# A universal approach to Krylov State and Operator complexities

Mohsen Alishahihaa and, Souvik Banerjeeb,c<EMAIL_ADDRESS><EMAIL_ADDRESS>a School of Physics, Institute for
Research in Fundamental Sciences (IPM),
P.O. Box 19395-5531, Tehran, Iran
b Institut für Theoretische Physik und Astrophysik, Julius-Maximilians-
Universität Würzburg,
Am Hubland, 97074 Würzburg, Germany
c Würzburg-Dresden Cluster of Excellence ct.qmat

###### Abstract

We present a general framework in which both Krylov state and operator
complexities can be put on the same footing. In our formalism, the Krylov
complexity is defined in terms of the density matrix of the associated state
which, for the operator complexity, lives on a doubled Hilbert space obtained
through the channel-state map. This unified definition of complexity in terms
of the density matrices enables us to extend the notion of Krylov complexity,
to subregion or mixed state complexities and also naturally to the Krylov
mutual complexity. We show that this framework also encompasses nicely the
holographic notions of complexity.

##  I Introduction

Quantum mechanics provides us with two apparently different yet natural ways
to understand the complexity of a system through time-evolution. In the
Schrödinger picture which allows the state to be time-dependent, complexity
measures the mixing of the initial state with other states through the time-
evolution, which for a time independent Hamiltonian $H$ takes the form

$|\psi(t)\rangle=e^{iHt}|\psi(0)\rangle=\sum_{n=0}^{\infty}\frac{(it)^{n}}{n!}|\psi^{(n)}\rangle,$
(1)

where $|\psi^{(n)}\rangle\equiv H^{n}|\psi(0)\rangle$. Thus, in this notation,
complexity of the final state amounts to understanding the spread of the
corresponding wavefunction in a fixed basis. The choice of an optimal basis
here is a bit tricky. One interesting way proposed in Balasubramanian  _et
al._ (2022) where the minimization of the spreading of the wavefunction
determines the optimal basis. This notion of state complexity was coined as
the “spread complexity”.

Alternatively, one may wish to switch to the Heisenberg picture where the time
evolution is attributed to the operator instead. This is realised by noting
the expectation value of an operator in the complexified state

$\langle\psi(t)|{\cal O}|\psi(t)\rangle=\langle\psi(0)|e^{-iHt}{\cal
O}e^{iHt}|\psi(0)\rangle$ (2)

and immediately recognizing the time dependent operator

${\cal O}(t)=e^{-iHt}{\cal
O}e^{iHt}\,=\sum_{n=0}^{\infty}\frac{(it)^{n}}{n!}{\cal O}^{(n)},$ (3)

where ${\cal O}^{(n)}$ are understood as nested structures of operators,
${\cal O}^{(n)}\equiv[H,\cdots,[H,{\cal O}]\cdots]$ and provide the notion of
operator complexity by interpreting ${\cal O}(t)$ as the operator wave
function evolved by a Liouvillian superoperator ${\cal L}$

${\cal O}(t)=e^{i{\cal L}t}{\cal O},$ (4)

with ${\cal L}=[H,\cdot]$. In this notation, ${\cal O}^{(n)}\equiv{\cal
L}^{n}{\cal O}$ which determines the mixing of operators. In order to compute
complexity corresponding to the growth of the operator, one uses the Lanczos
algorithm Lanczos (1950) to construct an optimal basis Viswanath and Müller
(1994), known in literature as the Krylov basis. The corresponding operator
complexity is termed as the Krylov complexity Parker  _et al._ (2019).

Last but not the least, in the present decade there has been one more entry in
the world of complexity in form of the holographic complexity. As its name
suggests, this notion of complexity arose out of the curiosity to understand
the interior of a black hole spacetime in the light of AdS/CFT correspondence
Maldacena (1998); Witten (1998); Gubser  _et al._ (1998). In particular, the
fact that the volume the interior of the black hole keeps growing even after
attaining thermal equilibrium Susskind (2016, 2018) is very much reminiscent
of the nature of complexity of a finite entropic fast-scrambling system.

Motivated by this striking similarity, a holographic definition of complexity
was proposed as the volume of the maximal slice in the interior of the black
hole. This proposal is celebrated in the name “Complexity = Volume” (CV)
conjecture Susskind (2016); Stanford and Susskind (2014). An efficient
formalism to study this interior volume in two dimensional theory of gravity
was developed in Iliesiu  _et al._ (2022); Alishahiha  _et al._ (2022);
Alishahiha and Banerjee (2022) which produces the expected behaviour of the
late-time linear growth and eventual saturation of complexity.

This observation was formalized in a more general context in Alishahiha
(2022), which, based on the Eigenstate Thermalization Hypothesis (ETH) Deutsch
(1991); Srednicki (1994), classified the possible candidates for complexity
which exhibit a linear growth at late time. It was also shown that this class
of observables naturally includes the expectation value of the quenched length
operator defined in Iliesiu  _et al._ (2022); Alishahiha  _et al._ (2022);
Alishahiha and Banerjee (2022).

It is a daunting task to bring all these apparently different notions of
complexity under the same umbrella. And this is precisely the motivation of
the present work.

In the process of doing so, we will first present, in section II, a general
framework to study complexity of a given state. This state can either be a
given autonomous state or could as well be a state evolved under the evolution
by a general Hermitian operator which is not necessarily the Hamiltonian of
the system. In either case, for a given orthonormal and ordered basis, we can
define a label operator and subsequently a number obtained by tracing the
former over the density matrix corresponding to the given or the evolved
state. This number is minimized over the space of operators used to generate
sets of the orthonormal basis through Gram-Schmidt process, to define the
complexity of the state.

These operators can be identified as different quantum gates in accordance
with the quantum informatic definition of circuit complexity Nielsen and
Chuang (2010). When the state is created through a unitary evolution, this
definition of complexity naturally yields the Krylov complexity.

We show that, for the Hamiltonian evolution, the class of observables
identified in Alishahiha (2022), with the desired pole structure to produce
the late-time linear growth of complexity, can be identified, naturally, with
the expectation value of the label operator mentioned above. We further show
that our formalism also naturally provides the late time saturation of
complexity following the linear growth.

In section III, we develop the formalism to study operator complexity. The
structure of the label operator dictates a mapping of the space of operators
to a doubled Hilbert space endowed with a inner product structure. This
enables us to recast the operator complexity associated with any given
operator to the complexity of a state in the doubled Hilbert space. This
definition is also consistent with the Liouvillian evolution for operator
complexity.

From the perspective of quantum information, the mapping to the doubled
Hilbert space is a realisation of the channel-state map. However, we
demonstrate that it has a beautiful interpretation in axiomatic quantum field
theory and reveals a deeper structure of entanglement in the Hilbert space
which turns out to be pivotal in understanding the holographic notion of
complexity.

In section IV we discuss a novel biproduct of our generalized formalism for
studying the state and the operator complexities. Since we defined complexity
in terms of density matrices, it fits in as the ideal candidate to study
subregion complexity using reduced density matrix of a given subregion.
Definition of subregion Krylov complexity reveals some more interesting and
fundamental aspects connecting quantum entanglement and the growth of
complexity.

Finally, we conclude in section V with some interesting open questions and
some works in progress.

##  II Complexity : The general framework

Be it for a state or for an operator, the general strategy to compute
complexity comprises of the following steps - i) start with an initial state
(operator), ii) allow it to spread over some state (operator) basis via an
evolution generated by a time-independent Hermitian operator, iii) define a
quantity that could probe the spreading while iv) the most efficient spreading
is quantified by the minimization of the afore-mentioned quantity which is
equivalent to finding a basis for the state (operator) so that the spreading
becomes minimum.

In this section we will develop this general framework for state complexity.
In the section to follow, we will extend this algorithm to study operator
complexity.

### Complexity of a generic given state

Let us consider a quantum system described by a time independent Hamiltonian
whose eigenstates and eigenvalues are denoted by $|E_{a}\rangle$ and $E_{a}$,
respectively. Here $a=1,2,\cdots{\cal D}$ with ${\cal D}$ being the dimension
of the associated Hilbert space ${\cal H}$.

For a given Hermitian operator, ${\cal A}:{\cal H}\rightarrow{\cal H}$, one
can construct an orthonormal and ordered basis associated with any state of
this Hilbert space. Denoting the corresponding state by $|\psi\rangle$, the
ordered orthonormal basis $\\{|n\rangle,n=0,1,2,\cdots,{\cal D}_{\psi}-1\\}$
can be constructed using the Gram-Schmidt process. The first element of the
basis is the given state of the Hilbert state $|0\rangle=|\psi\rangle$ which
we assume to be normalized. Then the other elements are constructed
recursively as follows

$|\widehat{n+1}\rangle=({\cal A}-a_{n})|n\rangle-b_{n}|n-1\rangle$ (5)

where $|n\rangle=b_{n}^{-1}|{\hat{n}}\rangle$ and

$a_{n}=\langle n|{\cal
A}|n\rangle,\;\;\;\;\;\;\;\;\;b_{n}=\sqrt{\langle{\hat{n}}|{\hat{n}}\rangle}$
(6)

This recursive procedure stops whenever $b_{n}$ vanishes which occurs for
$n={\cal D}_{\psi}$ defined as the dimension of subspace ${\cal H}_{\psi}$
expanded by the basis $\\{|n\rangle\\}$. The dimension of ${\cal H}_{\psi}$ is
in general smaller than the dimension of the full Hilbert space: ${\cal
D}_{\psi}<{\cal D}$. Note that this procedure produces an orthogonal basis
together with coefficients $a_{n}$ and $b_{n}$ known as the Lanczos
coefficients Lanczos (1950).

