Capacity of entanglement for scalar fields in squeezed states

M. Reza Mohammadi Mozaffar

Department of Physics, University of Guilan, P.O. Box 41335-1914, Rasht, Iran

E-mail<EMAIL_ADDRESS>
We study various aspects of capacity of entanglement in the squeezed states of
a scalar field theory. This quantity is a quantum informational counterpart of
heat capacity and characterizes the width of the eigenvalue spectrum of the
reduced density matrix. In particular, we carefully examine the dependence of
capacity of entanglement and its universal terms on the squeezing parameter in
the specific regimes of the parameter space. Remarkably, we find that the
capacity of entanglement obeys a volume law in the large squeezing limit. We
discuss how these results are consistent with the behavior of other
entanglement measures including entanglement and Renyi entropies. We also
comment on the existence of consistent holographic duals for a family of
Gaussian states with generic squeezing parameter based on the ratio of
entanglement entropy and the capacity of entanglement.

###### Contents

  1. 1 Introduction
  2. 2 Set-up
  3. 3 Preliminaries: $C_{E}$ for ground state
    1. 3.1 Two coupled harmonic oscillators
    2. 3.2 A lattice of oscillators and continuum limit
  4. 4 $C_{E}$ for harmonic systems in a squeezed state
    1. 4.1 Two coupled harmonic oscillators
    2. 4.2 A chain of oscillators
  5. 5 $C_{E}$ for a massless scalar in squeezed states
    1. 5.1 Squeezing a single mode
    2. 5.2 Squeezing all modes
  6. 6 Conclusions and discussions

##  1 Introduction

In recent years, surprising new connections have been developing between
quantum information theory, quantum many-body systems and quantum gravity. In
particular, understanding the entanglement structure of quantum systems in a
pure or mixed state has become an active area of research, e.g., see [1, 2, 3,
4, 5, 6] for reviews. Also in the context of gauge/gravity correspondence,
fascinating connections have been developing between quantum information and
quantum gravity including the holographic entanglement entropy proposals and
the notion of geometry from entanglement, e.g., see [7] and references
therein. Moreover, in order to quantify entanglement and quantum correlations
several measures has been studied so far including entanglement and Renyi
entropies. Indeed, the entanglement entropy is the unique measure which
assesses the amount of quantum entanglement between two subsystems for a given
pure state $|\psi\rangle$. In this case, assuming that $\mathcal{H}_{\rm
tot.}=\mathcal{H}_{A}\otimes\mathcal{H}_{\bar{A}}$, entanglement entropy can
be written in terms of von Nuemann entropy as

$\displaystyle S_{E}=-{\rm Tr}_{A}\left(\rho_{A}\log\rho_{A}\right),$ (1.1)

where $\rho_{A}$ is the reduced density matrix defined as $\rho_{A}={\rm
Tr}_{\bar{A}}\left(|\psi\rangle\langle\psi|\right)$. Further, the Renyi
entropy is a one-parameter generalization of entanglement entropy which is
given by

$\displaystyle S_{n}=\frac{1}{1-n}\log{\rm Tr}\rho_{A}^{n},$ (1.2)

where $n$ is a positive integer. It is then easy to show that entanglement
entropy follows by analytic continuation of the Renyi entropy down to $n=1$,
i.e., $S_{E}=\lim_{n\rightarrow 1}S_{n}$. Further, other interesting cases to
consider are the large and small $n$ limits, which yield

$\displaystyle
S_{\infty}\equiv\lim_{n\rightarrow\infty}S_{n}=-\log\lambda_{\rm max},\hskip
42.67912ptS_{0}\equiv\lim_{n\rightarrow 0}S_{n}=\log\mathcal{N},$ (1.3)

where $\lambda_{\rm max}$ is the largest eigenvalue of the reduced density
matrix and $\mathcal{N}$ denotes the number of nonvanishing eigenvalues of
$\rho_{A}$. Besides the already mentioned case of the entanglement and Renyi
entropies, there are many attempts to construct new information theoretic
measures for studying the entanglement structure in more general setting,
e.g., mutual information [8], logarithmic negativity [9] and entanglement of
purification [10]. However, in this paper, we focus on another measure that
has recently entered this discussion which is called the capacity of
entanglement [11]

$\displaystyle C_{E}\equiv\lim_{n\rightarrow 1}C_{n}=\lim_{n\rightarrow
1}n^{2}\frac{\partial^{2}}{\partial n^{2}}\left((1-n)S_{n}\right),$ (1.4)

where $C_{n}$ is the $n$-th capacity of entanglement. Indeed, considering $n$
as the inverse temperature, the above definition is similar to the
corresponding relation between the heat capacity and thermal entropy. Further,
using the definition of the modular Hamiltonian, i.e., $H_{A}=-\log\rho_{A}$,
it is straightforward to show that the entanglement entropy and capacity of
entanglement are the expectation value and variance (the second cumulant) of
$H_{A}$ respectively

$\displaystyle S_{E}={\rm Tr}\;\rho_{A}H_{A}=\langle H_{A}\rangle,$
$\displaystyle C_{E}={\rm Tr}\;\rho_{A}H_{A}^{2}-\left({\rm
Tr}\;\rho_{A}H_{A}\right)^{2}=\langle H_{A}^{2}\rangle-\langle
H_{A}\rangle^{2}.$ (1.5)

Thus the capacity of entanglement characterizes the width of the eigenvalue
spectrum of the reduced density matrix such that for a maximally entangled
state, i.e., $\rho_{A}\propto I$, it vanishes. Indeed, in this case the Renyi
entropies are independent of $n$ and we have a flat entanglement spectra.
Recently, there have been many attempts to investigate various properties of
this quantity in different setups both in the field theory and holography
which have led to a remarkably rich and varied range of new insights, e.g.,
[12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. In particular, an
interesting observation in [14] was that systems where are all entanglement is
carried by EPR pairs have zero capacity of entanglement. Thus, whenever we
find that $C_{E}\sim S_{E}$, e.g., $(1+1)$-dimensional CFTs at equilibrium,
EPR pairs are not a very good approximation of the quantum state and randomly
entangled pairs of qubits give a better description. Moreover, as proposed in
this reference, restricting to field theories with holographic duals without
higher derivative terms, the ratio $\frac{C_{E}}{S_{E}}$ turns out to be
exactly equal to one. An interesting question is that whether this result also
hold for more general QFTs without a holographic dual.

Clearly, the capacity of entanglement depends on the choice of the original
state of the system. The main aim of this article is to further investigate
the state dependence of the capacity of entanglement. We study this dependence
by considering a general class of Gaussian states which are solutions to the
time dependent Schrödinger equation, the so-called squeezed states. As we will
see the squeezed states are less classical than the vacuum state (which is a
special case of a coherent state) and thus study the entanglement structure in
a squeezed state may help us to gain a better insight into the quantum
features of the system in question. In the following we will focus on a scalar
field theory in $(1+1)$-dimensions to address this problem. Indeed, in order
to avoid the ultraviolet divergences in the continuum limit, we should
regulate the theory by placing it on a one dimensional spatial lattice. Thus
the model reduces to an infinite chain of quantum harmonic oscillators. Let us
add that the entanglement entropy for this setup in squeezed states has been
previously studied in [25]. These authors proposed an extension of the method
introduced in [26, 27] to more general cases including squeezed states.
Interestingly, using a systematic method they show that for a free scalar
field theory in $(1+1)$-dimensions the entanglement entropy obeys the volume
law rather than the area law in large squeezing limit. This is in contrast
with the behavior of entanglement entropy in typical ground states which
exhibits area law scaling where in $(1+1)$-dimensions is replaced with a
logarithmic scaling. Further, their result is in agreement with the Page’s
argument where for a typical pure quantum state of a joint system, the smaller
subsystem is almost maximally mixed, showing little sign that the total system
is pure [28]. Interestingly, the Page curve followed by the entropy of Hawking
radiation [29]. In this context, an especially interesting question concerns
how the scaling of the capacity of entanglement alters under a non-trivial
squeezing when the total system is in a random squeezed state.

