aainstitutetext: Asia Pacific Center for Theoretical Physics, 77 Cheongam-ro,
Nam-gu, Pohang-si, Gyeongsangbuk-do, 37673, Korea.bbinstitutetext: Ramakrishna
Mission Vivekananda Educational and Research Institute, Belur Math,
Howrah-711202, West Bengal, Indiaccinstitutetext:

# Krylov Complexity in $2d$ CFTs with SL$(2,\mathbb{R})$ deformed Hamiltonians

Vinay Malvimat b Somnath Porey b Baishali Roy<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this study, we analyze Krylov Complexity in two-dimensional conformal field
theories subjected to deformed SL$(2,\mathbb{R})$ Hamiltonians. In the Vacuum
state, we find that the K-complexity exhibits a universal phase structure. The
phase structure involves the K-complexity exhibiting an oscillatory behaviour
in the non-heating phase, which contrasts with the exponential growth observed
in the heating phase, while it displays polynomial growth at the phase
boundary. Furthermore, we extend our analysis to compute the K-complexity of a
light operator in excited states, considering both large-c CFT and free field
theory. In the free field theory, we find a state-independent phase structure
of K-complexity. However, in the large-c CFT, the behavior varies, with the
K-Complexity once again displaying exponential growth in the heating phase and
polynomial growth at the phase boundary. Notably, the precise exponent
governing this growth depends on the heaviness of the state under examination.
In the non-heating phase, we observe a transition in K-complexity behavior
from oscillatory to exponential growth, akin to findings in Kundu:2023hbk , as
it represents a special case within the non-heating phase.

##  1 Introduction

Quantum Chaos has emerged as a frontier field of intense research exploration
captivating the attention of both high energy as well as quantum many-body
physicists. Several interesting probes including the level spacing statistics,
the exponential growth of the Out-of -Time Order Correlators (OTOC) at late
times, dip-ramp-plateau behaviour of the Spectral Form Factor(SFF) have
provided illuminating insights into the understanding of thermalization,
ergodicity, and chaos in various quantum many-body systems Srednicki:1994mfb ;
Gogolin:2015gts ; DAlessio:2015qtq . In Roberts:2014ifa , it was shown that
the four-point OTOCs in the thermal state can distinguish between chaotic (eg.
large-c) and integrable (eg. Ising model) 2D CFTs. Subsequently, other studies
have provided evidence for this even in the non-equilibrium setup of globally
quenched Das:2021qsd  and driven Das:2022jrr  CFTs.

The interesting interplay between the onset of quantum chaos and the growth of
complexity is also a fascinating area of intense investigation. An interesting
measure that characterizes this growth is the so-called circuit complexity. It
is defined in quantum information theory as the minimum number of unitary
gates in a quantum circuit required to reach a specific state from a reference
state. Interestingly, the growth of complexity in holography is expected to be
dual to the growth of the volume of the black hole interior in the dual bulk
gravitational theory Stanford:2014jda ; Susskind:2018pmk ; Susskind:2018tei ;
Chattopadhyay:2023fob . While these studies have led to many interesting
insights, several questions regarding circuit complexity remain. These include
defining the precise notion of distance between states in quantum field
theories and finding the exact bulk dual of circuit complexity in holography
(See Jefferson:2017sdb ; Chapman:2017rqy ; Chagnet:2021uvi ; Belin:2021bga
and the references therein for many interesting discussions).

Recently, a new notion of complexity known as Krylov complexity characterizing
the growth of a local operator in operator Hilbert space (Krylov space) has
received considerable attention. This quantity was introduced in
Parker:2018yvk , where the authors proposed that the Lanczos coefficients
associated with Krylov space grow linearly (maximally) in a system that
exhibits quantum chaos. This is known as the operator growth hypothesis.
Following this development, Krylov complexity and the associated Lanczos
coefficients have been examined in varied systems including quantum and
conformal field theories Dymarsky:2019elm ; Avdoshkin:2019trj ; Barbon:2019wsy
; Dymarsky:2021bjq ; Caputa:2021sib ; Caputa:2021ori ; Avdoshkin:2022xuw ;
Camargo:2022rnt ; Khetrapal:2022dzy ; Rabinovici:2022beu , random matrix
theories Kar:2021nbm ; Erdmenger:2023wjg ; Tang:2023ocr , holography
Rabinovici:2023yex ; Anegawa:2024wov ; Basu:2024tgg , open quantum systems
Bhattacharjee:2022lzy , Matrix models Iizuka:2023pov ; Iizuka:2023fba , Scar
states Bhattacharjee:2022qjw , driven quantum systems Nizami:2023dkf  among
others111A related notion known as spread complexity has also been explored
extensively for various quantum systems Balasubramanian:2022tpr ;
Caputa:2022eye ; Caputa:2022yju ; Afrasiar:2022efk ; Chattopadhyay:2023fob  .
Furthermore, the Krylov complexity is conjectured to provide an upper bound
for the Lyapunov exponent obtained from OTOC Parker:2018yvk

$\displaystyle\lambda_{L}\leq\lambda_{K}\leq\frac{2\pi}{\beta}$ (1)

where $\lambda_{L}$ is the Lypunov exponent, $\lambda_{K}$ the Krylov exponent
and $\beta$ is the inverse temperature.

In Dymarsky:2021bjq , the authors computed K-Complexity in free scalar and
fermionic quantum field theories and 2d CFTs utilizing the Wightman inner
product for Krylov space. They demonstrated that the Lanczos coefficents grow
linearly and that the K-Complexity grows exponentially in all cases. Since for
2D CFTs, the auto-correlation function is related to the thermal two-point
function, the exponential behaviour of K-Complexity turns out to be universal.
In Kundu:2023hbk , the authors addressed this intriguing issue by choosing a
state-dependent innerproduct. In 2d CFTs, they obtained the K-Complexity for
large-c CFTs and Ising model for various states. They showed that while the
Lanczos coefficients grow linearly for large-n, the K-complexity exhibits a
transition from an oscillating to an exponentially growing behaviour beyond a
critical dimension of the primary operator, creating the state by acting on
the vacuum. However, in two-dimensional free field theory and Ising model the
K-complexity always showed an oscillatory behaviour, independent of the
dimension of the primary operator.

The main focus of this draft is to examine the dynamics of K-complexity in
out-of-equilibrium systems. We use quantum quenches in (1+1)D CFT defined on a
circle as our basic framework, which provides a tractable yet insightful setup
for this investigation. Different types of quench protocols have been studied
in the literature, in the context of 2D CFTs. For example, one such
extensively studied protocol is that of global quench setup cardy , in which
the system is initially prepared in the regulated boundary state of a CFT and
is then evolved with the usual CFT Hamiltonian. In this article, we will
consider yet another kind of quench, where the system is prepared in some
eigenstate of the usual CFT Hamiltonian and is then evolved with a deformed
CFT Hamiltonian.

The usual CFT Hamiltonian of 2D CFT on a ring is built out of only the
Virasoro modes $L_{0}$ and $\bar{L}_{0}$, which generate scaling in the radial
quantization. One can, however, consider a situation where Hamiltonian
consists of other Virasoro generators. Specifically one can study systems
under SL${}_{q}[2,\mathbb{R}]$ deformations where the Hamiltonian is made out
of the linear combination of generators of sl2 subalgebra of the Virasoro
algebra rendering the unitary evolution operators to be $su(1,1)$ valued that
generate Möbius transformations in the q-sheet Riemann SurfaceFan:2020orx . A
well-known fact about these classes of deformed CFTs is that there are three
distinct types of evolution operators, depending on the sign of the Casimir of
the underlying algebra, which results in three distinct classes of dynamics -
hyperbolic (heating phase), elliptic (non-heating phase) and parabolic (phase
boundary)Wen:2018agb ; Wen:2021mlv ; Fan:2020orx ; Wen:2020wee ; Wen:2022pyj
222 Understanding the spectrum of the Hamiltonians in each of these phases is
an interesting problem, see  Ishibashi:2015jba . In particular, it was shown
in Das:2022jrr  that OTOC in large-c driven CFTs shows qualitatively different
behavior in each of the three phases. While in the heating phase, it shows the
characteristic Lyapunov behaviour, in the non-heating phase and phase boundary
it shows oscillatory and polynomial behaviour respectively. In light of these
results, it would be interesting to understand whether K-complexity can
distinguish between different phases in such systems and also whether it can
demarcate between large-c and integrable CFTs. In this draft, we have tried to
address these questions. Our key findings are as follows:

  * •

In the vacuum state, the K-complexity is determined from a two-point function
and it shows exponential, oscillatory, and polynomial behaviour respectively
in the heating phase, non-heating phase, and at the phase boundary.

  * •

For large-c CFTs, in the non-heating phase, behaviour of K-complexity in the
excited state shows qualitatively different behaviour for states with
conformal dimension ($\Delta$) above and below a certain threshold
value($\Delta^{*}=\frac{c}{12})$. The temporal dependence is exponential and
oscillatory respectively above and below $\Delta^{*}$. For, the free massless
scalar field theory, however, the K-complexity shows only oscillatory
behaviour for all excited states.

  * •

On the other hand, in the heating phase, and at the phase boundary, the
K-complexity shows the same behavior in all excited states both for large-c
CFT as well as for free massless scalar field theory. In particular, it shows
exponential and polynomial growth respectively for the heating phase and phase
boundary dynamics.

The rest of the draft is organized as follows. We begin with a brief review of
K-complexity in section 2. In section 3, we compute the auto-correlation
functions and examine the behaviour of K-complexity in the deformed CFT setup.
In 3.1, we set up the computation of the K-complexity in the CFT ground state.
This is followed by the computation of the same in an excited state in 3.2. In
the excited state, we carry out this computation in large-c (3.2.1) as well as
in the free massless scalar field theory(3.2.2). We wrap up by discussing our
results and suggesting potential future directions in section 4. To enhance
the overall coherence and completeness of the draft, we also include two
appendices at the end. In appendix A, we have reviewed the details of deformed
CFTs, and in appendix B, we have discussed the details of the numerical
technique utilized in this paper.

##  2 Brief review of K-complexity in CFTs

The time evolution of a local operator $\mathcal{O}$ in the Heisenberg picture
can be expressed as a sum of nested commutators of the operator with the
Hamiltonian $H$

$\mathcal{O}(t)=e^{iHt}\mathcal{O}(0)e^{-iHt}=\sum_{k=0}^{\infty}\frac{(-it)^{k}}{k!}[H,[H\dots[H,\mathcal{O}(0)]\dots]]$
(2)

By defining the Liouvillian operator $\mathcal{L}:=[H,\cdot]$, the above
expression can be re-expressed as follows

$\mathcal{O}(t)=e^{i\mathcal{L}t}\mathcal{O}(0):=\sum_{k=0}^{\infty}\frac{(-it)^{k}}{k!}\tilde{\mathcal{O}_{k}}$

where, $\tilde{\mathcal{O}_{k}}=\mathcal{L}^{k}\mathcal{O}(0)$ and
$\mathcal{L}^{k}$ are nested commutators $[H,[H\dots[H,\mathcal{O}]\dots]]$
with the Hamiltonian. The set of operators $\\{\tilde{\mathcal{O}_{k}}\\}$
span a vector space known as the Krylov space associated with the operator
$\mathcal{O}$. One can construct an orthonormal basis in this space, known as
the Krylov basis $\\{\mathcal{O}_{k}\\}$, through the standard Gram-Schmidt
orthogonalization which is implemented through an iterative procedure known as
Lanczos algorithm. The Krylov space has to be endowed with a valid inner
product. The standard choice for this is the Wightmann inner product. However,
as pointed out in Dymarsky:2019elm  various other choices can be made. A class
of such valid inner products is of the form

$(\mathcal{O}_{1}|\mathcal{O}_{2})=Tr(\rho_{1}\mathcal{O}_{1}^{\dagger}\rho_{2}\mathcal{O}_{2})$

where $\rho_{1}$ and $\rho_{2}$ are positive Hermitian operators that commute
with $H$. A particular choice of $\rho_{1}=\rho_{\psi}$, which is the density
matrix associated with a pure eigen state $\ket{\psi}$ and $\rho_{2}=I$ was
chosen in Kundu:2023hbk

