# Interference of the signal from a local dynamical process with the quantum
state propagation in spin chains

Saikat Sur1 and V. Subrahmanyam2 Department of Physics, Indian Institute Of
Technology, Kanpur-208016, India<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
###### Abstract

The effect of a local instantaneous quantum dynamical process (QDP), either
unitary or non-unitary, on the quantum state transfer through a unitary
Hamiltonian evolution is investigated for both integrable and non-integrable
dynamics. There are interference effects of the quantum state propagation and
the QDP signal propagation. The state transfer fidelity is small for further
sites, from the site where the information is coded, indicating a finite speed
for the propagation of the quantum correlation. There is a small change in the
state transfer fidelity for the case of non-unitary QDP intervening the
background unitary dynamics. In the case of unitary QDP, the change is more
pronounced, with a substantial increase in the fidelity for appropriate sites
and times. For the non-integrable case, viz. a kicked Harper model, the state
transfer fidelity is quite large for further sites for short times, indicating
a finite speed for the propagation of the quantum correlation cannot be
defined.

PACS: 03.65.Ud 03.67.Bg 03.67.Hk 75.10.Pq

##  1  Introduction

Quantum spin chains have been investigated, over the last few years, from the
viewpoint of quantum information and communication. A quantum spin chain
serves as a possible channel for the quantum state transfer[1, 2, 3, 4]. These
systems have been studied extensively, as exactly-solvable condensed matter
physics systems, for studying spin ordering of novel spin states, and quantum
critical behaviour.[5, 6, 7]. These studies deal with the quantum dynamics of
initial many-body states through the Schroedinger time evolution[8], and the
statistical mechanics of the phase transitions[5, 8].

The spin chain dynamics can be treated viewing the spin chain as a closed
system, with a given initial state with a distribution of correlations.
Through the time evolution the quantum correlations can be redistributed and
the spin chain can exhibit a variety of multi-party quantum correlation and
entanglement structures. The wide spectrum of the dynamics investigation of
spin chains includes the study of magnon bound states and scattering[9, 10],
spin current and relativistic density wave dynamics[11, 12], quantum
correlations after a quench[13] and light-cone in entanglement spreading[14].
The effect of unitary evolution of the quantum spin chain can be viewed as, in
the quantum information language, various global and local multi-qubit gate
operations acting on the many-qubit system. The system will undergo a
redistribution of entanglement and quantum correlations[15, 16, 17, 18, 19]
through the unitary evolution. However, a multi-party qubit system in general
can experience non unitary or incoherent processes [20] acting on some qubits
locally or globally, that can lead to decoherence. Investigating the effect of
local decohering process on the quantum correlations and the entanglement
structure is difficult in general, as traditional approximation tools like the
time-dependent perturbation theory are not applicable for these systems. In a
simple example of local decohering process intervening the Schroedinger
evolution of quantum states is investigated[21], where a the Hamiltonian
unitary evolution of a many-qubit quantum state is interrupted by a local
instantaneous quantum dynamical process (QDP) on a certain qubit at a certain
epoch of time. The QDP can be a non-unitary decohering operation or a unitary
operation different from the back ground evolution. The QDP signal, the effect
of the occurrence of the process at a given time and location, and its
propagation have been investigated using both magnetisation conserving and
non-conserving dynamics for initial states with and without entanglement for
various model Hamiltonian dynamics [21]. The signal propagation speed in
general depends on the type of interaction of the qubits, the strength of the
interaction, the spatial range of the interaction and the initial state.

The quantum state transfer protocol[1, 2, 3, 22] studied for the spin chains
relies on the Hamiltonian evolution for a faithful communication and detection
of the state from an initial qubit location to a target qubit location. The
average fidelity, averaged over all possible input qubit state, of the state
transfer, is a figure of merit of the protocol. The fidelity depends on the
interaction parameters of the Hamiltonian, and in general it is oscillatory as
a function of the time of evolution as a result of quantum interference. In
general, the decoherence of individual spins, in the quantum spin channel
during the state transfer, reduces the fidelity; this has been discussed in
[23]. Protected quantum state transfer through a noisy channel by coupling the
end registers has been proposed in [24]. The state transfer fidelity can have
non trivial effects due to the QDP intervention of the unitary evolution, and
the fidelity can increase or decrease depending on the location, time and
parameters of the QDP; the detailed investigation of these effects is the main
focus of this paper. In Harper model, the transverse potential is a periodic
kick, that causes every qubit to undergo a coherent operation, during the
evolution with the background XY dynamics. Depending on the strength and the
kicking period, the dynamics is regular or chaotic[25, 26]. Thus, we cam
discuss the state transfer in a non integrable model, and contrast it with the
behaviour for the integrable model.

This paper is arranged as follows. In Section 2, we outline the quantum state
transfer protocol and the computation of the transfer fidelity. We will study
the effect of a local decohering process in Section 3, followed by a local
coherent process in Section 4, the computation of the fidelity using one and
two qubit Green’s functions. We will discuss a local decohering process in
non-integrable systems in Section 5.

##  2 Quantum State transfer through Unitary Dynamics

Faithful and faultless transmission of quantum states between two distant
locations is the main challenge for quantum communication and information
processing protocols. The quantum spin chains as channels of quantum state
transfer was proposed and investigated[1, 2, 22]. The spin of the electrons
are basic examples of qubits, viz. two-level quantum systems, and the spin
chains can be realised physically in magnetic systems. One-dimensional spin
models have been studied extensively using the Heisenberg-XY or transverse-
field Ising model Hamiltonians, due to their exact solvability using the Bethe
ansatz technique or Jordan-Wigner fermion technique respectively. In this
paper, we will be using the Heisenberg dynamics for the most part, and will
study a variant of transverse-XY model in the last section.

In a standard state transfer protocol, a direct-product many-qubit initial
state is prepared, at time $t=0$, with the first spin in a desired state (or
the information is coded into the state of the first spin). The many-qubit
state is evolved through time using the Hamiltonian Schroedinger dynamics to a
final state at time $t$. Then the desired state is to be recovered from a
certain target spin, the probability of the state transfer is measured by the
fidelity, normalised to unity, computed by taking the overlap of the target
state with the desired state. A large value of the state transfer fidelity for
a spin at a particular location at a particular time implies the efficacy of
the state transfer protocol. In this paper, we will deviate from the standard
protocol, and intervene the smooth Heisenberg-XY dynamics by a local quantum
dynamical process (QDP) at a given location and time, viz. an instantaneous
quantum operation is carried out on the many-qubit state at a time $t=t_{0}$,
between the coding time $t=0$ of the quantum information at the first spin,
and the recovery or readout time $t$ of the quantum information from a
different spin. In this paper the focus would be on the effect of QDP on
quantum state transfer fidelities. We will see that local QDPs can decrease or
increase the fidelity depending upon their locations and time. Also, we will
discuss the state transfer and the effect of local QDP with a non integrable
background Hamiltonian dynamics, and investigate the signature of quantum
chaos in Sec. 5.

Let us consider a one-dimensional array of N qubits interacting through a
nearest-neighbour exchange interaction, and a particular qubit state has to be
transferred from one end to the other. The protocol encodes the information to
be transferred in to the state $|\phi\rangle=\alpha|0\rangle+\beta|1\rangle$.
The first qubit is initialised to this state at one end, the many qubit
initial state is $|\psi(0)\rangle=|\phi 00...0\rangle$, a direct product of
the state $|\phi~{}\rangle$ for the first qubit and the state $|0\rangle$ for
all other qubit. The unitary time evolution will transform the many qubit
state to $|\psi(t)\rangle$ using the Hamiltonian dynamics. The desired qubit
$|\phi\rangle$ can spread out to other qubits during the evolution, i.e., the
desired qubit state can travel to the other locations with a some speed and
probability. The speed associated with such transfer is inversely related to
the interaction strength between the sites. For an efficient state transfer,
the target state $\rho_{l}$, is the reduced density matrix of the $l$’th
qubit, calculated by taking a trace over all other qubits from the many qubit
state, $\rho_{l}=tr^{\prime}|\psi(t\rangle\langle\psi(t)|$, should be the same
as the desired state $\rho_{\phi}=|\phi\rangle\langle\phi|$. The state
transfer fidelity gives a probabilistic measure of the state transferred from
one site to another and is quantified by taking overlap of the target and the
desired states, given by

$\mathscr{F}_{l,\phi}\equiv
Tr\rho_{\phi}\rho_{l}=\langle\phi|\rho_{l}|\phi\rangle.$ (1)

Now, the fidelity shown above depends on the initial state $\phi$ , apart from
the target spin location and time, and the interaction parameters of the
Hamiltonian dynamics. A figure of merit measure for the efficacy of the
protocol is to average the above over all possible initial qubit states, i.e.,
over the surface of the Bloch sphere representing all qubit pure states.

The first exactly-solvable and integrable non trivial models of interacting
quantum spins is a one-dimensional chain of spins interacting with their
nearest neighbour Heisenberg exchange interaction, known as the Heisenberg
model. We will use the Pauli operator $\vec{\sigma}_{l}$ to represent the
different components of the spin operator $\vec{S}_{i}$ at $i$’th site. Let us
consider a one-dimensional chain of $N$ spins interacting through the nearest-
neighbour anisotropic Heisenberg model. The Hamiltonian is given by,

$H=-J\sum_{i}({\sigma}^{x}_{i}{\sigma}^{x}_{i+1}+{\sigma}^{y}_{i}{\sigma}^{y}_{i+1}+\Delta{\sigma}^{z}_{i}{\sigma}^{z}_{i+1}).$
(2)

where $J$ is the exchange interaction strength for the nearest-neighbour
spins, and $\Delta$ is the anisotropy strength. As all the three Pauli spin
matrices appear in the Hamiltonian, an exchange interaction of neighbouring
spin is implied in all three spin dimensions. The model exhibits ferromagnetic
(antiferromagnetic) behaviour in the ground state for $\Delta>0(\Delta<0)$.
The ground state and all the excited states are known, and can be found using
the Bethe ansatz[7]. Let us use the basis states for the $i$’th spin as
$|0\rangle$ (up-spin state) and $|1\rangle$(down-spin) states, denoting the
eigenstates $\sigma_{i}^{z}$ with eigenvalues +1 and -1 respectively. The
basis states for the many-qubit system can be chosen to be the direct products
of the basis states of each spin. The z-component of the total spin
$\Sigma\sigma_{i}^{z}$ is a constant of motion, which implies that the
eigenstates will have a definite number of down spins. The many-qubit basis
states with $l$ down spins can also be labeled by the locations
($x_{1},x_{2}..x_{l}$) of the $l$ down spins, where the set is an ordered set
with $x_{1}<x_{2}$ and so on. An eigenstate with $l$ down spins, a $l-$magnon
state, can be written as a superposition of the basis states as,