Additionally, in [30, 31], using the holographic proposals, it was shown that
the capacity of entanglement is a useful probe of the Hawking radiation during
the black hole evaporation process. Interestingly enough, the authors argued
that this measure shows a discontinuity or a peak at the Page time while the
entanglement entropy varies smoothly. In this sense, one can consider $C_{E}$
as a convenient probe which can characterizes the order of a phase of the
quantum systems. Indeed a primary motivation for this work came from these
efforts to study the behavior of different entanglement measures including the
capacity of entanglement in random pure states and to compare our results to
those obtained in the context of holography. Let us mention that this
comparison involves two very different models, that is a free scalar theory
with a single degree of freedom versus a holographic quantum field theory
which is strongly coupled and has a large number of degrees of freedom.

The remainder of our paper is organized as follows: In section 2, we give the
general framework in which we are working, establishing our notation and the
general form of the squeezed states and harmonic models in question. Section 3
contains a summary about the behaviors of entanglement measures in ground
state of a scalar theory. We review old results for the entanglement entropy
and also find new ones for the case of capacity of entanglement. To get a
better understanding of the results, we will also compare the behavior of
different measures in specific scaling regimes. In section 4, we extend our
studies to the case of squeezed states. Specifically, we present a combination
of numerical and analytic results on the scaling of different entanglement
measures in the large squeezing limit. Next, we return to the field theory
problem by generalizing these results to the continuum limit in section 5.
Given the entanglement measures for the scalar field theory, we then ask how
our results compare to holographic ones. Finally, we close in section 6 with a
brief discussion of our results and directions for future work.

##  2 Set-up

We start by considering a simple quantum harmonic oscillator with the
following Hamiltonian111For simplicity of notation in what follows, we will
use hat symbol for the quantum operators only when there is a danger of
confusion. Also we will set $\hbar=1$.

$H=\frac{p^{2}}{2m}+\frac{1}{2}m\omega^{2}x^{2}.$ (2.1)

Defining the annihilation and creation operators

$a=\frac{1}{\sqrt{2m\omega}}(m\omega x+ip),\hskip
56.9055pta^{\dagger}=\frac{1}{\sqrt{2m\omega}}(m\omega x-ip),$ (2.2)

we have $H=\omega(a^{\dagger}a+\frac{1}{2})$. In this case a class of Gaussian
solutions to the time dependent Schrödinger equation which consists of the so-
called coherent states are given by

$\psi(x,t)=\left(\frac{m\omega}{\pi}\right)^{1/4}\exp\left(-\frac{m\omega}{2}(x-x_{0}(t))^{2}+ip_{0}(t)(x-x_{0}(t))-i\phi_{c}(t)\right),$
(2.3)

where

$x_{0}(t)=X_{0}{\rm Re}\left(e^{i\omega(t-t_{0})}\right),\hskip
34.14322ptp_{0}(t)=m\dot{x}_{0}(t),\hskip
34.14322pt\phi_{c}(t)=\frac{-x_{0}(t)p_{0}(t)}{2}+\frac{\omega(t-t_{0})}{2}+\phi_{0}.$
(2.4)

These states minimize the uncertainty relation with uncertainty equally
distributed between position and momentum such that

$\Delta x=\frac{\Delta p}{m\omega}=\frac{1}{\sqrt{2m\omega}},\hskip
56.9055pt\Delta x\,\Delta p=\frac{1}{2}.$ (2.5)

Indeed, they have the same minimal uncertainty value as found for the vacuum
state. Further, the mean values of position and momentum operators in coherent
state follow a classical orbit, e.g.,
$\langle\ddot{x}\rangle+\omega^{2}\langle x\rangle=0$, and hence these states
correspond to the most classical states of the harmonic oscillator. It is also
easy to show that any eigenstate of the annihilation operator, i.e.,
$a|\alpha\rangle=\alpha|\alpha\rangle$, where $\alpha$ is a complex parameter,
represents a coherent state. Clearly, these states include the vacuum state as
a special case with $\alpha=0$.

Let us now turn to another class of Gaussian solutions to the time dependent
Schrödinger equation which are more general, the so-called squeezed states.
The corresponding wave function reads

$\psi(x,t)=\left(\frac{m{\real}(w(t))}{\pi}\right)^{1/4}\exp\left(-\frac{mw(t)}{2}(x-x_{0}(t))^{2}+ip_{0}(t)(x-x_{0}(t))-i\phi_{s}(t)\right),$
(2.6)

where222We set the origin of time coordinate to be 0.

$\displaystyle w(t)=\omega\frac{1-i\sinh z\;\cos 2\omega t}{\cosh z+\sinh
z\;\sin 2\omega t},\hskip 28.45274pt\phi_{s}(t)=\phi_{c}(t)-\frac{\omega
t}{2}+\frac{1}{2}\tan^{-1}\frac{\tanh\frac{z}{2}+\tan\omega
t}{1+\tanh\frac{z}{2}\tan\omega t}.$ (2.7)

In the above expressions $z$ is the squeezing parameter such that for $z=0$,
we recover the coherent states. The key feature of a squeezed state in the
harmonic potential is that although its profile is still Gaussian, its width
is different from the vacuum state. Further, the corresponding uncertainty
relation is easily found to be

$\Delta x\,\Delta p=\frac{1}{2}\sqrt{1+\sinh^{2}z\;\cos^{2}2\omega t},$ (2.8)

which shows that the squeezed states are not minimal uncertainty states at all
times. Indeed, the time slices where the above relation becomes minimal are
easily found to be $t_{\rm min}=(2k+1)\frac{T}{8}$ where
$T=\frac{2\pi}{\omega}$ and $k$ is an integer. It is straightforward to show
that when the squeezed state is a minimal uncertainty state, we have

$(\Delta x)_{\rm min}=\frac{1}{\sqrt{2m\omega e^{\pm z}}},\hskip
42.67912pt(\Delta p)_{\rm min}=\sqrt{\frac{m\omega e^{\pm z}}{2}},$ (2.9)

which shows the possibility of arbitrary compression of the position
uncertainty at the expense of appropriate fluctuation in the momentum variable
and vice-versa.

In the next sections much of our discussion will focus on the different
aspects of entanglement measures when the corresponding state is given by eq.
(2.6) to examine how the squeezing parameter can affect different quantities.
To do so, we consider a free massive scalar field in $(1+1)$-dimensions with
Hamiltonian

$\displaystyle H=\frac{1}{2}\int
dx\left(\pi^{2}(x)+\left(\partial_{x}\phi(x)\right)^{2}+m^{2}\phi^{2}(x)\right).$
(2.10)

In order to circumvent the divergences so as to obtain finite results for the
measures, we should regulate the above model by placing it on a lattice, which
reduces the system to an infinite chain of coupled harmonic oscillators.
Indeed, the general Hamiltonian for a system of $N$ coupled (one-dimensional)
harmonic oscillators can be written as

$\displaystyle
H=\frac{1}{2}\sum_{i=1}^{N}\textbf{p}^{T}.\textbf{p}+\frac{1}{2}\sum_{i,j=1}^{N}\textbf{x}^{T}.K.\textbf{x},$
(2.11)

where

$\displaystyle\textbf{x}^{T}=\left(\begin{matrix}x_{1},\cdots,x_{N}\end{matrix}\right),\hskip
56.9055pt\textbf{p}^{T}=\left(\begin{matrix}p_{1},\cdots,p_{N}\end{matrix}\right),$
(2.12)

and $K$ is a real symmetric matrix. Now, this suggests that we begin with an
even simpler warm-up problem, namely, the case of two coupled harmonic
oscillators

$\displaystyle
H=\frac{p_{1}^{2}}{2}+\frac{p_{2}^{2}}{2}+\frac{k_{0}}{2}(x_{1}^{2}+x_{2}^{2})+\frac{k_{1}}{2}(x_{1}-x_{2})^{2},$
(2.13)

where $k_{1}$ determines the strength of the coupling between the two
oscillators such that weak and strong coupling regimes correspond to $k_{1}\ll
k_{0}$ and $k_{1}\gg k_{0}$ limits respectively. We see that this simple set-
up maintains some interesting features of our original problem which helps us
to better investigate the continuum scalar theory. Let us recall that related
investigations attempting to better understand the entanglement structure in
general Gaussian states have also appeared in [32, 33, 34].