Since the basis of the subspace ${\cal H}_{\psi}$ is an ordered basis by
construction, one can label any element of the subspace by a number which
amounts to defining a label operator as

$\ell=\sum_{n=0}^{{\cal D}_{\psi}-1}\,c_{n}|n\rangle\langle n|\,,$ (7)

for arbitrary functions $c_{n}$ which is the “label” associated with the state
$|n\rangle$. Note that for $n>n^{\prime}$ one assumes $c_{n}>c_{n^{\prime}}$.
Since the basis $\\{|n\rangle\\}$ is already ordered, a natural choice for the
coefficient $c_{n}$ is $c_{n}=n$.

By construction, the basis $\\{|n\rangle\\}$ defines a complete basis for the
subspace ${\cal H}_{\psi}$. Therefore, any state $|\phi\rangle\in{\cal
H}_{\psi}$ can be expanded as

$|\phi\rangle=\sum_{n=0}^{{\cal D}_{\psi}-1}\,\phi_{n}|n\rangle,\;\;\;\;{\rm
with}\,\;\;\sum_{n=0}^{{\cal D}_{\psi}-1}\,|\phi_{n}|^{2}=1\,.$ (8)

The expectation value of the label operator in this state $|\phi\rangle$ is
given by

$\displaystyle\langle\phi|\ell|\phi\rangle={\rm
Tr}(\ell\rho_{\phi})=\sum_{n=0}^{{\cal D}_{\psi}-1}\,n|\phi_{n}|^{2}\,,$ (9)

where $\rho_{\phi}=|\phi\rangle\langle\phi|$ is the density matrix associated
with the state $|\phi\rangle$.

Using this expression, one can assign a “spreading number” to a given state
$|\phi\rangle$ in terms of its density matrix as

${\cal C}_{\phi}={\rm Tr}(\ell\rho_{\phi})\,.$ (10)

As its name suggests, (10) can be thought of as a quantity that measures the
spreading of state $|\phi\rangle$ in the orthogonal basis $\\{|n\rangle\\}$.
Note that, in this notation, the spreading of the state
$|m\rangle\in\\{|n\rangle\\}$ is $m$. Since the above definition of spreading
number is given in terms of the density matrix, it can be naturally extended
for mixed states as well. We will come back to this point later in this paper.

For large ${\cal D}_{\psi}$ one would expect that for a typical state given by
(8) with maximum spreading, the spreading is distributed statistically over
all elements of the basis with equal probability:
$|\phi_{n}|^{2}\sim\frac{1}{{\cal D}_{\psi}}$Rabinovici  _et al._ (2021). From
this argument, the maximum value for spreading of a state can be estimated as

${\rm Tr}(\ell\rho_{\phi})\sim\frac{{\cal D}_{\psi}}{2}\,.$ (11)

Although already evident from our construction, it is worth emphasizing that
the spreading number we have associated with the state $|\phi\rangle\in{\cal
H}_{\psi}\subset{\cal H}$ depends on two ingredients: the original state
$|\psi\rangle$ and the operator ${\cal A}$ by which the ordered basis is
constructed. This is very reminiscent of the computational complexity in
quantum information theory Nielsen and Chuang (2010) in the sense that the
original state $|\psi\rangle$ plays the role of the reference state while
operator ${\cal A}$, or equivalently the ordered basis $\\{|n\rangle\\}$, can
be thought of as quantum gates.

Endowed with this interesting identification, we can go ahead to define
complexity using (10) as follows. After fixing the reference state,
$|\psi\rangle$, one can construct the ordered basis using different Hermitian
operators which can be thought of as considering different gates. If one can
find a Hermitian operator among all possible operators, the basis constructed
from which minimizes the spreading number (10), then the corresponding
spreading number we will define as the complexity of the state. Therefore, in
this context, finding complexity of a state boils down to the problem of
finding the optimal operator, $\cal A_{\rm opt}$.

We note, however, that in general, for given reference and target states, this
minimization procedure to find $\cal A_{\rm opt}$ is a complicated program.
However, we will show now that for particular cases in which the state is
obtained by a unitary transformation from the reference state, this can be
successfully achieved.

### Complexity following a unitary evolution

So far we considered a typical autonomous state $|\phi\rangle$ without knowing
apriori whether this state was obtained through any dynamical process from an
initial reference state. We will now focus on the case where the desired
(target) state is obtained from a reference state via a unitary
transformation. In this case it is straightforward to generalize the notion of
spreading number as follows.

Let us consider the following state

$|\phi\rangle\equiv|\psi(s)\rangle=U({\tilde{\cal A}},s)|\psi\rangle\,,$ (12)

whose evolution in the parameter space $s$ is governed by the Schrödinger-
like equation

$i\frac{d}{ds}|\psi(s)\rangle={\cal U}({\tilde{\cal A}},s)|\psi(s)\rangle$
(13)

with ${\cal U}({\tilde{\cal A}},s)=i\frac{d}{ds}U({{\tilde{\cal
A}}},s)\,U^{-1}({\tilde{\cal A}},s)$. Here the unitary operator $U$ which
evolves the state from an initial reference state
$|\psi\rangle\equiv|\psi(0)\rangle$, is a function of the Hermitian operator
${\tilde{\cal A}}$ and the parameter $s$ which defines the flow through (13).
The latter can be chosen such that at $s=0$ one has $U=1$. Following its
definition given in (10), the spreading number at any arbitrary point $s$ on
the flow is then given by

${\cal C}(s)={\rm Tr}(\ell\rho(s))=\sum_{n=0}^{{\cal D}_{\psi}-1}\,n\,\langle
n|\psi(s)\rangle\langle\psi(s)|n\rangle\,,$ (14)

where $\rho(s)=|\psi(s)\rangle\langle\psi(s)|$ is the density of state at $s$
and the label operator $\ell$ is constructed using a set of orthonormal basis
$\\{|n\rangle\\}$ which is complete in ${\cal H}_{\psi}$. This basis can in
principle be constructed from any Hermitian operator ${\cal A}$ acting on the
Hilbert space ${\cal H}_{\psi}$ using the Gram- Schmidt procedure as in (5).
One can expand the target state in the same basis, the coefficients of this
expansion being functions of $s$ as

$|\psi(s)\rangle=\sum_{n=0}^{{\cal D}_{\psi}-1}\,\psi_{n}(s)\,|n\rangle\,,$
(15)

which yields, from (14),

${\cal C}(s)=\sum_{n=0}^{{\cal D}_{\psi}-1}\,n|\psi_{n}(s)|^{2}\,.$ (16)

In principle, the coefficients $\psi_{n}(s)$ can be recursively read off from
the equation (12) if one is equipped the full knowledge of Lanczos
coefficients $a_{n}$ and $b_{n}$.

Let us now denote the eigentstates and eigenvalues of the operator
${\tilde{\cal A}}$ by $\\{|\alpha_{i}\rangle\\}$ 111$\\{|\alpha_{i}\rangle\\}$
do not in general form an ordered basis. and $\alpha_{i}$, respectively. Since
${\tilde{\cal A}}$ is Hermitian, these eigenstates form a complete set of
states in ${\cal H}_{\psi}$. Using this fact, the spreading number (14) can be
recast into the following form

${\cal
C}(s)=\sum_{\alpha_{1},\alpha_{2}}\,U(\alpha_{1},s)U^{*}(\alpha_{2},s)\rho_{0}(\alpha_{1},\alpha_{2})\langle\alpha_{1}|\ell|\alpha_{2}\rangle\,,$
(17)

where
$\rho_{0}(\alpha_{1},\alpha_{2})=\langle\alpha_{1}|\psi(0)\rangle\langle\psi(0)|\alpha_{2}\rangle$
is the density matrix in the $\alpha$-basis.
$\langle\alpha_{1}|\ell|\alpha_{2}\rangle$ are the matrix elements of the
label operator in eigenvectors of the Hermitian operator $\tilde{\cal A}$
given by

$\langle\alpha_{1}|\ell|\alpha_{2}\rangle=\sum_{n=0}^{{\cal
D}_{\psi}}\,n\langle\alpha_{1}|n\rangle\langle n|\alpha_{2}\rangle\,.$ (18)

It is worth noting here that these matrix elements can be computed directly in
the continuum limit using the recursion relation given in (5). Expanding
$|n\rangle$ in the basis of eigenvectors of the operator ${\tilde{\cal A}}$

$|n\rangle=\sum_{i}c_{n}(\alpha_{i})|\alpha_{i}\rangle\,,$ (19)

the equation (5) reads

$\alpha
c_{n}(\alpha)=a_{n}c_{n}(\alpha)+b_{n}c_{n-1}(\alpha)+b_{n+1}c_{n+1}(\alpha),$
(20)

where $c_{n}(\alpha)=\langle\alpha|n\rangle$, $|\alpha\rangle$ being a
particular eigenstate of $\tilde{\cal A}$ with non-degenerate eigenvalue
$\alpha$. This equation can be thought of as a time independent Schrödinger
equation for which one can find the wave functions $c_{n}(\alpha)$
recursively. Using these wave functions, one can further compute the matrix
elements (18).