$(\mathcal{O}_{1}|\mathcal{O}_{2})=\frac{Tr(\rho_{\psi}\mathcal{O}_{1}^{\dagger}\mathcal{O}_{2})}{Tr\rho_{\psi}}$

The orthonormal basis constructed utilizing Lanczos Algorithm) is given as

$\displaystyle|\mathcal{P}_{n}):$
$\displaystyle=\mathcal{L}|\mathcal{O}_{n-1})+b_{n-1}|\mathcal{O}_{n-2})-a_{n-1}|\mathcal{O}_{n-1})$
(3) $\displaystyle|\mathcal{O}_{n}):$
$\displaystyle=b_{n}^{-1}|\mathcal{P}_{n})$ $\displaystyle b_{n}:$
$\displaystyle=\sqrt{(\mathcal{P}_{n}|\mathcal{P}_{n})}$

The coefficients $a_{n}$ and $b_{n}$ so generated are called the Lanczos
coefficients. They can be expressed in terms of the matrix elements of the
$\mathcal{L}$ as,

$a_{n}:=(\mathcal{O}_{n}|\mathcal{L}|\mathcal{O}_{n})\ \ \ \ \text{and}\ \ \
b_{n}:=(\mathcal{O}_{n}|\mathcal{L}|\mathcal{O}_{n-1}).$ (4)

The Liouvillian can be expressed as a tridiagonal matrix in this basis:

$\mathcal{L}_{mn}=\begin{pmatrix}a_{0}&b_{1}&0&0&0&0&\dots&0\\\
b_{1}&a_{1}&b_{2}&0&0&0&\dots&0\\\ 0&b_{2}&a_{2}&b_{3}&0&0&\dots&0\\\
0&0&b_{3}&a_{3}&b_{4}&0&\dots&0\\\
\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\dots&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&b_{m}\\\
0&0&0&0&0&\dots&b_{m}&a_{m}\end{pmatrix}$ (5)

Having defined a Krylov space with a valid inner product, the operator
$\mathcal{O}(t)$ at any moment in time can be expanded in terms of the Krylov
basis elements $|\mathcal{O}_{n})$:

$|\mathcal{O}(t)\ )=\sum_{n}i^{n}\phi_{n}(t)|\mathcal{O}_{n})$

where $\phi_{n}(t)$’s are complex coefficients. The Heisenberg equation of
motion essentially reduces to a recursion relation of these $\phi_{n}$’s, as
given below

$\displaystyle\partial_{t}\phi_{n}(t)=b_{n}\phi_{n-1}(t)-b_{n+1}\phi_{n+1}(t)+ia_{n}\phi_{n}(t)$
(6)

with the boundary condition $\phi_{-1}(t)=0,b_{0}=0$ and
$\phi_{n}(0)=\delta_{n,0}$. $\phi_{0}$ in our case is going to be the unequal-
time correlation (or auto-correlation) function
$(\mathcal{O}(0)|\mathcal{O}(t))$ in some state $\psi$.

$C_{\psi}(t)=(\mathcal{O}(0)|\mathcal{O}(t))_{\psi}=\phi_{0}(t)=\frac{Tr(\rho_{\psi}\mathcal{O}(t)\mathcal{O}(0))}{Tr\rho_{\psi}}$

The Krylov-complexity is then defined as,

$K_{\mathcal{O}}(t)=(\mathcal{O}(t)|n|\mathcal{O}(t))=\sum_{n}n|\phi_{n}(t)|^{2}$

##  3 From auto-correlation function to K-complexity

We consider a two-dimensional CFT on a cylinder whose circumference is of
length L. The system is described by the coordinates $\omega=\tau_{c}+ix_{c}$
and $\bar{\omega}=\tau_{c}-ix_{c}$ where $\tau_{c}$ is the Euclidean time and
$x_{c}$ is the compact spatial direction. The standard CFT Hamiltonian in
terms of Virasoro generator is given by $H=L_{0}+\bar{L}_{0}$333Here, we
ignore the constant casimir term $-\frac{c}{24}$, since it does not play any
crucial role in our computation.. In the present article, we will instead
focus our attention on the quantum system which is evolved with a Hamiltonian
that is deformed by the presence of other Virasoro generators as described
below,

$\displaystyle
H^{cyl}_{d}=\alpha(L_{0}+\bar{L}_{0})+\beta(L_{q+}+\bar{L}_{q+})+\gamma(L_{q-}-\bar{L}_{q-})$
(7)

with, $L_{q+}=\frac{L_{q}+L_{-q}}{2}$, $L_{q-}=\frac{L_{q}-L_{-q}}{2i}$ and
$\alpha,\ \beta$ and $\gamma$ are some real numbers. The dynamics of such a
Hamiltonian can be classified into three distinct phases depending on the sign
of the parameter $\delta=-\alpha^{2}+\beta^{2}+\gamma^{2}$ Wen:2020wee . CFTs
with such deformed Hamiltonians have been investigated in great detail in
Wen:2018agb ; Wen:2021mlv ; Fan:2020orx ; Wen:2020wee ; Wen:2022pyj . We
direct the reader to a brief review of the same in Appendix A.

To compute the Krylov complexity of an operator evolving with the deformed
Hamiltonian, we consider the generic form of the auto-correlation function
$C_{\phi}(\tau)$:

$C_{\phi}(\tau)=\frac{\langle\phi|\mathcal{O}(\tau)\mathcal{O}(0)|\phi\rangle}{\langle\phi|\phi\rangle}$
(8)

where, the operator $\mathcal{O}(\tau)$ in the Heisenberg picture is given by,

$\displaystyle\mathcal{O}(\tau)=e^{H^{cyl}_{d}\tau}\mathcal{O}(0)e^{-H^{cyl}_{d}\tau}$
(9)

where $H^{cyl}_{d}$ is the deformed Hamiltonian on the ring defined in (7) and
$\tau$ is the corresponding Euclidean time generated by $H^{cyl}_{d}$. In the
following, we aim to analyze the K-complexity of such systems for different
cases.

###  3.1 Vacuum state of a 2d CFT

In this section, we consider the situation, when the auto-correlation function
is evaluated in the vacuum state of the CFT. For computational convenience, we
work with the deformed Hamiltonian with $q=1$ in (7) which is as follows

$\displaystyle
H^{cyl}_{d}=\alpha(L_{0}+\bar{L}_{0})+\beta(L_{+}+\bar{L}_{+})+\gamma(L_{-}-\bar{L}_{-})$
(10)

where, $L_{+}=\frac{L_{1}+L_{-1}}{2}$, $L_{-}=\frac{L_{1}-L_{-1}}{2i}$. Since,
the above deformed Hamiltonian is made out of the global Virasoro generators
$L_{0},L_{1},L_{-1}$ and their corresponding complex conjugates, they generate
a Möbius transformation in the complex plane:

$\displaystyle z^{\prime}_{i}=\frac{az_{i}+b}{cz_{i}+d}$

where, $z_{i}$ are complex coordinates on the plane and $z^{\prime}_{i}$ are
the corresponding coordinates after the Möbius transformation. Note that the
vacuum state (ground state of $H_{0}=L_{0}+\bar{L}_{0}$) is invariant under
the action of these generators and hence, only the unequal time correlation
functions have access to the dynamics of the drive. To compute the auto-
correlation function, we first map the correlator from
cylinder($\omega,\bar{\omega}$) to the plane($z,\bar{z}$) via the coordinate
transformations 444We have taken $L=2\pi$ and in the rest of the paper we
follow this convention.,

$\displaystyle z=e^{\omega},\hskip 14.22636pt\bar{z}=e^{\bar{\omega}}.$ (11)

We will also consider the operators $1\ \text{and}\ 2$ to be initially placed
at coordinate points $x_{c\ 1,2}=\tau_{c\ 1,2}=0$ on the cylinder.

The auto-correlation function is given by,

$\displaystyle\langle 0|\mathcal{O}(\tau)\mathcal{O}(0)|0\rangle_{cyl}$
$\displaystyle=|z_{1}|^{\Delta}|z_{2}|^{\Delta}\bigg{|}\frac{\partial
z_{1\tau}}{z_{1}}\bigg{|}^{\Delta}\langle
0|\mathcal{O}(z_{1\tau},\bar{z}_{1\tau})\mathcal{O}(z_{2},\bar{z}_{2})|0\rangle_{pl}$
(12) $\displaystyle=\frac{1}{|c_{\tau}+d_{\tau}|^{2\Delta}}\langle
0|\mathcal{O}\bigg{(}\frac{a_{\tau}+b_{\tau}}{c_{\tau}+d_{\tau}},\frac{\bar{a}_{\tau}+\bar{b}_{\tau}}{\bar{c}_{\tau}+\bar{d}_{\tau}}\bigg{)}\mathcal{O}(0)|0\rangle_{pl}$
$\displaystyle=\frac{1}{|a_{\tau}+b_{\tau}-c_{\tau}-d_{\tau}|^{2\Delta}}$

where, in the second line of the equation, we write555The subscript $\tau$ in
$a_{\tau},b_{\tau},c_{\tau},d_{\tau}$ indicates that the parameters of the
Möbius transformation vary with time, as elaborated in Appendix A.
$z_{1\tau}=g(z_{1},\tau)=\frac{a_{\tau}z_{1}+b_{\tau}}{c_{\tau}z_{1}+d_{\tau}}$
and
$\bar{z}_{1\tau}=\bar{g}(\bar{z}_{1},\tau)=\frac{\bar{a}_{\tau}\bar{z}_{1}+\bar{b}_{\tau}}{\bar{c_{\tau}}\bar{z}_{1}+\bar{d}_{\tau}}$.
$\Delta$ in $\eqref{dacf1}$ is the conformal dimension of the operator
$\mathcal{(}O)$, and $\Delta=2h$ since we are considering scalar operators
here. Now, following the earlier discussion, depending on the different values
of the parameter $\delta$ one can distinguish the three phases and
correspondingly one can also find three different expressions for the
$(a_{\tau},b_{\tau},c_{\tau},d_{\tau})$ for the three different phases. We
discuss the details of this derivation in appendix A.

Using the expressions for $(a_{\tau},b_{\tau},c_{\tau},d_{\tau})$ from (A.11),
(A.12) and (A.13), one can write down the analytic expression for the auto-
correlation functions in the heating phase, non-heating phase and phase
boundary respectively. Once the auto-correlation functions are known, one can
find out the Lanczos Coefficients in each phase using the Toda-Chain (or
Hirota’s bilinear) technique Dymarsky:2019elm . As a first step to this
technique, one uses the autocorrelation function $C_{\phi}$ to initialize the
toda-function $\nu_{0}$ and solve the Toda Chain recursion relation

$\nu_{n}\ddot{\nu}_{n}-\dot{\nu}^{2}_{n}=\nu_{n-1}\nu_{n+1}$ (13)

with $\nu_{-1}=0$ and $\nu_{0}=C_{\phi}(\tau)$, to find an expression for
toda-function after nth iteration i.e. $\nu_{n}$. This then allows us to
compute the Lanczos coefficients using (13) and its derivatives666A more
detailed review of this technique can be found in the appendix of
Kundu:2023hbk . The knowledge of $a_{n}\ \text{and}\ b_{n}$ then facilitates
one to compute the normalized $\phi_{n}$ using (6).

Below we write down the analytic expressions for the auto-correlation
function, Lanczos coefficients, and normalized $\phi_{n}$777After analytically
continuing to real-time t by $\tau\rightarrow\tau_{0}+it$ in the three phases.