$|\psi\rangle=\sum_{x_{1},x_{2}..x_{l}}\psi({x_{1},x_{2}..x_{l}})|x_{1},x_{2}..x_{l}\rangle$
(3)

where the eigenfunction $\psi({x_{1},x_{2}..x_{l}})$ denotes the wave function
amplitude for the corresponding basis state. The eigenfunction is given by the
Bethe Ansatz[27], labeled by the the set of momenta $(p_{1},p_{2}..p_{l})$ of
the down spins, which are determined by solving algebraic Bethe ansatz
equations, with periodic boundary conditions. There is only one zero-magnon
state $|F\rangle=|00..0\rangle$, which is just a ferromagnetic ground state
with all the spins polarized along one direction. It is straightforward to see
that it an eigenstate of the above Hamiltonian with energy $\epsilon_{0}=-NJ$
for periodic boundary conditions (closed chain), and $\epsilon_{0}=-J(N-1)$
for open boundary conditions (open chain). Starting from $|F\rangle$, one-
magnon excitations can be created by turning any one of the spins, giving $N$
localised one-magnon states, which can be labeled by the location of the down
spin. One-magnon eigenstates are labeled by the momentum of the down spin, the
eigenfunction is given by,

$\displaystyle\psi_{p}^{x}=$ $\displaystyle\sqrt{\frac{1}{N}}e^{ipx};p={2\pi
I\over N},{\rm for~{}a~{}closed~{}chain}$ $\displaystyle\psi_{p}^{x}=$
$\displaystyle\sqrt{\frac{2}{N+1}}\sin(px);p={\pi I\over N+1},{\rm
for~{}an~{}open~{}chain},$ (4)

where the momentum $p$ is determined by an integer $I=1,2,..N$ for both cases.
The one-magnon eigenvalue is given by
$\epsilon_{1}(p)=\epsilon_{0}-2J\cos{p}$. The interaction strength $J$
determines the hopping of the down spins to neighbouring sites, and the
interaction of the two down spins is determined by $\Delta$. The one-magnon
eigen energies are independent of $\Delta$ as the states carry only one down
spin. The two-magnon (l=2) and other eigenstates (l $>$ 2) include both
scattering states and bound states of the down spins. The eigenfunction for
the two-magnon eigenstate is labeled by two momenta $p_{1}$ and $p_{2}$, is
given by

$\psi_{p_{1},p_{2}}^{x_{1},x_{2}}=A(p_{1},p_{2})(e^{ip_{1}x_{1}}e^{ip_{2}x_{2}}+e^{i\theta(p_{1},p_{2})}e^{ip_{1}x_{2}}e^{ip_{2}x_{1}}),$
(5)

where $A(p_{1},p_{2})$ is a normalisation factor. The two terms in the wave
function differ by a permutation of the positions $x_{1}$ and $x_{2}$. The
phase factor $\theta(p_{1},p_{2})$ depends on the moment and the interaction
strengths, and is given by

$\tan{\theta(p_{1},p_{2})\over
2}={\frac{\Delta\sin[(p_{1}-p_{2})/2]}{\cos[(p_{1}+p_{2})/2]-\Delta\cos[(p_{1}-p_{2})/2]}}.$
(6)

The two momenta are determined by the Bethe ansatz equations,

$p_{1}N=2\pi I_{1}+\theta(p_{1},p_{2}),~{}p_{2}N=2\pi
I_{2}-\theta(p_{1},p_{2}),$ (7)

where $I_{1},I_{2}$ are any integers. The two-magnon eigenvalues are given by
$\epsilon_{2}(p_{1},p_{2})=\epsilon_{0}+2(\Delta-\cos{p_{1}})+2(\Delta-\cos{p_{2}})$.
The two-magnon scattering states have been studied for the spin-independent
scattering[28]. The detail of solving the above Bethe ansatz equations for the
two-magnon scattering and bound states is discussed in the Appendix A, along
with the computation of the time-dependent Green’s functions.

Figure 1:  The evolution of the initial state $\rho(0)$ in three steps: The
state evolves to $\rho(t_{0}^{-})$ through a unitary process using the
Heisenberg Hamiltonian till $t=t_{0}^{-}$ , the state become
$\tilde{\rho}(t_{0}^{+})$ through an instantaneous local QDP at $t=t_{0}$, and
finally it evolves to $\tilde{\rho}(t)$ through the same unitary process.

We will see no dynamics effects if the initial state is this eigenstate of the
Hamiltonian. To study the dynamics of an arbitrary initial state, we will
consider a linear combination of zero-magnon and one-magnon states that are
not eigenstates of the Hamiltonian, in conjunction with the QDP dynamics
below. Let us consider a spin chain with open boundary condition and the
initial state given by,

$|\psi(0)\rangle=\alpha|00...0\rangle+\beta|100..0\rangle=\alpha|F\rangle+\beta|x=1\rangle,$
(8)

where $x$ denotes location of the down spin. The above state is a linear
superposition of an eigenstate of the Hamiltonian with eigenvalue
$\epsilon_{0}$ and a one magnon state which can be written as a linear
combination of one-magnon momentum eigenstates $|p\rangle$ of the Hamiltonian
with eigenvalues $\epsilon_{1}(p)$, we have

$|\psi(0)\rangle=\alpha|F\rangle+\beta\sum_{p}\psi_{p}(1)|p\rangle,$ (9)

Now, the time evolution of the system will transport the single down spin from
the first site to other sites and the state after a time $t$ becomes,

$|\psi(t)\rangle=\alpha
e^{-i\epsilon_{0}t}|F\rangle+\beta\sum_{y}G^{y}_{1}(t)|y\rangle,$ (10)

where the time-dependent function $G^{x^{\prime}}_{x}(t)$ [21] is given in
terms of the wave functions defined above as,

$G^{x^{\prime}}_{x}(t)=\sum_{p}\psi_{p}^{x}\psi_{p}^{x^{\prime}*}e^{-it\epsilon_{1}(p)}.$
(11)

Using the one-magnon eigenfunctions, shown in Eq.4, we can express the Green’s
function as,

$G^{x^{\prime}}_{x}(t)=\frac{N}{2\pi}\int_{0}^{\pi}\psi^{x}_{p}\psi^{x^{\prime}*}_{p}e^{-it\epsilon(p)}dp=(-i)^{(x-x^{\prime})}J_{x-x^{\prime}}(2t)-(-i)^{x+x^{\prime}}J_{x+x^{\prime}}(2t),$
(12)

for an open chain, and we have

$G^{x^{\prime}}_{x}(t)=(-i)^{x-x^{\prime}}J_{x-x^{\prime}}(2t),$ (13)

for a closed chain. $J_{x}(y)$ is the Bessel function of integer order $x$ and
argument $y$. The reduced density matrix (RDM)of the $l^{th}$ qubit is defined
by taking trace of the whole quantum state over all qubits except the $l^{th}$
one, as $\rho_{l}=Tr^{\prime}\rho(t)$, where
$\rho(t)=|\psi(t)\rangle\langle\psi(t)|$, is the time-evolved many-qubit
state. Using the basis $|0\rangle,|1\rangle$ for the $l$’th qubit, the reduced
density matrix is given by,

$\rho_{l}(t)=\left(\begin{array}[]{cc}1-x_{l}(t)&y_{l}(t)\\\
y^{*}_{l}(t)&x_{l}(t)\\\ \end{array}\right),$ (14)

where the elements of the RDM are time dependent functions that can be
calculated from Eq.10 as,

$\displaystyle x_{l}(t)=\langle
1|\rho_{l}(t)|1\rangle=|\beta|^{2}|G^{l}_{1}(t)|^{2},$ $\displaystyle
y_{l}(t)=\langle
0|\rho_{l}(t)|1\rangle=\alpha\beta^{*}e^{-i\epsilon_{0}t}G^{*l}_{1}(t).$ (15)

The state transfer fidelity for the $l^{th}$ site as a function of time is
given by,

$\mathscr{F}_{l,\phi}(t)=|\alpha|^{2}(1-x_{l}(t))+|\beta|^{2}x_{l}(t)+2Re(\alpha^{*}\beta
y_{l}(t)).$ (16)

The quantity $\mathscr{F}_{l,\phi}(t)$ depends on the parameters $\alpha$ and
$\beta$ of the initial state. The averages over the parameters in the above
equations are given as
$\overline{|\alpha|^{2}}=\overline{|\beta|^{2}}=\frac{1}{2}$,
$\overline{|\alpha|^{2}|\beta|^{2}}=\frac{1}{6}$and
$\overline{|\beta|^{4}}=\frac{1}{3}$. Taking the average over all possible
pure states on the Bloch sphere characterised by $\alpha$ and $\beta$ the
average state transfer fidelity[1] becomes,

$\mathscr{F}_{l}(t)=\frac{1}{2}+\frac{1}{6}|G^{l}_{1}(t)|^{2}+\frac{1}{3}Re(e^{i\epsilon_{0}t}G^{l}_{1}(t)).$
(17)

The state transfer fidelity $\mathscr{F}_{l}(t)$ with unitary dynamics is
shown in Fig. 2(a) as a function of time for different site indices. Since the
function $G^{l}_{1}(t)$ is a Bessel function with time as argument it falls
inversely with time and the fidelity of state transfer decreases with time for
a particular site and over a large time saturates to the value 0.5. It also
deceases with the increase in the distance between the first site and the
decoding site l. The maximum fidelity for the site $l$ is obtained around time
$l/2$ (here, we have taken interaction strength $J=1/2$). So, the speed of
state transfer is equal to the value of interaction strength $J$ between the
spins.

Figure 2: (a) State transfer fidelity $\mathscr{F}_{l}(t)$ as function of site
index($l$) and time ($t$) for Heisenberg dynamics with with open boundary
condition without any decohering process. (b) Differences of state transfer
fidelities with and without QDP ($\Delta\mathscr{F}_{l},m=1(t;t_{0}=0.0)$) as
function of site index($l$) and time ($t$); where the QDP occurs at the site
($m=1$) and time ($t_{0}=0.0$).(c) $\Delta\mathscr{F}_{l},m=20(t;t_{0}=10.0)$
as function of site index($l$) and time ($t$); where the QDP occurs at the
site ($m=20$) and time ($t_{0}=10.0$). (d)
$\Delta\mathscr{F}_{l},m=1(t;t_{0}=10.0)$ as function of site index($l$) and
time ($t$); where the QDP occurs at the site ($m=1$) and time
($t_{0}=10.0$).(e)Time dependence of the fidelities with QDP
$\tilde{\mathscr{F}}_{l,m}(t;t_{0})$ and without QDP $\mathscr{F}_{l}(t)$ for
the site $l=100$, site of QDP $m=100$ and time of QDP $t_{0}=50.0$. (f)
Maximum fidelities with and without QDP as a function of site index $l$; the
site of QDP $m=l$ and time of QDP $m=l/2$ Number of sites taken $N=100$.