In order to obtain an expression for the matrix elements in the continuum
limit, one can first rescale $c_{n}(\alpha)\rightarrow(i)^{n}c_{n}(\alpha)$
and then set $x=n\epsilon,\,b(x)=2\epsilon b_{n},a(x)=a_{n}$ and
$c_{n}(\alpha)=c(x,\alpha)$. Thereafter, expanding (20) upto the leading order
in $\epsilon$ yields a much simpler equation

$-i(\alpha-a(y))f(y,\alpha)=\partial_{y}f(y,\alpha),$ (21)

where $\partial_{y}=b(x)\partial_{x}$ and
$f(y,\alpha)=\sqrt{b(x)}c(x,\alpha)$. The equation (21) can be readily solved
to obtain

$f(y,\alpha)=f(0,\alpha)\,e^{-i\alpha
y+i\int_{0}^{y}a(y^{\prime})dy^{\prime}}\,.$ (22)

Using this solution, finally one arrives at an expression for the matrix
element (18) as

$\langle\alpha_{1}|\ell|\alpha_{2}\rangle=\frac{1}{\epsilon^{2}}\int
dy\,x(y)\,e^{-i\alpha_{12}y}$ (23)

where $\alpha_{12}=\alpha_{1}-\alpha_{2}$. As its form suggests, to evaluate
the matrix elements explicitly, one needs to know $x$ as a function of $y$
which can be obtained from the relation $\frac{dx}{b(x)}=dy$. Given $b(x)$,
the integral can be performed to find $x(y)$ and subsequently, the explicit
expression for the desired matrix elements which determine the spread in the
continuum limit.

### Minimising the spreading and complexity:

Since the flow itself is generated in terms of a particular Hermitian operator
$\tilde{\cal A}$, it is natural to consider a special case when ${\tilde{\cal
A}}={\cal A}$ that is when ordered basis is constructed using the same
Hermitian operator that generates the unitary flow (12). One can show that in
this case the spreading number is minimum compared to any other choice of
Hermitian operator or equivalently, for any other choice of basis. This
follows directly from theorem 1 of Balasubramanian  _et al._ (2022), by
setting $n=1$.

Therefore the resultant spreading number obtained with the choice of
${\tilde{\cal A}}={\cal A}$ can be considered as the complexity of the
s-evolved target state. Indeed, looking at equation (14) one finds that the
minimum of this with $\\{|n\rangle\\}$ being the orthonormal basis generated
by the Hermitian operator ${\cal A}$ is of the same form as that of the Krylov
complexity Parker  _et al._ (2019); Barbón  _et al._ (2019); Jian  _et al._
(2021); Rabinovici  _et al._ (2021) (see also Fan (2022a); Mück and Yang
(2022); Fan (2022b)).

It is worth mentioning that due to the property of the trace, the complexity
defined above at given $s$ may also be given by

${\cal C}(s)={\rm Tr}(\ell(s)\rho),$ (24)

that means, we keep the initial reference state fixed and construct the
ordered basis using the Hermitian operator ${\cal A}$ appearing in the
evolution operator ${\cal U}({\cal A},s)$.

### The time-evolution as an example:

As a particularly interesting and rather physical example of the generalized
framework developed above, let us consider the case where the Hermitian
operator ${\cal A}$ is the Hamiltonian of the quantum system. This is the
example mainly studied in the literature to compute Krylov complexity. In this
case, starting with an initial state $|\psi(0)\rangle$, the dynamics is given
by the Schrödinger equation

$|\psi(t)\rangle=e^{iHt}|\psi(0)\rangle\,.$ (25)

Then the density matrix associated with this state at any time $t$ is given by

$\rho(t)=|\psi(t)\rangle\langle\psi(t)|=e^{iHt}\,\rho(0)\,e^{-iHt}$ (26)

where $\rho(0)=|\psi(0)\rangle\langle\psi(0)|$, by which the complexity is

${\cal C}(t)={\rm Tr}(\ell\rho(t))=\sum_{n=0}^{{\cal
D}_{\psi}-1}\,n\,|\psi_{n}(t)|^{2}\,,$ (27)

where $\psi_{n}(t)$ is given by the following expression where the state is
expanded in terms of the ordered basis $\\{|n\rangle\\}$ (known as Krylov
basis in this case)

$|\psi(t)\rangle=\sum_{n=0}^{{\cal D}_{\psi}-1}\,\psi_{n}(t)|n\rangle,$ (28)

Using the Schrödinger equation, one can deduce an equation for $\psi_{n}(t)$
that can be solved recursively using the knowledge of Lanczos coefficients
$a_{n}$ and $b_{n}$. Actually for a chaotic system these coefficients, as
functions of $n$, follow a universal behavior that includes a linear growth
Parker  _et al._ (2019) followed by a saturation Barbón  _et al._ (2019); Jian
_et al._ (2021); Rabinovici  _et al._ (2021). These behaviors give an early-
time exponential and the late times linear growths for complexity respectively
Barbón  _et al._ (2019); Jian  _et al._ (2021); Rabinovici  _et al._ (2021).

In order to proceed further to explore the complexity associated with the
Hamiltonian evolution, let us assume that the Hamiltonian of the system has a
continuous spectrum. Therefore, using the energy eigenstates, the Krylov
complexity (27) may be recast into the following form

${\cal C}(t)=\int
dE_{a}\,dE_{b}\,e^{i(E_{a}-E_{b})t}\rho_{0}(E_{a},E_{b})\,A(E_{a},E_{b})\,,$
(29)

where

$\displaystyle\rho_{0}(E_{a},E_{b})=\langle
E_{a}|\rho(0)|E_{b}\rangle,\;\;\;\;A(E_{a},E_{b})=\langle
E_{a}|\ell|E_{b}\rangle\,,$

which is the same expression proposed for complexity in Alishahiha (2022). In
Alishahiha (2022), the left hand side of (29) was identified as complexity
provided the function $A(E_{a},E_{b})$ had a double pole structure at late-
time, i.e. in the coincident limit $E_{a}\rightarrow E_{b}$, thereby
restricting the function $A(E_{a},E_{b})$ to a class of quantum expectation
values which violates the ETH. We shall argue below that this is naturally the
case when we consider the expectation value of the label operator which turns
out to be the atypical non-local operator existence whereof was postulated in
Alishahiha (2022).

As we have already mentioned the saturation of Lanczos results in a linear
growth at late times. In the notation of (29) this behavior at late times
imposes a condition on the function $A(E_{a},E_{b})$ to have a double pole
structure

$A(E_{a},E_{b})=-\frac{a(E)}{\omega^{2}}+{\rm
local\,terms},\;\;\;\;\;\;\;\;{\rm for}\,\,\omega\rightarrow 0\,,$ (31)

where $\omega=E_{a}-E_{b}$ and $2E=E_{a}+E_{b}$, in agreement with the
proposal of Alishahiha (2022). We can derive the surprising connection between
the double pole structure of the function $A(E_{a},E_{b})$ and the saturation
of Lanczos coefficients from the expectation value of the label operator in
the continuous limit given in (23). When the Lanczos coefficient $b(x)$
saturates to a constant, one gets $x=y$. This yields, from the expression
(23),

$\langle
E_{a}|\ell|E_{b}\rangle=\frac{1}{\epsilon^{2}}\int_{0}^{\Lambda}dy\,x(y)\,e^{-i\omega
y}=-\frac{1}{\omega^{2}}\,(1-e^{-i\omega\Lambda})$ (32)

where $\Lambda$ is a cutoff. This is the double pole structure we expect to
get from $A$-function that generates linear growth for complexity at late
time. For $\Lambda\rightarrow\infty$ the above integral may be recast into the
following form

$\langle
E_{a}|\ell|E_{b}\rangle=\frac{1}{\epsilon^{2}}\int_{0}^{\infty}dy\,x(y)\,e^{-i\omega
y}=\frac{1}{\epsilon^{2}}\,i\frac{d}{d\omega}\delta(\omega)\,.$ (33)

On the other hand, in the regime when the Lanczos coefficient $b(x)$ exhibits
a linear growth, say, $b(x)=2\lambda x$, one obtains the relation
$x=e^{2\lambda y}$, which upon inserting in equation (23) yields

$\langle
E_{a}|\ell|E_{b}\rangle=\frac{1}{\epsilon^{2}}\delta(\omega+2i\lambda)\,.$
(34)

Plugging this result in (29) one finds early time exponential growth ${\cal
C}(t)\sim e^{2\lambda t}$ as the case for chaotic system.

For a more general case of $x=y^{m}$ for $m>1$, one gets

$\langle
E_{a}|\ell|E_{b}\rangle=\frac{1}{\epsilon^{2}}\,i^{m}\frac{d^{m}}{d\omega^{m}}\delta(\omega)\,,$
(35)

which can happen at early times in a non-chaotic system. Therefore, from (29)
one gets ${\cal C}(t)\sim t^{m}$ which can be interpreted as the early time
power-law growth for complexity.

An advantage to study the matrix elements of the label operator is that the
universal behavior of Lanczos coefficients associated with a Hermitian
operator may be studied without refereeing to a particular dynamics by which a
state evolves.

To explore the significance of the label operator better, we note that for an
arbitrary operator
$\Lambda=\sum_{a}\lambda_{a}|\lambda_{a}\rangle\langle\lambda_{a}|$ where
$\lambda_{a}$ and $|\lambda_{a}\rangle$ are its eigenvalues and eigenstates,
respectively, one can compute ${\rm Tr}(\Lambda\rho(t))$ which has the same
form as that of (29), though in this case it is not guaranteed that the
functions $A(E_{a},E_{b})=\langle E_{a}|\Lambda|E_{b}\rangle$ exhibit double
pole structures at late times. Actually we would expect that for a typical
operator $\Lambda$ this satisfies the ETH ansatz and thus ${\rm
Tr}(\Lambda\rho(t))$ gives a time independent quantity interpreted as the
average of the operator $\Lambda$. Therefore in order to have the notion of
complexity it is important to compute (29) specifically for the label operator
of an ordered basis. This justifies further the uniqueness of our definition
of complexity in terms of the label operator.