Non-heating Phase ($\delta<0$):

In this case, substitution of (A.11) in (12) leads to the auto-correlation
function,

$\displaystyle{}C_{NHP}(\tau)$
$\displaystyle=\left(\frac{2(\alpha+\beta)\sinh\left(\frac{\sqrt{|\delta|}\tau}{2}\right)}{\sqrt{|}\delta|}\right)^{-2\Delta}$
(14)

Then as described earlier, using the Toda chain technique, the Lanczos
coefficients can be obtained as,

$\displaystyle a_{n}$
$\displaystyle=-\frac{2(\Delta+n)\sqrt{|\delta|}}{tanh(\sqrt{|\delta|}\tau_{0})}$
(15) $\displaystyle b_{n}$ $\displaystyle=\frac{\sqrt{-n+2\Delta
n+n^{2}}\sqrt{|\delta|}}{sinh(\sqrt{|\delta|}\tau_{0})}$ (16)

Here, we notice that the Lanczos coefficients grow linearly with n which is
expected in such computations. Using $a_{n}$ and $b_{n}$, we obtain the
normalized wave function in this case as,

$\displaystyle\phi_{n}(t)$
$\displaystyle=\sqrt{\frac{\Gamma(2\Delta+n)}{\Gamma(2\Delta)\Gamma(n+1)}}\sin^{n}(\sqrt{|\delta|}t)csch^{n+2\Delta}(\sqrt{|\delta|}(\tau_{0}+it))\left(-\frac{\sqrt{|\delta|}}{\alpha+\beta}\right)^{2\Delta}$
(17)

wave function is oscillatory in time and we will see that the K-complexity for
this case will turn out to be oscillatory as well (27).

Heating Phase ($\delta>0)$:

In this phase, substitution of (A.12) in (12) leads to the auto-correlation
function,

$\displaystyle{}C_{HP}(\tau)=\left(\frac{2(\alpha+\beta)\sin\left(\frac{\sqrt{|\delta|}\tau}{2}\right)}{\sqrt{|\delta|}}\right)^{-2\Delta}$
(18)

This leads to the corresponding Lanczos coefficients (that are linear
functions in n), as follows:

$\displaystyle a_{n}$
$\displaystyle=-\frac{2(\Delta+n)\sqrt{|\delta|}}{tan(\sqrt{|\delta|}\tau_{0})}$
(19) $\displaystyle b_{n}$ $\displaystyle=\frac{\sqrt{-n+2\Delta
n+n^{2}}\sqrt{|\delta|}}{sin(\sqrt{|\delta|}\tau_{0})}$ (20)

The analytically continued normalized wave function, in this case, becomes a
hyperbolic function in t,

$\displaystyle\phi_{n}(t)$
$\displaystyle=\sqrt{\frac{\Gamma(2\Delta+n)}{\Gamma(2\Delta)\Gamma(n+1)}}\sinh^{n}(\sqrt{|\delta|}t)\csc^{n+2\Delta}(\sqrt{|\delta|}(\tau_{0}+it))\left(-\frac{\sqrt{|\delta|}}{\alpha+\beta}\right)^{2\Delta}$
(21)

and therefore, leads to the exponential k-complexity in the late time limit.

Phase Boundary ($\delta=0$):

Upon substitution of (A.13) in (12), the one finds the correlation function
for this phase to be

$\displaystyle{}C_{PB}(\tau)$ $\displaystyle=(\tau(\alpha+\beta))^{-2\Delta}$
(22)

which leads to corresponding Lanczos coefficients,

$\displaystyle a_{n}$ $\displaystyle=-\frac{2(\Delta+n)}{\tau_{0}}$ (23)
$\displaystyle b_{n}$ $\displaystyle=\frac{\sqrt{-n+2\Delta
n+n^{2}}}{\tau_{0}}$ (24)

and subsequently, to the analytically continued normalized wave function

$\displaystyle\phi_{n}(t)$
$\displaystyle=(-\tau_{0})^{4h}\frac{t^{n}}{(\tau_{0}+it)^{2\Delta+n}}\sqrt{\frac{\Gamma(2\Delta+n)}{\Gamma(2\Delta)\Gamma(n+1)}}.$
(25)

Notice that these expressions of the auto-correlation functions do not involve
the parameter $\gamma$ in any phase. It is also particularly relevant for the
heating phase because one possible choice of parameters for the heating phase
is when $\alpha$ and $\beta$ are both zero but $\gamma$ is non-zero since this
choice will also yield $\delta>0$. However, it seems from (18) that the auto-
correlation function diverges when $\alpha+\beta=0$. This issue gets resolved
once we note that for this particular choice of $\alpha,\beta,\gamma$, initial
position $x=0$ corresponds to a fixed point of the problem, where z equals 1.
In practical terms, this implies that under the deformed Hamiltonian
evolution, the position at $x=0$ remains unchanged. As a result, two of the
operators essentially stay in the same position, leading to the observed
divergence in the correlator.

Note that, in $\phi_{n}(t)$, we have analytically continued the Euclidean time
$\tau$ to $\tau_{0}+it$. We also find that the coefficients $a_{n}$ and
$b_{n}$ can be written in terms of $\tau_{0}$. one can then easily find the
exact analytic expression for each phase as given below

$\displaystyle K_{HP}$
$\displaystyle=\frac{2\Delta}{\sin^{2}\bigg{(}\frac{\sqrt{|\delta|}\tau_{0}}{2}\bigg{)}}\sinh^{2}\bigg{(}\frac{\sqrt{|\delta|}t}{2}\bigg{)}\
\ \ \ \ \ \ \ \ \ \textbf{(Heating Phase)}$ (26) $\displaystyle K_{NHP}$
$\displaystyle=\frac{2\Delta}{\sinh^{2}\bigg{(}\frac{\sqrt{|\delta|}\tau_{0}}{2}\bigg{)}}\sin^{2}\bigg{(}\frac{\sqrt{|\delta|}t}{2}\bigg{)}\
\ \ \ \ \ \ \ \textbf{(Non-heating Phase)}$ (27) $\displaystyle K_{PB}$
$\displaystyle=\frac{2\Delta}{\tau_{0}^{2}}t^{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \
\hskip 93.89418pt\ \ \textbf{(Phase Boundary)}$ (28)

Surprisingly, we find that the behaviour of K-complexity for the heating and
the non-heating phases that we have found in this section is similar to the
behaviour of the two-point function in the generic state above and below the
black hole threshold respectively found in Kundu:2023hbk . Next, we study the
behaviour of the K-complexity for deformed CFTs in a generic state.

###  3.2 Excited state of a 2d CFT

In this section, we define the auto-correlation function $C_{\phi}(\tau)$ in
an excited state, following Kundu:2023hbk  as,

$C_{\psi}(\tau)=\frac{\langle\psi_{out}|\mathcal{O}_{L}(\tau)\mathcal{O}_{L}(0)|\psi_{in}\rangle}{\langle\psi_{out}|\psi_{in}\rangle}$
(29)

where the $in$ and $out$ states are defined as,
$|\psi_{in}\rangle=\lim\limits_{T\to\infty}\mathcal{O}_{H}(T)|0\rangle$ and
$\langle\psi_{out}|=\lim\limits_{T\to\infty}\langle 0|\mathcal{O}_{H}(T)$ and
$|0\rangle$ is the ground state of the 2d CFT on a circle 888This is the
ground state of the deformed CFT as well. Since the deformed Hamiltonian is
made out of global conformal generators only.. Noting that,
$\mathcal{O}_{L}(\tau)=e^{H^{cyl}_{d}\tau}\mathcal{O}_{L}(0)e^{-H^{cyl}_{d}\tau}$,
we can write (29) explicitly as:

$C_{\psi}(\tau)=\lim_{\begin{subarray}{c}\omega_{1}\to\infty\\\ \omega_{4}\to
0\end{subarray}}\frac{\langle
0|\mathcal{O}_{H}(\omega_{1},\bar{\omega_{1}})e^{H^{cyl}\tau}\mathcal{O}_{L}(\omega_{2},\bar{\omega_{2}})(0)e^{-H^{cyl}\tau}\mathcal{O}_{L}(\omega_{3},\bar{\omega_{3}})\mathcal{O}_{H}(\omega_{4},\bar{\omega_{4}})|0\rangle}{\langle
0|\mathcal{O}_{H}(\omega_{1},\bar{\omega_{1}})\mathcal{O}_{H}(\omega_{4},\bar{\omega_{4}})|0\rangle}$
(30)

Under the above coordinate transformations, the correlation function
transforms as,

$C_{\psi}(\tau)=\left|\frac{dz^{\prime}_{2}}{dz_{2}}\right|^{\Delta_{L}}\left|z_{14}\right|^{2\Delta_{H}}\left|z_{2}z_{3}\right|^{2\Delta_{L}}\langle
0|\mathcal{O}_{H}(z_{1})\mathcal{O}_{L}(z^{\prime}_{2})(0)\mathcal{O}_{L}(z_{3})\mathcal{O}_{H}(z_{4})|0\rangle\times
C.C$ (31)

In the above, we have used the fact that, in this case, the time evolution
becomes a conformal transformation in the complex plane.999For details of the
calculation see appendix A . Our problem is now reduced to the computation of
a four-point correlation function, which depends on the dynamical input of the
CFT in consideration and is in general a complicated function
$\mathcal{F}(\eta,\bar{\eta})$ of the cross-ratio $\eta$ and its complex
conjugate $\bar{\eta}$ defined below,

$\displaystyle\eta=\frac{z_{12}z_{34}}{z_{13}z_{24}},\hskip
14.22636pt\bar{\eta}=\frac{\bar{z}_{12}\bar{z}_{34}}{\bar{z}_{13}\bar{z}_{24}}$

To compute $C_{\psi}$, we again fix the initial positions of the operators on
the cylinders as $\omega_{1}=T,\omega_{2}=0,\omega_{3}=0,\omega_{4}=-T$ and
hence the positions on the plane are,

$\displaystyle z_{1}=e^{T},\hskip 8.5359ptz_{2}=1,\hskip
8.5359ptz_{3}=1,\hskip 8.5359ptz_{4}=e^{-T}$

In the $T\to\infty$ limit, the cross-ratio is

$\displaystyle\eta=\frac{c_{\tau}+d_{\tau}}{a_{\tau}+b_{\tau}},\hskip
14.22636pt\bar{\eta}=\frac{\bar{c}_{\tau}+\bar{d}_{\tau}}{\bar{a}_{\tau}+\bar{b}_{\tau}}$

Putting the values of $z$, the auto-correlation function becomes,

$\displaystyle C_{\psi}(\tau)$ $\displaystyle=$
$\displaystyle\lim\limits_{T\to\infty}\frac{1}{\left|c_{\tau}+d_{\tau}\right|^{2\Delta_{L}}}\left(e^{T}-e^{-T}\right)^{2\Delta_{H}}\langle
0|\mathcal{O}_{H}(e^{T})\mathcal{O}_{L}\left(\frac{a_{\tau}+b_{\tau}}{c_{\tau}+d_{\tau}}\right)\mathcal{O}_{L}(1)\mathcal{O}_{H}(e^{-T})|0\rangle\times
C.C$ (32) $\displaystyle=$
$\displaystyle\frac{1}{\left|c_{\tau}+d_{\tau}\right|^{2\Delta_{L}}}\langle
0|\mathcal{O}_{H}(\infty)\mathcal{O}_{L}\left(\frac{1}{\eta}\right)\mathcal{O}_{L}(1)\mathcal{O}_{H}(0)|0\rangle\times
C.C$ $\displaystyle=$
$\displaystyle\frac{1}{\left|a_{\tau}+b_{\tau}\right|^{2\Delta_{L}}}\mathcal{F}(\eta,\bar{\eta})$

In the second line of the above, we have used the fact that
$\langle\mathcal{O}(\infty)|=\lim\limits_{\begin{subarray}{c}z\to\infty\\\
\bar{z}\to\infty\end{subarray}}\langle\mathcal{O}(z,\bar{z})|$ and we have
redefined101010We have used the identity
$\langle\mathcal{O}_{H}(\infty)\mathcal{O}_{L}(\frac{1}{\eta},\frac{1}{\bar{\eta}})\mathcal{O}_{L}(1)\mathcal{O}_{H}(0)\rangle=|\eta|^{2\Delta_{L}}\langle\mathcal{O}_{H}(\infty)\mathcal{O}_{L}(1)\mathcal{O}_{L}(\eta,\bar{\eta})\mathcal{O}_{H}(0)\rangle$
$\eta$ as $\frac{1}{\eta}$. A closed-form analytic expression for
$\mathcal{F}(\eta,\bar{\eta})$ is not known generally, except, in certain
specific limits. One such example is when we consider that the central charge
of the CFT is large ($c\to\infty$). We’ll consider this scenario here.

