# Entanglement detection in postquench nonequilibrium states:
thermal Gibbs vs. generalized Gibbs ensemble

Ferenc Iglói  0000-0001-6600-7713<EMAIL_ADDRESS>Wigner Research
Centre for Physics, Institute for Solid State Physics and Optics, H-1525
Budapest, Hungary  Institute of Theoretical Physics, University of Szeged,
H-6720 Szeged, Hungary  Csaba Zoltán Király<EMAIL_ADDRESS>Institute of Theoretical Physics, University of Szeged, H-6720 Szeged, Hungary

###### Abstract

We use entanglement witnesses related to the entanglement negativity of the
state to detect entanglement in the $XY$ chain in the postquench states in the
thermodynamic limit after a quench when the parameters of the Hamiltonian are
changed suddenly. The entanglement negativity is related to correlations,
which in the postquench stationary state are described by a generalized Gibbs
ensemble, in the ideal case. If, however, integrability breaking perturbations
are present, the system is expected to thermalize. Here we compare the
nearest-neighbor entanglement in the two circumstances.

##  I Introduction

Entanglement is a fundamental ingredient of quantum theory and has a central
role in quantum information theory[1, 2, 3]. For pure states this is directly
related to correlations, while for mixed states entanglement has a more
complex meaning. A quantum state is entangled, if its density matrix cannot be
written as a mixture of product states. Deciding whether a state is entangled
or not is a difficult problem. However, there are conditions that are
necessary and sufficient for small systems., e.g. for $2\times 2$ (two-qubit)
and $2\times 3$ bipartite systems [4, 5] and for multi-mode Gaussian states
[6]. There are also conditions that are sufficient conditions for entanglement
for larger systems, but does not detect all entangled states.

Considering experiments usually only limited information about the quantum
state is available and this is true for theoretical calculations for very
large systems. Only those approaches for entanglement detection can be applied
that require the measurement of a few observables. There are entanglement
conditions that are linear in operator expectation values, these are the
entanglement witnesses.They are operators that have a positive expectation
value for all separable states. Thus, a negative expectation value signals the
presence of entanglement. The theory of entanglement witnesses has recently
been rapidly developing [7, 8, 9, 10]. It is also known how to optimize a
witness operator in order to detect the most entangled states [9].

Apart from determining optimal entanglement witnesses, it is also important to
find witnesses that are easy to measure in an experiment or possible to
evaluate in a theoretical calculation. From both point of views, witnesses
based on spin chain Hamiltonians attracted considerable attention [11, 12, 13,
14, 15, 16, 17]. The energy-based witnesses have been used in various physical
systems [18, 19, 20, 21, 22, 23]. However, it has been shown that the optimal
witness for the thermal state of the chain is not necessarily the Hamiltonian
[17]. Therefore it is recommended to consider another approach, based on a
family of witnesses that detect entanglement whenever the entanglement
negativity of the nearest-neighbor two-spin density matrix is nonzero [24],
i.e., when the state violates the entanglement criterion based on the
positivity of the partial transpose (PPT) [4, 5].

For mixed states one generally considers thermal states and calculates a
temperature bound, below which the state is entangled[11, 12, 13, 14, 15, 16,
17]. Recently, however, the entanglement of other type of mixed states has
also been considered, which are postquench nonequilibrium states. These states
are obtained through such a protocol, when the quantum system is first placed
to its ground state, then a quench is performed, when the parameters of the
Hamiltonian change suddenly[25, 26, 27, 28, 29]. Since the state is not an
eigenstate of the new Hamiltonian, dynamics start. In the infinite time limit,
the system approaches a stationary state, which is some mixture of the states
appearing during the dynamics. If the Hamiltonian is non-integrable, the
system is expected to be thermalized and the stationary state is described by
a Gibbs ensemble with an effective temperature [30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40], see however Refs. [41, 42, 43, 44]. For integrable systems,
such as the transverse Ising chain, $XY$ and $XXY$ chains, the stationary
state is assumed to be described by a so-called Generalized Gibbs Ensemble
(GGE) [45, 46, 47, 48, 49, 50, 51, 52, 53], for which different effective
temperatures are assigned to each conserved quantities. This type of
description has been exactly calculated for the quantum Ising chain [54], and
a similar formalism is conjectured for the $XY$ chain [55].