The other important element of the formula given in (29) is the density matrix
$\rho_{0}(E_{a},E_{b})$. At the leading order in the dimension of the Hilbert
space, the density matrix $\rho_{0}(E_{a},E_{b})$ is factorized, though in
general it has a form

$\rho_{0}(E_{a},E_{b})=\rho(E_{a})\rho(E_{b})+\rho_{c}(E_{a},E_{b})\,.$ (36)

Here $\rho_{\rm c}$ represents the connected term meaning that it cannot be
written in a factorized form of $g_{1}(E_{1})g_{2}(E_{2})$ with $g_{1,2}$
being arbitrary functions of energy. While this object remains a silent
spectator in the discussion of the late time linear growth of complexity
above, we will now argue that it plays a pivotal role in understanding
saturation phase of complexity at later times.

### On the saturation of complexity at late time:

In what follows we would like to present a general form of complexity in
$\tau$ scaling limit in which we take $\\{t,{\cal
D}_{\psi}\\}\rightarrow\infty$ while keeping $\tau=t\,{\cal D}_{\psi}^{-1}$
fixed.

We proceed by rescaling $\rho_{0}(E_{a},E_{b})={\cal
D}_{\psi}^{2}\tilde{\rho}_{0}(E_{a},E_{b})$ and switching to the $(E,\omega)$
coordinates. The latter is convenient for studying the coincident limit
$E_{a}\rightarrow E_{b}$ relevant for the late time limit
$t\rightarrow\infty$. With this, the complexity (29), up to an appropriate
normalization, reads

${\cal C}(t)=\int_{0}^{\infty}dE\int_{-\infty}^{\infty}d\omega\,e^{i\omega
t}\tilde{\rho}_{0}(E,\omega)A(E,\omega)\,.$ (37)

As we have already demonstrated above, the saturation phase of the Lanczos
coefficients, which corresponds to the $\tau$ scaling limit, results in a
double pole structure for $A(E,\omega)$. Assuming to have an expression for
complexity growth at leading order consistent with Lloyld’s bound one arrives
at Alishahiha (2022)

$A(E,\omega)=-\frac{\sqrt{E}}{\tilde{\rho}(E)\,\omega^{2}}+{\rm
Local\,terms}\,.$ (38)

On the other hand, at the $\tau$ scaling limit, the physics is dominated by
correlations between nearby energy levels. In this limit, the connected part
of the matrix elements of the density matrix denoted by
$\rho_{c}(E_{a},E_{b})$ in (36), are described by the universal sine-kernel
formula Mehta (1991). More precisely, one has

$\tilde{\rho}_{0}(E,\omega)=\tilde{\rho}(E)^{2}+\frac{{\tilde{\rho}}(E)}{{\cal
D}_{\psi}}\delta(\omega)-\frac{\sin^{2}({\cal
D}_{\psi}{\tilde{\rho}}(E)\omega)}{({\cal D}_{\psi}\omega)^{2}}\,.$ (39)

Putting everything together, one arrives at

$\displaystyle{\cal C}(t)=C_{0}-{\cal
D}^{-1}_{\psi}\int_{0}^{\infty}dE\sqrt{E}\int_{-\infty}^{\infty}d\omega\,\frac{e^{i\omega
t}}{\omega^{2}}\,\delta(\omega)$ (40) (41)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;-\int_{0}^{\infty}dE\sqrt{E}\tilde{\rho}(E)\int_{-\infty}^{\infty}d\omega\,\frac{e^{i\omega
t}}{\omega^{2}}$ (42) (43)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\left(1-\frac{\sin^{2}({\cal
D}_{\psi}{\tilde{\rho}}(E)\omega)}{({\cal
D}_{\psi}\tilde{\rho}(E)\omega)^{2}}\right),$ (44)

where $C_{0}$ is a constant. It is exactly the same expression which was
obtained for (super)JT gravity in which the density $\tilde{\rho}$ is known
explicitly Iliesiu  _et al._ (2022); Alishahiha  _et al._ (2022); Alishahiha
and Banerjee (2022). In this equation, the first line which is divergent in
general and needs to be regularized by a cutoff, is nevertheless time
independent and does not contribute at late times. Therefore, in what follows
we only need to consider the last term. Setting $\omega t=\xi$, this term
yields

$\displaystyle{\cal C}(t)=-{\cal
D}_{\psi}\,\tau\int_{0}^{\infty}dE\sqrt{E}\tilde{\rho}(E)\int_{-\infty}^{\infty}d\xi\,\frac{e^{i\xi}}{\xi^{2}}$
(45) (46)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\left(1-\frac{\sin^{2}(\frac{{\tilde{\rho}}(E)}{\tau}\,\xi)}{(\frac{{\tilde{\rho}}(E)}{\tau}\,\xi)^{2}}\right),$
(47)

which vanishes for $\tau>\tilde{\rho}(E)$ for any density function
$\tilde{\rho}$ Iliesiu  _et al._ (2022). This behaviour is quite universal in
the sense that the saturation phase does not depend on the details of the
model. On the other hand, for $\tau<\tilde{\rho}(E)$ one can expand the sine
function in terms of exponential functions and performing the complex integral
over a contour excluding the poles in the lower half plane one obtains Iliesiu
_et al._ (2022); Alishahiha  _et al._ (2022)

$\displaystyle{\cal C}(t)\approx-\frac{2\pi{\cal
D}_{\psi}}{3}\,\int_{E_{\tau}}^{\infty}dE\sqrt{E}\,\tilde{\rho}^{2}(E)\left(1-\frac{\tau}{2\tilde{\rho}(E)}\right)^{3}\;\;\;$
(48)

where $E_{\tau}$ can be read off from the equation
$\tau=\tilde{\rho}(E_{\tau})$. Although to perform the integration over $E$
one needs to know the explicit form of the density $\tilde{\rho}$, one can
still extract certain universal behavior for the complexity. In particular for
$\tau\ll 1$ one gets ${\cal C}(t)\sim-\alpha_{0}{\cal D}_{\psi}+\alpha_{1}t$
with $\alpha_{0,1}$ being order one constants. Although the saturation phase
occurs at $\tau\sim 1$ limit is also universal, the actual way the complexity
approaches the saturation phase is model dependent and is fixed as soon as the
density is fixed.

To conclude, we note that for a chaotic system, the complexity has a universal
behavior starting with early times exponential growth and linear growth at
late times followed by a saturation. It is worth noting that while the early
exponential and the late times linear growths are described by the behavior of
the $A(E,\omega)$ function, already at leading order, the saturation phase is
the consequence of the contribution of the connected part of the density-
density short range correlation known as sine-kernel.

This has to be compared with the numerical computations done e.g. in
Rabinovici  _et al._ (2022), where the saturation of complexity was due to the
descent phase of the Lanczos coefficients. It would be interesting to
understand and compare these two different approaches by which the complexity
reaches the saturation phase. An early saturation of complexity due to the
breaking of ETH might actually signal the chaotic nature of the system. Work
in this direction is in progress Alishahiha and Banerjee (2023) and we hope to
report on this soon.

##  III Complexity: The Operator-State correspondence

In this section we will extend the algorithm elaborated in the previous
section, to the study of complexity corresponding to growth of an operator. To
achieve this, generally, one starts with a reference operator and constructs
an ordered basis of operators using a Hermitian operator. Then one looks for
the spreading of the desired operator over this ordered basis.

Typically, in order to go through this procedure, one needs to define a proper
inner product in the space of operators. However, in what follows, we will
take a different approach so that the inner product will arise naturally.

### The doubled Hilbert space:

To compute complexity associated with the growth of an operator we will
essentially use the same procedure as that of the state complexity simply by
making use of the channel-state map Nielsen and Chuang (2010) which maps an
operator to a state.

Let us consider a quantum system described by the Hamiltonian $H$ with
eigenstates and eigenvalues $|E_{a}\rangle$ and $E_{a}$ respectively. The
corresponding Hilbert space is also denoted by ${\cal H}$ with dimension
${\cal D}$. In this set up, we will consider a generic operator ${\cal
O}:{\cal H}\rightarrow{\cal H}$ and a complete basis spanning the Hilbert
space, denoted by $\\{|i\rangle,i=1,2,\cdots{\cal D}\\}$. The matrix elements
of the operator in this basis is ${\cal O}_{ij}=\langle i|{\cal O}|j\rangle$.
Following the channel-state duality, one can then define an associated state,
through a linear bijection 222This is known in literature as the Choi-
Jamiolkowski isomorphism Jiang  _et al._ (2013)., in an auxiliary doubled
Hilbert space ${\cal H}_{d}={\cal H}\otimes{\cal H}$ as follows

$|\psi_{\cal
O}\rangle=\sum_{i,j=1}^{N}\varrho_{ij}\,|i\rangle\otimes|j\rangle\equiv\sum_{i,j=1}^{N}\varrho_{ij}\,|i,j\rangle\,.$
(49)

Here the density of state $\varrho_{ij}$ is proportional to the matrix
elements ${\cal O}_{ij}$ such that $\sum_{i,j}|\varrho_{ij}|^{2}=1$. More
precisely, one has

$\varrho_{ij}=\frac{{\cal O}_{ij}}{\sqrt{\sum_{i^{\prime},j^{\prime}}{\cal
O}_{i^{\prime}j^{\prime}}{\cal O}_{i^{\prime}j^{\prime}}}}\;.$ (50)

This map would naturally define an inner product between two operators

${\cal O}_{1}\cdot{\cal O}_{2}\equiv\langle\psi_{{\cal O}_{1}}|\psi_{{\cal
O}_{2}}\rangle.$ (51)

Once we have the state in the doubled Hilbert space, we can simply adopt our
previously developed formalism for state complexity, now for a state in the
doubled Hilbert space. For a given Hermitian operator ${\cal A}:{\cal
H}\rightarrow{\cal H}$ one can then readily construct an orthonormal and
ordered basis in the doubled Hilbert space with the identification of the
first state as $|0,0\rangle=|\psi_{\cal O}\rangle$. All other elements are
obtained using the algorithm given in (5), though, now in the doubled Hilbert
space

$|\widehat{n+1},\widehat{n+1}\rangle=({\cal A}_{d}-a_{n})|n,n\rangle-
b_{n}|n-1,n-1\rangle$ (52)

where $|n,n\rangle=b_{n}^{-1}|{\hat{n},\hat{n}}\rangle$ and

$a_{n}=\langle n,n|{\cal
A}_{d}|n,n\rangle,\;\;\;\;\;\;\;\;\;b_{n}=\sqrt{\langle{\hat{n}},\hat{n}|{\hat{n}},\hat{n}\rangle}\,.$
(53)

where ${\cal A}_{d}={\cal A}\otimes{\cal A}$ is an operator acting on the
doubled Hilbert space ${\cal A}_{d}:{\cal H}_{d}\rightarrow{\cal H}_{d}$. The
resulting ordered basis defines a subspace of the doubled Hilbert space
denoted by ${\cal H}_{d,\psi_{\cal O}}$, ${\cal D}_{\psi_{\cal O}}$ being the
dimension of this sub Hilbert space.