##  3 Preliminaries: $C_{E}$ for ground state

As a first step towards understanding different entanglement measures
including capacity of entanglement in squeezed states, we would like to study
the same quantities in a simpler set-up where the state is vacuum. We begin
with the case of two coupled harmonic oscillators and then, having built up
some intuition, we return to the more involved problem by generalizing these
results to a lattice of coupled oscillators.

###  3.1 Two coupled harmonic oscillators

In this case the corresponding Hamiltonian is given by eq. (2.13). Indeed, the
entanglement entropy in this model was studied in [26, 27]. Here we would like
to apply the techniques developed in these references to examine the behavior
of Renyi entropy and capacity of entanglement. To do so, one simply rewrites
the Hamiltonian in terms of the canonical coordinates,

$\displaystyle
H=\frac{1}{2}\left(p_{+}^{2}+\omega_{+}^{2}x_{+}^{2}+p_{-}^{2}+\omega_{-}^{2}x_{-}^{2}\right),$
(3.1)

where

$\displaystyle x_{\pm}=\frac{x_{1}\pm x_{2}}{\sqrt{2}},\hskip
28.45274pt\omega_{+}^{2}=k_{0},\hskip
28.45274pt\omega_{-}^{2}=\omega_{+}^{2}+2k_{1}.$ (3.2)

Now we have two decoupled harmonic oscillators, and thus the vacuum state can
be written as the product of the vacuum state wave functions for the two
individual oscillators as follows

$\displaystyle\psi_{0}(x_{+},x_{-})=\left(\frac{\omega_{+}\omega_{-}}{\pi^{2}}\right)^{1/4}\exp\left(-\frac{\omega_{+}x_{+}^{2}+\omega_{-}x_{-}^{2}}{2}\right).$
(3.3)

The above expression can be rewritten in terms of the physical coordinates of
the two oscillators and then the corresponding reduced density matrix can be
found by integrating out the $x_{1}$ coordinate, i.e.,
$\rho_{2}(x_{2};x^{\prime}_{2})=\int
dx_{1}\psi_{0}(x_{1},x_{2})\psi_{0}^{*}(x_{1},x^{\prime}_{2})$, which yields

$\displaystyle\rho_{2}(x_{2};x^{\prime}_{2})=\left(\frac{\gamma-\beta}{\pi}\right)^{1/2}\exp\left(-\frac{\gamma}{2}\left(x_{2}^{2}+{x^{\prime}}_{2}^{2}\right)+\beta{x}_{2}{x^{\prime}}_{2}\right),$
(3.4)

where

$\displaystyle\beta=\frac{\left(\omega_{+}-\omega_{-}\right)^{2}}{4\left(\omega_{+}+\omega_{-}\right)},\hskip
56.9055pt\gamma=\frac{\left(\omega_{+}+\omega_{-}\right)^{2}+4\omega_{+}\omega_{-}}{4\left(\omega_{+}+\omega_{-}\right)}.$
(3.5)

Moreover, solving the eigenvalue problem for the reduced density matrix, i.e.,

$\displaystyle\int dy\;\rho_{2}(x;y)f_{k}(y)=p_{k}f_{k}(x),$ (3.6)

one finds an infinite tower of eigenvalues as follows

$\displaystyle p_{k}=(1-\xi)\xi^{k},\hskip
56.9055pt\xi=\frac{\beta}{\gamma+\alpha}=\left(\frac{\sqrt{\omega_{+}}-\sqrt{\omega_{-}}}{\sqrt{\omega_{+}}+\sqrt{\omega_{-}}}\right)^{2},\hskip
56.9055ptk=0,\cdots,\infty,$ (3.7)

where $\alpha=\sqrt{\gamma^{2}-\beta^{2}}$. Note that based on the above
result the weak and strong coupling regimes correspond to $\xi\rightarrow 0$
and $\xi\rightarrow 1$ limits respectively. Now using eq. (1.2), we can
evaluate the Renyi entropy which yields

$\displaystyle
S_{n}=\frac{1}{1-n}\log\sum_{k=0}^{\infty}p_{k}^{n}=\frac{1}{1-n}\left(n\log(1-\xi)-\log(1-\xi^{n})\right).$
(3.8)

Combining the above result with eqs. (1.1) and (1.4), we obtain entanglement
entropy and capacity of entanglement

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\frac{\xi}{\xi-1}\log\xi-\log(1-\xi),$ $\displaystyle C_{E}$
$\displaystyle=$ $\displaystyle\xi\left(\frac{\log\xi}{1-\xi}\right)^{2}.$
(3.9)

Further the $n$-th capacity of entanglement is obtained by the replacement
$\xi\rightarrow\xi^{n}$ in $C_{E}$. Remarkably, in the weak coupling regime a
perturbative expansion yields

$\displaystyle S_{E}=-\xi\log\xi+\xi+\cdots,\hskip
56.9055ptC_{E}=\xi\left(\log\xi\right)^{2}+2\xi^{2}\left(\log\xi\right)^{2}+\cdots,$
(3.10)

which shows that in this limit both quantities vanish. This is consistent with
the idea that as $k_{1}/k_{0}$ decreases, the reduced density matrix becomes
more and more separable. On the other hand, in the strong coupling limit we
obtain

$\displaystyle S_{E}=-\log(1-\xi)+1+\cdots\hskip
56.9055ptC_{E}=1-\frac{(\xi-1)^{2}}{12}+\cdots.$ (3.11)

Interestingly, we see that in this limit $C_{E}/S_{E}\ll 1$ which is
consistent with the idea that the reduced density matrix becomes more and more
maximally mixed as one increases the coupling.

In figures 1 and 2 we summarize the numerical results for different
entanglement measures. Figure 1 presents various quantities as functions of
the coupling between the two harmonic oscillators. The left panel presents the
dependence of entanglement entropy and capacity of entanglement on $\xi$. We
note a number of key features: First, both these measures start at the same
value (which is equal to zero corresponds to a separable state) and then
increase as one increases the coupling. Second, although the entanglement
entropy diverges in the strong coupling limit, the capacity of entanglement
saturates to unity in agreement with eq. (3.11). Interestingly, for a specific
value of the coupling these measures coincide, i.e.,
$C_{E}(\xi_{*})=S_{E}(\xi_{*})$ where $\xi_{*}\sim 0.31$. The middle panel
demonstrates the Renyi entropy as a function of the coupling for several
values of $n$. Although $S_{n}$ is a decreasing function of the Renyi index,
it increases with the coupling as expected. In the right panel we show the
$n$-th capacity of entanglement for the same values of the parameters. We see
that $C_{n}$ decreases with $n$ and also saturates from below to unity.

Figure 1: $S_{E}$ and $C_{E}$ (left), $S_{n}$ (middle) and $C_{n}$ (right) as
functions of the coupling between the two harmonic oscillators.

We present the $n$-dependence of $S_{n}$ and $C_{n}$ for several values of the
coupling in figure 2. Based on these plots, we see that the qualitative
dependence of these measures on the coupling is similar to $n=1$ case. Also
both measures decrease as we increase the Renyi index. Finally, as this simple
example illustrates, the entanglement entropy and its fluctuations,
characterized by the capacity of entanglement have no reason to be equal,
except for a specific coupling. This result is different from what happens for
QFTs with a holographic gravity dual (without higher derivative terms) where
the ratio $C_{E}/S_{E}$ turns out to be exactly equal to one [14].

Figure 2: $S_{n}$ (left) and $C_{n}$ (right) as functions of the Renyi index
for several values of the coupling between the two harmonic oscillators.