In experiments one cannot realize such systems, which are purely integrable,
since weak integrable breaking perturbations are always unavoidable. In the
presence of a perturbation that breaks integrability, usual thermalization is
again expected to take place. However, if the perturbation is small, the
process may require a long time. On a finite time scale, the dynamics is
approximately described by the evolution under the integrable unperturbed
Hamiltonian. The system initially relaxes to a stationary state of the
unperturbed Hamiltonian, which is called prethermalization, while genuine
thermalization only occurs at later times [56, 57, 58]. This later
thermalization is typically involes a thermalization time
$\tau\sim\lambda^{-2}$, where $\lambda$ is the perturbation strength [59, 60].
However, for specific Hamiltonians this timescale can be much longer[61, 62].

In this paper, we aim to compare the entanglement properties of a
prethermalized and a genuine thermalized state. For this purpose we use the
$XY$ model, which is exactly solvable and consider some type of integrable
breaking perturbation. For the unperturbed model several entanglement based
properties have been studied recently [63, 64, 65, 66, 67] and also the
postquench nonequilibrium stationary state is analysed by energy-based and
negativity-based entanglement witnesses[68]. Here we let switch on an
integrable breaking perturbation and repeat the calculation.

Our paper is organized as follows. In Sec. II, we introduce the $XY$ model,
present its free-fermion representation, calculate thermal averages and
present its conjectured GGE after a global quench. In Sec. IV, the negativity-
based entanglement witness is described. In Sec. V, the bounds for postquench
states are calculated and the entangled areas are compared for prethermalized
and genuine thermalized states. In Secs. VI we close our paper with a
discussion.

##  II Model and methods

Here we consider the $XY$ spin-chain defined by the Hamiltonian

$\displaystyle\displaystyle{\cal H}_{XY}=$
$\displaystyle-\sum_{l=1}^{L}\left[\dfrac{1+\gamma}{2}\sigma^{x}_{l}\sigma^{x}_{l+1}+\dfrac{1-\gamma}{2}\sigma^{y}_{l}\sigma^{y}_{l+1}\right]$
$\displaystyle-h\sum_{l=1}^{L}\sigma^{z}_{l}+\lambda\displaystyle{\cal
V}(\\{\pmb{\sigma}\\})\;,$ (1)

in terms of the $\sigma_{l}^{x},$ $\sigma_{l}^{y},$ and $\sigma_{l}^{z}$ Pauli
spin operators acting on the spin at site $l,$ and
$\sigma_{L+1}^{\alpha}\equiv\sigma_{1}^{\alpha}$ for $\alpha=x,y,z.$ We
mention that the special case $\gamma=1$ represents the transverse Ising
model, and for $h=0$, $\gamma=0$ the Hamiltonian reduces to the $XX$ chain.

Later we extend the Hamiltonian with a general integrability breaking term:

$\displaystyle{\cal H}=\displaystyle{\cal H}_{XY}+\lambda\displaystyle{\cal
V}(\\{\pmb{\sigma}\\})\;,$ (2)

This term, $\displaystyle{\cal V}(\\{\pmb{\sigma}\\})$ can c.f. contain
interaction between more distant neighbours, but we do not specify its form,
the rǒle of this perturbation for $\lambda\ll 1$ is to ensure thermalization
of the model after a quench for sufficiently long time, $\sim\lambda^{-2}$.

###  II.1 Integrable model - $\lambda=0$

In detail we consider the integrable model with $\lambda=0$ and using standard
techniques [69, 70] it is expressed in terms of fermion creation and
annihilation operators $\eta^{{\dagger}}_{p}$ and $\eta_{p}$ as

${\cal
H}_{XY}=\sum_{p}\varepsilon\left(p\right)\left(\eta^{{\dagger}}_{p}\eta_{p}-\frac{1}{2}\right),$
(3)

where the sum runs over $L$ quasi-momenta, which are equidistant in
$[-\pi,\pi]$ for periodic boundary conditions . The energy of the modes is
given by [26, 25, 55]