####  3.2.1 For 2d CFT with large central charge

((a))

((b))

Figure 1: [a]. The k-complexity for Heating(•), Non-Heating(•), and Phase-
boundary(•) when the weight of the heavy operator is above the black hole
threshold. As is evident, in the Heating phase the complexity grows
exponentially in time, while it remains polynomial for the Phase-boundary.
However, in the Non-Heating phase, the behavior becomes exponential compared
to the oscillatory behavior below the threshold. [b]. Below the BH threshold,
the k-complexity for the heating phase(•), non-heating phase (•), and phase
boundary (•) is exponential, oscillatory, and polynomial respectively similar
to the vacuum state computation.

((a))

((b))

Figure 2: [a]. The logarithm of the complexity for the heating phase(•) and
phase boundary(•) above the threshold. As we can see, it saturates in the
phase boundary but keeps increasing in the heating phase. [b]. The logarithm
of the complexity for the heating phase(•) and phase boundary(•) below the
threshold. As we can see, it saturates in the phase boundary but keeps
increasing in the heating phase, though the rate of increment decreases
compared to the above threshold case.

((a))

((b))

Figure 3: [a]. The logarithm of the K-complexity in the phase boundary above
the threshold. It saturates in a long time limit, confirming that the behavior
is polynomial. [b]. The logarithm of the K-complexity in the phase boundary
above the threshold. It saturates in a long time limit, which again confirms
that the behavior is polynomial.

It is known that any correlation function in CFT can be written as the sum of
the Virasoro conformal blocks. However, when we consider $c$ to be large with
$\frac{h_{L}}{c}$ is fixed and small and $h_{H}$ fixed and large, only the
vacuum block dominates Fitzpatrick:2014vua ; Fitzpatrick:2015zha . In this
limit,

$\displaystyle\mathcal{F}(\eta,\bar{\eta})=\nu^{2\Delta_{L}}\left(\frac{\eta^{\frac{\nu-1}{2}}}{1-\eta^{\nu}}\right)^{\Delta_{L}}\left(\frac{\bar{\eta}^{\frac{\nu-1}{2}}}{1-\bar{\eta}^{\nu}}\right)^{\Delta_{L}}+\mathcal{O}\left(\frac{1}{c}\right)\hskip
5.69046pt\text{where}\hskip 5.69046pt\nu=\sqrt{1-12\frac{\Delta_{H}}{c}}$ (33)

Using (33) in (32), the auto-correlation function becomes,

$\displaystyle
C_{\psi}(\tau)=\left(\nu\right)^{2\Delta_{L}}\left(\frac{(c_{\tau}+d_{\tau})^{\frac{\nu-1}{2}}(a_{\tau}+b_{\tau})^{\frac{\nu-1}{2}}}{(a_{\tau}+b_{\tau})^{\nu}-(c_{\tau}+d_{\tau})^{\nu}}\right)^{\Delta_{L}}\left(\frac{(\bar{c}_{\tau}+\bar{d}_{\tau})^{\frac{\nu-1}{2}}(\bar{a}_{\tau}+\bar{b}_{\tau})^{\frac{\nu-1}{2}}}{(\bar{a}_{\tau}+\bar{b}_{\tau})^{\nu}-(\bar{c}_{\tau}+\bar{d}_{\tau})^{\nu}}\right)^{\Delta_{L}}$
(34)

The behavior of the auto-correlation function depends on
$a_{\tau},b_{\tau},c_{\tau},d_{\tau}$, and the parameter $\nu$. The parameters
of the Möbius transformation depend on the phase we are considering, and
details of the expressions are given in appendix A. In each phase, there are
two different scenarios to consider separately: one when the weight of the
heavy operator is above the critical value ($\Delta_{H}>\frac{c}{12}$) and the
other case when it’s below the critical value ($\Delta_{H}<\frac{c}{12}$). In
the previous case, we could write the analytical expressions of K-complexity
in the three phases using the Lanczos algorithm and recursion relation in
$\phi_{n}$. For this case, however, it is difficult to find out the analytic
expression for Lanczos coefficients and therefore of the K-complexity. One can
however turn to numerical approaches to obtain some important observations in
such cases.

I: Heating Phase ($\delta>0$)

We consider a particular choice of $\alpha,\beta$, and $\gamma$s that belong
to the heating phase. We considered $\beta=1,\alpha=\gamma=0$. However, since
all the other points within the heating phase are related to each other by
some unitary transformation, the behavior would be similar for other choices
as well.

With this particular choice, when the weight of the heavy operator is below
the threshold 111111In this case and all the other cases, we have taken
$\Delta_{L}=1$. ($\Delta_{H}=\frac{3c}{48}$), the auto-correlation function
is,

$C_{\psi}(\tau)=\frac{\left(\tan\left(\frac{\tau}{2}\right)+1\right)^{3/2}\cot\left(\frac{\tau}{2}\right)}{2(\sin(\tau)+1)\sqrt{\tan\left(\frac{\tau}{2}\right)-1}}$
(35)

Figure 4: K-complexity in Heating phase above and below threshold

With the above auto-correlation function, we find numerically that the Lanczos
coefficients grow linearly in n. The Krylov complexity grows exponentially in
time, as shown in Fig.[4]. On the other hand, when the weight of the heavy
operator is above the critical value ($\Delta_{H}=\frac{c}{6}$), the
correlation function is

$C_{\psi}(\tau)=\frac{\sin(\tau)+\cos(\tau)+1}{2(\sin(\tau)+1)(\sin(\tau)-\cos(\tau)-1)\left(\cos\left(\frac{1}{2}\log\left(\frac{2}{\sin(\tau)+1}-1\right)\right)-\cosh(\pi)\right)}$
(36)

In this case also the behavior of the Lanczos coefficients and the Krylov
complexity is similar to the above case. However, we notice that the rate of
the growth of the complexity increases in this scenario (Fig.[4]). To confirm
that the behavior of the K-complexity is indeed exponential both above and
below the threshold, we also plot the logarithm of the K-complexity in both
cases, and we find that it keeps increasing linearly in time (Fig.[2(a),
2(b)].

II: Non-heating phase($\delta<0$)The simplest example of the non-heating phase
is when $\alpha=1$ and $\beta=\gamma=0$. This choice leads to the usual CFT
Hamiltonian. In this case, the expressions of the auto-correlation functions
above and below-threshold are much simpler and hence we can get the exact
analytic expressions of Lanczos coefficients and the complexity Kundu:2023hbk
.

Figure 5: K-complexity in non-heating phase above and below threshold

When the weight of the heavy operator is below the threshold, the auto-
correlation function is

$C_{\psi}(\tau)=\frac{(\frac{\gamma}{2})^{2\Delta_{L}}}{\sinh{\left(\frac{\gamma\tau}{2}\right)}^{2\Delta_{L}}}$
(37)

In this case, the k-complexity is given by

$K_{L}(t)=2\Delta_{L}csch^{2}(\frac{\gamma\tau_{0}}{2})\sin^{2}{\left(\frac{\gamma
t}{2}\right)}$ (38)

While above the threshold, the auto-correlation function is,

$C_{\psi}(\tau)=\frac{(\frac{\gamma}{2})^{2\Delta_{L}}}{\sin{\left(\frac{\gamma\tau}{2}\right)}^{2\Delta_{L}}}$
(39)

With the above auto-correlation function, the complexity is,

$K_{L}(t)=2\Delta_{L}csc^{2}(\frac{\gamma\tau_{0}}{2})\sinh^{2}{\left(\frac{\gamma
t}{2}\right)}$ (40)

So, as one can see, there is a change in the behavior of the complexity from
oscillatory to exponential when we go from below to above the threshold limit.
As discussed in Kundu:2023hbk , this change in the behavior is related to the
fact that two phases correspond to thermal AdS3 and black hole geometries in
the bulk.

III: Phase boundary ($\delta=0$)To study the behavior in the phase boundary,
we choose $\alpha=\beta=1$ and $\gamma=0$.

Figure 6: K-complexity in phase boundary above and below threshold

With this choice, we find that in the case when we are below threshold, the
auto-correlation function is:

$C_{\psi}(\tau)=\frac{1}{2\tau\sqrt{\tau^{2}-1}}$ (41)

With the above expression, we find that again the Lanczos coefficients grow
linearly in n as expected, but this time the complexity grows polynomial in
time. The auto-correlation function above the threshold value is,

$C_{\psi}(\tau)=-\frac{\cos\left(\log\left(\frac{\tau+1}{\tau-1}\right)\right)+\cos\left(\log\left(\frac{\tau-1}{\tau+1}\right)\right)-2\cosh(\pi)}{2\left(\tau^{2}-1\right)\left((4\cosh(\pi)-1)\cos\left(\log\left(\frac{\tau-1}{\tau+1}\right)\right)-2-\cosh(2\pi)\right)}$
(42)

In this scenario, we once again notice a polynomial increase in K-complexity.
However, it is noteworthy that the growth rate intensifies during this phase
(Fig.[6]). To confirm that the behavior in both cases is polynomial, we also
plot the logarithm of the K-complexity in both cases, and we observe that it
saturates to a particular value after a certain time (Fig.[3(a)], Fig.[3(b)]).
Due to limitations in numerical computations, we couldn’t plot it for longer
times, but the plots clearly indicate that the behavior is polynomial.
To compare the behavior of the K-complexity in all three phases, both above
and below the threshold, we have plotted them together in Fig.[1(a)] and
Fig.[1(b)].

####  3.2.2 For massless free scalar field theory

Once more, we examine the auto-correlation function in the excited state. This
time, our focus is on the two-point function of two scalar operators with
$\Delta_{1}=1$. The state is generated by applying the vertex operator
$e^{ik\phi}$ to the vacuum. With this specific operator configuration, the
auto-correlation function in the deformed CFT case is derived as follows:

${}C_{\psi}(\tau)=\lim\limits_{T\to\infty}\frac{1}{\left|c_{\tau}z_{2}+d_{\tau}\right|^{2\Delta_{L}}}(z_{2}z_{3})\left(\frac{1}{z_{12\tau}z_{34}}+\frac{1}{z_{13}z_{2\tau
4}}-\frac{1}{4\pi}\frac{1}{z_{2\tau 3}^{2}}\right)$ (43)

Where, $z_{ij}=z_{i}-z_{j}$ and
$z_{i\tau}=\frac{a_{\tau}z_{i}+b_{\tau}}{c_{\tau}z_{i}+d_{\tau}}$. The exact
form of (43) depends on the form of $a_{\tau},b_{\tau},c_{\tau}$ and
$d_{\tau}$ and hence on the phase in which we are computing. For the sake of
simplicity, we consider specific choices of $\alpha,\beta$, and $\gamma$s as
representatives of different phases. For example, we choose $\alpha\neq
0,\beta=\gamma=0$ for non-heating phase and $\beta\neq 0,\alpha=\gamma=0$ and
$\alpha=\beta\neq 0,\gamma=0$ for heating phase and phase boundary
respectively.
Non-heating phase: With the above choices of $\alpha,\beta,\gamma$, the auto-
correlation function is,

$C_{\psi}(\tau)=-\frac{1}{16\pi}\left(\frac{1}{\sinh{\frac{\alpha\tau}{2}}}\right)^{2}$
(44)

Heating phase: In the heating phase, the auto-correlation function is

$C_{\psi}(\tau)=-\frac{1}{16\pi}\left(\frac{1}{\sin{\frac{\beta\tau}{2}}}\right)^{2}$
(45)

Phase boundary: In the phase boundary, the auto-correlation function is

$C_{\psi}(\tau)=-\frac{1}{16\pi}\left(\frac{1}{\beta\tau}\right)^{2}$ (46)

As we can see from the above expression, the auto-correlation function matches
with the auto-correlation functions in the vacuum state ((18)-(22)), with
$\Delta=1$. Therefore, the Lanczos coefficients and the Krylov complexities
will be similar to those in the three phases. The important point to notice
here is that, even in the free field theory case, Krylov complexity grows
exponentially in time in the heating phase. Hence, the K-complexity can’t
differentiate between a chaotic (large-c) and non-chaotic CFT (free field
theory) in the heating phase and on the phase boundary.