For a given operator ${\cal O}$, the spreading is now defined as

${\cal C}_{{\cal O}}={\rm Tr}(\ell\rho_{{\cal O}})$ (54)

where $\ell=\sum_{n=0}^{{\cal D}_{\psi_{\cal O}}}n|n,n\rangle\langle n,n|$ and
$\rho_{\cal O}=|\psi_{\cal O}\rangle\langle\psi_{\cal O}|$ is the density
matrix of the state $|\psi_{\cal O}\rangle$ associated with the operator
${\cal O}$.

Following this definition, it is then a straightforward task to compute
complexity of an operator that is obtained from a reference operator ${\cal
O}$ via a unitary evolution. Let us consider the following unitary
transformation for an operator

${\cal O}(s)={\cal U}({\cal A},s)\;{\cal O}\;{\cal U}^{-1}({\cal A},s)$ (55)

so that

$\rho_{\cal O}(s)=\sum_{i,j=1}^{{\cal D}}\sum_{i^{\prime}j^{\prime}=1}^{\cal
D}\;\varrho_{ij}(s)\varrho^{*}_{i^{\prime}j^{\prime}}(s)\;|i,j\rangle\langle
j^{\prime},i^{\prime}|$ (56)

where $\varrho_{ij}(s)=\langle i|{\cal O}(s)|j\rangle/\sqrt{\sum|{\cal
O}_{ij}|^{2}}$.

From this, using the definition of spreading in (54), one gets

${\cal C}(s)={\rm Tr}(\ell\rho_{{\cal O}}(s))=\sum_{n=0}^{{\cal D}_{\psi_{\cal
O}}-1}n|{\cal O}_{n}(s)|^{2}$ (57)

where

${\cal O}_{n}(s)=\sum_{i,j}^{{\cal D}}\varrho_{ij}(s)\langle n,n|i,j\rangle\,$
(58)

which implies to have the following expansion for the state in terms of
ordered basis in the doubled Hilbert space

$|\psi_{\cal O}(s)\rangle=\sum_{n=0}^{{\cal D}_{\psi_{\cal O}}-1}{\cal
O}_{n}(s)\,|n,n\rangle\,.$ (59)

One can also take the basis $\\{|i\rangle\\}$ to be the eigenstates of the
Hermitian operator ${\cal A}$. Note that, here we considered the evolution of
the operator by the same operator $\cal A$ used to construct the ordered basis
in the doubled Hilbert space. Following our argument in the case of state
complexity, this ensures that (57) represents minimum spreading and therefore
can be interpreted as the complexity of the target operator ${\cal O}_{s}$.

### The time evolution revisited:

As before, our main interest lies in the study of growth of an operator
following the evolution with the Hamiltonian of the system

${\cal O}(t)=e^{iHt}{\cal O}e^{-iHt}\,.$ (60)

In this case, it is natural to use the energy eigenstates for the channel-
state map by which the corresponding reference space is given by (49) with the
replacement $|i,j\rangle\rightarrow|E_{a},E_{b}\rangle$

$|\psi_{\cal O}\rangle=\sum_{a,b}^{\cal
D}\varrho_{ab}\,|E_{a},E_{b}\rangle\,,$ (61)

where $\varrho_{ab}$ is the normalized matrix elements of the operator in the
energy basis. At a given time $t$, the corresponding matrix elements are

$\varrho_{ab}(t)=e^{i(E_{a}-E_{b})t}\varrho_{ab},$ (62)

so that the associated state in doubled Hilbert space is transformed as
follows

$|\psi_{\cal O}(t)\rangle=e^{iH_{-}t}\,|\psi_{\cal O}\rangle$ (63)

where $H_{-}=H\otimes 1-1\otimes H$. In order to compute the complexity
through minimizing the spreading, following our earlier discussion, one needs
to construct the ordered basis, specifically using the Hermitian operator
${\cal A}=H_{-}$. In this case, the general algorithm mentioned above, yields
the operator complexity

${\cal C}(t)={\rm Tr}(\ell\rho_{{\cal O}}(t))=\sum_{n=0}^{{\cal D}_{\psi_{\cal
O}}-1}n|{\cal O}_{n}(t)|^{2}\,,$ (64)

where ${\cal O}_{n}(t)=\sum_{a,b}\varrho_{ab}(t)\langle
n,n|E_{a},E_{b}\rangle$.

In the case when the Hamiltonian possesses a continuum spectrum, one can write
down an expression for complexity similar to (29) in the doubled Hilbert
space. This will lead to a late time saturation of complexity followed by a
linear growth. The saturation will be guaranteed through the perturbative
contact terms and non-perturbative contributions appearing in the density
correlation $\langle E_{a},E_{a^{\prime}}|\rho_{{\cal
O}}(t))|E_{b},E_{b^{\prime}}\rangle$ in the coincident limits
$E_{a}-E_{b}\rightarrow 0$ and $E_{a^{\prime}}-E_{b^{\prime}}\rightarrow 0$.

To summarize, we note here that in order to study operator complexity for an
operator evolving with the Heisenberg equation, one can equivalently run the
algorithm developed to compute the state complexity, but in the doubled
Hilbert space with the Hamiltonian $H_{-}$.

It is worth noting that, in this case, the ordered basis, generated using the
particular effective Hamiltonian $H_{-}$ which governs the time evolution of
the state, renders the Lanczos coefficients $a_{n}$ to be zero. This is of
course expected due to the fact that the evolution of an operator here is
given by the Liouvillian which amounts to have vanishing $a_{n}$.

Let us now discuss one interesting subtlety of this construction. For quantum
system defined on the doubled Hilbert space ${\cal H}_{d}$ whose dynamics is
given by $H_{-}$, the average energy $H_{+}=H\otimes 1+1\otimes H$ is a
conserved charge. This follows from the fact that $[H_{-},H_{+}]=0$. In other
words, for an operator whose associated state is defined by (61), one should
impose the following condition

$H_{+}|\psi_{\cal O}(t)\rangle=E|\psi_{\cal O}(t)\rangle,$ (65)

where $E=E_{a}+E_{b}$ is the average energy which is kept fixed.

The average energy should remain constant during the state evolution (63) and
therefore it does not mix states with different energies. In other words the
complexity can be computed for each individual sector with fixed average
energy Kar  _et al._ (2022).

Precisely due to this particular dynamics of the operator growth, the diagonal
part of the matrix elements ${\cal O}_{ab}$ does not contribute to the
operator growth. In other words, restricting ourselves to an operator whose
corresponding state in doubled Hilbert space is given by

$|\psi_{\cal O}^{(0)}\rangle=\sum_{a}\varrho_{a}|E_{a},E_{a}\rangle$ (66)

one gets trivial dynamics under a unitary time evolution given by $H_{-}$.
Here the index $(0)$ indicates that this state belongs to a subspace with
constant $H_{-}$ (which could be zero).

Actually this defines rather atypical states whose dynamics are, rather
naturally, given by $H_{+}$,

$|\psi_{\cal O}^{(0)}(t)\rangle=e^{iH_{+}t}|\psi_{\cal O}^{(0)}\rangle\,.$
(67)

Note that for $\varrho_{a}=e^{-\beta E_{a}/2}/\sqrt{\sum_{a}e^{-\beta E_{a}}}$
this state corresponds to a thermofield double state with inverse temperature
$\beta$. In general $\varrho_{a}$ could be a complex function having a phase
$\varrho_{a}=|\varrho_{a}|e^{i\alpha_{a}}$ which could be thought of the
generalized thermofield double state Verlinde (2020); Papadodimas and Raju
(2015).

This apparently confusing role reversal of $H_{\pm}$, mentioned above in the
context of the atypical state, is actually not that surprising. In fact, in
the construction of the doubling of the Hilbert space in (61), it is assumed
that energy eigenstates of the Hamiltonians participating in the doubling are
also eigenstates of the time reversal operators, as is the case of most
quantum mechanical system. This is, however, not generically true. Taking this
into account, explicitly, (61) can be rewritten as

$|\psi_{\cal O}\rangle=\sum_{a,b}^{\cal D}\varrho_{ab}\,{\cal
T}|E_{a},E_{b}\rangle\,,$ (68)

where ${\cal T}$ is the time-reversal operator, which being an anti-linear
operator, results in the normalized matrix elements

$\varrho_{ab}(t)=e^{i(E_{a}+E_{b})t}\varrho_{ab}.$ (69)

Then the associated evolution of the state in the doubled Hilbert space is
given by

$|\psi_{\cal O}(t)\rangle=e^{iH_{+}t}\,|\psi_{\cal O}\rangle\,.$ (70)

When $|\psi_{\cal O}\rangle$ is the TFD state, such time evolutions give rise
to phase-shifted thermofield doubled states mentioned above.