###  3.2 A lattice of oscillators and continuum limit

In this section we generalize our studies to a lattice of $N$ coupled harmonic
oscillators, where the corresponding Hamiltonian is given by eq. (2.11), in
specific directions. Similar to the previous case, the ground state wave
function is described by a factorized Gaussian in the corresponding normal
mode space. To see this we recall that as shown in [27] the matrix $K$ can be
diagonalize by a real orthogonal similarity transformation, i.e.,
$K_{D}=UKU^{T}$. Then writing the Hamiltonian in terms of the normal modes
reduces the problem to a chain of decoupled quantum harmonic oscillators. Thus
the total vacuum state can be written as

$\displaystyle\psi_{0}(\tilde{\textbf{x}})=\prod_{i=1}^{N}\left(\frac{\omega_{i}}{\pi}\right)^{1/4}\exp\left(-\frac{\omega_{i}\tilde{x}_{i}^{2}}{2}\right),$
(3.12)

where $\tilde{x}_{i}$ and $\omega_{i}$ denote the normal coordinates and
eigenfrequencies respectively. Further, in order to find the reduced density
matrix we would like to express the above wave function in terms of the
original variables $x_{i}$ in the position basis. It is relatively simple to
show that in this case the above expression becomes

$\displaystyle\psi_{0}(\textbf{x})=\left(\frac{{\rm
det}\;\Omega}{\pi^{N}}\right)^{1/4}\exp\left(-\frac{\textbf{x}^{T}.\Omega.\textbf{x}}{2}\right),$
(3.13)

where $\Omega$ is the square root of $K$, i.e.,

$\displaystyle\Omega=U^{T}K_{D}^{1/2}U,\hskip
56.9055pt(K_{D})_{ij}=\omega_{i}\;\delta_{ij}.$ (3.14)

We would like to evaluate the reduced density matrix by tracing over the first
$\widetilde{N}$ oscillators. To do so, we consider the following decomposition

$\displaystyle\Omega=\left(\begin{matrix}A&B\\\ B^{T}&C\end{matrix}\right),$
(3.15)

where $A$ is an $\widetilde{N}\times\widetilde{N}$ matrix and $C$ is an
$(N-\widetilde{N})\times(N-\widetilde{N})$ matrix. Now it is straightforward
to explicitly evaluate the reduced density matrix as333Note that $x$ now has
$N-\widetilde{N}$ components, so we do not use the bold face notation which
refers to eq. (2.12).

$\displaystyle\rho_{\rm
red.}(x_{\widetilde{N}+1},\cdots,x_{N};x^{\prime}_{\widetilde{N}+1},\cdots,x^{\prime}_{N})=\mathcal{N}\exp\left(-\frac{x^{T}.\gamma.x+{x^{\prime}}^{T}.\gamma.x^{\prime}}{2}+x^{T}.\beta.x^{\prime}\right),$
(3.16)

where $\beta=\frac{B^{T}A^{-1}B}{2}$, $\gamma=C-\beta$ and $\mathcal{N}$ is a
normalization factor. Moreover, as shown in [27] the above expression can be
rewritten as follows

$\displaystyle\rho_{\rm
red.}(z_{\widetilde{N}+1},\cdots,z_{N};z^{\prime}_{\widetilde{N}+1},\cdots,z^{\prime}_{N})=\mathcal{N}\;\prod_{i=\widetilde{N}+1}^{N}\exp\left(-\frac{z_{i}^{2}+{z^{\prime}}_{i}^{2}}{2}+\tilde{\beta}_{i}z_{i}z^{\prime}_{i}\right),$
(3.17)

where

$\displaystyle\tilde{\beta}=\gamma_{D}^{-1/2}V\beta
V^{T}\gamma_{D}^{-1/2},\hskip 42.67912ptz=W^{T}\gamma_{D}^{1/2}Vx.$ (3.18)

Also $V$ and $W$ defined as the corresponding similarity transformations that
diagonalize $\gamma$ and $\tilde{\beta}$ respectively, i.e.,
$\gamma_{D}=V\gamma V^{T}$ and $\tilde{\beta}_{D}=W\tilde{\beta}W^{T}$.
Comparing eq. (3.17) to eq. (3.4) we see that the Renyi entropy can be
calculated by summing the contribution for each of the remaining modes as
follows

$\displaystyle S_{n}=\frac{1}{1-n}\sum_{j=1}^{j_{\rm
max}}\left(n\log(1-\xi_{j})-\log(1-\xi_{j}^{n})\right),$ (3.19)

where $\xi_{j}=\frac{\tilde{\beta}_{j}}{1+(1-\tilde{\beta}_{j}^{2})^{1/2}}$
and $j_{\max}={\min}(\widetilde{N},N-\widetilde{N})$. Note that as long as we
are dealing with pure states, the spectrum of the reduced density matrices for
$A$ and $\bar{A}$ is the same and hence $S_{n}(A)=S_{n}(\bar{A})$. Further,
the corresponding expressions for entanglement entropy and capacity of
entanglement are similar to eq. (3.1) where again we should consider the
contribution for each of the modes, i.e.,

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\sum_{j}\left(\frac{\xi_{j}}{\xi_{j}-1}\log\xi_{j}-\log(1-\xi_{j})\right),$
(3.20) $\displaystyle C_{E}$ $\displaystyle=$
$\displaystyle\sum_{j}\xi_{j}\left(\frac{\log\xi_{j}}{1-\xi_{j}}\right)^{2}.$
(3.21)

We now wish to apply this approach to the problem of finding different
entanglement measures in the continuum limit. For example considering a free
scalar theory given by eq. (2.10) and replacing the space continuum with a
discrete mesh of lattice points the Hamiltonian can be transformed into a
discrete counterpart as follows

$\displaystyle
H=\frac{1}{2}\sum_{i=1}^{N}\left(\pi_{i}^{2}+\left(\phi_{i+1}-\phi_{i}\right)^{2}+m^{2}\phi_{i}^{2}\right),$
(3.22)

where without loss of generality we set the lattice spacing equal to unity,
i.e., $\epsilon=1$. We see that the above expression and eq. (2.11) will be in
complete agreement if we choose

$\displaystyle
K_{ij}=(2+m^{2})\delta_{i,j}-\left(\delta_{i+1,j}+\delta_{i,j+1}\right).$
(3.23)

It is then possible to extract $\Omega$ using eq. (3.14) and to use eqs.
(3.19) and (3.20) to determine the behavior of the entanglement measures.
Generally, it is not possible to find the quantities analytically and thus in
the following we will employ a numerical treatment.

The corresponding numerical results for different quantities are summarized in
figures 3 and 4. Note that we will mainly consider $N=60$, because the
interesting qualitative features of the entanglement measures are independent
of the total system size. Also this choice facilitates a comparison to the
analogous results for entanglement entropy in [27]. In the left panel of
figure 3, the entanglement entropy is plotted as a function of $\widetilde{N}$
for several values of the mass parameter. Further, we show the behavior of
capacity of entanglement for the same values of the parameters in the right
panel of this figure. We can see that both measures have qualitatively similar
behavior and decrease with $m$. Also in all cases $C_{E}$ is slightly larger
than $S_{E}$. Notice that as long as we are dealing with pure states, the
entanglement measures are symmetric, e.g., $S_{E}(A)=S_{E}(\bar{A})$ where
$\bar{A}$ is the complement of $A$.

Figure 3: $S_{E}$ (left) and $C_{E}$ (right) as functions of $\widetilde{N}$
for several values of the mass parameter.

In the left panel of figure 4, we present the behavior of the Renyi entropy as
a function of $\widetilde{N}$ for several values of $n$ in the massless
regime. From this plot, one can infer that the qualitative features of the
Renyi entropy are similar to $S_{E}$. Again, we see that the Renyi entropy is
a decreasing function of $n$ such that the rate of change of $S_{n}$ is a
monotonically decreasing function of the Renyi index and saturates from above
to a constant in the large $n$ limit. Let us add that we found similar results
for $C_{n}$, although we do not explicitly show the corresponding figures
here. In order to investigate how these quantities approach the large $n$
limit, in the right panel of figure 4, we plot the maximum values of $S_{n}$
and $C_{n}$ (which occurs at $\widetilde{N}=\frac{N}{2}$) as functions of the
Renyi index. The dashed line indicates the asymptotic value of the Renyi
entropy given by eq. (1.3) where the largest eigenvalue of the reduced density
matrix can be evaluated numerically to be $\lambda_{\rm max}\sim 0.83$.
Finally, the asymptotic behavior of $n$-th capacity is consistent with the
large $n$ expansion, i.e.,
$C_{\infty}\sim\lim_{n\rightarrow\infty}n^{2}\lambda_{\rm max}^{-n}\sim 0$.