##  3 Local quantum decohering process

We now consider the effect of a local QDP intervening the smooth unitary
dynamics, thus effect the state transfer protocol. The time evolution of a
general state involves, as we have seen in the previous section, the coherent
movement of the down spins, due to the unitary dynamics generated by the
Heisenberg Hamiltonian we are considering. A local incoherent process or QDP
can be any single qubit quantum gate operation, for example a local projective
measurement[20, 29]. The QDP we consider below occurs instantaneously at a
time $t=t_{0}$, operating on a certain spin, thus changing the whole many-
qubit state instantaneously. That is is the QDP occurs between the time
$t=t_{0}^{-}$ and the time $t=t_{0}^{+}$. The dynamics is unitary for
$t<t_{0}^{-}$, and again for $t>t_{0}^{+}$, and it is given by the Heisenberg
Hamiltonian. The QDP can be non-unitary, which causes explicit decoherence of
the many-qubit state, considered in this section, or the QDP can be another
coherent quantum operation that will be discussed in the next section. First,
we consider a non-unitary QDP that is a local decohering process, as an
example, a projective measurement of $\sigma_{m}^{z}$ which has two outcomes,
corresponding projectors $P_{0}=(1+\sigma_{m}^{z})/2$ and
$P_{1}=(1-\sigma_{m}^{z})/2$. A more general measurement operators, for
example measuring an arbitrary component of $\vec{\sigma}_{m}$, have been
considered and seen to be qualitatively similar[21]. Thus, now we have marked
three qubits, at three different locations, namely, the first qubit where the
initial information is coded, the $l$’th qubit where we recover the desired
state transfer, and the $m$’th qubit where a local QDP occurs. We will see
below the case of $l>m$, in which both the desired state from the first spin
and the effect of QDP will have to travel in the same direction to reach the
recovery site, and the case of $m>l$ in which the two effects travel to the
recovery site from opposite directions. There will be interesting quantum
interferences effects, particularly if the QDP is a coherent operation that
will be discussed in the next section.

The initial state $\rho(0)=|\psi(0)\rangle\langle\psi(0)|$ undergoes three
different evolutions here in three steps, as depicted in Fig.1 In the first
step, the state is evolved through the unitary operation using the Heisenberg
Hamiltonian. The evolution of the state up to $t=t_{0}^{-}$ results in the
state, $\rho(t_{0}^{-})=U_{t_{0},0}\rho(0)U_{t_{0},0}^{\dagger}$, where the
unitary operator is given by $U_{t_{2},t_{1}}=\exp{-iH(t_{2}-t_{1})}$, where
we have absorbed the Planck constant in the time itself or set $\hbar=1$. The
state is given by,

$\rho(t_{0}^{-})=e^{-iHt_{0}}|\psi(0)\rangle\langle\psi(0)|e^{iHt_{0}}=|\psi(t_{0})\rangle\langle\psi(t_{0})|.$
(18)

We have discussed the computation of $|\psi(t)\rangle$ in the last section,
along with the state transfer fidelity. In the second step, an instantaneous
operation of the local QDP at the $l$’th site, the state transforms from
$\rho(t_{0}^{-})$ to $\tilde{\rho}(t_{0}^{+})$. Using the Kraus operator
formalism with the two operators $P_{0}$ and $P_{1}$, the state is now given
by

$\tilde{\rho}(t_{0}^{+})=P_{0}\rho(t_{0}^{-})P_{0}^{\dagger}+P_{1}\rho(t_{0}^{-})P_{1}^{\dagger}.$
(19)

As we can see from the above, the state here is a mixed state with two
different pure-state components, suffering decoherence due to the occurrence
of the QDP at $m$’th site. Just after the QDP, the state can be written in
terms of two pure states as,

$\tilde{{\rho}}(t_{0^{+}})=|\tilde{\psi}_{+}(t_{0})\rangle\langle\tilde{\psi}_{+}(t_{0})|+|\tilde{\psi}_{-}(t_{0})\rangle\langle\tilde{\psi}_{-}(t_{0})|,$
(20)

where, $|\tilde{\psi}_{\pm}(t_{0})\rangle\equiv{1\pm\sigma_{m}^{z}\over
2}|\psi(t_{0})\rangle$. In the third step, the state undergoes a unitary time
evolution from $t=t_{0}^{+}$ to time $t$ using the Heisenberg Hamiltonian, we
have

$\tilde{\rho}(t)=U_{t,t_{0}}\tilde{\rho}(t_{0}^{+})U_{t,t_{0}}^{\dagger}.$
(21)

In this step, the unitary Hamiltonian evolution occurs for an interval
$t-t_{0}$. Finally, we can rewrite the state using the two pure states
$|\tilde{\psi}^{\pm}\rangle$, as

$\tilde{{\rho}}(t)=|\tilde{\psi}_{+}(t)\rangle\langle\tilde{\psi}_{+}(t)|+|\tilde{\psi}_{-}(t)\rangle\langle\tilde{\psi}_{-}(t)|,$
(22)

where,

$\displaystyle|\tilde{\psi}_{+}(t)\rangle=\alpha
e^{-i\epsilon_{0}t}|00...0\rangle+\beta\sum_{y^{\prime}}H^{y^{\prime}}_{1}(m,t,t_{0})|y^{\prime}\rangle,$
$\displaystyle|\tilde{\psi}_{-}(t)\rangle=\beta\sum_{y^{\prime}}K^{y^{\prime}}_{1}(m,t,t_{0})|y^{\prime}\rangle,$
(23)

Here, the time-dependent wave functions[21] are given by,

$\displaystyle H^{y^{\prime}}_{y}(m,t,t_{0})=\sum_{y^{\prime\prime}\neq
m}G^{y^{\prime\prime}}_{y}(t_{0})G^{y^{\prime}}_{y^{\prime\prime}}(t-t_{0}),$
$\displaystyle
K^{y^{\prime}}_{y}(m,t,t_{0})=G^{m}_{y}(t_{0})G^{y^{\prime}}_{m}(t-t_{0}).$
(24)

Now, the elements of the time dependent RDM (given in Eq. 14) of the $l^{th}$
site are given by,

$\displaystyle\tilde{x}_{l}(t)=\langle
1|\tilde{\rho}_{l}(t)|1\rangle=|\beta|^{2}(|H^{l}_{1}(m,t,t_{0})|^{2}+|K^{l}_{1}(m,t,t_{0})|^{2}),$
$\displaystyle\tilde{y}_{l}(t)=\langle
0|\tilde{\rho}_{l}(t)|1\rangle=\alpha\beta^{*}e^{-i\epsilon_{0}t}H^{*l}_{1}(m,t,t_{0}).$
(25)

The average state transfer fidelity averaged over all pure states on the Bloch
sphere for $l^{th}$ site is given by,

$\displaystyle\tilde{\mathscr{F}}_{l,m}(t;t_{0})=\frac{1}{2}+\frac{1}{6}(|H^{l}_{1}(m,t,t_{0})|^{2}+|K^{l}_{1}(m,t,t_{0})|^{2})+\frac{1}{3}Re(e^{i\epsilon_{0}t}H^{l}_{1}(m,t,t_{0}))$
$\displaystyle=\frac{1}{2}+\frac{1}{6}|G^{l}_{1}(t)|^{2}+\frac{1}{3}Re(e^{i\epsilon_{0}t}(G^{l}_{1}(t)-2K^{l}_{1}(m,t,t_{0}))-G^{*l}_{1}(t)K^{l}_{1}(m,t,t_{0}))$
$\displaystyle+\frac{5}{6}|K^{l}_{1}(m,t,t_{0})|^{2}.$ (26)

The maximum fidelity saturates to the value $1/2$ over a long time $t$ because
the function $G^{l}_{1}(t)$ is basically a Bessel function which falls as
$1/t$ . In case of a local decohering process the dynamics is not very
different or in other words the QDP does not ”disturb” the system
significantly. Hence, the difference in fidelity with and without QDP for
$l^{th}$ site is given by
$\Delta\mathscr{F}_{l,m}(t;t_{0})=\tilde{\mathscr{F}}_{l,m}(t;t_{0})-\mathscr{F}_{l}(t)$,
we have

$\displaystyle\Delta\mathscr{F}_{l,m}(t;t_{0})=\frac{5}{6}|K^{l}_{1}(m,t,t_{0})|^{2}-\frac{1}{3}Re((e^{i\epsilon_{0}t}+G^{*l}_{1}(t))K^{l}_{1}(m,t,t_{0})).$
(27)

However, the quantum state transfer process (which is a coherent or unitary
process) and the effect of the QDP (which is a decohering process) can
interfere with each other and change the fidelity slightly. If the QDP is
performed at a particular site at a time when the target state is at site
while moving the fidelity changes slightly. From Eq. 27 maximum difference of
fidelity with and without QDP can be estimated to be
$\frac{5}{6}|K^{l}_{1}(m,t,t_{0})|^{2}$ at time $t=l/2$, because the second
term is merely a oscillating term.

The Fig. 2(b) depicts the change in fidelity for one extreme case where the
QDP occurs at the first site just after the evolution starts. In that case the
maximum difference in fidelity is around 0.4. However, when the QDP occurs at
some other site at a later time this difference is much less. For example in
Fig. 2(c) maximum change in fidelity is around 0.15 when the QDP is occurred
at the site $m=20$ at the time $t_{0}=10$. If the QDP occurs at a site where
where the basic fidelity without the QDP itself is very small the interference
effect is negligible and the fidelity does not change significantly. In
Fig.2(d) depicts the case where the QDP occurs at the first site but at much
later time $t_{0}=10.0$, the difference is seen to be negligible. In Fig. 2(e)
the dependence of the fidelity for the site $l=100$ is plotted with and
without the QDP, where the location $m$ of the QDP is same, and the time of
QDP is $t_{0}$ is $50.0$. Both the fidelities show peaks at a time slightly
after $t=50.0$, but the maximum fidelity with the QDP is slightly greater than
that without the QDP. The time at which the fidelity is maximum for is around
$t=l/2$, for the QDP occurring at the location $l$. In Fig. 2(f) the maximum
fidelity is shown as a function of site index $l$ obtained with and without
QDP, with the location and the time of QDP being $m=l$ and $t_{0}=l/2$. The
difference of fidelities is small for sites close to the first site, from
where the state transfer has been started. The difference of fidelities is
almost constant for further sites, as seen in the figure, and the difference
is around $0.1$.