##  4 Discussion

In this article, we studied the evolution of operator complexity within the
framework of $SL(2,\mathbb{R})$ deformed Conformal Field Theories (CFTs) in
1+1 dimensions. This investigation is equally applicable to the study of
complexity in the context of periodically driven/floquet CFTs, given that the
dynamics of such systems can be effectively described by the Hamiltonian of
these deformed CFTs. Our findings reveal distinct qualitative behaviors of
K-complexity in different phases.

During the non-heating phase, K-complexity demonstrates oscillatory behavior.
In contrast, within the phase boundary and heating phases, particularly when
considering the vacuum state of the CFT, we observe both polynomial and
exponential growth. However, when examining a generic state in the non-heating
phase of large c CFT, K-complexity behaves differently above and below the
threshold. Above the threshold, complexity exhibits exponential growth, while
below the threshold, it displays oscillatory behavior. In the non-heating
phase of a free-field theory, the dynamics of K-complexity always remain
oscillatory.

Interestingly, it appears that in some generic states during the non-heating
phase, K-complexity can differentiate between chaotic and non-chaotic CFTs.
Nevertheless, in the heating phase and at the phase boundary, the situation
changes. Notably, in large c, the behavior of complexity remains exponential
and polynomial both above and below the threshold in these phases. Even in the
massless free field theory, complexity displays exponential and polynomial
growth in the heating phase and at the phase boundary.

The inability of K-complexity to differentiate between chaotic and non-chaotic
CFTs in the heating phase and at the phase boundary may be attributed to the
fact that distinct phases correspond to different quantizations of 2D CFT. For
instance, the sine-squared deformed Hamiltonian, which is an example of phase
boundary, corresponds to ‘dipolar quantization,’ where the spectrum of
Hamiltonian is continuous due to the infinite effective length of the system
Ishibashi:2015jba . A similar continuous spectrum is observed in the heating
phase as well Tada:2019rls . This contrasts with radial quantization, which
yields a discrete spectrum. Thus, K-complexity seems to differentiate between
chaotic and non-chaotic systems only when the theory exhibits a discrete
spectrum, aligning with the observations in Iizuka:2023pov .

As a future direction, it would be interesting to extend this study to other
quench protocols as well. For instance, one immediate extension could be to
analyze a global quench protocol where one initially starts from a regulated
boundary state and then evolves the system with the usual CFT Hamiltonian. It
was discussed in cardy , that such a system thermalizes at late times.
Therefore, it would be interesting to study how the K-complexity evolves in
this system. In Rabinovici:2023yex , the authors conjectured a holographic
bulk dual description of K-complexity in terms of the length of a two-sided
wormhole. Since, the holographic dual geometries of $SL(2,\mathbb{R})$
deformed CFTs are known Das:2022pez , it would be worthwhile to study the
complexity from the bulk and match it with the boundary results.

## Acknowledgments

The authors thank Suman Das for the initial collaboration on some topics
related to this project. The authors are thankful to Bobby Ezhuthachan, Arnab
Kundu, Ritam Sinha, and Junggi Yoon for several useful discussions and
comments. BR would like to thank YITP, Kyoto University for their warm
hospitality at the long-term conference (YITP-T-23-01), during which a part of
this work was done. SP would like to thank Prof. Junggi Yoon for the month-
long visit to APCTP, Pohang, where, some final works of this project were
completed. The work of VM was supported by the NRF grant funded by the Korea
government (MSIT) (No. 2022R1A2C1003182) and by the Brain Pool program funded
by the Ministry of Science and ICT through the National Research Foundation of
Korea (RS-2023-00261799). The work of SP and BR is supported by the Senior
Research Fellowship(SRF) funded by the University Grant Commission(UGC) under
the CSIR-UGC NET Fellowship.

##  Appendix A  Conformal transformation generated by the deformed Hamiltonian

In this section, we briefly review the dynamics of a deformed conformal field
theory. Consider a 1+1D CFT defined on an infinite cylinder of circumference
L($=2\pi$). The Hamiltonian of the system generates translations along the
length of the cylinder and therefore is defined by integrating the energy
density on a time slice $t_{c}=0$ over the spatial circle $x_{c}=x$121212The
subscript c denotes the coordinates on the cylinder. as

$H=\oint T_{00}(x)dx$ (A.1)

Using analytic continuation to Euclidean space on a complex plane defined by
$\zeta=\tau_{c}+ix_{c}$ and $\bar{\zeta}=\tau_{c}-ix_{c}$ ) the above equation
can be rewritten in terms of the left(right) moving components $T(\bar{T})$ of
the energy-momentum tensor and the corresponding Virasoro modes

$\displaystyle H$
$\displaystyle=\frac{1}{2\pi}\oint(T(\zeta)d\zeta+\bar{T}(\bar{\zeta})d\bar{\zeta})$
(A.2) $\displaystyle=L_{0}+\bar{L}_{0}-\frac{c}{12}$

using the usual constructions of operator formalism in radial quantization.

In the case of a Deformed CFT, instead of evolving the system with an ordinary
CFT Hamiltonian H, one studies the operator evolution under a deformed CFT
Hamiltonian

$H_{def}=\oint f(x)T_{00}(x)+g(x)T_{01}(x)dx$ (A.3)

where the energy density $T_{00}$ and momentum density $T_{01}$ are changed
using some control function $f(x)$ and $g(x)$. For simplicity, one can make
the choice $f(x)=g(x)$ and therefore

$H_{def}=\frac{1}{2\pi}\oint
f(\zeta)T(\zeta)d\zeta+\bar{f}(\bar{\zeta})\bar{T}(\bar{\zeta})d\bar{\zeta}$
(A.4)

.

This deformed Hamiltonian $H_{def}$ in general can be written as a linear
combination of Virasoro generators $L_{n}$s and $\bar{L}_{n}$s.

$H=\sum_{n}p_{n}L_{n}+\sum_{n}q_{n}\overline{L}_{n}-\frac{c}{12}$

One can simplify the problem by restricting to a certain choice of functions
such that the deformed Hamiltonian is only made out of the generators
SL${}_{q}(2,\mathcal{R})$,i.e. $(L_{0},L_{q},L_{-q})$ and its conjugates, as
described in (7)

$\displaystyle
H_{def}=\alpha(L_{0}+\bar{L}_{0})+\beta(L_{q+}+\bar{L}_{q+})+\gamma(L_{q-}+\bar{L}_{q-})-\frac{c}{12}$
(A.5)

where, $L_{q+}=\frac{L_{q}+L_{-q}}{2}$, $L_{q-}=\frac{L_{q}-L_{-q}}{2i}$ and
$\alpha,\ \beta$ and $\gamma$ are some real numbers. This linear combination
of generators can be classified into three distinct classes, depending on the
quantity $\delta=-\alpha^{2}+\beta^{2}+\gamma^{2}$ defined using the
coefficients $\alpha,\ \beta,\ \gamma$. This can be understood as the adjoint
action on $\alpha L_{0}+\beta L_{q+}+\gamma L_{q-}$ changes the coefficients
to some other values, $\alpha^{\prime},\ \beta^{\prime},\ \gamma^{\prime}$
keeping the value of $\delta$ same.

$\delta=-\alpha^{2}+\beta^{2}+\gamma^{2}=-\alpha^{\prime 2}+\beta^{\prime
2}+\gamma^{\prime 2}$

The three classes of Hamiltonians for $\delta\ (=0,\ >0,\ <0)$ are known
respectively as parabolic, hyperbolic, and elliptic in the literature. It has
been shown that two-point and one-point correlation functions e.g.
entanglement entropy and energy density show different dynamical behaviour
when evolved with these three types of Hamiltonians. The EE shows linear,
oscillatory, and logarithmic behavior whereas the Energy density shows
exponential, oscillatory, and linear behaviour for $\delta>0,<0$, and
$\delta=0$ respectively. Following the authors in Wen:2018agb ; Wen:2020wee
we will be referring to the three classes as heating $(\delta>0)$ phase and
non-heating phase $(\delta<0)$ separated by a phase boundary $(\delta=0)$.

In Das:2022jrr  it has also been shown that even non-trivial higher point
correlators such as four-point OTOC show different behaviors in these three
phases.

The fact that we can now write the Hamiltonian in $\eqref{defh3}$ as a linear
combination of Virasoro generators, the dynamics get mapped to a conformal
transformation and therefore the problem becomes analytically tractable.
Specifically, in the Heisenberg picture, the time evolution (with the
Euclidean time $s$131313Note that, here we refer to this Euclidean time ”s”
(instead of $\tau$ )) of a primary operator (with conformal weights
$\mathbf{h}$, $\mathbf{\overline{h}}$), gets translated into the dynamics of
the operator under conformal transformation

$\displaystyle e^{H_{def}s}O(z,{\overline{z}})e^{-H_{def}s}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial z^{\prime}}{\partial
z}\right)^{h}\left(\frac{\partial{\overline{z}^{\prime}}}{\partial{\overline{z}}}\right)^{\overline{h}}O(z^{\prime},{\overline{z}^{\prime}})$
(A.6)

in that maps the coordinates $z=e^{\frac{2\pi
q\zeta}{L}},\bar{z}=e^{\frac{2\pi q\bar{\zeta}}{L}}$ to some
$(z^{\prime},\bar{z}^{\prime})$ by generating Möbius transformation in the
q-sheet Riemann surface141414 The Conformal map $z=e^{\frac{2\pi
q\zeta}{L}},\bar{z}=e^{\frac{2\pi q\bar{\zeta}}{L}}$ maps the cylinder
coordinates $(\zeta,\bar{\zeta})$ to the coordinates $(z,\bar{z})$ of the
q-sheet surface defined by $z$. For q=1, this corresponds to the usual map
from cylinder to the complex plane $(z,\bar{z})$151515we have only written the
holomorphic part

$\displaystyle z\to z^{\prime}=\frac{a_{s}z+b_{s}}{c_{s}z+d_{s}}\ \
\text{with}\ \ \left(\begin{array}[]{cc}a_{s}&b_{s}\\\
c_{s}&d_{s}\end{array}\right)\in{SL}(2,{\mathbb{R}})$ (A.9)

This enables us to now use the usual rules of 2D CFT at our disposal to find
out how correlation functions behave under such conformal transformations. Our
next task, therefore would be to find out the analytic expressions for
$a_{s},b_{s},c_{s},d_{s}$ in terms of the Hamiltonian parameters
$\alpha,\beta,\gamma,\delta$ and Euclidean time $s$.