Figure 4: Left: Renyi entropy as a function of $\widetilde{N}$ for several
values of $n$. Right: Maximum values of entanglement entropy and capacity of
entanglement as a functions of the Renyi index. The dashed line indicates the
asymptotic value of the Renyi entropy given by eq. (1.3). In both plots we
consider the massless regime.

##  4 $C_{E}$ for harmonic systems in a squeezed state

In this section, we wish to return to our original problem which is computing
the capacity of entanglement of a chain of coupled oscillators in squeezed
states. Again, as a warm up problem, we first consider the case of a single
pair of coupled harmonic oscillators and then generalize the problem to more
involved case consisting of $N$ degrees of freedom in the next sections.

###  4.1 Two coupled harmonic oscillators

Let us begin with the case of two oscillators as in section 3.1, but now the
overall system lying in a squeezed state given by eq. (2.6). Again, we can
write the total wave function as the product of the wave functions for the two
individual oscillators as follows444Based on the results of [25] in the
following we will set $\phi_{s+}=\phi_{s-}=0$ without loss of generality.

$\displaystyle\psi(x,t)$ $\displaystyle=$
$\displaystyle\left(\frac{{\real}(w_{+}){\real}(w_{-})}{\pi^{2}}\right)^{\frac{1}{4}}\times$
(4.1)
$\displaystyle\exp\left(-\frac{w_{+}(x_{+}-x_{0+})^{2}+w_{-}(x_{-}-x_{0-})^{2}}{2}+ip_{0+}(x_{+}-x_{0+})+ip_{0-}(x_{-}-x_{0-})\right),$

where $x_{\pm}$ were defined in eq. (3.2) and we have also introduced the new
variables $x_{0\pm}=\frac{x_{01}\pm x_{02}}{\sqrt{2}}$ and
$p_{0\pm}=\frac{p_{01}\pm p_{02}}{\sqrt{2}}$. The corresponding reduced
density matrix can be found by rewriting the wave function in terms of the
original coordinates and integrating out the $x_{1}$ variable. The calculation
follows straightforwardly from the considerations of the previous section. We
have then

$\displaystyle\rho_{2}(x_{2};x^{\prime}_{2})=\left(\frac{{\rm
Re}(\gamma)-\beta}{\pi}\right)^{1/2}\exp\left(-\frac{\gamma
y_{2}^{2}+\gamma^{*}{y_{2}^{\prime}}^{2}}{2}+\beta{y}_{2}y^{\prime}_{2}+ip_{02}(y_{2}-y_{2}^{\prime})\right),$
(4.2)

where we have defined $y_{2}=x_{2}-x_{02}$,
$y_{2}^{\prime}=x_{2}^{\prime}-x^{\prime}_{02}$ and

$\displaystyle\beta=\frac{|w_{+}-w_{-}|^{2}}{4{\rm
Re}\left(w_{+}+w_{-}\right)},\hskip
56.9055pt\gamma=\frac{|w_{+}+w_{-}|^{2}+4w_{+}w_{-}}{4{\rm
Re}\left(w_{+}+w_{-}\right)}.$ (4.3)

Interestingly, although $\gamma$ is complex, the eigenvalues of the reduced
density matrix depend only on the real part of this parameter [25]. Indeed, as
shown in this reference the calculation proceeds essentially as the ground
state case and the resulting spectrum is given by

$\displaystyle p_{k}=(1-\xi)\xi^{k},$ (4.4)

where $\xi=\frac{\beta}{{\rm Re}(\gamma)+\alpha}$ and $\alpha=\sqrt{{\rm
Re}(\gamma)^{2}-\beta^{2}}$. Again we can evaluate different entanglement
measures using eqs. (3.8) and (3.1). It is worthwhile to mention that based on
eq. (2.7) the spectrum of the reduced density matrix depends on time and thus
we expect that the measures are also time-dependent.

With these tools in hand, let us examine the dependence of the measures on the
squeezed parameter in more detail. The numerical results for different
quantities are summarized in figures 5 and 6.

Figure 5: Left: Evolution of entanglement entropy and capacity of
entanglement. Right: $S_{n}$ and $C_{n}$ as a function of time for $n=2$. In
both panels we set $\omega_{-}=2\omega_{+}=2$, $z_{+}=0.5$ and $z_{-}=0$. Also
the dashed lines are the corresponding results for the ground state where both
squeezing parameter vanish.

Figure 5 shows the time evolution of entanglement measures when only the
symmetric mode is squeezed. From these plots, one can infer that different
measures exhibit qualitatively similar behaviors. Clearly, in this case the
evolution is periodic with period
$\tau_{+}\equiv\frac{T_{+}}{2}=\frac{\pi}{\omega_{+}}$. We see that $C_{E}$
reaches it maximum values at times $t_{\rm max}=(4k+1)\frac{\tau_{+}}{4}$
where $k$ is an integer. Interestingly, the capacity of entanglement is always
greater than the entanglement entropy such that the difference of these
quantities becomes maximal when they reach their maximum values. Moreover, the
right panel shows that for larger values of $n$ this difference becomes less
pronounced.

Figure 6: Evolution of entanglement entropy (left) and capacity of
entanglement (right) for several values of $z_{+}$ for
$\omega_{-}=2\omega_{+}=2$ and $z_{-}=0$.

Now we proceed further by examining in more detail the $z$ dependence of the
entanglement measures, as shown in figure 6. The key observation to note here
is that while both $S_{E}$ and $C_{E}$ are monotonically increasing functions
of the squeezing parameter, only the entanglement entropy goes on to grow
indefinitely as we increase $z$. Indeed, the right panel illustrates that
$C_{E}$ saturates to unity in the large $z$ limit. To gain some insights into
this behavior, let us turn our attention to the computation of the mean
quantities which is defined as
$\overline{\mathcal{M}}\equiv\frac{1}{T}\int_{0}^{T}\mathcal{M}(t)dt$. Before
examining the full $z$ dependence of the mean capacity of entanglement, we
would like to study its asymptotic behaviors in small and large $z$ limit. In
[25] it was shown that in these limits the expansions for the mean
entanglement entropy become

$\displaystyle\overline{S_{E}}=\Bigg{\\{}\begin{array}[]{lcr}S_{E0}-\frac{1}{16}\left(1+\frac{1+4\xi_{0}+\xi_{0}^{2}}{1-\xi_{0}^{2}}\ln\xi_{0}\right)z_{+}+\cdots,&\,\,\,z_{+}\ll
1,\\\ \frac{z_{+}}{2}+1-3\ln
2+\ln\left(\sqrt{\frac{\omega_{+}}{\omega_{-}}}+\sqrt{\frac{\omega_{-}}{\omega_{+}}}\right)+\mathcal{O}(e^{-z_{+}}),&\,\,\,z_{+}\gg
1,\end{array}$ (4.7)

where $S_{E0}$ and $\xi_{0}$ denote the corresponding vacuum values with
$z_{+}=0$. This shows that for a large squeezing parameter the mean
entanglement entropy grows linearly with $z$. A similar derivation holds in
the present case and it is also easy to find that the expression for the mean
capacity of entanglement reduces to

$\displaystyle\overline{C_{E}}=\Bigg{\\{}\begin{array}[]{lcr}C_{E0}+\frac{1}{8}\left(1+2\frac{1+3\xi_{0}+\xi_{0}^{2}}{1-\xi_{0}^{2}}\ln\xi_{0}+\frac{1}{2}\frac{1+6\xi_{0}+\xi_{0}^{2}}{(1-\xi_{0}^{2})^{2}}\left(\ln\xi_{0}\right)^{2}\right)z_{+}^{2}+\cdots,&\,\,\,z_{+}\ll
1,\\\ 1-\mathcal{O}(e^{-2z_{+}}),&\,\,\,z_{+}\gg 1,\end{array}$ (4.10)

which shows that in the large $z$ limit, the capacity of entanglement
saturates to unity.