##  4 Local quantum coherent process

In this section, we will turn our attention to the case of an instantaneous
local unitary QDP intervening the unitary dynamics of the Heisenberg
Hamiltonian. Similar to the last section, the QDP operates on the $m$’th spin
at time $t=t_{0}$. But unlike the previous section, the unitary QDP does not
cause decoherence. The three steps of evolution that are involved here, as
shown in Fig.1, are all unitary evolutions, as a result the initial pure state
evolves into another pure state. The intervening QDP being unitary, it can be
generated using a Hamiltonian. The instantaneous local unitary QDP intervening
the unitary Hamiltonian evolution can be viewed as an evolution with a kicked
Heisenberg Hamiltonian. The new Hamiltonian $\tilde{H}$ can be written as a
sum of two terms, the Heisenberg Hamiltonian $H$, given in Eq. 2 that
generates the background unitary evolution, and a magnetic field term
$H^{\prime}=\vec{\sigma}_{m}.\hat{n}$ with a delta-function kick, where
$\hat{n}$ is the direction of the magnetic field at the $m$’th site. We can
consider $\hat{n}$ to be a unit vector in the x-y plane. Thus, the total
Hamiltonian covering all the three steps can be written as,

$\tilde{H}=H+H^{\prime}\delta(t/t_{0}-1).$ (28)

The second term in the Hamiltonian represents the instantaneous QDP operating
on the given spin at $t=t_{0}$. The unitary evolution operator for $t>t_{0}$,
is a product of three unitary operators corresponding to the three time steps,
i.e., $U_{t_{0},0}=e^{-iHt_{0}}$ for the evolution up to $t=t_{0}^{-}$,
followed by $V_{m}=e^{-t_{0}\vec{\sigma}_{m}.\hat{n}}$ for the instantaneous
unitary QDP, and $U_{t,t_{0}}=e^{-i(t-t_{0})H}$ for the evolution from
$t=t_{0}^{+}$ up to time $t$. Thus, an initial state $|\psi(0)\rangle$
prepared at time $t=0$ evolves to the state
$|\tilde{\psi}(t)\rangle=\tilde{U}_{t,0}|\psi(0)\rangle$, the evolution
operator is given by,

$\tilde{U}_{t,0}=U_{t,t_{0}}V_{m}U_{t_{0},0}.$

Now, similar to the non-unitary QDP, we start with an initial state
$|\psi(0)\rangle=\alpha|0..0\rangle+\beta|100.0\rangle$. In the first step,
the state evolves unitarily upto a time $t_{0}^{-}$ with the Hamiltonian $H$,
yielding $|\psi(t_{0}^{-})\rangle$. This part is the same as discussed in the
last section, as shown in Eq. 10. In the second step, the state is changed by
an operation of $V_{m}$, the $m^{th}$ qubit undergoes a local unitary
operation. This coherent or unitary process, which is a local quantum gate
operation, is represented by an instantaneous unitary operation that acts on
the state of the given qubit between time $t=t_{0}^{-}$ and time $t_{0}^{+}$.
The operation of the unitary operator ${V}_{m}$ on the basis states of $m$’th
spin is given by ,

${V}_{m}|0\rangle=\gamma|0\rangle+\delta|1\rangle,~{}{V}_{m}|1\rangle=-\delta^{*}|0\rangle+\gamma|1\rangle,$
(29)

where the various amplitudes are related to $t_{0}$, the components
$n_{x},n_{y}$ of the unit vector $\hat{n}$ as, $\gamma=\cos{t_{0}}$ and
$(n_{y}+in_{x})\sin{t_{0}}=\delta$. The state
$|\tilde{\psi}(t_{0}^{+})\rangle$, immediately after the operation at
$t=t_{0}$ is given by,

$\displaystyle|\tilde{\psi}(t_{0^{+}})\rangle={V}_{m}[\alpha
e^{-i\epsilon_{0}t_{0}}|F\rangle+\beta\sum_{y}G^{y}_{1}(t_{0})|y\rangle]$
$\displaystyle=(\alpha\gamma
e^{-i\epsilon_{0}t_{0}}-\beta\delta^{*}G^{m}_{1}(t_{0}))|F\rangle+\alpha\delta
e^{-i\epsilon_{0}t_{0}}|m\rangle$
$\displaystyle+\beta\gamma\sum_{y^{\prime}}G^{y^{\prime}}_{1}(t_{0})|y^{\prime}\rangle+\beta\delta\sum_{y^{\prime}\neq
m}G^{y^{\prime}}_{1}(t_{0})|m,y^{\prime}\rangle.$ (30)

Here, after the coherent operation, we can see that two-magnon states are also
generating from one magnon states, that were absent in the last section. Now,
the state is a mixture of zero, one and two magnon states.This is the simplest
possible operation where three different magnon sectors are obtained. Each of
these sectors will have their own dynamics, during the further evolution using
the Heisenberg Hamiltonian, in the third step. The state at a later time $t$
becomes,

$\displaystyle|\tilde{\psi}(t)\rangle=U_{t,t_{0}}|\tilde{\psi}(t_{0^{+}})\rangle=(\alpha\gamma
e^{-i\epsilon_{0}t}-\beta\delta^{*}G^{m}_{1}(t_{0})e^{-i\epsilon_{0}(t-t_{0})})|F\rangle+$
$\displaystyle\sum_{y}[\alpha\delta
e^{-i\epsilon_{0}t_{0}}G^{y}_{m}(t-t_{0})+\beta\gamma
X^{y}_{1}(m,t,t_{0})]|y\rangle+\beta\delta\sum_{y_{1},y_{2};y_{1}<y_{2}}L^{y_{1},y_{2}}_{1,m}(m,t,t_{0})|y_{1},y_{2}\rangle.$

where we have used
$X_{1}^{n}(m,t,t_{0})=K_{1}^{n}(m,t,t_{0})+H_{1}^{n}(m,t,t_{0})$ to simplify
the expression. where, the new time-dependent two-magnon sector wave function
is given by,

$\displaystyle
L^{y_{1},y_{2}}_{y^{\prime},m}(m,t,t_{0})=\sum_{y^{\prime\prime}>m}G^{y^{\prime\prime}}_{y^{\prime}}(t_{0})G^{y_{1},y_{2}}_{m,y^{\prime\prime}}(t-t_{0})+\sum_{y^{\prime\prime}<m}G^{y^{\prime\prime}}_{y^{\prime}}(t_{0})G^{y_{1},y_{2}}_{y^{\prime\prime},m}(t-t_{0}),$

in terms of the two-particle Green’s function. The new Green’s function is
defined in terms of the two-magnon eigenfunctions
$\psi_{p_{1},p_{2}}(x_{1},x_{2})$, shown in Eq.5, and the eigenvalues
$\epsilon_{2}(p_{1},p_{2})$, we have,

$G^{x^{\prime}_{1},x^{\prime}_{2}}_{x_{1},x_{2}}(t)=\sum_{p_{1},p_{2}}\psi_{p_{1},p_{2}}^{x_{1},x_{2}}\psi_{p_{1},p_{2}}^{x^{\prime}_{1},x^{\prime}_{2}*}e^{-it\epsilon_{2}(p_{1},p_{2})}.$
(33)

We have given the details of calculation of this Green’s function both for
two-magnon bound and scattering states in Appendix A.

Now, the matrix elements of the RDM of the $l^{th}$ site, as given in
Equation(12) , are modified with
$\tilde{x}_{l}(t)=\frac{1}{2}\langle\tilde{\Phi}(t)|\mathbf{1}-\sigma^{z}_{l}|\tilde{\Phi}(t)\rangle$
and
$\tilde{y}_{l}(t)=\langle\tilde{\Phi}(t)|\sigma^{+}_{l}|\tilde{\Phi}(t)\rangle$
of the RDM can be calculated from Equation(28) and are given by,

$\displaystyle\tilde{x}_{l}(t)=|\alpha\delta
e^{-i\epsilon_{0}t_{0}}G^{l}_{m}(t-t_{0})+\beta\gamma
X^{l}_{1}(m,t,t_{0})|^{2}+|\beta|^{2}|\delta|^{2}\sum_{y^{\prime}\neq
l}|L^{l,y^{\prime}}_{1,m}(m,t,t_{0})|^{2},$

and the off-diagonal matrix element is given by

$\displaystyle\tilde{y}_{l}(t)=e^{-i\epsilon_{0}(t-t_{0})}(\alpha\gamma-\beta\delta^{*}G^{m}_{1}(t_{0})e^{i\epsilon_{0}t_{0}})[\alpha^{*}\delta^{*}G^{*l}_{m}(t-t_{0})+\beta^{*}\gamma^{*}e^{-i\epsilon_{0}t_{0}}X^{*l}_{1}(m,t,t_{0})]$
$\displaystyle+\sum_{y\neq
l}(\alpha\beta^{*}|\delta|^{2}e^{-i\epsilon_{0}t_{0}}G^{y}_{m}(t-t_{0})+|\beta|^{2}\gamma\delta^{*}X^{y}_{1}(m,t,t_{0}))L^{*l,y}_{1,m}(m,t,t_{0}).$
(35)

After averaging over all pure states on the Bloch sphere average state
transfer fidelity for $l^{th}$ site is given by,

$\displaystyle\tilde{\mathscr{F}}_{l,m}(t;t_{0})=\frac{1}{2}+\frac{|\gamma|^{2}}{6}[|X^{l}_{1}(m,t,t_{0})|^{2}+2Re(e^{-i\epsilon_{0}t}X^{*l}_{1}(m,t,t_{0})]+\frac{|\delta|^{2}}{6}[\sum_{y\neq
l}|L^{l,y}_{1,m}(m,t,t_{0})|^{2}$
$\displaystyle-|G^{l}_{m}(t-t_{0})|^{2}+2Re(\sum_{y\neq
l}e^{-i\epsilon_{0}t_{0}}G^{y}_{m}(t-t_{0})L^{*l,y}_{1,m}(m,t,t_{0}))].$ (36)

Unlike the previous case discussed in Section 3 here quantum state transfer
will be contributed by both the magnon bound states and scattering states
sectors. We can try to analyse how much contribution to the fidelity comes
from two magnon scattering states and two magnon bound states. Hence, the time
dependent function $G^{x^{\prime}_{1},x^{\prime}_{2}}_{x_{1},x_{2}}(t)$ can be
split into two parts $G^{x^{\prime}_{1},x^{\prime}_{2}}_{(B);x_{1},x_{2}}(t)$
and $G^{x^{\prime}_{1},x^{\prime}_{2}}_{(S);x_{1},x_{2}}(t)$ corresponding to
contributions from bound states and scattering states respectively. It is
known that for a chain with $N$ sites number of two magnon bound states is of
the order of $N$ where as, number of two magnon scattering states is
$O(N^{2})$.