$\varepsilon\left(p\right)=2\sqrt{\gamma^{2}\sin^{2}p+\left(h-\cos
p\right)^{2}}$ (4)

and the Bogoliubov angle $\varTheta_{p}$ diagonalizing the Hamiltonian is
given by

$\tan\varTheta_{p}=-\gamma\sin p/\left(h-\cos p\right).$ (5)

The ground state is the fermionic vacuum, its energy being

$E_{0}=-\sum_{p}\frac{\varepsilon\left(p\right)}{2}\;.$ (6)

The model has a so called disorder line at

$h_{d}(\gamma)=\sqrt{1-\gamma^{2}}.$ (7)

along which the ground state of the model is a product state, i.e. not
entangled. For $h<h_{d}(\gamma)$ ($h>h_{d}(\gamma)$) the long-range two-point
correlation functions have an (don’t have) oscillatory behavior, while at
$h=h_{d}(\gamma)$ they are constant [25, 26].

At finite temperature, $T>0$ the average value of the energy is given by

$\langle{\cal
H}\rangle_{T}=-\sum_{p}t(p,T)\frac{\varepsilon\left(p\right)}{2},$ (8)

with

$t(p,T)=\tanh\left(\frac{\varepsilon\left(p\right)}{2T}\right).$ (9)

(Here and in the following we set $k_{B}=1$.)

In the thermodynamic limit, $L\to\infty$, the two-point correlation functions
are calculated in Refs. [26, 25] and the nearest-neighbor correlations are
given by:

$\displaystyle\langle\sigma_{l}^{x}\sigma_{l+1}^{x}\rangle_{T}$
$\displaystyle=g_{c}-g_{s},$
$\displaystyle\langle\sigma_{l}^{y}\sigma_{l+1}^{y}\rangle_{T}$
$\displaystyle=g_{c}+g_{s},$
$\displaystyle\langle\sigma_{l}^{z}\sigma_{l+1}^{z}\rangle_{T}$
$\displaystyle=g_{0}^{2}-g_{c}^{2}+g_{s}^{2},$ (10)

with

$\displaystyle g_{c}$ $\displaystyle=\frac{1}{\pi}\int_{-\pi}^{\pi}{\rm
d}p\cos p(\cos p-h)~{}t(p,T)\varepsilon^{-1}\left(p\right),$ $\displaystyle
g_{s}$ $\displaystyle=-\gamma\frac{1}{\pi}\int_{-\pi}^{\pi}{\rm
d}p\sin^{2}p~{}t(p,T)\varepsilon^{-1}\left(p\right),$ $\displaystyle g_{0}$
$\displaystyle=\frac{1}{\pi}\int_{-\pi}^{\pi}{\rm d}p(h-\cos
p)~{}t(p,T)\varepsilon^{-1}\left(p\right).$ (11)

##  III Nonequilibrium stationary states after a quench

We consider global quenches at zero temperature, which suddenly change the
parameters of the Hamiltonian from $\gamma_{0}$, $h_{0}$ for $t<0$ to
$\gamma$, $h$ for $t>0$, keeping however the value of. For $t<0$ the system is
assumed to be in equilibrium, i.e., in the ground state
$\left|\Phi_{0}\right\rangle$ of the Hamiltonian ${\cal H}_{0}$ with
parameters $\gamma_{0}$ and $h_{0}.$ After the quench, for $t>0$, the state
evolves coherently according to the new Hamiltonian ${\cal H}$ as

$\left|\Phi_{0}(t)\right\rangle=\exp(-i{\cal H}t)\left|\Phi_{0}\right\rangle.$
(12)

Correspondingly, the time evolution of an operator in the Heisenberg picture
is

$\sigma_{l}\left(t\right)=e^{i{\cal H}t}\sigma_{l}e^{-i{\cal H}t}.$ (13)

After large enough time and in the thermodynamic limit the system is expected
to reach a stationary state,

${\bm{\rho}}_{q}=\lim_{\tau\rightarrow\infty}\frac{1}{\tau}\int_{0}^{\tau}e^{-i{\cal
H}t}|\Phi_{0}\rangle\langle\Phi_{0}|e^{+i{\cal H}t}{\rm d}t,$ (14)