An easy method to do so is to re-write the Hamiltonian as a generator of
translation in a coordinate system labeled by $(\tilde{z},\bar{\tilde{z}})$,
such that,

$\displaystyle
H_{def}=\tilde{\mathcal{L}}_{-1}+\bar{\tilde{\mathcal{L}}}_{-1}\ \ \ \
\text{and}\ \ \ \ \
e^{sH_{def}}\tilde{z}\implies\tilde{z}^{\prime}=\tilde{z}+s$ (A.10)

Using this and by expressing the coordinate transformations written in terms
of $(\tilde{z},\bar{\tilde{z}})$ and
$(\tilde{z}^{\prime},\bar{\tilde{z}^{\prime}})$ to write in terms of original
coordinates $(z,\bar{z})$ and $(z^{\prime},\bar{z^{\prime}})$, one can find
the expressions for $a_{s},b_{s},c_{s},d_{s}$ to be as follows:

For heating Phase

$\displaystyle
a_{s}=-\frac{\alpha}{\sqrt{|\delta|}}\sin\left(\frac{\sqrt{|\delta|}qs}{2}\right)-\cos\left(\frac{\sqrt{|\delta|}qs}{2}\right),\
\ \ \ \ \
b_{s}=-\frac{(\beta+i\gamma)\sin\left(\frac{\sqrt{|\delta|}qs}{2}\right)}{\sqrt{|\delta|}}$
(A.11) $\displaystyle
c_{s}=\frac{(\beta-i\gamma)\sin\left(\frac{\sqrt{|\delta|}qs}{2}\right)}{\sqrt{|\delta|}},\
\ \ \ \ \ \
d_{s}=-\frac{\alpha}{\sqrt{|\delta|}}\sin\left(\frac{\sqrt{|\delta|}qs}{2}\right)+\cos\left(\frac{\sqrt{|\delta|}qs}{2}\right)$

For non-heating Phase

$\displaystyle
a_{s}=-\frac{\alpha}{\sqrt{|\delta|}}\sinh\left(\frac{\sqrt{|\delta|}qs}{2}\right)-\cosh\left(\frac{\sqrt{|\delta|}qs}{2}\right),\
\ \ \ \ \
b_{s}=-\frac{(\beta+i\gamma)\sinh\left(\frac{\sqrt{|\delta|}qs}{2}\right)}{\sqrt{|\delta|}}$
(A.12) $\displaystyle
c_{s}=\frac{(\beta-i\gamma)\sinh\left(\frac{\sqrt{|\delta|}qs}{2}\right)}{\sqrt{|\delta|}},\
\ \ \ \
d_{s}=-\frac{\alpha}{\sqrt{|\delta|}}\sinh\left(\frac{\sqrt{|\delta|}qs}{2}\right)+\cosh\left(\frac{\sqrt{|\delta|}qs}{2}\right)$

For Phase boundary

$\displaystyle a_{s}=-(1+\frac{\alpha\sqrt{|\delta|}qs}{2})\ \ \ \ \ \
b_{s}=-\frac{(\beta+i\gamma)\sqrt{|\delta|}qs}{2})$ (A.13) $\displaystyle
c_{s}=\frac{(\beta-i\gamma)\sqrt{|\delta|}qs}{2})\ \ \ \ \ \ \
d_{s}=(\frac{\alpha\sqrt{|\delta|}qs}{2}-1)$

where for each case $a_{s}d_{s}-b_{s}c_{s}=1$ is satisfied. In this paper, we
have simplified our problem further by considering the q=1 case, such that,
the Hamiltonian is built out of only the global Virasoro generators
$L_{0},L_{\pm 1}$ and their conjugates. One must note that these generators
keep the vacuum state invariant, and therefore for the initial vacuum state
only unequal time correlators can capture the dynamics of the drive.

##  Appendix B Numerical Technique utilized to compute the K-Complexity

In this section, we briefly outline the numerical technique utilized to
compute K-Complexity. In order to do this we first determine the Lanczos
coefficients numerically from the respective auto-correlations utilizing
either the Toda chain method Dymarsky:2019elm  (also see appendix of
Kundu:2023hbk ) or a modified version of moment method for non-zero $a_{n}$s
described in VMR:2008vsg . Since the Lanczos coefficients grow linearly up to
some large-n we extrapolate the results and utilize them to obtain $\phi_{n}$s
through the method described below. In all the cases we have considered, we
find that the Lanczos grow linearly in n as expected in a QFT. We have given
plots for some of the examples in Fig.[7] and Fig.[8].

((a))

((b))

Figure 7: The above plots show the behavior of the Lanczos coefficients in the
Heating phase above the threshold value. Plot [a]. shows Lanczos coefficient
$b_{n}$ increases linearly in $n$ and [b]. shows Lanczos coefficient $a_{n}$
decreases with $n$. The red dots are the numerical data points, and the blue
line represents the best fit, determined by specific values of the parameter
sets (p, q) and (r, s).

((a))

((b))

Figure 8: The above plots show the behavior of the Lanczos coefficients at the
phase boundary below the threshold value. Plot [a]. shows Lanczos coefficient
$b_{n}$ increases linearly in $n$ and [b]. shows Lanczos coefficient $a_{n}$
decreases with $n$. The red dots are the numerical data points, and the blue
line represents the best fit, determined by specific values of the parameter
sets (p, q) and (r, s).

Determining the $\phi_{n}$s by directly solving the recursion equation in (6)
is not very tenable numerically. Hence, we resort to an interesting technique
described in Bhattacharya:2023zqt . The method involves re-expressing the
recursion equation in (6) as a matrix differential equation expressed below

$\displaystyle\frac{d\Phi(t)}{dt}={\tilde{\cal L}}\Phi(t)$ (B.1)

where $\tilde{{\cal L}}$ is given by

$\displaystyle\tilde{\mathcal{L}}=\left(\begin{array}[]{cccccc}ia_{0}&-b_{1}&&&&0\\\
b_{1}&ia_{1}&-b_{2}&&&\\\ &b_{2}&ia_{2}&\ddots&&\\\ &&\ddots&\ddots&-b_{m}&\\\
&&&b_{m}&ia_{m}&\ddots\\\ 0&&&&\ddots&\ddots\end{array}\right).$ (B.8)

and $\Phi$ is a column vector made up of the amplitudes $\phi_{n}$

$\displaystyle\Phi(t)=\left(\begin{array}[]{c}\phi_{0}(t)\\\ \phi_{1}(t)\\\
\phi_{2}(t)\\\ \vdots\\\ \phi_{n_{max}-2}(t)\\\
\phi_{n_{max}-1}(t)\end{array}\right)$ (B.15)

Note the norm of the vector $\Phi(t)$ gives the sum of the probabilities. The
complexity on the other hand is given by the norm of the following column
vector

$\displaystyle\sqrt{\mathcal{K}(t)}=\left(\begin{array}[]{c}0\\\
\sqrt{1}\phi_{1}(t)\\\ \sqrt{2}\phi_{2}(t)\\\ \vdots\\\ \sqrt{n_{max}-2}\
\phi_{n_{max}-2}(t)\\\ \sqrt{n_{max}-1}\
\phi_{n_{max}-1}(t)\end{array}\right)$ (B.22)

We then solved the matrix differential equation numerically to obtain the
results we have shown in our article. Although this method is more suitable
for quantum systems with finite-dimensional Krylov space, it turns out to be
very efficient in accurately reproducing the dynamics of the infinite-
dimensional case up to a certain $t_{max}$ for a large enough matrix (For the
results we have obtained we have gone upto a matrix of dimensions as high as
2000 $\times$ 2000). This $t_{max}$ can be determined approximately by
comparing $\phi_{0}(t)$ obtained from numerical computation with the exact
auto-correlation we derived. We have also verified that this method accurately
reproduces the known results.

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# High key rate quantum conference key agreement with unconditional security

Xiao-Yu Cao  Yu-shuo Lu  Zhao Li  Jie Gu  Hua-Lei Yin<EMAIL_ADDRESS>Zeng-
Bing Chen<EMAIL_ADDRESS>National Laboratory of Solid State
Microstructures, School of Physics and Collaborative Innovation Center of
Advanced Microstructures, Nanjing University, Nanjing 210093, China

###### Abstract

Quantum cryptography is a major ingredient of the future quantum internet that
promises various secure communication tasks. Quantum conference key agreement
(CKA) is an important cryptographic primitive of quantum cryptography, which
provides the conference key among multiple users simultaneously. However,
quantum CKA is currently far from practical application due to the low
conference key rate. Here, we propose a quantum CKA protocol of three users
with information-theoretic security. Our protocol only requires phase-
randomized weak coherent sources and threshold single-photon detectors, and is
anticipated to be experimentally demonstrated over 600 km under current
technology. Our scheme can be widely implemented in the approaching large-
scale quantum network.

##  I INTRODUCTION

Security of various communication tasks can be guaranteed by quantum internet,
which consists of numerous quantum cryptographic networks. Quantum key
distribution (QKD) allows two remote users to share unconditionally secure key
based on quantum laws [1, 2], which has stepped into a stage where an
integrated network from satellite to ground is waiting to be built.
Nevertheless, there are kinds of multiparty cryptographic primitives beyond
two users, e.g., conference key agreement (CKA) [3, 4]. Quantum CKA aims to
allow at least three users to share the conference key with unconditional
security, which can be applied in various practical situations, such as
netmeeting, online education and telemedicine. The conference key promises
group encryption for legitimate users, in which any member of that group can
decipher the message. As it generalizes the QKD to multiple users, it is also
called multipartite QKD [5].

Quantum CKA has been a research hot topic [6]. At the very beginning, it
enables multiple users to share the conference key with information-theoretic
security by employing multipartite entanglement [7, 8, 9]. Besides, the
measurement-device-independent CKA protocol [10] has been proposed via post-
selected Greenberger-Horne-Zeilinger (GHZ) entanglement [11, 12]. It has been
generalized extensively to various cases including finite size [13, 14],
continuous variables [15, 16] and four users with $W$ state [17]. Moreover,
the finite-key analysis with composable security [18], device independence
[19, 20] and other special cases [21, 22, 23] are also considered in quantum
CKA. The experimental demonstration of quantum CKA has been implemented
recently over 50 km fiber using the state-of-the-art four-photon GHZ
entanglement source [24]. By directly distributing GHZ entanglement states,
quantum CKA has been proved to drastically outperform QKD in resource expense
for group communication [25].

Although progress on quantum CKA has been made, the low conference key rate,
short transmission distance, and excessive resource cost seriously restrict
its real life applications. It has been shown recently that the conference key
rate is rigorously limited by the entanglement state distribution capacity of
quantum networks [26, 27, 28]. In order to increase the conference key rate,
one may utilize quantum repeaters [29] and multi-user scheme [10] of adaptive
measurement-device-independence [30]. However, these approaches remain
difficult to be implemented.

Introducing an intermediate node to perform a single-photon interference,
twin-field QKD [31] and its variants [32, 33, 34, 35, 36, 37, 38, 39, 40] are
able to break the limit of the secret key capacity of quantum channels [41,
42], specifically, the Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound [42].
Exploiting the twin-field theory [31], some new quantum CKA protocols [43, 44,
45] are proposed to improve the conference key rate and transmission distance.
However, the scheme of Ref. [43] can not be experimentally demonstrated with
current technology and the key rate of Ref. [44] is relatively low. In
addition, zero-error attack on coherent one-way QKD has been proposed [46],
therefore the security of Ref. [45] employing the idea of coherent one-way QKD
[47], can not be guaranteed.

Here, we propose a three-party quantum CKA protocol inspired by the twin-field
theory with vacuum state and single-photon encoding [32, 35], i.e., the
sending-or-not-sending scheme [32]. The conference key rate of our protocol
scales with the square-root of the total channel transmittance, which enables
our scheme to break the limit of entanglement state distribution capacity of
quantum networks, and our protocol can be experimentally demonstrated over 600
km with developed technology in twin-field QKD [48, 49, 50, 51, 52, 53]. The
key rate of our protocol is three orders of magnitude higher than that of Ref.
[44] within a range of 600 km and can remain at a high level even with a large
misalignment rate of quantum channels.

Figure 1: The setup of quantum CKA protocol. Alice and Bob exploit continuous-
wave lasers to generate the global phase stabilized coherent light. They
employ intensity modulator (IM), phase modulator (PM) and attenuator (Att) to
implement the pulse preparation, decoy-state modulation, phase randomization,
phase encoding and weak-light modulation. Charlie directly measures the pulses
sent by Alice and Bob with detectors D1 and D2, respectively, as the $Z$ basis
measurement. Only detector D1 (D2) click represents logic bit value 0 (1),
which means that a quantum state $\ket{1}_{a}\ket{0}_{b}$
($\ket{0}_{a}\ket{1}_{b}$) has been detected. Charlie performs an interference
with beam splitter (BS) and detectors D3 and D4, as the $X$ basis measurement.
Only detector D3 (D4) click denotes logic bit value 0 (1), which means that an
entangled state
$\ket{\psi^{+}}_{ab}=\frac{1}{\sqrt{2}}(\ket{1}_{a}\ket{0}_{b}+\ket{0}_{a}\ket{1}_{b})(\ket{\psi^{-}}_{ab}=\frac{1}{\sqrt{2}}(\ket{1}_{a}\ket{0}_{b}-\ket{0}_{a}\ket{1}_{b}))$
has been acquired. To simplify, Charlie applies a passive-basis choice.