The dependence of the state transfer fidelity on the target site location and
time has been shown as density plots in Fig.3 for the Heisenberg dynamics with
a unitary QDP. Fig. 3(a) and Fig. 3(b) show the contribution to the state
transfer fidelity from the two-magnon bound states and the scattering states
respectively. We can see that the contribution of the two-magnon scattering
states is larger than the contribution coming from the two-magnon bound
states. This is according to our expectation, as the number of bound states is
$N$, whereas there are $O(N^{2})$ number of scattering states. The difference
in the fidelities $\Delta{\mathscr{F}}$ of the two states, $\tilde{\rho}(t)$
and $\rho(t)$, is shown as a density plot in Fig.3(c) as a function of $l$ and
$t$. It can be seen from here that it can take both positive and negative
values, corresponding to constructive and destructive interference of the
propagation of the the coded information at $t=0$ and the unitary QDP
occurring at $t=t_{0}$. The more interesting regime is where the fidelity
difference is positive and is around 0.2, implying a 20 per cent increase in
the probability of the state transfer, coming from the implementation of the
QDP at a single site. It is important to state here that this increase occurs
only for the case, the coded state arrives at the location of QDP, viz. site
$m$ at the time $t=t_{0}$, to take the maximum benefit of the interference. We
have shown the difference in the fidelities in Fig. 3(d) as a function of the
target site $l$, for various values of the interaction parameter $\Delta$ of
the Hamiltonian. We can see it is fluctuating around 0.1 for all parameter
values, implying a similar behaviour. The amplitude of the fluctuations is
maximum for the case of $\Delta=0$, corresponding to the XY model.

Figure 3: (a) $\frac{1}{6}\sum_{y^{\prime}\neq
l}|L^{l,y^{\prime}}_{1,m}(m,t,t_{0})|^{2}$ as function of site index($l$) and
time ($t$) for Heisenberg dynamics with closed boundary condition with local
unitary operation on the tenth site at $t_{0}=5.0$. without taking two magnon
scattering states into account for anisotropy constant $\Delta=1.0$.(b)
$\frac{1}{6}\sum_{y^{\prime}\neq l}|L^{l,y^{\prime}}_{1,m}(m,t,t_{0})|^{2}$ as
function of site index($l$) and time ($t$) for Heisenberg dynamics with closed
boundary condition with local unitary operation on the tenth site at
$t_{0}=5.0$. without taking two magnon bound states into account for
anisotropy constant $\Delta=1.0$.(c) $\Delta\mathscr{F}_{l,m=15}(t;t_{0}=7.5)$
as function of site index($l$) and time ($t$) for Heisenberg dynamics with
closed boundary condition with local unitary operation on the tenth site at
$t_{0}=7.5$ for anisotropy constant $\Delta=1.0$.
(d)$\Delta\mathscr{F}_{l,m=l}(t=l/2;t_{0}=l/2)$ as function of site index $l$
for different values of anisotropy constants $\Delta$.

An interesting case can be studied when the first qubit undergoes the local
operation just after the evolution starts. As a consequence it can be seen
from Eq. 33 that the contribution from two magnon sector becomes zero because
of the term $G^{y}_{1}(0)$. The Eq. 37 and Eq. 38 then reduce to respectively,

$\displaystyle\tilde{\mathscr{F}}_{l,m=1}(t;t_{0}=0_{+})=\frac{1}{2}+\frac{1}{6}(|\gamma|^{2}-|\delta|^{2})|G^{l}_{1}(t)|^{2}+\frac{|\gamma|^{2}}{3}Re((e^{i\epsilon_{0}t}G^{l}_{1}(t)),$
$\displaystyle\tilde{\mathscr{F}}_{l,m=1}(t;t_{0}=0_{+})=\frac{1}{2}+\frac{1}{6}Re((e^{i\epsilon_{0}t}G^{l}_{1}(t)).$
(37)

The last terms of the above equations are merely oscillating terms. The second
equation suggests that state transfer will not take place. Setting
$|\gamma|=|\delta|$ (Hadamard gate operation)in the first one also shows the
same.

Figure 4:  The state transfer fidelity $\mathscr{F}_{l}(t)$ as function of
site index($l$) and time ($t$) for kicked Harper dynamics with open boundary
condition with parameters (a)$g=1.0$ and $\tau=0.1$, (b)$g=3.0$ and $\tau=0.1$
, (c)$g=1.0$ and $\tau=0.4$, (d)$g=1.0$ and $\tau=0.9$ , $\eta=\sqrt{2}$ for
each case. The initial state in all the cases is
$|\psi(0)\rangle=\sqrt{0.75}|00...0\rangle+\sqrt{0.25}|100..0\rangle$.

The speed of state transfer is if the order of $J$ (here $J$ is taken to be
$1/2$). Another interesting case can be studied where $t_{0}=Jl=l/2$, which
means that the evolution is interrupted at a particular site when the target
state is passing through. In context of quantum state transfer where the
target state is unknown at the receiver end one can vary the parameters
$\gamma$ and $\delta$ to get the maximum fidelity. The values of $m$, $t_{0}$
and $t$ is set $l$, $l/2$ and $l/2$ respectively in Eq.37. Maximum value of
the fidelity can be obtained when $\gamma=0$ and $\delta=1$ (bit flip or X
gate)for large values of $l$ and the difference in fidelity with and without
the local operation is given by,

$\displaystyle\Delta\mathscr{F}_{l,m=l}(t={l\over 2};t_{0}={l\over
2})=\tilde{\mathscr{F}}_{l,m=l}(t={l\over 2};t_{0}={l\over
2})-\mathscr{F}_{l}(t={l\over 2})$ $\displaystyle=\frac{1}{6}[\sum_{y\neq
l}|L^{l,y}_{1,l}(m=l,t={l\over 2},t_{0}={l\over 2})|^{2}-|G^{l}_{m}(t={l\over
2})|^{2}-2Re(e^{i\epsilon_{0}l/2}G^{l}_{1}(t={l\over 2}))-1].$

##  5 Local decohering process in non-integrable quantum systems

In the previous sections, we have considered the effect of an intervening QDP,
either non-unitary or unitary, on the background unitary Hamiltonian
evolution, governed by the Heisenberg Hamiltonian, that belongs to the
integrable class of dynamical systems. It will be interesting to see the
effect of a QDP intervention on a unitary evolution, corresponding to non-
integrable Hamiltonian systems. The signal propagation from a local QDP and
its detection with a background non-integrable dynamics has been investigated
in XY model with transverse and longitudinal magnetic fields[21].Moreover,
there are sharp differences in the eigenvalue spacing distribution and the
structure of the eigenfunctions in the two cases, that have been widely
investigated.

To see the effect of QDP in non integrable systems we consider a simple model
Hamiltonian with a tuneable parameter, to go continuously from completely
integrable to completely non-integrable regimes. We use a one-dimensional
periodically-kicked Harper model, a simple model of fermions hopping on a
chain with an inhomogeneous site potential, appearing as a kick at regular
intervals. The Hamiltonian is given by

$H(t)=\sum_{j=1}^{N}[{\hat{c}}^{\dagger}_{j}{\hat{c}}_{j+1}+H.C.]+g\cos(\frac{2\pi
j\eta}{N}){\hat{c}}^{\dagger}_{j}{\hat{c}}_{j}\sum_{n=-\infty}^{\infty}\delta({t}/\tau-n)],$
(39)

where $c_{j}^{\dagger}$ is a creation operator at site $j$, $g$ is a potential
strength parameter, $\eta$ is a parameter to make the potential either
commensurate or incommensurate with the lattice, and $\tau$ is the kicking
interval. The first term represents the kinetic energy of the fermion or
hopping term, and the second term is the kicked potential energy operator. The
Hilbert space for a site is two-dimensional, either occupied site or
unoccupied site, thus it can be mapped to the spin language. The first term
will turnout to be XY term, shown in In the Eq.2, and the second term becomes
a transverse field that is inhomogeneous. The fermion occupation can be mapped
to the down spin occupation in the spin states. Unlike the Heisenberg
Hamiltonian that incorporates an interaction of down spins on neighbouring
sites, there are no many-body interaction effects here. The kicked Harper
model has been investigated for the entanglement distribution and
dynamics[25]. The effect of the magnetic field or the potential is through a
train of kicking pulses with an interval $\tau$. For $\tau\rightarrow 0$ the
dynamics of the Harper Hamiltonian is integrable, and for large values of
$\tau$ the dynamics is completely chaotic (see Figure 4 in the work of
Lakshminarayan and Subrahmanyam[25] for further details)[26].

Figure 5:  The detector function $f_{l}(t)$ as function of site index($l$) and
time ($t-t_{0}$) for kicked Harper dynamics with open boundary condition with
local unitary operation on the first site after $5$ kicks ($t_{0}=5\tau$) with
parameters (a)$g=1.0$ and $\tau=0.1$, (b)$g=3.0$ and $\tau=0.1$ , (c)$g=10.0$
and $\tau=0.1$, (d)$g=1.0$ and $\tau=0.4$, (e)$g=2.5$ and $\tau=0.4$,
(f)$g=1.0$ and $\tau=0.9$. $\eta=\sqrt{2}$ for each case. The initial state in
all the cases is
$|\psi(0)\rangle=\sqrt{0.5}|00...0\rangle+\sqrt{0.5}|100..0\rangle$.

Similar to the previous sections, we will consider an initial state
$|\psi_{0}\rangle=\alpha|00..0\rangle+\beta|10..0\rangle$, a linear
combination of zero-particle state and one-particle state localised at first
site. Through the time evolution, the particle can hop around to other sites.
We will consider evolution at discrete times, viz. $t=\tau^{+},2\tau^{+}$ etc,
that is at instant just after a kick. The evolution operator $U_{n}$ at a time
just after $n$ kicks (corresponding to $t=n\tau$), is given by

$U(n)=\left(e^{-i\tau\sum_{j}c_{j}^{\dagger}c_{j+1}+H.C.}e^{-i\tau
g\sum_{j}\cos{\frac{2\pi j\eta}{N}}{c_{j}^{\dagger}c_{j}}}\right)^{n},$ (40)

where the two operator factors appearing above do not commute. This evolution
operator evolves the initial state to give
$|\psi_{n}\rangle=U(n)|\psi_{0}\rangle$.