The numerical results for the Renyi entropy and $n$-th capacity of
entanglement as functions of the squeezing parameter for different values of
the Renyi index with specific values of $\omega_{\pm}$ are summarized in
figure 7. Note that the case of $n=1$ corresponds to $\overline{S_{E}}$ and
$\overline{C_{E}}$. In this case our numerical results coincide with the
analytical expansions given by eqs. (4.7) and (4.10). Interestingly, we see
that in the large $z$ limit $\overline{C_{E}}/\overline{S_{E}}\ll 1$ which
shows that the reduced density matrix becomes more and more maximally mixed as
one increases the squeezing parameter. The left panel illustrates that all
curves for $\overline{S_{n}}$ have the same behavior in large $n$ limit. In
particular, one gets asymptotic linear growth with the same slope for
different values of $n$ and $z$. Moreover, the right panel also shows that the
asymptotic behavior of the mean value of the $n$-th capacity of entanglement
in large squeezing limit is almost independent of the Renyi index. Indeed, it
is straightforward to show that the corresponding scaling of
$\overline{S_{n}}$ and $\overline{C_{n}}$ with $z$ at leading order is the
same as eqs. (4.7) and (4.10) respectively.

Figure 7: Mean values of the Renyi entropy (left) and $n$-th capacity of
entanglement (right) as functions of the squeezing parameter for several
values of the Renyi index. The asymptotic behavior of $\overline{C_{n}}$ in
large $z$ limit is almost independent of $n$. Here we set
$\omega_{-}=2\omega_{+}=2$ and $z_{-}=0$.

To close this subsection, let us comment on extending this discussion to cases
where both modes are squeezed. Indeed, in this case the evolution of the
entanglement measures is in general not periodic, as the ratio of the
frequencies of the two modes may be irrational. Again we found similar results
for small and large squeezing parameter, although we do not explicitly show
the corresponding figures here.

###  4.2 A chain of oscillators

In this section we again consider the system of $N$ coupled harmonic
oscillators which was defined in eq. (2.11) to compute several entanglement
measures, but now for the squeezed states given by eq. (2.6). The main
question of interest is what are the additional technicalities involved in
computing different measures in squeezed states compared to the vacuum state.
In particular, we would like to investigate to what extent the squeezing
parameter modifies the behavior of capacity of entanglement. Here, it is worth
mentioning that, the influence of this parameter on the entanglement entropy
was studied in [25] and we will follow the discussion there closely. Indeed,
the spectrum of the reduced density matrix corresponding to a squeezed state
can be evaluated as detailed in this reference. These authors proposed three
equivalent scenarios to compute the eigenvalues of $\rho_{A}$. In what follows
we briefly review one of the approaches used in the reference which is fairly
simple. To do so, assume that all the corresponding normal modes lie in a
squeezed state whose wave function is given as follows

$\displaystyle\psi(\textbf{x})=\left(\frac{{\rm det}\;{\rm
Re}\;(\Omega)}{\pi^{N}}\right)^{1/4}\exp\left(-\frac{(\textbf{x}-\textbf{x}_{0}).\Omega.(\textbf{x}-\textbf{x}_{0})}{2}+i\textbf{p}_{0}.(\textbf{x}-\textbf{x}_{0})-i\sum_{j}\phi_{sj}\right),$
(4.11)

where $\Omega$ was defined in eq. (3.14), but now $\omega_{i}$’s are given by
the application of eq. (2.7) for each normal mode and hence it is a complex
symmetric matrix. It is straightforward to show that in this case the reduced
density matrix for the remaining $(N-\widetilde{N})$ oscillators becomes

$\displaystyle\rho_{\rm
red.}(x_{\widetilde{N}+1},\cdots;x^{\prime}_{\widetilde{N}+1},\cdots)=\left(\frac{{\rm
det}\,{\rm
Re}(\gamma-\beta)}{\pi^{N-\widetilde{N}}}\right)^{\frac{1}{2}}\exp\left(\frac{-y.\gamma.y-y^{\prime}.\gamma^{*}.y^{\prime}}{2}+y^{\prime}.\beta
y+ip_{0}.(y_{2}-y^{\prime}_{2})\right),$ (4.12)

where

$\displaystyle\beta=\frac{1}{2}B^{\dagger}{\rm Re}(A)^{-1}B,\hskip
42.67912pt\gamma=C-\frac{1}{2}B^{T}{\rm Re}(A)^{-1}B,$ (4.13)

and $y=x-x_{0}$. Note that we consider the same decomposition for $\Omega$ as
in eq. (3.15). Also note that here $\gamma$ is a real symmetric matrix and
$\beta$ is a Hermitian matrix. Indeed, this result is different from what
happens for the ground state where $\beta$ is a real and symmetric matrix.
Hence in this case it cannot be diagonalized via a real orthogonal
transformation. Fortunately, as shown in [25] for our purposes a general
solution is not required and in order to find the eigenvalues of eq. (4.12),
it is sufficient to compute the spectrum of a simpler matrix $\tilde{\Omega}$
defined by

$\displaystyle\tilde{\Omega}={\rm Re}(\Omega)^{-1}\left(\begin{matrix}-{\rm
Re}(A)&i{\rm Im}(B)\\\ -i{\rm Im}(B)^{T}&{\rm Re}(C)\end{matrix}\right).$
(4.14)

The relation between the two spectra is
$\xi_{j}=\frac{\tilde{\xi}_{j}-1}{\tilde{\xi}_{j}+1}$ where the normalization
condition for the density matrix allows us to neglect $\tilde{\xi}_{j}<1$.
Further, the eigenstates can be written in terms of the Hermite polynomials.
We skip over the details of the calculation and we refer the interested reader
to [25] for further details. Having the corresponding eigenvalues, the
entanglement measures can be found using the same formulae as in the previous
section, e.g., eq. (3.20). Now we are equipped with all we need to study the
behavior of the entanglement spectrum and thereby other related quantities in
a squeezed state. To do so, we employ a numerical treatment in the next
section.

Before we proceed further, we would like to study the asymptotic behaviors of
the entanglement measures in large squeezing limit. Indeed, this study plays
an important role in our analysis in the next section. Here for simplicity we
restrict our analysis to a specific case where all modes lie in a squeezed
state with the same squeezing parameter which is very large, i.e., $z\gg 1$.
Of course, a similar analysis has been previously done for the entanglement
entropy in [25]. Indeed, as shown in this reference in the large $z$ limit the
corresponding eigenvalues can be written as

$\displaystyle\xi_{j}=1-e^{-z}\xi^{(1)}_{j}+\cdots,$ (4.15)

where $\xi^{(1)}_{j}$ are some $z$ independent positive coefficients. Using
the above expression one finds the following expansion for the entanglement
entropy

$\displaystyle S_{E}={\rm
min}(\widetilde{N},N-\widetilde{N})(z+1)-\sum_{j=1}^{j_{\rm
max}}\log\xi^{(1)}_{j}-\mathcal{O}\left(e^{-z}\right).$ (4.16)

Thus in the large $z$ limit the entanglement entropy has a linear dependence
on the squeezing parameter. In a similar manner, one can show that

$\displaystyle S_{n}={\rm
min}(\widetilde{N},N-\widetilde{N})\left(z+\frac{\log
n}{n-1}\right)-\sum_{j=1}^{j_{\rm
max}}\log\xi^{(1)}_{j}+\mathcal{O}\left(e^{-z}\right),$ (4.17)

which shows that the Renyi entropy also has a linear behavior in this limit.
Further, the corresponding expression for the capacity of entanglement can be
determined in a similar way and the result is

$\displaystyle C_{E}={\rm
min}(\widetilde{N},N-\widetilde{N})-\mathcal{O}\left(e^{-2z}\right).$ (4.18)

Thus asymptotically, it approaches a constant value which is exactly the
number of degrees of freedom of the smaller subsystem. Indeed, based on the
above results we see that at leading order all the measures are time-
independent and proportional to the volume of the smaller subsystem. Again, we
see that in this limit $C_{E}\ll S_{E}$ which shows that the reduced density
matrix becomes more and more maximally mixed as one increases the squeezing
parameter. It is also easy to show that in this case the expansion for $C_{n}$
is the same as $C_{E}$. In the next section, we numerically evaluate the
entanglement measures for a free scalar theory in a squeezed state which
enables us to more explore the validity of these expressions.