##  II Protocol description

As depicted in Fig. 1, our scheme employs the same devices as twin-field QKD
[31, 48, 52, 53]. We choose the decoy-state quantum CKA with three intensities
to simplify the experiment. Hereafter, $\ket{0}$ and $\ket{1}$ form the photon
number Hilbert space for vacuum and single photon. We exploit the post-
selected phase-matching method to acquire the effective events with
$-\delta+r\pi\leq\theta_{a}-\theta_{b}-\varphi_{ab}\leq\delta+r\pi(\rm{mod}\
2\pi)$, where $\theta_{a}$ (for Alice) and $\theta_{b}$ (for Bob) are the
random global phases and $r=0,1$. $\delta$ is a selected value and
$\varphi_{ab}$ is the phase difference between the reference frames of Alice
and Bob, which can be obtained by strong light reference [52, 53]. The
successful probability of post-selected phase-matching is $p_{\rm
pm}=\frac{2\delta}{\pi}$ under the assumption of continuous phase
randomization. Our protocol has three legitimate users, Alice, Bob and
Charlie. Here, we consider a symmetric case. The detailed protocol is
described as follows.

1\. Preparation. Alice and Bob independently and randomly choose the $Z$ and
$X$ bases. Alice prepares phase-randomized weak coherent pulses using
intensities 0 and $\mu$, with probability $1-t$ and $t$, for logic bit $0$ and
$1$ in the $Z$ basis. Bob does the same. Alice (Bob) generates phase-
randomized weak coherent pulses $\ket{e^{i(\theta_{a}+x_{a}\pi)}\sqrt{k_{a}}}$
($\ket{e^{i(\theta_{b}+x_{b}\pi)}\sqrt{k_{b}}}$) for logic bit $x_{a}$
($x_{b}$) in the $X$ basis, with $x_{a},x_{b}\in\\{0,1\\}$,
$\theta_{a},\theta_{b}\in[0,2\pi)$ and $k_{a},k_{b}\in\\{\mu,\nu,0\\}$. They
send the optical pulses to Charlie through the insecure quantum channels.

2\. Measurement. Charlie randomly measures the received pulse pairs in the $Z$
and $X$ bases. For the $Z$ basis, one and only one click between detectors D1
and D2 reveals a successful event. For the $X$ basis, one and only one click
between detectors D3 and D4 indicates a successful outcome.

3\. Reconciliation. For each successful measurement event, Alice, Bob, and
Charlie announce the basis information. Alice and Bob publish the intensity
information unless Alice, Bob, and Charlie all select the $Z$ basis. Alice and
Bob disclose the global phase and perform post-selected phase-matching when
they both choose intensity $\nu$ and Charlie selects the $X$ basis. All
classical communications among Alice, Bob, and Charlie are transmitted via the
authenticated classical channels. Alice always flips her logic bit if choosing
the $Z$ basis. She will flip her logic bit if selecting the $X$ basis and
$r=1$.

4\. Parameter estimation. Alice, Bob, and Charlie adopt the data of the $Z$
basis as the raw key. The data of the $X$ basis is exploited to estimate the
leaked information. They exploit the decoy-state method [54, 55] to the yield
and phase error rate of joint single-photon state
$\rho_{AB}^{1}=\frac{1}{2}(\ket{01}_{ab}\bra{01}+\ket{10}_{ab}\bra{10})$.

5\. Postprocessing. They extract the common conference key using classical
error correction, error verification and privacy amplification.

##  III Security analysis

For the case of the $Z$ basis, the joint quantum states $\ket{01}_{ab}$ and
$\ket{10}_{ab}$ form the joint single-photon state
$\rho_{AB}^{1}=\frac{1}{2}(\ket{01}_{ab}\bra{01}+\ket{10}_{ab}\bra{10})$ with
probability $2t(1-t)\mu e^{-\mu}$, which can be utilized to extract key [56]
since the coherent states prepared by Alice and Bob are phase-randomized. The
joint quantum state $\ket{01}_{ab}$ ($\ket{10}_{ab}$) can be acquired by Alice
sending vacuum (single-photon) state and Bob sending single-photon (vacuum)
state.

For the case of $X$ basis, Alice and Bob implement post-selected phase-
matching and only consider the case of single photon component [35], such as

$\displaystyle\ket{e^{i\theta_{a}}\sqrt{\nu}}_{a}\ket{\pm
e^{i\theta_{b}}\sqrt{\nu}}_{b}\xrightarrow{\textrm{one
photon}}\frac{\ket{10}_{ab}\pm
e^{i(\theta_{b}-\theta_{a})}\ket{01}_{ab}}{\sqrt{2}},$ (1)

where $\theta_{a},\theta_{b}\in[0,2\pi)$. The joint quantum states
$\ket{\psi^{\pm}(\theta)}_{ab}=(\ket{10}_{ab}\pm
e^{i\theta}\ket{01}_{ab})/\sqrt{2}$ can also form the joint single-photon
state
$\rho_{AB}^{1}=\frac{1}{2}(\ket{\psi^{+}(\theta)}_{ab}\bra{\psi^{+}(\theta)}+\ket{\psi^{-}(\theta)}_{ab}\bra{\psi^{-}(\theta)})$.

Here, we will exploit the entanglement distillation argument of the GHZ state
[11, 12, 10] to provide security against coherent attacks. We introduce a
virtual protocol to prove the security of practical protocol as shown in Fig.
1. For the joint single-photon state case, no one can distinguish the
practical protocol and the virtual protocol P1, which is similar to the twin-
field QKD [32, 35].

Virtual protocol P1. Let P1 represent a virtual entanglement-based protocol.
(i) Alice prepares an entangled state

$\displaystyle\ket{\Psi}_{Aa}=$
$\displaystyle\sqrt{1-t}\ket{0}_{A}\ket{0}_{a}+\sqrt{t}\ket{1}_{A}\ket{1}_{a}$
(2) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}[\ket{+}_{A}(\sqrt{1-t}\ket{0}_{a}+\sqrt{t}\ket{1}_{a})$
$\displaystyle+\ket{-}_{A}(\sqrt{1-t}\ket{0}_{a}-\sqrt{t}\ket{1}_{a})]$

where $A$ and $a$ are the qubit system and optical mode, respectively. Note
that $\ket{\pm}_{A}=(\ket{0}_{A}+\ket{1}_{A})/\sqrt{2}$. Likewise, Bob does
the same. A sifting step is needed to be performed, i.e., Alice and Bob only
reserve the case that the total photon number is one between optical modes $a$
and $b$. After this step, the normalized joint state is

$\displaystyle\ket{\phi}_{ABab}=$
$\displaystyle\frac{1}{\sqrt{2}}(\ket{10}_{AB}\ket{10}_{ab}+\ket{01}_{AB}\ket{01}_{ab})$
(3) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\frac{\ket{++}_{AB}-\ket{--}_{AB}}{\sqrt{2}}\frac{\ket{10}_{ab}+\ket{01}_{ab}}{\sqrt{2}}$
$\displaystyle+\frac{1}{\sqrt{2}}\frac{\ket{+-}_{AB}-\ket{-+}_{AB}}{\sqrt{2}}\frac{\ket{10}_{ab}-\ket{01}_{ab}}{\sqrt{2}}.$

Then, Alice (Bob) sends optical mode $a$ ($b$) to Charlie and keeps the qubit
system $A$ ($B$). Let optical modes $\ket{10}_{ab}$ and $\ket{01}_{ab}$ denote
the qubit system $\ket{0}_{C}$ and $\ket{1}_{C}$. Therefore, the joint quantum
state among Alice, Bob and Charlie can be written as a GHZ state,
$\ket{\phi}_{ABC}=\frac{1}{\sqrt{2}}(\ket{100}_{ABC}+\ket{011}_{ABC})$. (ii)
Alice, Bob and Charlie independently and randomly choose the $Z$ basis or the
$X$ basis to measure the received optical modes or kept qubit system. (iii)
They disclose the basis information. (iv) They utilize data of the $Z$ basis
as raw key while data of the $X$ basis are used to estimate Eve’s
eavesdropping [10]. (v) They generate final key with classical post-
processing.

Practical protocol. The asymptotic conference key rate of the practical
protocol, as shown in Fig. 1, can be written as [32, 35, 40]

$\displaystyle R=2t(1-t)\\{e^{-\mu}Y_{0}^{z}+\mu
e^{-\mu}Y_{1}^{z}[1-h(e_{1}^{x})]\\}-\lambda_{\rm EC},$ (4)

where $Y_{0}^{z}$ is the yield when both Alice and Bob choose the $Z$ basis
and send vacuum state. $\lambda_{\rm EC}=Q^{z}fh(E^{z})$ is the revealed
information in classical error correction. The gain $Q^{z}$ and quantum bit
error rate $E^{z}$ of the $Z$ basis can be directly acquired in experiment,
where $E^{z}$ is the maximum value between the marginal bit error rates of
Alice-Charlie and Bob-Charlie when the raw key of Charlie is set as the
reference key. The yields $Y_{0}^{d}=Q_{00}^{d}$ and $Y_{1}^{d}$ can be
estimated by exploiting the decoy-state method [54, 55],

$\displaystyle
Y_{1}^{d}\geq\frac{\mu/2}{\mu\nu-\nu^{2}}\left(e^{\nu}Q_{\nu}^{d}-\frac{\nu^{2}}{\mu^{2}}e^{\mu}Q_{\mu}^{d}-\frac{\mu^{2}-\nu^{2}}{\mu^{2}}Q_{0}^{d}\right),$
(5)

where we define $Q_{k}^{d}=Q_{k0}^{d}+Q_{0k}^{d}$. $Q_{k_{a}k_{b}}^{d}$ is the
gain of Charlie selecting the $d$ basis given that Alice and Bob send
intensities $k_{a}$ and $k_{b}$ and can be given by

$\displaystyle Q_{k_{a}k_{b}}^{z}=$
$\displaystyle(1-p_{d})(e^{-k_{a}\sqrt{\eta}}+e^{-k_{b}\sqrt{\eta}})$ (6)
$\displaystyle-2(1-p_{d})^{2}e^{-(k_{a}+k_{b})\sqrt{\eta}}.$

Besides, the gain $Q_{k_{a}k_{b}}^{x}$ can be written as

$\displaystyle Q_{k_{a}k_{b}}^{x}=$ $\displaystyle
2(1-p_{d})e^{-\frac{k_{a}+k_{b}}{2}\sqrt{\eta}}\times[I_{0}(\sqrt{k_{a}k_{b}}\sqrt{\eta})$
(7) $\displaystyle-(1-p_{d})e^{-\frac{k_{a}+k_{b}}{2}\sqrt{\eta}}],$

where $I_{0}(x)$ is the modified Bessel function of the first kind.

For $Y_{1}^{x}$, we employ the fact that
$\frac{1}{2}(\ket{10}_{ab}\bra{10}+\ket{01}_{ab}\bra{01})=\frac{1}{2}(\ket{\psi^{+}}_{ab}\bra{\psi^{+}}_{ab}+\ket{\psi^{-}}_{ab}\bra{\psi^{-}}_{ab})$.
The bit error rate $e_{1}^{x}$ can be bounded by

$\displaystyle e_{1}^{x}\leq\frac{1}{2\nu Y_{1}^{x}p_{\rm
pm}}\left(e^{2\nu}\mathsf{E}_{\nu\nu}^{x}\mathsf{Q}_{\nu\nu}^{x}-\frac{p_{\rm
pm}}{2}Y_{0}^{x}\right),$ (8)

where $\mathsf{E}_{\nu\nu}^{x}$ and $\mathsf{Q}_{\nu\nu}^{x}$ are the bit
error rate and gain of Charlie selecting the $X$ basis when Alice and Bob both
send intensity $\nu$ and they successfully perform the post-selected phase-
matching.