The above unitary evolution is intervened by a local instantaneous non-unitary
QDP, as considered in Section 3, at time $t=n_{0}\tau^{+}$. Thus, the
evolution proceeds in three steps as shown in Fig. 1. In the first step, the
initial state will become the state $|\psi(n_{0})\rangle$ at time
$t=n_{0}\tau$. In the second step, the state $\rho(n_{0})$ , in analogy with
Eq. 19, will result in the state
$\tilde{\rho}(n_{0})=P_{0}\rho(n_{0})P_{0}^{\dagger}+P_{1}\rho(n_{0})P_{1}^{\dagger}$.
The third step, the state evolves unitarily to become,

$\tilde{\rho}(n)=U(n-n_{0})(P_{0}\rho(n_{0})P_{0}^{\dagger}+U(n-n_{0})P_{1}\rho(n_{0})P_{1}^{\dagger})U^{\dagger}(n-n_{0}).$
(41)

The quantum state transfer fidelity can also be calculated numerically, by
calculating the matrix elements of the reduced density matrix $\rho_{l}(n)$
for the $l$’th site, noting that $\tilde{x}_{l}$ is just the expectation value
of the fermion number operator. We have, analogous to Eq. 25,

$\tilde{x}_{l}(n)=Tr~{}c_{l}^{\dagger}c_{l}~{}\tilde{\rho}(n),~{}\tilde{y}_{l}(n)=Tr~{}c^{\dagger}_{l}~{}\tilde{\rho}(n),$
(42)

where we have written the matrix elements as expectation value of operators.
The state transfer fidelity can be calculated numerically using the above. The
average fidelity for an initial state
$|\psi_{0}\rangle=\sqrt{0.75}|0\rangle+\sqrt{0.25}|1\rangle$ has been shown in
Fig. 4(a)-4(d) for a different values of $\tau$, $g$ as a density plot, for
various target sites and times. For a smaller values of $\tau$, the dynamics
is still closer to an integrable one which can be seen in Fig. 4(a). For small
times, we can see larger fidelity only for small $l$. As the time increases,
for larger values of $l$ get larger fidelity. This is as expected, the state
is spreading at a given speed. For larger value of potential strength
parameter $g$ the spreading of the state does not take place. This can be seen
in Fig. 4(b) where the density plot for fidelity has been plotted for
$\tau=0.1$ and $g=3.0$. However, for larger value of $\tau$ as shown in 4(d)
even for small times too there are larger fidelities for far away sites. This
implies, we cannot define a speed of propagation for the non-integrable case.
This transition occurs at some intermediate value near $\tau=0.4$ as shown in
Fig. 4(c). Similar features were seen in XY model with transverse and
longitudinal fields[21].

We will define and study a detector function given by,

$f_{l}(t=n\tau)=\tilde{x}_{l}(t)-x_{l}(t).$ (43)

Here, the two quantities are the diagonal matrix elements of the RDM from the
two states $\tilde{\rho}(n)$ and $\rho(n)$ evolved with and without the QDP
occurrence. This is a useful detector function to see the effect of the QDP,
similar to the Loschmidt echo, but a lot simpler to calculate. This function
has been studied in detail for various Hamiltonians in conjunction with a non-
unitary QDP[21]. One can also study similar detector functions constructed
from the off-diagonal matrix elements, and the von-Neumann entropy of the RDM.
It has been see that they all show similar structure. The results for the
above detector function are shown in Fig. 5(a) to Fig. 5(f) for various
combination of values for $g$ and $\tau$.

For smaller values of kicking interval $\tau$ and potential strength $g$ the
detector function $f_{l}(t)$ shows behaviour similar to that of integrable
models like Heisenberg model, XY model with transverse magnetic field [21].
The signal of the local QDP propagates through the chain with a finite speed
which is shown in Fig. 5a). But the qualitative nature of the detector
function depends on both the parameters $\tau$ and $g$. If the value of $g$ is
increased the signal propagation does not take place or in other words
$f_{l}(t)$ is zero for farther sites. This is depicted in Fig. 5(b), Fig. 5(c)
and Fig. 5(e). For larger values of $\tau$ the speed of propagation abruptly
becomes very high, which can taken as a signature of transition from
integrability to non integrability. Fig. 5(d) shows this for $\tau=0.4$ and
$g=1$. For $\tau=0.9$ the signal travels very quickly to the other end of the
chain and, thus, it is difficult to define a finite speed in this case as it
is shown in Fig. 6(f). Similar behaviour we have seen for the state transfer
fidelity in Fig. 5. We can say that for $g>1$ and smaller value of $\tau$, the
signal does not propagate to larger distances, and we cannot define a speed,
as shown in Fig.5(b), (c) and (e). On the other hand, for $g=1$, and larger
values of $\tau$, as seen in Fig. 5(d) and 5(f), the signal spreads out almost
instantaneously to larger distances, and we cannot get a finite speed.

##  6 Conclusions

In this paper, we have investigated the effect of a local quantum dynamical
process (QDP), both unitary and non-unitary processes, intervening the process
of quantum state transfer through a unitary evolution for both integrating and
non integrating models. For Heisenberg model, that represents an integrable
dynamics, we have analytically calculated the state transfer fidelity in terms
of time-dependent Green functions. The information coded in the state of the
first spin, spreads out to other location through the unitary dynamics, We
have seen that the signal from the local QDP can interfere with the dynamics
of the state transfer, and it changes the fidelity depending on the time and
the location of the QDP. In the case of a non-unitary QDP, there is a small
change in the state transfer fidelity, even if the QDP occurs at the location
of the particular qubit where the coded state arrives at the time of QDP.
However, for a coherent or unitary QDP, the fidelity can increase or decrease
depending on the constructive or destructive interference of the propagation
of the coded state, and the signal from the QDP. For appropriate location and
the time, the state transfer fidelity can increase substantially. In the case
of unitary QDP, an initial state with a combination of zero and one-magnon
states, becomes a superposition of zero, one and two magnon eigenfunctions.
The state transfer has contributions from both two-magnon bound states and
scattering states, which are quantitatively calculated separately using the
two-magnon Bethe ansatz wave functions.

Finally, we have investigated the dynamics of Kicked Harper model in the
context of quantum state transfer. The dynamics of this model changes from
integrable to non integrable by increasing the kicking interval time. We show
that the spreading of the coded state is dependent on the kicking interval
time. For smaller values of $\tau$ the dynamics is similar to the integrable
model and larger values of $\tau$ the spreading takes place quickly which is a
possible signature of non integrability.The signal propagation also depends on
the potential strength parameter $g$ and $\tau$. The signal propagates with a
finite speed below a certain value of $\tau$ and above that value the
propagation takes place too quickly to define a speed. Whereas, for larger
value of $g$ the signal does not reach the farther sites. So, the signal of
the QDP gets localised.

Acknowledgement: SS acknowledges the financial support from CSIR, India. We
thank Professor A. Lakshminarayan for useful discussion of non-integrable
systems.

Appendix A

The time-dependent two-particle Green’s function
$G^{x^{\prime}_{1},x^{\prime}_{2}}_{x_{1},x_{2}}(t)$ has contributions from
both scattering and bound states. We can write the function as two parts, as

$G^{x^{\prime}_{1},x^{\prime}_{2}}_{x_{1},x_{2}}(t)=G^{x^{\prime}_{1},x^{\prime}_{2}}_{(B);x_{1},x_{2}}(t)+G^{x^{\prime}_{1},x^{\prime}_{2}}_{(S);x_{1},x_{2}}(t).$
(44)

In Bethe Ansatz solution for the two-magnon eigenfunctions for a chain with
periodic boundary conditions, the momenta can be parametrised by the relation,

$p_{i}=2\cot^{-1}(2\lambda_{i}).$ (45)

The S-matrix in the Bethe Ansatz wave function is given in the form,

$S(p_{1},p_{2})=e^{i\theta(p_{1},p_{2})},$ (46)

Where, the phase factor $\theta$ is given by,

$\theta(p_{1},p_{2})=2\tan^{-1}\Big{[}\frac{\Delta\sin[(p_{1}-p_{2})/2]}{\cos[(p_{1}+p_{2})/2]-\Delta\cos[(p_{1}-p_{2})/2]}\Big{]}.$
(47)

Bound state solutions are complex and are given by the form,

$\lambda_{1}=q+i/2;\lambda_{2}=q-i/2.$ (48)

Here, we consider the case of infinitely long isotropic ferromagnetic chain
where $\Delta=1$ to calculate the bound state contribution to state transfer
fidelity. In this case the the number $q$ varies continuously within the
limits $-\infty<q<+\infty$ . The S-matrix for bound state wave function can be
calculated using the solutions given above using Eq. 49,

$e^{i\theta(\lambda_{1},\lambda_{2})}=e^{2i\tan^{-1}(\lambda_{1}-\lambda_{2})}=e^{\log\frac{i-\lambda_{1}+\lambda_{2}}{i+\lambda_{1}-\lambda_{2}}}=0.$
(49)

Two magnon bound state wave function apart from a normalisation factor is
given by,

$\psi^{x_{1},x_{2}}_{p_{1},p_{2}}=e^{i(p_{1}x_{1}+p_{2}x_{2})}$ (50)

Since the momenta $p_{1}$ and $p_{2}$ are related as shown in Eq. 49, the wave
function for the bound state can be written in terms of the quantity $q$ as,

$\displaystyle\psi^{x_{1},x_{2}}_{q}=A(q)e^{2i[x_{1}\cot^{-1}(2q+i)+x_{2}\cot^{-1}(2q-i)]}=A(q)e^{x_{1}\log(\frac{iq-1}{iq})+x_{2}\log(\frac{iq}{iq+1})}$
$\displaystyle=A(q)\big{(}\frac{q^{2}}{1+q^{2}}\big{)}^{(x_{2}-x_{1})/2}e^{i(x_{1}+x_{2})\tan^{-1}(\frac{1}{q})}.$
(51)

Where, $A(q)$ is the normalisation factor.The normalisation factor for the
wave function can be calculated as,

$|A(q)|^{-2}={\sum_{r=1}^{N-1}r(\frac{q^{2}}{1+q^{2}})^{N-r}}$ (52)

