleportation in the SYK thermofield double_ , _JHEP_ 02 (2023) 095, [2211.07658].
  * (83) E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, _A bulk manifestation of Krylov complexity_ , _JHEP_ 08 (2023) 213, [2305.04355].