Therefore, we note that there are two different ways to arrive at the same
generalized TFD states, either as a simple one-sided time evolution (67),
preserving the time-reversal symmetry of the basis states, or in terms of the
doubled Hilbert space structure using energy eigenstates which are not time-
reversal symmetric as we derived in (70).

These two points of view correspond to the boundary and bulk perspectives,
respectively, in the context of AdS/CFT. The former approach remains faithful
to the time-reversal symmetry of the CFT while creating an enlarged phase
space of complexified quantum states with asymptotic charges Harlow (2016),
and the latter creates a complexified bulk states through evolution with
$H_{+}$ identified as the bulk Hamiltonian. Accordingly, the additional phase
appearing in the generalized TFD state assumes interpretations either due to
boundary charges or as topological quantum phases arising as a consequence of
bulk non-locality.

This non-locality is quantified through the symplectic structure in the bulk,
which although can have local interpretation of time evolution with $H_{+}$,
are globally non-exact, thereby giving rise to the additional phases appearing
in the generalized TFD states Nogueira  _et al._ (2022); Banerjee  _et al._
(2022).

### A little more on doubling and the holographic interpretation:

Our construction of doubled basis follows a secret rendition of the Reeh-
Schlieder-Theorem. By construction, the domain ${\Gamma}_{\psi}$ of any
initial given state $|\psi\rangle$ is dense in the full Hilbert space of the
theory once we are successfully able to construct the ordered basis using an
algebra of operators $\Omega$ that minimize the spreading. Therefore, this
state can be dubbed as a cyclic vector. Furthermore, to have a non-vanishing
complexity, one needs to start with a state which is not an eigenstate of the
operator, which means there cannot be an annihilation operator in the small
algebra of operators, ${\Omega}$. This makes the initial state $|\psi\rangle$
a separating vector.

With these two defining conditions of the Reeh-Schlieder Theorem Haag (1992),
one can visualize the full algebra of operators acting on the Hilbert space
$\Gamma_{\psi}$ as an entangled algebra
${\Omega}_{d}={\Omega}\otimes{\bar{\Omega}}$ where ${\bar{\Omega}}$ is the
commutant algebra which can be obtained using modular automorphism via the
Tomita- Takesaki theorem Haag (1992). Furthermore, given the emerging
structure of entanglement, it is natural to invoke a doubled basis as in (49)
to expand the state and consequently, a doubled ordered basis as in (52) to
analyse the growth of the state.

This justifies why the doubling, introduced in (49) using the channel- state
map, is essential to cast the operator complexity as a corresponding state
complexity333The realisation we achieve through this unification is quite in
line with the Gelfand-Naimark-Segal (GNS) construction discussed in the
context of studying operator growths in large $\cal N$ theories Magán and
Simón (2020). .

In the context of holography, such constructions were instrumental in
understanding the state-dependent reconstruction of black hole interior
Papadodimas and Raju (2014)444However, in this case the small algebras were
only approximate algebra at large $\cal N$ with edge effects which were
important to realize the existence of the black hole interior. In the language
of Lanczos, it would corresponding to only approximate breaking of the
algorithm when $b_{n}$ becomes smaller to a given hierarchy scale, as $\cal N$
is, in the case of holographic CFT’s.. We note here, following our unified
interpretation of the state and the operator complexities, that the same
construction is also pivotal to connect to the the notion of holographic
complexity which measures the growing volume of the interior of a black hole.

It is worth mentioning that in the holographic context, there can actually be
an infinitely many quantities which possess the same late time behaviour in
terms of a linear growth followed by a saturation and in principle, all of
them can be dubbed as complexities Alishahiha (2022); Belin  _et al._ (2022).
Identifying the $A$-functions in terms of the matrix elements of the label
operator potentially removes this ambiguity. Therefore, the interpretation of
both, in terms of operator growth, should follow from the same doubling
algorithm.

An alert reader might wonder about the identification of the matrix elements
of the label operator with the expectation values of a non-local holographic
position operator. However, we would like to emphasize that the entanglement
between two spacelike separated regions can have an equivalent description in
terms of entangling algebras of operators defined on the Hilbert space. While
the previous description is more geometric and visually pleasing, the latter
one provides a more general notion of entanglement independent of spacetime.
Following the interpretation of the doubled Hilbert space in terms of the
modular automorphism mentioned above, this generalized notion of entanglement
justifies the matrix elements of the label operator being identified with the
holographic non-local operator. More details regarding this identification
will be elaborated in Alishahiha and Banerjee (2023).

##  IV Subregion Krylov complexity

Entanglement entropy or other measures of entanglements are given in terms of
the reduced density matrix. This is unlike the complexity the usually is
defined for a pure state or an operator in entire space. Nevertheless,
subregion complexity and complexities for mixed states have also been studied
in the context of holographic complexity Alishahiha (2015); Carmi  _et al._
(2017); Agón  _et al._ (2019); Alishahiha  _et al._ (2019). The circuit
complexity for mixed states in open systems has been also studied in
Bhattacharyya  _et al._ (2021, 2022a); Bhattacharya  _et al._ (2022a, b);
Bhattacharyya  _et al._ (2022b); Bhattacharjee  _et al._ (2022).

Since the approach used in the literature to study complexity, mainly relied
on the state, its generalizations to mixed states or to subregions are not
straightforward. On the other hand, in this paper, our construction for
complexity (of pure state) is given in terms of a particular trace over the
density matrix. An advantage to define complexity in terms of density matrix
is that it may be extended for the cases where we are dealing with subregion
or mixed states. In these cases one would expect that the definition would be
the same and we just need to use reduced density matrix.

Let us consider a quantum system whose Hilbert space can be decomposed into
two parts ${\cal H}={\cal H}_{A}\otimes{\cal H}_{B}$. The dynamics of the
system is given by a Hamiltonian which, in general, may not be decomposed into
two parts acting on ${\cal H}_{A}$ and ${\cal H}_{B}$ separately. Therefore,
even if we start with a reference state which is separable, as times goes the
state spreading makes it very complicated.

Let us start with a reference state $|\psi\rangle$ and construct the
orthonormal, ordered basis using the Hamiltonian of the system. Then it is
rather straightforward to compute the reduced label operator by taking trace
over subsystem $B$ $\ell_{A}={\rm Tr}_{B}(\ell)$. One may also compute reduced
density matrix at given time $\rho_{A}(t)={\rm Tr}_{B}(\rho(t))$. Then the
subregion Krylov complexity is defined by

${\cal C}_{A}(t)={\rm Tr}_{A}(\ell_{A}\rho_{A}(t))\,.$ (71)

It is important to note that although the time evolution of the density matrix
is simple and follows from the Schrödinger equation

$\rho(t)=e^{iHt}\rho(0)e^{-iHt},$ (72)

for the case of reduced density matrix it is rather involved even for an
initial density matrix which is factorized
$\rho(0)=\rho_{A}(0)\otimes\rho_{B}(0)$ Lidar (2019). More precisely, using
the fact that the density matrix is positive and normalized, one may write
$\rho_{B}(0)=\sum_{\mu}\lambda_{\mu}|\mu\rangle\langle\mu|$ which yields Lidar
(2019)

$\rho_{A}(t)=\sum_{\mu,\nu}K_{\mu\nu}(t)\,\rho_{A}(0)\,K^{\dagger}_{\mu\nu}(t)$
(73)

where $K_{\mu\nu}(t)$ known as the Kraus operators are

$K_{\mu\nu}(t)=\sqrt{\lambda_{\nu}}\langle\mu|e^{iHt}|\nu\rangle\,.$ (74)

In general it is not an easy task to compute complexity for reduced density
matrix, though for a special cases where the dynamics of the two subsystem is
separable, one can expect to make some progress.

More generally, motivated by the operator-state mapping of the previous
section, let us consider the following time dependent density matrix

$\rho(t)=\sum_{a,b,a^{\prime},b^{\prime}}\varrho_{ab}(t)\varrho^{*}_{a^{\prime}b^{\prime}}(t)\,|E_{a},E_{b}\rangle\langle
E_{b^{\prime}},E_{a^{\prime}}|\,.$ (75)

whose initial density is found by setting $t=0$. Then the reduced density
matrix at given time reads

$\rho_{r}(t)=\sum_{a,a^{\prime},c}\varrho_{ac}(t)\varrho^{*}_{a^{\prime}c}(t)\,|E_{a}\rangle\langle
E_{a^{\prime}}|\,,$ (76)

by which the complexity is

${\cal C}(t)={\rm
Tr}(\ell_{r}\rho_{r}(t))=\sum_{a,a^{\prime},c}\varrho_{ac}(t)\varrho^{*}_{a^{\prime}c}(t)A(E_{a},E_{a^{\prime}})\,.$
(77)

For the case of $\varrho_{ab}(t)=e^{i(E_{a}\pm E_{b})t}\varrho_{ab}$ one finds

${\cal
C}(t)=\sum_{a,a^{\prime}}e^{i(E_{a}-E_{a^{\prime}})t}\rho(E_{a},E_{a^{\prime}})A(E_{a},E_{a^{\prime}})\,.$
(78)

which has essentially the same form as that we have found for pure state
complexity. Here
$\rho(E_{a},E_{a,})=\sum_{c}\varrho_{ac}\varrho^{*}_{a^{\prime}c}$. Note that
in this expression, the functions $A(E_{a},E_{a^{\prime}})$ are matrix
elements of the reduced label operator.