The function $G^{x^{\prime}_{1},x^{\prime}_{2}}_{(B);x_{1},x_{2}}(t)$ for an
infinitely long ferromagnetic isotropic chain then becomes (using
$x_{12}=x_{2}+x_{2}^{\prime}-x_{1}-x_{1}^{\prime}$ and
$\tilde{x}_{12}=x_{2}-x_{2}^{\prime}+x_{1}-x_{1}^{\prime}$),

$\displaystyle
G^{x^{\prime}_{1},x^{\prime}_{2}}_{(B);x_{1},x_{2}}(t)=\lim_{N\to\infty}\frac{N}{2\pi}\int_{-\infty}^{+\infty}\psi_{q}^{x_{1},x_{2}}\psi_{q}^{*x^{\prime}_{2},x^{\prime}_{2}}e^{-it\epsilon(q)}\frac{\partial
p}{\partial q}dq$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dq\frac{\partial
p}{\partial
q}\frac{1}{q^{2}}\Big{(}\frac{q^{2}}{1+q^{2}}\Big{)}^{x_{12}/2}e^{i\tilde{x}_{12}\tan^{-1}(1/q)-it\epsilon(q)},$
(53)

Where, $\epsilon(q)$ is the energy associated with the state and given in the
form,

$\epsilon(q)=\epsilon_{0}+\frac{2}{1+q^{2}}$ (54)

Two magnon scattering state wave function is given by,

$\psi^{x_{1},x_{2}}_{p_{1},p_{2}}=A(p_{1},p_{2})e^{i\theta(p_{1},p_{2})/2}\big{(}e^{i[p_{1}x_{1}+p_{2}x_{2}-\theta(p_{1},p_{2})/2]}\\\
-e^{i[p_{1}x_{2}+p_{2}x_{1}+\theta(p_{1},p_{2})/2]}\big{)}.$ (55)

Where, the normalisation factor $A(p_{1},p_{2})$ for the wave function is
given by,

$|A(p_{1},p_{2})|^{-2}={4\sum_{r=1}^{N-1}r\cos^{2}\big{[}\frac{(N-r)(p_{2}-p_{1})-\theta(p_{1},p_{2})}{2}\big{]}}$
(56)

The function $G^{x^{\prime}_{1},x^{\prime}_{2}}_{(S);x_{1},x_{2}}(t)$ becomes,

$G^{x^{\prime}_{1},x^{\prime}_{2}}_{(S);x_{1},x_{2}}(t)=\sum_{p_{1},p_{2}}|A(p_{1},p_{2})|^{2}\psi^{x_{1},x_{2}}_{p_{1},p_{2}}\psi^{*x^{\prime}_{1},x^{\prime}_{2}}_{p_{1},p_{2}}e^{-i\epsilon(p_{1},p_{2})t}.$
(57)

Where, the energy eigenvalues for Heisenberg chain with anisotropy constant
$\Delta$ are given by,

$\epsilon(p_{1},p_{2})=\epsilon_{0}+2(\Delta-\cos p_{1})+2(\Delta-\cos
p_{2}).$ (58)

For an infinitely long chain it can be shown that the magnon momenta $p_{1}$
and $p_{2}$ become independent and takes continuous values in an interval
depending on the value of anisotropy constant $\Delta$. For an infinitely long
chain with anisotropy constant $\Delta=1$ and $\Delta$ ($-1<\Delta<1$) Eq. 58
becomes,

$G^{x^{\prime}_{1},x^{\prime}_{2}}_{(S);x_{1},x_{2}}(t)=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}dp_{1}dp_{2}\psi^{x_{1},x_{2}}_{p_{1},p_{2}}\psi^{*x^{\prime}_{1},x^{\prime}_{2}}_{p_{1},p_{2}}e^{-i\epsilon(p_{1},p_{2})t}.$
(59)

This can rewritten as, using $\phi=\pi-\cos^{-1}(-\Delta)$, as

$G^{x^{\prime}_{1},x^{\prime}_{2}}_{(S);x_{1},x_{2}}(t)=\frac{1}{4\phi^{2}}\int_{-\phi}^{\phi}dp_{1}\int_{-\phi}^{\phi}dp_{2}\psi^{x_{1},x_{2}}_{p_{1},p_{2}}\psi^{*x^{\prime}_{1},x^{\prime}_{2}}_{p_{1},p_{2}}e^{-i\epsilon(p_{1},p_{2})t}.$
(60)

##  7 References

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# Linear Truncations Package for Macaulay2

Lauren Cranton Heller  Department of Mathematics, University of California,
970 Evans Hall, Berkeley, CA 94720, USA<EMAIL_ADDRESS>and  Navid
Nemati  Université Côte d’Azur, Inria, 2004 route des Lucioles, 06902 Sophia
Antipolis, France<EMAIL_ADDRESS>
###### Abstract.

We introduce the Macaulay2 package $\mathtt{LinearTruncations}$ for finding
and studying the truncations of a multigraded module over a standard
multigraded ring that have linear resolutions.

##  1\. Introduction and preliminaries
$\mathtt{multigradedPolynomialRing}$, $\mathtt{isLinearComplex}$,
$\mathtt{supportOfTor}$

Castelnuovo–Mumford regularity is a fundamental invariant in commutative
algebra and algebraic geometry. Roughly speaking, it measures the complexity
of a module or sheaf. Let $S$ be a polynomial ring with the standard grading
and $M$ a finitely generated $S$-module. In this case Castelnuovo–Mumford
regularity is typically defined in terms of either the graded Betti numbers of
$M$ or the vanishing of local cohomology modules $H^{i}_{\mathfrak{m}}(M)$,
where $\mathfrak{m}$ is the maximal homogeneous ideal of $S$. Eisenbud and
Goto show in [EG84] that the Castelnuovo–Mumford regularity of $M$ is the
minimum degree where the truncation of $M$ has a linear resolution.

An extension of Castelnuovo–Mumford regularity to multigraded modules was
introduced by Hoffman and Wang [HW04] in a special case, and later by Maclagen
and Smith [MS04] and Botbol and Chardin [BC17] in a more general setting. The
multigraded regularity of a module is a region in
$\operatorname{\mathbb{Z}}^{r}$ rather than an integer. It is invariant under
positive translations and thus can be described by its minimal elements. An
affirmative answer to the following open question would reduce this to a
finite computation.

######  Question 1.1.

Can the minimal elements of the regularity of $M$ be bounded in terms of $S$
and the Betti numbers of $M$?

In analogy to the singly graded case, one may ask about the relation between
multigraded Castelnuovo–Mumford regularity and the multidegrees where the
truncation of a module has a linear resolution, which we call the linear
truncation region. (See Definitions 1.2 and 1.3.) In the multigraded setting
these regions can differ, but a bound on the linear truncations would answer
Question 1.1 by [EES15, Proposition 4.11]. Our goal is to compute the minimal
elements of the linear truncation region within a specified finite region of
$\operatorname{\mathbb{Z}}^{r}$.

We introduce the LinearTruncations package for [M2], which provides tools for
studying the resolutions of truncations of modules over rings with standard
multigradings. Given a module and a bounded range of multidegrees, our package
can identify all linear truncations in the range. The algorithm uses a search
function that is also applicable to other properties of modules described by
sets of degrees. The examples here were computed using version 1.18 of
Macaulay2 and version 1.0 of LinearTruncations.

In section 2 we describe the main algorithms of this package,
$\mathtt{findRegion}$ and $\mathtt{linearTruncations}$. In Section 3, we
discuss the relation between the linear truncation region and the multigraded
regularity and we introduce $\mathtt{regularityBound}$ and
$\mathtt{linearTruncationsBound}$ as faster methods for calculating subsets of
the multigraded regularity and linear truncation regions, respectively.

To set our notation, let $k$ be a field and

$S=k\left[x_{i,j}\mid 1\leq i\leq r,\,0\leq j\leq n_{i}\right]$

a $\operatorname{\mathbb{Z}}^{r}$-graded polynomial ring with $\deg
x_{ij}=\mathbf{e}_{i}$, the $i$-th standard basis vector in
$\operatorname{\mathbb{Z}}^{r}$, for all $j$ (so that $S$ is the coordinate
ring of a product $\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{r}}$ of
projective spaces). The function $\mathtt{multigradedPolynomialRing}$ produces
such rings:

        i1 : needsPackage "LinearTruncations"
        o1 = LinearTruncations
        o1 : Package
        i2 : S = multigradedPolynomialRing {1,2}
        o2 = S
        o2 : PolynomialRing
        i3 : degrees S
        o3 = {{1, 0}, {1, 0}, {0, 1}, {0, 1}, {0, 1}}
        o3 : List

Let $M$ be a finitely generated $\operatorname{\mathbb{Z}}^{r}$-graded
$S$-module. For a multidegree
$\mathbf{d}=(d_{1},\dots,d_{r})\in\operatorname{\mathbb{Z}}^{r}$, write
$\bar{\mathbf{d}}$ for the total degree $d_{1}+\cdots+d_{r}$ of $\mathbf{d}$
and $M_{\geq\mathbf{d}}$ for the truncation
$\oplus_{\mathbf{d}^{\prime}\geq\mathbf{d}}M_{\mathbf{d}^{\prime}}$ of $M$ at
$\mathbf{d}$, where $\mathbf{d}^{\prime}\geq\mathbf{d}$ if this inequality is
true for each coordinate.

######  Definition 1.2.

A homogeneous chain complex

$0\leftarrow G_{0}\leftarrow G_{1}\leftarrow\cdots\leftarrow G_{k}\leftarrow
0$

of free $S$-modules is _linear_ if $G_{0}\simeq\bigoplus S(-\mathbf{d})$ for
some $\mathbf{d}\in\operatorname{\mathbb{Z}}^{r}$ and for each free summand
$S(-\mathbf{d}^{\prime})$ of $G_{i}$ we have
$\bar{\mathbf{d}}^{\prime}=\bar{\mathbf{d}}+i$.