Note that quantum bit error rate $E^{z}$ is the maximum value between the
marginal error rates of Alice-Charlie and Bob-Charlie as Charlie’s raw key is
the reference key in the error correction step. For the case of our practical
protocol, we have the gain $Q^{z}=Q_{c}^{z}+Q_{e}^{z}$ and quantum bit error
rate $E^{z}=Q_{e}^{z}/Q^{z}$. Therein, $Q_{c}^{z}$ and $Q_{e}^{z}$ are the
correct and incorrect gains between Alice and Bob,

$\displaystyle Q_{c}^{z}$ $\displaystyle=t(1-t)(Q_{\mu 0}^{z}+Q_{0\mu}^{z}),$
(9) $\displaystyle Q_{e}^{z}$
$\displaystyle=(1-t)^{2}Q_{00}^{z}+t^{2}Q_{\mu\mu}^{z},$

For the case of Alice and Bob both choosing intensity $\nu$, they should
perform the post-selected phase-matching. The corresponding gain and error
rate can be written as
$\mathsf{Q}_{\nu\nu}^{x}=\mathsf{Q}_{c,\nu\nu}^{x}+\mathsf{Q}_{e,\nu\nu}^{x}$
and
$\mathsf{E}_{\nu\nu}^{x}=[e_{d}^{x}\mathsf{Q}_{c,\nu\nu}^{x}+(1-e_{d}^{x})\mathsf{Q}_{e,\nu\nu}^{x}]/\mathsf{Q}_{\nu\nu}^{x}$.
Therein, the correct gain $\mathsf{Q}_{c,\nu\nu}^{x}$ and incorrect gain
$\mathsf{Q}_{e,\nu\nu}^{x}$ are

$\displaystyle\mathsf{Q}_{c,\nu\nu}^{x}=$ $\displaystyle\frac{p_{\rm
pm}(1-p_{d})}{\delta}\int_{0}^{\delta}e^{-\nu\sqrt{\eta}(1-\cos\theta)}d\theta$
(10) $\displaystyle-p_{\rm pm}(1-p_{d})^{2}e^{-2\nu\sqrt{\eta}}$
$\displaystyle\approx$ $\displaystyle\frac{p_{\rm
pm}(1-p_{d})}{\delta}\sqrt{\frac{\pi}{2\nu\sqrt{\eta}}}{\rm
erf}\left(\sqrt{\frac{\nu\sqrt{\eta}}{2}}\delta\right)$ $\displaystyle-p_{\rm
pm}(1-p_{d})^{2}e^{-2\nu\sqrt{\eta}},$

and

$\displaystyle\mathsf{Q}_{e,\nu\nu}^{x}=$ $\displaystyle\frac{p_{\rm
pm}(1-p_{d})}{\delta}\int_{0}^{\delta}e^{-\nu\sqrt{\eta}(1+\cos\theta)}d\theta$
(11) $\displaystyle-p_{\rm pm}(1-p_{d})^{2}e^{-2\nu\sqrt{\eta}}$
$\displaystyle\approx$ $\displaystyle\frac{p_{\rm
pm}(1-p_{d})}{\delta}e^{-2\nu\sqrt{\eta}}\sqrt{\frac{\pi}{2\nu\sqrt{\eta}}}{\rm
erfi}\left(\sqrt{\frac{\nu\sqrt{\eta}}{2}}\delta\right)$ $\displaystyle-p_{\rm
pm}(1-p_{d})^{2}e^{-2\nu\sqrt{\eta}},$

where the approximate results are always true for $0<\delta\leq\frac{\pi}{4}$.
Therein, ${\rm erf}(x)$ and ${\rm erfi}(x)$ are the error function and
imaginary error function.

Single-photon protocol. Here, we also present an ideal protocol, called the
single-photon protocol, where ideal single-photon sources are required to
replace the phase-randomized weak coherent pulses in the $Z$ basis. Alice
(Bob) randomly sends optical modes $\ket{0}$ and $\ket{1}$ by using the vacuum
and single-photon source with probabilities $1-t$ and $t$ in the $Z$ basis.

Following the entanglement distillation argument [10], the asymptotic
conference key rate of the single-photon protocol is given by

$\displaystyle\widetilde{R}=2t(1-t)Y_{1}^{z}[1-h(e_{1}^{x})]-\widetilde{\lambda}_{\rm
EC},$ (12)

where $Y_{1}^{d}$ and $e_{1}^{d}$ are the yield and bit error rate of the
joint single-photon state given that all users choose the $d$ basis, with
$d\in\\{Z,X\\}$. For example, $Y_{1}^{z}$ is the probability that Charlie has
a successful detection in the $Z$ basis given that Alice and Bob send the
state $\rho_{ab}^{1}$. Let $x_{a}\oplus x_{b}\neq x_{c}$ denote the bit error
of the $X$ basis, where $x_{c}$ is the logic bit value of Charlie in the $X$
basis. $\widetilde{\lambda}_{\rm EC}$ is the leaked information in classical
error correction. $h(x)=-x\log_{2}(x)-(1-x)\log_{2}(1-x)$ is the binary
Shannon entropy. There is always a marginal bit error rate in the $Z$ basis
between every two users. Taking the raw key of Charlie as the reference key,
the $Z$ basis bit error rate of this protocol is the maximum value between the
marginal bit error rates of Alice-Charlie and Bob-Charlie.

For the case of the scheme with single-photon source, Alice and Bob directly
prepare optical mode $\ket{1}$ with single photon. We also have the gain
$Q^{z}=Q_{c}^{z}+Q_{e}^{z}$ and quantum bit error rate
$E^{z}=Q_{e}^{z}/Q^{z}$. Therein, $Q_{c}^{z}$ and $Q_{e}^{z}$ are the correct
and incorrect gains between Alice and Bob,

$\displaystyle Q_{c}^{z}$ $\displaystyle=t(1-t)(Y_{10}^{z}+Y_{01}^{z}),$ (13)
$\displaystyle Q_{e}^{z}$ $\displaystyle=(1-t)^{2}Y_{00}^{z}+t^{2}Y_{11}^{z},$

where these yields can be expressed as

$\displaystyle Y_{10}^{z}$
$\displaystyle=Y_{01}^{z}=1-(1-p_{d})(1-2p_{d})(1-\sqrt{\eta}),$ (14)
$\displaystyle Y_{11}^{z}$
$\displaystyle=2(1-p_{d})(1-\sqrt{\eta})[1-(1-p_{d})(1-\sqrt{\eta})],$

and $Y_{00}^{z}=2p_{d}(1-P_{d})$. Thereby, the revealed information in
classical error correction is $\widetilde{\lambda}_{\rm EC}=Q^{z}fh(E^{z})$.
Besides, we have the yields $Y_{1}^{z}=\frac{1}{2}(Y_{10}^{z}+Y_{01}^{z})$.
Note that $Y_{1}^{x}=\frac{1}{2}(Y_{10}^{x}+Y_{01}^{x})$ due to
$\frac{1}{2}(\ket{10}_{ab}\bra{10}+\ket{01}_{ab}\bra{01})=\frac{1}{2}(\ket{\psi^{+}}_{ab}\bra{\psi^{+}}_{ab}+\ket{\psi^{-}}_{ab}\bra{\psi^{-}}_{ab})$,
where $Y_{10}^{x}$ and $Y_{01}^{x}$ can be given by

$\displaystyle Y_{10}^{x}=Y_{01}^{x}=1-(1-p_{d})(1-2p_{d})(1-\sqrt{\eta}).$
(15)

Figure 2: Conference key rate (per pulse) in logarithmic scale as a function
of the total transmission distance using the experimental parameters in Table
1. For comparison, we also draw the curves of the single-photon protocol, Ref.
[44] and distributing GHZ states among three parties [25], where the
transmitting distance to Charlie is zero while the transmitting distances to
Alice and Bob are $L/2$.

##  IV Performance

We assume that the efficiencies and dark count rates of Charlie’s detectors
are the same and the distances of Alice-Charlie and Bob-Charlie are both
$L/2$. We numerically optimize the conference key rate over the free
parameters $t$, $\mu$ and $\nu$ by utilizing Genetic Algorithm. To display the
conference key rate of our protocol scaling with the square-root of the total
channel transmittance, we define $\sqrt{\eta}=\eta_{d}\times 10^{-\alpha
L/20}$, where $\eta_{d}$ is the detector efficiency. In order to show the
performance of our quantum CKA protocol, we utilize the practical parameters
in Table 1 for simulation, which has been recently realized over 500 km fiber
in twin-field QKD [52]. The conference key rate of our practical quantum CKA
protocol as a function of the total transmission distance $L$ among three
users is shown in Fig. 2. The transmission distance of our practical protocol
reaches over 600 km and its key rate can surpass that of distributing GHZ
states among 3 parties [25] at the distance of 160 km. For the case of
distributing GHZ states [25], we set the misalignment rate of its channels is
0 and only dark counts contribute to its error rate, which aims to make its
key rate close to the GHZ entanglement distribution capacity. Compared with
the scheme of Ref. [44], although our scheme is not a measurement-device-
independent protocol, unlike Ref. [44], the key rate of our protocol has
improved significantly and is three orders of magnitude higher within 600 km.
Additionally, the single-photon protocol can be theoretically demonstrated
over 800 km, which also shows the importance to develop ideal single-photon
source.

Besides, the bit error rate $E^{z}$ can be bounded by adjusting probability
$t$, which is irrelevant to the misalignment of channel. Therefore, our
practical protocol can always extract conference key as long as the
misalignment rate of $X$ basis $e_{d}^{x}<50\%$ in principle. As shown in Fig.
3, the secure transmission distance is larger than 500 km in the case of
$e_{d}^{x}=18\%$. Besides, the conference key rate of the practical protocol
can surpass that of distributing GHZ states over the networks [25] even when
the misalignment rate is larger than $25\%$. These results indicate that our
quantum CKA is practical and feasible even in the field environment.

Figure 3: Conference key rates of the practical protocol in logarithmic scale as a function of the total distance with different misalignment rates of the $X$ basis. The conference key rate can also beat that of distributing GHZ states among 3 parties [25] even when the misalignment rate is up to $e_{d}^{x}=25\%$.  Table 1: Simulation parameters [52]. $\eta_{d}$ and $p_{d}$ are the detector efficiency and dark count rate. $e_{d}^{x}$ is the misalignment rate of the $X$ basis. $\alpha$ is the attenuation coefficient of the ultralow-loss fiber. $f$ is the error correction efficiency. $\eta_{d}$ | $p_{d}$ | $e_{d}^{x}$ | $\alpha$ | $f$ |  $\delta$
---|---|---|---|---|---
$56\%$ | $10^{-8}$ | $3.5\%$ | $0.167$ | $1.1$ |  $\pi/18$

##  V Conclusion

In summary, by employing a special GHZ-class state, we have proposed a
practical quantum CKA protocol that allows three users to share the
information-theoretically secure conference key. Its conference key rate
scales as $O(\sqrt{\eta})$ rather than the total transmittance, which can beat
the fundamental linear limit on the private capacity of quantum network [26].
This protocol exploits the same devices and technology as twin-field QKD [31,
48, 52, 53] and as such, it can be demonstrated over 600 km. New results are
significantly beyond what one could expect for quantum CKA in previous works,
whose key rate is three orders of magnitude higher within 600 km compared to
Ref. [44] and keeps at a high level even with large misalignment rate. We
believe that this quantum CKA protocol can be widely implemented to build the
large-scale quantum cryptographic network.

Here, we would like to clarify that our quantum CKA is not a measurement-
device-independent scheme and will suffer from the standard quantum hacking
attacks, including blinding attack [57]. In order to treat double-click
attack, one should exploit the squashing model [56], i.e., random basis and
random measurement outcomes should be assigned. Our protocol also requires the
phase stability of two independent pulses, which can be circumvented by using
active phase stabilisation [49, 48], phase tracking and post-selection [51].
Different from the scheme with post-selected W state [43], our scheme with
arbitrary $N$ parties is an interesting and nontrivial work.

## Acknowledgments

We gratefully acknowledge support from the National Natural Science Foundation
of China (under Grant No. 61801420); the Key-Area Research and Development
Program of Guangdong Province (under Grant No. 2020B0303040001); the
Fundamental Research Funds for the Central Universities (under Grant No.
020414380182).

## Disclosures

The authors declare no conflicts of interest.

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