So that, for an observable ${\cal O}$ the stationary value is given by

$\langle{\cal O}\rangle_{\rm st}={\rm Tr}({\bm{\rho}}_{q}{\cal O})\;.$ (15)

###  III.1 Stationary values in the integrable model

In the integrable model with $\lambda=0$ the energy of the system after the
quench is given as

$\langle\Phi_{0}|\displaystyle{\cal
H}|\Phi_{0}\rangle=\sum_{p}\varepsilon\left(p\right)\left(\langle\Phi_{0}|\eta^{{\dagger}}_{p}\eta_{p}|\Phi_{0}\rangle-\frac{1}{2}\right),$
(16)

where the occupation probability of mode $p$ in the initial state
$\left|\Phi_{0}\right\rangle$ is given as

$f_{p}=\left\langle\Phi_{0}\right|\eta^{{\dagger}}_{p}\eta_{p}\left|\Phi_{0}\right\rangle.$
(17)

For the $XY$ model it is expressed through the difference
$\Delta_{p}=\varTheta^{0}_{p}-\varTheta_{p}$ of the Bogoliubov angles as

$f_{p}=\tfrac{1}{2}\left(1-\cos\Delta_{p}\right)$ (18)

with the cosine of the difference $\Delta_{p}$ given as

cosΔ_p=4(cosp -h0)(cosp -h)+γγ0sin2pε(p)ε0(p),

(19)

where the index $0$ refers to quantities before the quench [55]. In the
thermodynamic limit, Eq. (16) can be rewritten as

$\frac{\langle\Phi_{0}|\displaystyle{\cal
H}|\Phi_{0}\rangle}{L}=-\frac{1}{4\pi}\int_{-\pi}^{\pi}\varepsilon(p)\cos\Delta_{p}{\rm
d}p.$ (20)

The fermions with occupation probability $f_{p}$ are quasiparticles, which are
created homogeneously in space and the corresponding wave-packets move
ballistically with constant velocity. Such a wave packet is well described by
a sharp kink excitation, if the quench is performed deep into the ordered
phase. For quenches close to the critical point the kinks are not sharply
localized and the domain walls have a finite extent of the order of the
equilibrium correlation length. In the thermodynamic limit this effect can be
taken into account by using an effective occupation probability:

$f_{p}\to{\tilde{f}}_{p}=-\frac{1}{2}\ln|\cos\Delta_{p}|\;,$ (21)

so that in leading order ${\tilde{f}}_{p}=f_{p}+{\cal O}(f_{p}^{2})$.

In the stationary state, due to conserved symmetries, averages of correlations
are described by a Generalized Gibbs Ensemble (GGE)[45, 46, 47, 48, 50, 51,
52, 53]. In this case to each fermionic mode an effective temperature, $T_{\rm
eff}(p)$ is attributed through the relation [55]

$\tanh\left(\frac{\varepsilon\left(p\right)}{2T_{\rm
eff}(p)}\right)=e^{-2{\tilde{f}}_{p}}=|2f_{p}-1|=|\cos\Delta_{p}|.$ (22)

In this way the nearest-neighbor correlations in the stationary state can be
obtained as in section II.1, just replacing $t(p,T)$ defined in Eq. (9) by
$|\cos\Delta_{p}|$

$t(p,T)\to|\cos\Delta_{p}|.$ (23)

In particular, we have to apply Eq. (23) for the correlation functions in Eqs.
(10) and (11).

###  III.2 Stationary values in the non-integrable model

In the non-integrable model with $0<\lambda\ll 1$ on finite time scale
prethermalization takes place and the quasi-stationary state is described by a
GGE, as explained in the previous subsection III.1. After sufficiently long
time, however, the system is expected to be thermalized and the genuine
stationary state is expected to be described by a Gibbs ensemble. Possible
ways to define a characteristic thermalization temperature, $T_{th}$, have
been discussed in several papers[32, 71, 72, 73, 74, 75]. After the quench at
the prethermalization state the different fermionic modes in the system are
characterised by a set of effective temperatures in Eq.(22) and at later times
in the thermalised stationary state some average of these effective
temperatures is expected. For small $\lambda$ the average value of the energy
remains the same, from which the following condition for the thermalization
temperature, $T_{th}$ follows:

$\int_{-\pi}^{\pi}\varepsilon(p)\tanh\left(\frac{\varepsilon\left(p\right)}{2T_{\rm
eff}(p)}\right){\rm
d}p=\int_{-\pi}^{\pi}\varepsilon(p)\tanh\left(\frac{\varepsilon\left(p\right)}{2T_{th}}\right){\rm
d}p\;.$ (24)

##  IV Negativity-based entanglement witness

Generally an operator ${\cal W}$ is called an entanglement witness, if its
expectation value, $\langle{\cal W}\rangle$ satisfies the following
requirements [76, 77].

(i) $\langle{\cal W}\rangle\geq 0$ for all separable states,

(ii) $\langle{\cal W}\rangle<0$ for some entangled state.

Such a state is detected by the witness as entangled. Entanglement witnesses
have been used in various physical systems to verify the presence of
entanglement [78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88]. We mention that a
single entanglement witness cannot detect all entangled states.

Here we consider one of the most important entanglement witness, which is
connected to the partial transpose of the density matrix[4, 5] and to the
entanglement negativity[24]. For a bipartite density matrix given as

${\bm{\rho}}=\sum_{kl,mn}{\bm{\rho}}_{kl,mn}\ket{k}\bra{l}\otimes\ket{m}\bra{n}$
(25)

the partial transpose according to first subsystem is defined by exchanging
subscripts $k$ and $l$ as

${\bm{\rho}}^{T_{A}}=\sum_{kl,mn}{\bm{\rho}}_{lk,mn}\ket{k}\bra{l}\otimes\ket{m}\bra{n}.$
(26)

It has been shown that for separable quantum states [4, 7]

${\bm{\rho}}^{T_{A}}\geq 0$ (27)

holds. Thus, if ${\bm{\rho}}^{T_{A}}$ has a negative eigenvalue then the
quantum state is entangled. For $2\times 2$ and $2\times 3$ systems, the PPT
condition detects all entangled states [7]. For systems of size $3\times 3$
and larger, there are PPT entangled states [5, 89]. The entanglement
negativity [24] is defined as

${\cal N}(\bm{\rho})=2{\rm max}(0,-{\rm min}(\mu_{\nu})),$ (28)

where $\mu_{\nu}$ are the eigenvalues of the partial transpose
$\bm{\rho}^{T_{A}}.$

Let us turn to $XY$ chains and consider the nearest-neighbor reduced density
matrix, $\bm{\rho}$, which is defined in the $\sigma^{z}$ basis. As described
in details in Ref.[17, 68] due to symmetries of the problem $\bm{\rho}$ is a
direct sum of two $2\times 2$ matrices and the same property holds also for
the partial tranpose ${\bm{\rho}}^{T_{A}}$. The minimal eigenvalues of the
$2\times 2$ submatrices can be calculated by second order quadrature in terms
of the matrix-elements of $\bm{\rho}$. The later for the XY and Heisenberg
spin chains can be expressed through nearest-neighbor correlations [64]. The
final results for the minimal eigenvalues are given by[17, 68]:

$\displaystyle\mu_{\rm
min}^{(1)}=\frac{\langle\sigma_{l}^{z}\sigma_{l+1}^{z}\rangle+1}{4}$
$\displaystyle-\frac{1}{4}\sqrt{(\langle\sigma_{l}^{z}\rangle+\langle\sigma_{l+1}^{z}\rangle)^{2}+(\langle\sigma_{l}^{x}\sigma_{l+1}^{x}\rangle+\langle\sigma_{l}^{y}\sigma_{l+1}^{y}\rangle)^{2}}\;,$
(29)

and

$\mu_{\rm
min}^{(2)}=-\frac{1}{4}(\langle\sigma_{l}^{x}\sigma_{l+1}^{x}\rangle-\langle\sigma_{l}^{y}\sigma_{l+1}^{y}\rangle+\langle\sigma_{l}^{z}\sigma_{l+1}^{z}\rangle-1).$
(30)

The entanglement witness related to $\mu_{\rm min}^{(2)}$ is given by:

${\cal
W}_{N}=-\frac{1}{4}(\sigma_{l}^{x}\sigma_{l+1}^{x}-\sigma_{l}^{y}\sigma_{l+1}^{y}+\sigma_{l}^{z}\sigma_{l+1}^{z}-\openone),$
(31)

whereas the same for $\mu_{\rm min}^{(2)}$ is more complicated and derived
in[68].