The function $\mathtt{isLinearComplex}$ checks this condition. To print the
degrees appearing in the complex use $\mathtt{supportOfTor}$.

        i4 : B = irrelevantIdeal S
        o4 = ideal (x   x   , x   x   , x   x   , x   x   , x   x   , x   x   )
                     0,1 1,2   0,0 1,2   0,1 1,1   0,0 1,1   0,1 1,0   0,0 1,0
        o4 : Ideal of S
        i5 : F = res comodule B
               1      6      9      5      1
        o5 = S  <-- S  <-- S  <-- S  <-- S  <-- 0

             0      1      2      3      4      5
        o5 : ChainComplex
        i6 : netList supportOfTor F

             +------+------+
        o6 = |{0, 0}|      |
             +------+------+
             |{1, 1}|      |
             +------+------+
             |{2, 1}|{1, 2}|
             +------+------+
             |{2, 2}|{1, 3}|
             +------+------+
             |{2, 3}|      |
             +------+------+
        i7 : isLinearComplex F
        o7 = false

######  Definition 1.3.

The _linear truncation region_ of $M$ is

$\\{\mathbf{d}\mid M_{\geq\mathbf{d}}\,\text{has a linear resolution with
generators in degree $\mathbf{d}$}\\}\subset\operatorname{\mathbb{Z}}^{r}.$

######  Remark 1.4.

Our definitions imply that the nonzero entries in the differential matrices of
a linear resolution will have total degree 1. We also require that the
generators have degree $\mathbf{d}$ so that a linear resolution for
$M_{\geq\mathbf{d}}$ implies the existence of a linear resolution for
$M_{\geq\mathbf{d}^{\prime}}$ whenever $\mathbf{d}^{\prime}\geq\mathbf{d}$. We
can thus describe the linear truncation region by giving its minimal elements.

##  2\. Finding linear truncations
$\mathtt{linearTruncations}$, $\mathtt{findMins}$, $\mathtt{findRegion}$

Eisenbud, Erman, and Schreyer proved in [EES15] that the linear truncation
region of $M$ is non-empty. In particular it contains the output of the
function $\mathtt{coarseMultigradedRegularity}$ from their package
_TateOnProducts_ [EES]. However, in general this degree is neither a minimal
element itself nor greater than all the minimal elements. (See Example 2.1.)

The function $\mathtt{linearTruncations}$ searches for multidegrees where the
truncation of $M$ has a linear resolution by calling the function
$\mathtt{findRegion}$, which implements Algorithm 1. Since we do not know of a
bound on the total degree of the minimal elements in the linear truncation
region given the Betti numbers of $M$, $\mathtt{linearTruncations}$ is not
guaranteed to produce all semigroup generators. By default it searches above
the componentwise minimum of the degrees of the generators of $M$ and below
the degree with all coordinates equal to $r+1$, where $r$ is the output of
$\mathtt{regularity}$. Otherwise the range is taken as a separate input.

######  Example 2.1.

Let $S=k[x_{0,0},x_{0,1},x_{0,2},x_{1,0},x_{1,1},x_{1,2},x_{1,3}]$ be the Cox
ring of $\mathbb{P}^{2}\times\mathbb{P}^{3}$. For each $d\geq 2$, let
$\phi_{d}\colon S(-d,-d)^{6}\rightarrow S(0,-d)^{2}\oplus S(-d,0)^{4}$ be
given by

$\begin{pmatrix}x_{0,0}^{d}&x_{0,1}^{d}&x_{0,2}^{d}&0&0&0\\\
0&0&0&x_{0,1}^{d}&x_{0,0}^{d}&x_{0,2}^{d}\\\
x_{1,0}^{d}&0&0&x_{1,0}^{d}&0&0\\\ 0&x_{1,1}^{d}&0&0&x_{1,1}^{d}&0\\\
0&0&x_{1,2}^{d}&0&0&x_{1,2}^{d}\\\ 0&0&0&x_{1,3}^{d}&0&0\end{pmatrix},$

and define $M^{(d)}:=\operatorname{coker}\phi_{d}$. The
$\mathtt{coarseMultigradedRegularity}$ of $M^{(3)}$ is $\\{3,3\\}$, the
$\mathtt{regularity}$ of $M^{(3)}$ is 5, and $\\{3,3\\}$ and $\\{8,2\\}$ are
minimal elements of the linear truncation region. Since $\\{8,2\\}$ is not
below $\\{5+1,5+1\\}$ it will not be returned by the
$\mathtt{linearTruncations}$ function with the default options.

Based on the computations from $M^{(d)}$ for $2\leq d\leq 10$ we expect that
for $d\geq 2$ the module $M^{(d)}$ will have
$\mathtt{coarseMultigradedRegularity}$ equal to $\\{d,d\\}$, with $\\{d,d\\}$
and $\\{3d-1,d-1\\}$ both minimal elements of the linear truncation region.

Input : a module $M$, a Boolean function $f$, and a range
$(\mathbf{a},\mathbf{b})$

Output : minimal elements between $\mathbf{a}$ and $\mathbf{b}$ where $M$
satisfies $f$

$A:=\emptyset$;

$K:=\\{\mathbf{a}\\}$;

while  _$K\neq\emptyset$_ do

$\mathbf{d}:=$ first element of $K$;

$K=K\setminus\\{\mathbf{d}\\}$;

if  _$\mathbf{d}\notin A+\operatorname{\mathbb{N}}^{r}$_ then

if  _$M$ satisfies $f$ at $\mathbf{d}$_ then

$A=A\cup\\{\mathbf{d}\\}$;

else

for  _$1\leq i\leq r$_ do

if  _$\mathbf{d}+\mathbf{e}_{i}\leq\mathbf{b}$_ then

$K=K\cup\\{\mathbf{d}+\mathbf{e}_{i}\\}$;

end if

end for

end if

end if

end while

return _$A$_

Algorithm 1 $\mathtt{findRegion}$

At each step of Algorithm 1 the set $A$ contains degrees satisfying $f$ and
the set $K$ contains the minimal degrees remaining to be checked. There are
options to initialize $A$ and $K$ differently—degrees in $A$ will be assumed
to satisfy $f$, and degrees below those in $K$ will be excluded from the
search (and thus assumed not to satisfy $f$). Supplying such prior knowledge
can decrease the length of the computation by limiting the number of times the
algorithm calls $f$.

The pseudocode in Algorithm 1 masks the fact that $A$ and $K$ are stored as
monomial ideals in a temporary standard graded polynomial ring. Similarly, the
function $\mathtt{findMins}$ will convert a list of multidegrees to a monomial
ideal in order to calculate its minimal elements via a Gröbner basis.

##  3\. Relation to regularity
$\mathtt{partialRegularities}$, $\mathtt{linearTruncationsBound}$,
$\mathtt{regularityBound}$

As discussed above, the minimal element of the linear truncation region of a
singly graded module agrees with its Castelnuovo–Mumford regularity, which can
be determined from its Betti numbers. In the multigraded case these concepts
are still linked, but their relationship is more complicated. For instance,
the following inclusion is strict:

######  Theorem 3.1.

If $H^{0}_{B}(M)=0$, then the linear truncation region of $M$ is a subset of
the multigraded regularity region $\operatorname{reg}M$ of M, as defined in
[MS04].

###### Proof.

See [BES20, Theorem 2.9] or [EES15, Proposition 4.11]. ∎

Unfortunately the multigraded Betti numbers of $M$ do not determine either its
regularity or its linear truncations. However, the functions
$\mathtt{regularityBound}$ and $\mathtt{linearTruncationsBound}$ compute
subsets of these regions using only the twists appearing in the minimal free
resolution of $M$. In many examples they produce the same outputs as
$\mathtt{multigradedRegularity}$ (from the package _VirtualResolutions_
[ABLS20]) and $\mathtt{linearTruncations}$, respectively, without computing
sheaf cohomology or truncating the module.

The algorithms for $\mathtt{linearTruncationsBound}$ and
$\mathtt{regularityBound}$ are based on the following theorem (from [BCHS21]):

######  Theorem 3.2.

If $H^{0}_{B}(M)=0$ and $H^{1}_{B}(M)=0$ then

$\bigcap_{\operatorname{Tor}_{i}(M,k)_{\mathbf{b}}\neq
0}\bigcup_{\sum\lambda_{j}=i}\left[\mathbf{b}-\lambda_{1}\mathbf{e}_{i}-\cdots-\lambda_{r}\mathbf{e}_{r}+\mathbb{N}^{r}\right]$

is a subset of the $\mathtt{linearTruncations}$ of $M$, and

$\bigcap_{\operatorname{Tor}_{i}(M,k)_{\mathbf{b}}\neq
0}\bigcup_{\sum\lambda_{j}=i-1}\left[\mathbf{b}-\mathbf{1}-\lambda_{1}\mathbf{e}_{i}-\cdots-\lambda_{r}\mathbf{e}_{r}+\mathbb{N}^{r}\right]$

is a subset of the $\mathtt{multigradedRegularity}$ of $M$.

The function $\mathtt{partialRegularities}$ calculates the Castelnuovo–Mumford
regularity in each component of a multigrading.

######  Remark 3.3.

In the bigraded case, Theorem 3.2 implies that $\mathbf{d}$ is in
$\mathtt{linearTruncations\ M}$ if $\mathbf{d}\geq\mathtt{partialRegularities\
M}$ and $\bar{\mathbf{d}}\geq\mathtt{regularity\ M}$.

For some modules $\mathtt{linearTruncationsBound}$ gives a proper subset of
the linear truncations:

        i8 : S = multigradedPolynomialRing 2;
        i9 : M = coker(map(S^{{-1,0},{0,-1},{0,-1}},S^{{-1,-1},{-1,-1}},
        {{x_(1,0),x_(1,1)},{-x_(0,0),0},{0,-x_(0,1)}}));
        i10 : multigraded betti res M
                    0   1
        o10 = 1: a+2b   .
              2:    . 2ab
        i11 : linearTruncations M
        o11 = {{0, 2}, {1, 1}}
        i12 : linearTruncationsBound M
        o12 = {{1, 1}}

##  4\. Acknowledgments

The authors would like to thank David Eisenbud for his comments. The second
author would like to thank the organizers of the Macaulay2 workshop in Leipzig
(June 2018), during which the algorithms were first implemented.

## References

  * [ABLS20] Ayah Almousa, Juliette Bruce, Michael Loper, and Mahrud Sayrafi.  The virtual resolutions package for Macaulay2.  J. Softw. Algebra Geom., 10(1):51–60, 2020.
  * [BES20] Christine Berkesch, Daniel Erman, and Gregory G. Smith.  Virtual resolutions for a product of projective spaces.  Algebr. Geom., 7(4):460–481, 2020.
  * [BC17] Nicolás Botbol and Marc Chardin.  Castelnuovo Mumford regularity with respect to multigraded ideals.  J. Algebra, 474:361–392, 2017.
  * [BCHS21] Juliette Bruce, Lauren Cranton Heller, and Mahrud Sayrafi.  Characterizing multigraded regularity on products of projective spaces.  Preprint, 2021.
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