##  5 $C_{E}$ for a massless scalar in squeezed states

In this section we proceed our previous analysis in a scalar theory, whose
Hamiltonian is given by eq. (3.22), to more investigate the behavior of the
capacity of entanglement in a squeezed state. First, we provide a simple
analysis where just a single mode is squeezed and examine the various regimes
in the evolution of entanglement measures. Next, we will extend this study to
a more general case where all the normal modes lie in a squeezed state which
enables us to directly extract some interesting features of the $z$ dependence
of the capacity of entanglement.

###  5.1 Squeezing a single mode

To begin, we consider the system lying in a state where only one normal mode
is squeezed and thus its wave function is given by eq. (2.6). Also the
remaining modes are put in their ground states. In the following we will use
$i_{s}$ to denote the index of the squeezed mode. Further, based on the
results of section 4.1 we expect that the evolution of the measures is
periodic with period $\tau_{s}\equiv\frac{T_{s}}{2}=\frac{\pi}{\omega_{s}}$
where $\omega_{s}$ is the normal frequency of the corresponding squeezed mode.
The examples depicted in figure 8 exhibit this behavior for the capacity of
entanglement. In the figure, $C_{E}$ as a function of $\widetilde{N}$ is
presented for several values of $i_{s}$ with $z=3$. The black markers
correspond to the vacuum state.

Figure 8: The capacity of entanglement as a function of $\widetilde{N}$ for
several time slices when only a single mode has been squeezed with $z=3$. The
black markers correspond to the vacuum state.

Based on this figure, it is evident that squeezing mostly increases $C_{E}$ in
comparison to that in the vacuum state. Nevertheless, we notice that for
$i_{s}=1$ and $t=0.75\,\tau_{s}$ the capacity of entanglement becomes smaller
than that of the vacuum state. It is expected that this behavior also happens
for other values of $z$ and $i_{s}$. Further, the oscillations of $C_{E}$ can
be traced back to the trigonometric functions appearing in the wave function
corresponding to the squeezed state. The larger $i_{s}$, the shorter the
period of oscillation. Also as this parameter increases, the amplitude of
oscillation becomes smaller. Moreover, one can see that in this regime the
oscillation seems more or less the same independent of $i_{s}$. Let us add
that similar behavior was found for the entanglement entropy in [25].

Figure 9: Mean capacity of entanglement as a function of $\widetilde{N}$
(left) and $i_{s}$ (right) for several values of the parameters when a single
mode has been squeezed with $z=3$.

To gain some insights into this behavior, let us turn our attention to the
computation of the mean entanglement measures in this setup. The corresponding
results for the mean capacity of entanglement are summarized in figure 9. The
left panel illustrates $\overline{C_{E}}$ as a function of $\widetilde{N}$ for
several values of the index of the squeezed mode. We see that the mean
capacity of entanglement is always larger than that in the vacuum state of the
system. Further, the right panel shows this quantity as a function of $i_{s}$
for several divisions of the system in two subsystems. We note a number of key
features: First, $\overline{C_{E}}$ is not monotonous with the index of the
squeezed mode. Indeed, for small values of $\widetilde{N}$ the mean capacity
of entanglement first increases approximately linearly with $i_{s}$ and then
after some fluctuations decreases to a constant value. Note that the
intermediate fluctuations become less pronounced in the large $\widetilde{N}$
limit. Second, for large values of $\widetilde{N}$ the mean capacity of
entanglement first increases very sharply with $i_{s}$ and then suddenly
saturates to a constant value.

To close this subsection, we examine the $z$ dependence of mean entanglement
measures in figures 10 and 11. We focus our analysis on the special case of
$i_{s}=1$ because the interesting qualitative features of the measures do not
depend strongly on which mode is squeezed. Note that we have also included the
corresponding results for the mean entanglement entropy, which was previously
reported in [25], to allow for a meaningful comparison between the different
measures. Based on figure 10, we observe that both $\overline{S_{E}}$ and
$\overline{C_{E}}$ are monotonically increasing functions of the squeezing
parameter, as expected. Further, the left panel illustrates that for large
squeezing, the entanglement entropy has a linear dependence on $z$.
Remarkably, from the right panel we can deduce that the mean capacity of
entanglement saturates to a constant value in this regime. Hence, we expect
that the corresponding reduced density matrix becomes more and more maximally
mixed as one increases the squeezing parameter. Moreover, the saturation value
is a monotonically increasing function of both $\widetilde{N}$ and $i_{s}$.

Figure 11 illustrates the behavior of mean values of the Renyi entropy and
$n$-th capacity of entanglement as functions of the squeezing parameter for
several values of the Renyi index. The left panel shows that
$\overline{S_{n}}$ exhibits a linear dependence on the squeezing parameter
with the same slope for different values of $n$. Further, in the right panel
we see that $\overline{C_{n}}$ with $n\geq 2$ saturate from below to unity
independent of the Renyi index. Indeed, for larger values of $n$, the mean
value of the $n$-th capacity of entanglement vanishes initially but then
rapidly rises to the final constant value corresponding to the large squeezing
regime.

Figure 10: Entanglement entropy (left) and capacity of entanglement (right) as
functions of $z$ for $i_{s}=1$ and several values of $\widetilde{N}$.

Figure 11: Mean values of the Renyi entropy (left) and $n$-th capacity of
entanglement (right) as functions of the squeezing parameter for several
values of the Renyi index with $i_{s}=1$ and $\widetilde{N}=10$.

###  5.2 Squeezing all modes

We turn now to a more realistic case where all modes lie in a squeezed state
with the same squeezing parameter. As in the previous section, unlike the case
where we had squeezed a single mode, the evolution of entanglement measures
are not necessarily periodic. Moreover, as we have shown in section 4.2, the
large $z$ behavior of the measures are completely independent of time. In
particular, in this regime the scaling of the capacity of entanglement is not
sensitive to the details of the time evolution of the reduced density matrix.

The corresponding numerical results are summarized in figures 12 and 13. The
capacity of entanglement for various random instants as function of
$\widetilde{N}$ when all modes lie in a squeezed state with the same squeezing
parameter is depicted in figure 12. Once again, the black markers correspond
to the vacuum state. Note that the qualitative behavior of the capacity of
entanglement is more or less the same independent of the random instant that
we have chosen. Clearly, $C_{E}$ is a monotonically increasing function of the
squeezing parameter. Further, as we increase the squeezing parameter the
scaling of the capacity of entanglement tends asymptotically to eq. (4.18) as
expected. Let us recall that according to this equation the leading term is
time-independent and thus the fluctuations of the capacity of entanglement
with time decrease as the squeezing parameter increases.

Figure 12: Capacity of entanglement for various random instants as functions
of $\widetilde{N}$ when all modes lie in a squeezed state with the same
squeezing parameter. The black markers correspond to the vacuum state.

To get a better understanding of the results, we compare the behavior of mean
entanglement entropy and mean capacity of entanglement in figure 13. The
markers in this figure show the numerical results which in the large $z$ limit
coincide with the leading term of the asymptotic expansions given by eqs.
(4.16) and (4.18) (represented by the continuous red line). Further, based on
our numerical results we see that at leading order $\overline{S_{E}}$ and
$\overline{C_{E}}$ are proportional to the volume of the smaller subsystem.
Indeed, for an infinite chain of oscillators in the continuum limit we have

$\displaystyle\overline{S_{E}}=\frac{z\ell}{\epsilon}+\cdots,\hskip
56.9055pt\overline{C_{E}}=\frac{\ell}{\epsilon}+\cdots,$ (5.1)

where $\ell\equiv\widetilde{N}\epsilon$ and $L\equiv
N\epsilon\rightarrow\infty$. Thus in this limit
$\frac{\overline{S_{E}}}{\overline{C_{E}}}=z\gg 1$ which is a feature of a
maximally mixed state. Moreover, the measures are both sensitive to the UV
cutoff, but the ratio is finite. We conclude that at least for the free
massless scalar field theory, the ratio of the leading terms is scheme
independent. Of course, the numerical results depicted in the left panel for
the mean entanglement entropy are consistent with the Page’s argument that in
an arbitrary quantum state the entropy is close to maximal. Hence we expect
entanglement entropy to scale with the volume (instead of the area) of the
entangling region.