In order to proceed, we need to know that the partial transposition of a two-
qubit state has at most one negative eigenvalue and all the eigenvalues lie in
$[-1/2,1]$[90, 91]. Thus, only one of the $\mu_{\rm min}^{(1)}$ and $\mu_{\rm
min}^{(2)}$ can be negative, and when they are equal to each other, they must
be non-negative and the state must be separable.

##  V Entanglement in nonequilibrium postquench states

In this section, we consider global quenches in the system, as described in
Sec. III and study the entanglement properties of nonequilibrium stationary
states, which are obtained in the large-time limit after the quench. First we
consider the prethermalized state, which is obtained by setting formally
$\lambda=0$ in Eq.(2) and having the integrable model. In this part to
calculate averages we use the GGE protocol and assign different effective
temperatures to each fermionic modes, as described in Eqs. (22) and (23).
Afterwards we let to switch on a small perturbation, $\lambda\ll 1$, and
consider the genuine thermalized state in which there is a unique
thermalization temperature as defined in Eq.(24).

We have calculated of postquench states detected as entangled by the
entanglement negativity-based witness using the two minimal eigenvalues,
$\mu_{\rm min}^{(1)}$ and $\mu_{\rm min}^{(2)}$ in Eqs(29) and (30),
respectively. These are presented in Fig. 1 having the entangled regions where
$\mu_{\rm min}^{(2)}<0$, and in Fig. 2 where in the entangled regions
$\mu_{\rm min}^{(1)}<0$. Entanglement detected areas in the prethermalized
states are coloured by violet and in the genuine thermalized states by green.
Overlapping areas are striped and contain both colours.

|
---|---
(a) |  (b)
|
(c) |  (d)
|
(e) |  (f)
Figure 1: Postquench states after a sudden quench protocol
$(h_{0},\gamma)\to(h,\gamma)$ in the $XY$ chain. Entanglement is detected in
the postquench state by the negativity-based method using $\mu^{(1)}_{\rm
min}$ in Eq. (29): in the prethermalized state (violet) and in the genuine
thermalized state (green). Overlapping areas are striped.

In Fig. 1(a), we consider the case with $\gamma=1$ and hence we have a quantum
Ising chain. In this case, the negativity-based witness with $\mu^{(2)}_{\rm
min}$ is applicable in the whole phase diagram, consequently in Fig. 2 (a)
there is no entangled region detected. In Fig. 1(b), (c), (d),(e) and (f), we
have $\gamma<1,$ and the condition $\mu_{\rm min}^{(1)}<\mu_{\rm min}^{(2)}$
is fulfilled in a part of the phase diagram. Thus, entangled postquench states
are also detected based on $\mu^{(1)}_{\rm min}$ and there are entangled
regions in Fig. 2(b), (c), (d),(e) and (f).

Let us now consider first Fig. 1 and compare the detected entangled regions of
the prethermalised and the genuine thermalised states. Here most of the
entangled regions are overlapping, but at the surfaces there are extra regions
of the genuine thermalized states. These regions are quite considerable for
$\gamma$ close to $1$, see the results in Fig. 1 (a) close to $h_{0}=0$. But
even in this case there is an extra prethermalized region close to the
critical point, $h_{0}=1$. We note that the detected entangled region shows
non analytical behaviour, both for the prethermalized and the genuine
thermalized states. By reducing the value of $\gamma$ the extra thermalized
region shrinks, and in the case $\gamma\ll 1$ there is even an extra entangled
prethermalized region detected by the negativity-based witness with
$\mu^{(2)}_{\rm min}$, see in Fig. 1 (f). If we look at Fig. 2, the trend is
rather the opposite. For larger values of $\gamma$ there are extra regions
belonging to the prethermalized state. These extra regions start to shrink by
decreasing $\gamma$ and for very small values of $\gamma$ the thermalized
states have larger entangled regions. It is interesting to compare the
entangled regions in the limit $\gamma\to 0^{+}$. In this limit in the
prethermalized state just a part of the area is detected entangled, while in
the thermalized state the complete area is entangled.

|
---|---
(a) |  (b)
|
(c) |  (d)
|
(e) |  (f)
Figure 2: The same as in Fig. 1 but using the negativity-based method with
$\mu^{(2)}_{\rm min}$ in Eq. (30).