For the maximally entangled case, given by the state

$|\psi\rangle=\sum_{a}\varrho_{a}|E_{a},E_{a}\rangle,$ (79)

$\varrho_{a}$ being a function of $E_{a}$ with a possible phase, one can
assume to have a separable dynamics as follows

$|\psi(t)\rangle=f_{1}(H_{1},t)f_{2}(H_{2},t)\,|\psi\rangle,$ (80)

where $f_{i}$’s are unitary transformations representing the time evolutions
of each subsystem. Then the reduced density matrix turns out to be independent
of time and is given by

$\rho_{1}(t)={\rm
Tr}_{2}(\rho(t))=\sum_{a}|\varrho_{a}|^{2}|E_{a}\rangle\langle E_{a}|\,.$ (81)

Clearly, even though the subsystem specified by the reduced density matrix are
complex, the complexity remains constant

${\cal C}(t)=\sum_{a}\rho(E_{a})A(E_{a},E_{a}),$ (82)

where $\rho(E_{a})=|\varrho_{a}|^{2}$. Therefore one may conclude that having
non- zero entanglement entropy is not enough to crate complexity under unitary
separable time evolutions and a direct interaction is needed. In other words
to get complexity growth for subsystem the dynamics of entanglement is matter
(see (73)).

This feature of complexity demonstrated above hints at the interesting fact
that in order to have a growth of complexity, it is not sufficient to be
content with the factorized structure of the density matrix, but rather it
needs some interaction. This can alternatively be attributed to a
fundamentally non-factorized structure of the Hilbert space as well. In the
context of the entanglement structure of a generic quantum system, this
connection was discussed in Nogueira  _et al._ (2022); Banerjee  _et al._
(2022).

This underlines the precise realization of the ER = EPR paradigm Maldacena and
Susskind (2013). In the context of complexity, one can expect a very similar
conclusion following the discussion above. This opens up the possibility of
describing the growth of complexity in terms of connected saddles like replica
wormholes which in turn could in principle provide the precise geometric
interpretation of the modified replica trick for complexity advocated in
Alishahiha  _et al._ (2022); Alishahiha and Banerjee (2022). Work in this
direction is in progress and we hope to report on this soon.

As a final comment, we note that the holographic subregion complexity for a
time-dependent geometry, given by the Vaidya metric, has been computed in Chen
_et al._ (2018); Ling  _et al._ (2019); Auzzi  _et al._ (2019). In the context
of gauge-gravity duality, the Vaidya metric provides a holographic description
for a thermal quench. It was then possible to explore the time-dependence of
holographic subregion complexity in this context. It was shown that the
holographic subregion complexity exhibits linear growth up to a maximum value
after which it shows a decreasing phase and finally saturates to a constant.

Intuitively, one can see that the Krylov subregion complexity should exhibit
similar features. In fact, from equation (78) one can note that the linear
growth can indeed be understood from the pole structure of the $A$-function,
much similar to what we studied in the context of Krylov complexity. On the
other hand, since the information about the Lanczos coefficients is encoded in
the $A$-function, the saturation phase can be understood from the fact that
the Lanczos coefficients vanish at late times.

It would, of course, be interesting to study Krylov subregion complexity more
rigorously and compare it with that of holographic one explicitly.

##  V Conclusions

To summarize our progress in this paper, we developed a unified formalism to
study the state and the operator complexities. The key observation towards the
unification is to realize that, within the realm of quantum mechanics, one can
think of a channel-state correspondence which helps us to cast the the problem
of studying operator complexity, to a corresponding state complexity obtained
through the evolution of the dual state living on a doubled Hilbert space.

While this connection itself is quite interesting and thought-provoking, it
reveals, on its way to development, a couple of more surprises. First, it
connects very naturally to the notion of complexity discussed in the context
of holography following two routes - i) realizing the doubling of the Hilbert
space for a more generic quantum system in terms of the entanglement structure
arising from the Reeh-Schlieder theorem in an axiomatic quantum field theory,
thus connecting this procedure to the state-dependent bulk
reconstructionsPapadodimas and Raju (2014). This shows that the state-channel
map which connects the state and operator complexities through a doubling of
the Hilbert space, does secretly exploit the entanglement algebra imposed
through the Lanczos algorithm. This hidden structure of entanglement
underlying the Krylov construction, to our knowledge, had not been spelt out
in the literature; ii) through a direct identification of the matrix elements
of the label operator in the energy basis defined to quantify the spreading in
a generic quantum system to the expectation values of the holographic position
operator appearing in the quenched geodesic length as in Alishahiha and
Banerjee (2022); Alishahiha  _et al._ (2022); Alishahiha (2022). This provides
a support for a general definition of complexity in terms of the pole
structure of the $A$-function as advocated in Alishahiha (2022). Our
construction establishes that this definition of complexity is universally
applicable for any quantum system, with or without gravity, so long as the
complexity has a late time linear growth. Such behaviour of complexity is
indeed expected for a large class of physical systems, both integrable and
chaotic and unlike the saturation phase, it is solely governed by the
saturation of the Lanczos coefficients, captured by the pole structure of the
$A$-function.

While these two routes lead to a further unification, now also incorporating
the holographic complexity along with the state and the operator complexities,
there is also a second but very important practical advancement in terms of
developing a formalism to study subregion complexity and complexity for mixed
quantum states, which had so far been quite illusive in the existing
literature. In our formalism, this appears, rather naturally, since our
universal definition of complexity is given in terms of a particular trace
over the density matrix. In particular, the subregion complexity can be
obtained simply by replacing the density matrix with the reduced density
matrix corresponding to any given subregion. Furthermore, while generalizing
the notion of complexity for a mixed state, we realise that this naturally
hints at having a replica wormhole saddles in the dual gravitational
spacetime, in line with the expectation coming from the modified replica trick
proposed in Alishahiha and Banerjee (2022); Alishahiha  _et al._ (2022).

Following this short summary let us now conclude with a few ongoing
progresses.

### The mutual complexity

Since we are dealing with subregion complexity, one can naturally define the
Krylov mutual complexity for two subregions $A$ and $B$ as follows

${\cal M}_{AB}={\cal C}_{A}+{\cal C}_{B}-{\cal C}_{A\cup B}\,,$ (83)

where ${\cal C}$’s are the Krylov subregion complexities associated with the
mentioned regions. In the context of the holographic complexity, the mutual
complexity has been defined in Alishahiha  _et al._ (2019); Caceres  _et al._
(2020).

### Complexity in open quantum systems

Our universal formalism for state and operator complexities is applicable as
well for open quantum systems which lack a Hermitian Hamiltonian which in turn
means that the ordered basis defined in (5) does not span the full doubled
Hilbert space ${\cal H}_{\psi}$. In this case, one needs two independent sets
of basis vectors, $|n\rangle$ and $\langle{\bar{n}}|$ which are not in general
orthogonal to each other. Nevertheless, given a non-Hermitian operator $\cal
A$, in many cases 555Although this formalism was used and tested for pseudo-
Hermitian Hamiltonians with real eigenvalues Mostafazadeh (2002); Grüning  _et
al._ (2011), we expect this procedure to work for more general non-Hermitian
operators., it is possible to construct an anti-linear operator $\cal L$ such
that ${\cal L}^{-1}{\cal A}\,{\cal L}$ is Hermitian with respect to the inner
product $\langle n|{\bar{n}}\rangle_{\cal L}\equiv\langle n|{\cal
L}\,{\bar{n}}\rangle$. One example of such an anti-linear operator is the CPT
operator. Adopting this formalism for computing complexity due to a non-
Hermitian operator $\cal A$ amounts to a generalization of the definition of
complexity (10) as

${\cal C}_{\phi}={\rm Tr}(\ell\rho_{\phi})_{\cal L}\,,$ (84)

for any state $|\phi\rangle\in{\cal H}_{\psi}$. Here the subscript $\cal L$
reminds us that the trace operation here should be performed taking into
account the modified inner product $\langle n|{\bar{n}}\rangle_{\cal L}$.

### The transition matrix

As an aside comment, let us mention that there is an other operator, beside
the density matrix, which plays an important role in this context that is
known as the transition matrix

${\tau}=|\psi\rangle\langle\phi|\,.$ (85)

It is natural to consider the case where the state $|\phi\rangle$ is given by
(12). Although the information of Lanczos coefficients is encoded in the trace
of the transition matrix ${\rm Tr}(\tau)=\langle\psi|U^{\dagger}({\cal
A},s)|\psi\rangle=\psi_{0}^{*}(s)$, its evolution does not lead to any
spreading as defined by (14). More precisely one has

${\rm Tr}(\ell\tau)=0.$ (86)

It is important to note that the vanishing of the above equation is a direct
consequence of our definition of the label operator in which we set $c_{n}=n$
that is zero for $n=0$. If one assumes a general $c_{n}$ for $c_{0}\neq 0$ one
has ${\rm Tr}(\ell\tau)=c_{0}{\rm Tr}(\tau)$.

Transition matrix is used to define the pseudo-entropy which is the
generalization of entanglement entropy with post-selection Mollabashi  _et
al._ (2021a, b). Taking into account that the fact that the autocorrelation
can be re-expressed in terms of the the Lanczos coefficients, one would expect
that the pseudo-entropy exhibits certain universal behavior, at least for
chaotic systems. We hope to report on this soon.

###### Acknowledgements.

We would like to thank Rathindra Nath Das, Moritz Dorband and Johanna
Erdmenger for several useful and inspiring discussions.