The right panel shows that this special behavior also holds for capacity of
entanglement. In particular, we see that $\overline{C_{E}}$ is always
continuous and varies smoothly. This feature contrasts with the holographic
results for the capacity of entanglement corresponding to the black hole
evaporation process as previously noted in [30, 31]. Of course, their results
should be hold in a strongly coupled theory with a large number of degrees of
freedom which is completely different from our free scalar model. Thus there
is no a priori reason to expect that the results should agree in these cases.
Interestingly enough, based on eq. (5.1) we see that in large squeezing limit
the relationship $C_{E}=S_{E}$ is completely broken. Indeed, as proposed in
[14] perhaps such a relation in QFTs is a hint of a dual gravitational
interpretation. Therefore, at least in this regime the corresponding squeezed
states can not have a solution of a classical gravity theory as a holographic
dual. This is the main clue that forbids us to compare our results with some
previous studies on capacity of entanglement in the context of holography. Let
us emphasize that we found similar results for $\overline{S_{n}}$ and
$\overline{C_{n}}$ which are consistent with the asymptotic expansions we
reported in the previous section, although we do not explicitly show the
corresponding figures here.

Figure 13: Mean entanglement entropy (left) and mean capacity of entanglement
(right) as functions of $\widetilde{N}$ for several values of $z$. The red
lines indicate the asymptotic behavior in large $z$ limit given by eqs. (4.16)
and (4.18). The mean has been calculated as the average of 300 random times.

##  6 Conclusions and discussions

In this paper, we explored the evolution and scaling of the entanglement
measures in squeezed states which are the most general Gaussian states. We
have mainly studied the behavior of capacity of entanglement, the quantum
information theoretic counterpart of heat capacity, for a specific harmonic
model which is a discrete counterpart of a $(1+1)$-dimensional free scalar
field theory. To gain some intuition for the problem, we began by studying the
simple case of a pair of harmonic oscillators. In this case we have studied
different aspects of the capacity of entanglement in various setups and with
different parameters numerically.

In particular, the time evolution of entanglement measures when only a single
mode has been squeezed is periodic with period equal to half the period of the
corresponding mode. Further, the capacity of entanglement is always greater
than the entanglement entropy such that the difference of these quantities
becomes maximal when they reach their maximum values. A key observation to
note here is that while both $S_{E}$ and $C_{E}$ are monotonically increasing
functions of the squeezing parameter, only the entanglement entropy goes on to
grow indefinitely as we increase $z$. Indeed, the capacity of entanglement
saturates to unity in the large $z$ limit. Hence, we see that in this limit
$\frac{C_{E}}{S_{E}}\ll 1$ which shows that the corresponding reduced density
matrix becomes more and more maximally mixed as one increases the squeezing
parameter. Moreover, the asymptotic behavior of the mean value of the Renyi
entropy and $n$-th capacity of entanglement in large squeezing limit is almost
independent of the Renyi index. In addition, considering more general cases
where both modes are squeezed, although the evolution of the entanglement
measures is in general not periodic, the qualitative features of the results
are the same.

We have also extended these studies to the system of $N$ coupled harmonic
oscillators to investigate to what extent the squeezing parameter modifies the
behavior of the entanglement measures including the capacity of entanglement.
In particular, we have shown that where all modes lie in a squeezed state the
measures are time-independent and proportional to the volume of the smaller
subsystem in large squeezing limit. Again, we have found that in this limit
the ratio of capacity over entropy is negligible and thus the reduced density
matrix behaves as a maximally mixed state.

In order to gain further insights into certain properties of these quantities
in squeezed states, we have also considered a scalar field theory, which is
the continuum counterpart of our harmonic chain. In this case when only a
single mode has been squeezed, we found a number of key features: First, the
mean capacity of entanglement is not monotonous with the index of the squeezed
mode such that for large values of $\widetilde{N}$, it first increases very
sharply with $i_{s}$ and then suddenly saturates to a constant value. Second,
we observed that $C_{E}$ is a monotonically increasing function of the
squeezing parameter and saturates to a constant value in the large $z$ limit.
Further, the mean Renyi entropy exhibits a linear dependence on the squeezing
parameter with the same slope for different values of the Renyi index. Also we
have shown that the mean value of $n$-th capacity saturates from below to
unity independent of $n$. Indeed, for larger values of the Renyi index,
$\overline{C_{n}}$ vanishes initially but then rapidly rises to a constant
value corresponding to the large squeezing regime.

Once again, considering the more realistic case where all modes lie in a
squeezed state, the large $z$ behavior of the measures are completely
independent of time. In particular, in this regime the scaling of the capacity
of entanglement is not sensitive to the details of the time evolution of the
reduced density matrix. We have also found numerically that at leading order
$C_{E}$ is proportional to the volume of the smaller subsystem which is
consistent with asymptotic scaling given by eq. (4.18). Interestingly, for an
infinite chain of oscillators in the continuum limit the corresponding
behaviors of $S_{E}$ and $C_{E}$ are given by eq. (5.1) which is a feature of
a maximally mixed state. Here we recall that this equation holds in the large
squeezing limit and thus $C_{E}/S_{E}\sim z^{-1}\rightarrow 0$. As we have
mentioned before this means that the corresponding reduced density matrix can
be approximated as proportional to the identity operator to the extent that
its Renyi entropies are independent of $n$ and thus we have a flat
entanglement spectra. Remarkably, in the holographic context one can produce
the $n$-independent Renyi entropies by considering specific semiclassical
states, the so-called fixed-area states [35, 36, 37, 38].555We thank Mohammad
Hasan Vahidinia for bringing this to our attention and raising the following
question. It is an interesting question whether or not a more concrete
connection can be found between the squeezing states in the filed theory and
fixed-area states of quantum gravity.

Of course, eq. (5.1) is consistent with the Page’s argument that in an
arbitrary quantum state the entropy is close to maximal. Hence we expect
entanglement entropy to scale with the volume (instead of the area) of the
entangling region. Our results show that this interesting behavior also holds
for the capacity of entanglement. Moreover, the measures are both sensitive to
the UV cutoff, but the ratio is finite and thus at least for the free massless
scalar field theory, the ratio of the leading terms is scheme independent. We
found similar results for other entanglement measures which are consistent
with the asymptotic expansions in the large squeezing limit.

To close our discussion, we would like to recall that our result for the
capacity of entanglement for a family of random states, where
$\overline{C_{E}}$ is continuous, contrasts with the holographic computations
correspond to the black hole evaporation process reported in [30, 31]. Indeed,
we expect that a meaningful comparison between the field theory and
holographic results is achieved by considering strongly coupled theories with
a large number of degrees of freedom. This investigation is beyond the scope
of the present paper. It would be interesting to explore the behavior of
$C_{E}$ in general setups including rational and holographic CFTs [39, 40, 41,
42]. Some additional topics to explore include generalizing our results to
higher dimensions [43] or to non-relativistic models [44, 45, 46, 47]. Another
interesting direction is to study the scaling of capacity of entanglement in
theories which exhibit a volume law scaling of the entanglement entropy [48,
49, 50, 51]. In addition, we have been focused on the exploration of the
capacity only in a free scalar theory, but it should be feasible to extend our
work to more realistic cases with non-trivial interactions where we believe
the capacity will exhibit more interesting features [52]. We leave the details
of some interesting problems for future study [53].

### Acknowledgements

The author would like to thank Ali Mollabashi and Mohammad Hasan Vahidinia for
reading the draft and fruitful discussions. We are also very grateful to
Dimitrios Katsinis, Georgios Pastras and Nikolaos Tetradis for clarifications
about aspects of ref. [25]. This work is based upon research funded by Iran
National Science Foundation (INSF) under project No. 4013637.

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