##  VI Discussion

Entanglement in mixed quantum states is a difficult problem, in particular
when the degrees of freedom is large and we approach the thermodynamic limit.
The possible systems of investigation are usually quantum spin chains, which
could have experimental realizations in condensed matter systems [92] or they
could be engineered artificially through ultracold atomic gases in an optical
lattice. Recently, this latter type of technique is very well developed and
different intriguing questions could be studied experimentally [93, 94, 95,
96, 97, 98, 99, 100, 101].

In this paper we consider the $XY$ chain, which is integrable through free-
fermionic techniques and several exact results are available, mainly in the
ground state but there are some known results even at finite temperature [25,
26]. We consider the entanglement properties of mixed states of the $XY$
chain, which are non-equilibrium stationary states after a quantum quench
protocol. To detect entanglement we use a family of entanglement witnesses
that detect states with a nonzero bipartite entanglement negativity. In
practice this witness can detect all states that have nearest-neighbor
entanglement.

The mixed states we consider are due to a quench, when parameters of the
Hamiltonian of the system are changed abruptly and the time evolution of the
system is governed by the new Hamiltonian. After a sufficiently long time the
system will approach a nonequilibrium stationary state, which is a mixed
quantum state. For integrable systems, such as the $XY$ chain, the postquench
state is described by a so-called Generalized Gibbs Ensemble, while for
general, non-integrable systems it is expected to be a thermal state, which is
described by an appropriate Gibbs ensemble. In expeiments one can not realize
such systems, which are purely integrable, since weak integrable breaking
perturbations are always present. If this perturbation is small the relaxation
takes part in two steps. The system initially relaxes to a stationary state of
the unperturbed Hamiltonian, which is the prethermalized state, while for
later times genuine thermalization takes part. In the present paper we studied
the entanglement properties of the two non-equilibrium stationary states, in
particular we want to clarify the difference between the areas detected to be
entangled by the entanglement negativity witness.

The entanglement negativity in Eq.(28) for the $XY$-chain can be non-zero due
to one of the two eigenvalues of the partial transpose, which are defined in
Eqs.(29) and (30). Therefore the areas which are detected entangled are
indicated separately in Figs. 1 and 2 for the two cases. We observed, that the
areas corresponding to the prethermalized and the the genuine thermalized
states are mainly overlapping, however there are extra regions at the
boundaries of the overlapping areas. For the negativity witness in Eq.(29),
corresponding to Fig. 1 generally the areas to be detected entangled increase
during the thermalization process. Even in this case there are opposite
tendencies close to the critical point of the initial state. Considering the
other witness in Eq.(30) and the corresponding Fig. 2 here in the
prethermalized state are larger entangled areas for larger values of $\gamma$,
which trend however reverse for small values of $\gamma$. We mention that it
would be interesting to check the entanglement properties of postquench states
of other (Bethe-Ansatz) integrable models.

While we studied the nearest-neighbor entanglement of the postquench state,
other properties uncovering hidden criticality of the initial system not
detectable by local quantities have recently been considered [102]. The method
has been based on efficient lower bounds on the negativity in $XY$ chains
[103, 104]. We have shown that the criticality of the initial state can still
be seen in the boundaries of the regions with nearest-neighbor entanglement.

###### Acknowledgements.

We thank G. Tóth for previous cooperation in the project and for discussions.
This work was supported by the Hungarian Scientific Research Fund under Grants
No. K128989 and No. K146736 and by the National Research, Development and
Innovation Office of Hungary (NKFIH) within the Quantum Information National
Laboratory of Hungary.

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