## References

  * Balasubramanian  _et al._ (2022) V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Phys. Rev. D 106, 046007 (2022), arXiv:2202.06957 [hep-th] .
  * Lanczos (1950) C. Lanczos, J. Res. Natl. Bur. Stand. B 45, 255 (1950).
  * Viswanath and Müller (1994) V. Viswanath and G. Müller,  _The Recursion Method: Application to Many Body Dynamics_, Lecture Notes in Physics Monographs (Springer Berlin Heidelberg, 1994).
  * Parker  _et al._ (2019) D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, Phys. Rev. X 9, 041017 (2019), arXiv:1812.08657 [cond-mat.stat-mech] .
  * Maldacena (1998) J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998), arXiv:hep-th/9711200 .
  * Witten (1998) E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998), arXiv:hep-th/9802150 .
  * Gubser  _et al._ (1998) S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B 428, 105 (1998), arXiv:hep-th/9802109 .
  * Susskind (2016) L. Susskind, Fortsch. Phys. 64, 24 (2016), [Addendum: Fortsch.Phys. 64, 44–48 (2016)], arXiv:1403.5695 [hep-th] .
  * Susskind (2018) L. Susskind (Springer, 2018) arXiv:1810.11563 [hep-th] .
  * Stanford and Susskind (2014) D. Stanford and L. Susskind, Phys. Rev. D 90, 126007 (2014), arXiv:1406.2678 [hep-th] .
  * Iliesiu  _et al._ (2022) L. V. Iliesiu, M. Mezei, and G. Sárosi, JHEP 07, 073 (2022), arXiv:2107.06286 [hep-th] .
  * Alishahiha  _et al._ (2022) M. Alishahiha, S. Banerjee, and J. Kames-King, JHEP 08, 181 (2022), arXiv:2205.01150 [hep-th] .
  * Alishahiha and Banerjee (2022) M. Alishahiha and S. Banerjee, “On the saturation of late-time growth of complexity in supersymmetric JT gravity,” (2022), arXiv:2209.02441 [hep-th] .
  * Alishahiha (2022) M. Alishahiha, “On Quantum Complexity,” (2022), arXiv:2209.14689 [hep-th] .
  * Deutsch (1991) J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).
  * Srednicki (1994) M. Srednicki, Phys. Rev. E 50, 888 (1994).
  * Nielsen and Chuang (2010) M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information: 10th Anniversary Edition_ (Cambridge University Press, 2010).
  * Rabinovici  _et al._ (2021) E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, JHEP 06, 062 (2021), arXiv:2009.01862 [hep-th] .
  * Barbón  _et al._ (2019) J. L. F. Barbón, E. Rabinovici, R. Shir, and R. Sinha, JHEP 10, 264 (2019), arXiv:1907.05393 [hep-th] .
  * Jian  _et al._ (2021) S.-K. Jian, B. Swingle, and Z.-Y. Xian, JHEP 03, 014 (2021), arXiv:2008.12274 [hep-th] .
  * Fan (2022a) Z.-Y. Fan, Phys. Rev. A 105, 062210 (2022a), arXiv:2202.07220 [quant-ph] .
  * Mück and Yang (2022) W. Mück and Y. Yang, Nucl. Phys. B 984, 115948 (2022), arXiv:2205.12815 [hep-th] .
  * Fan (2022b) Z.-Y. Fan, JHEP 08, 232 (2022b), arXiv:2206.00855 [hep-th] .
  * Mehta (1991) M. L. Mehta,  _Random Matrices (Revised and Enlarged Second Edition)_, revised and enlarged second edition ed. (Academic Press, San Diego, 1991) p. xv.
  * Rabinovici  _et al._ (2022) E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, JHEP 07, 151 (2022), arXiv:2207.07701 [hep-th] .
  * Alishahiha and Banerjee (2023) M. Alishahiha and S. Banerjee, To appear soon (2023).
  * Jiang  _et al._ (2013) M. Jiang, S. Luo, and S. Fu, Phys. Rev. A 87, 022310 (2013).
  * Kar  _et al._ (2022) A. Kar, L. Lamprou, M. Rozali, and J. Sully, JHEP 01, 016 (2022), arXiv:2106.02046 [hep-th] .
  * Verlinde (2020) H. Verlinde, “ER = EPR revisited: On the Entropy of an Einstein-Rosen Bridge,” (2020), arXiv:2003.13117 [hep-th] .
  * Papadodimas and Raju (2015) K. Papadodimas and S. Raju, Phys. Rev. Lett. 115, 211601 (2015), arXiv:1502.06692 [hep-th] .
  * Harlow (2016) D. Harlow, JHEP 01, 122 (2016), arXiv:1510.07911 [hep-th] .
  * Nogueira  _et al._ (2022) F. S. Nogueira, S. Banerjee, M. Dorband, R. Meyer, J. v. d. Brink, and J. Erdmenger, Phys. Rev. D 105, L081903 (2022), arXiv:2109.06190 [hep-th] .
  * Banerjee  _et al._ (2022) S. Banerjee, M. Dorband, J. Erdmenger, R. Meyer, and A.-L. Weigel, JHEP 08, 162 (2022), arXiv:2202.11717 [hep-th] .
  * Haag (1992) R. Haag, “Local quantum physics: Fields, particles, algebras,” (1992).
  * Magán and Simón (2020) J. M. Magán and J. Simón, JHEP 05, 071 (2020), arXiv:2002.03865 [hep-th] .
  * Papadodimas and Raju (2014) K. Papadodimas and S. Raju, Phys. Rev. D 89, 086010 (2014), arXiv:1310.6335 [hep-th] .
  * Belin  _et al._ (2022) A. Belin, R. C. Myers, S.-M. Ruan, G. Sárosi, and A. J. Speranza, Phys. Rev. Lett. 128, 081602 (2022), arXiv:2111.02429 [hep-th] .
  * Alishahiha (2015) M. Alishahiha, Phys. Rev. D 92, 126009 (2015), arXiv:1509.06614 [hep-th] .
  * Carmi  _et al._ (2017) D. Carmi, R. C. Myers, and P. Rath, JHEP 03, 118 (2017), arXiv:1612.00433 [hep-th] .
  * Agón  _et al._ (2019) C. A. Agón, M. Headrick, and B. Swingle, JHEP 02, 145 (2019), arXiv:1804.01561 [hep-th] .
  * Alishahiha  _et al._ (2019) M. Alishahiha, K. Babaei Velni, and M. R. Mohammadi Mozaffar, Phys. Rev. D 99, 126016 (2019), arXiv:1809.06031 [hep-th] .
  * Bhattacharyya  _et al._ (2021) A. Bhattacharyya, S. S. Haque, and E. H. Kim, JHEP 10, 028 (2021), arXiv:2011.04705 [hep-th] .
  * Bhattacharyya  _et al._ (2022a) A. Bhattacharyya, T. Hanif, S. S. Haque, and M. K. Rahman, Phys. Rev. D 105, 046011 (2022a), arXiv:2112.03955 [hep-th] .
  * Bhattacharya  _et al._ (2022a) A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, JHEP 12, 081 (2022a), arXiv:2207.05347 [quant-ph] .
  * Bhattacharya  _et al._ (2022b) A. Bhattacharya, A. Bhattacharyya, and S. Maulik, Phys. Rev. D 106, 086010 (2022b), arXiv:2209.00049 [hep-th] .
  * Bhattacharyya  _et al._ (2022b) A. Bhattacharyya, T. Hanif, S. S. Haque, and A. Paul, “Decoherence, Entanglement Negativity and Circuit Complexity for Open Quantum System,” (2022b), arXiv:2210.09268 [hep-th] .
  * Bhattacharjee  _et al._ (2022) B. Bhattacharjee, X. Cao, P. Nandy, and T. Pathak, “Operator growth in open quantum systems: lessons from the dissipative SYK,” (2022), arXiv:2212.06180 [quant-ph] .
  * Lidar (2019) D. A. Lidar, “Lecture notes on the theory of open quantum systems,”  (2019).
  * Maldacena and Susskind (2013) J. Maldacena and L. Susskind, Fortsch. Phys. 61, 781 (2013), arXiv:1306.0533 [hep-th] .
  * Chen  _et al._ (2018) B. Chen, W.-M. Li, R.-Q. Yang, C.-Y. Zhang, and S.-J. Zhang, JHEP 07, 034 (2018), arXiv:1803.06680 [hep-th] .
  * Ling  _et al._ (2019) Y. Ling, Y. Liu, C. Niu, Y. Xiao, and C.-Y. Zhang, JHEP 11, 039 (2019), arXiv:1908.06432 [hep-th] .
  * Auzzi  _et al._ (2019) R. Auzzi, G. Nardelli, F. I. Schaposnik Massolo, G. Tallarita, and N. Zenoni, JHEP 11, 098 (2019), arXiv:1908.10832 [hep-th] .
  * Caceres  _et al._ (2020) E. Caceres, S. Chapman, J. D. Couch, J. P. Hernandez, R. C. Myers, and S.-M. Ruan, JHEP 03, 012 (2020), arXiv:1909.10557 [hep-th] .
  * Mostafazadeh (2002) A. Mostafazadeh, J. Math. Phys. 43, 205 (2002), arXiv:math-ph/0107001 .
  * Grüning  _et al._ (2011) M. Grüning, A. Marini, and X. Gonze, Computational Materials Science 50, 2148 (2011).
  * Mollabashi  _et al._ (2021a) A. Mollabashi, N. Shiba, T. Takayanagi, K. Tamaoka, and Z. Wei, Phys. Rev. Lett. 126, 081601 (2021a), arXiv:2011.09648 [hep-th] .
  * Mollabashi  _et al._ (2021b) A. Mollabashi, N. Shiba, T. Takayanagi, K. Tamaoka, and Z. Wei, Phys. Rev. Res. 3, 033254 (2021b), arXiv:2106.03118 [hep-th] .