###### Abstract

The black hole singularity plays a crucial role in formulating Hawking’s
information paradox. The global spacetime analysis may be reconciled with
unitarity by imposing a final state boundary condition on the spacelike
singularity. Motivated by the final state proposal, we explore the effect of
final state projection in two dimensional conformal field theories. We
calculate the time evolution under postselection by employing the real part of
pseudo-entropy to estimate the amount of quantum entanglement averaged over
histories between the initial and final states. We find that this quantity
possesses a Page curve-like behavior.

††preprint: YITP-21-158; IPMU21-0086

_1\. Introduction.—_ The process of postselection is useful in studies of
dynamical properties of quantum many-body systems and quantum field theories
(QFTs). The nonunitary dynamics of projective measurements, for instance,
provides a new tool for controlling many-body systems, giving rise to
measurement-induced phase transitions [1, 2]. Postselection also plays a key
role in the black hole final state proposal [3] that suggests a possible
resolution to the black hole information puzzle. Namely, even though the
evaporation process due to Hawking radiation [4, 5] seems to change the
initial pure state forming the black hole by gravitational collapse into a
mixed state [6], the final state is still a pure state due to the state
projection imposed on the spacelike singularity, cf. the left panel of Fig. 1.
However, it has been pointed out that the black hole final state has to be of
a particular type to preserve information [7].

On the other hand, it has earlier been argued by Page that black hole
unitarity should be reflected in the entanglement spectrum of Hawking
radiation [8, 9]. The basic idea is that the Bekenstein-Hawking entropy [10,
5] gives the leading term for the logarithm of the number of black hole
microstates. Accordingly, when the number of states of the radiation becomes
larger than that of the remaining microstates, the entanglement entropy
between the two should be upper bounded by the Bekenstein-Hawking entropy of
the black hole. The curve that describes the time evolution of the
entanglement entropy is known as the Page curve.

It is essential to mention that studies of postselection in QFTs have been
quite limited until so far. One reason for this may be the lack of universal
and calculable quantities that can characterize the evolution of quantum
states in the corresponding processes. For example, quantum quenches are often
studied as a typical class of time-dependent systems, and entanglement entropy
is important in probing how the systems thermalize [11].

However, in the presence of postselection, the use of entanglement entropy is
limited—nevertheless, see [12, 13, 14, 15]—since it only depends on a single
quantum state, namely either on the initial state or the final state. Instead,
it is desirable to consider a universal quantity reflecting histories from the
initial state to the final state under postselection.

Figure 1: Final state projection in an evaporating black hole (left) and its
field theory simplification (right).

Recently, one such candidate, called pseudo-entropy, has been introduced in
[16] and defined as follows. Let us consider two pure states
$|\psi_{1}\rangle$ and $|\psi_{2}\rangle$, and decompose the total system into
the subsystems $A$ and $B$ such that the entire Hilbert space becomes
factorized into ${\cal H}_{\rm tot}={\cal H}_{A}\otimes{\cal H}_{B}$. Taking
the reduced transition matrix as

$\tau^{1|2}_{A}=\mbox{Tr}_{B}\left[{|\psi_{1}\rangle\langle\psi_{2}|\over\langle\psi_{2}|\psi_{1}\rangle}\right]\,.$
(1)

the pseudo-entropy is given by

$S^{1|2}_{A}=-\mbox{Tr}\,[\tau^{1|2}_{A}\log\tau^{1|2}_{A}]\,.$ (2)

This quantity arises as a straightforward extension of holographic
entanglement entropy [17, 18] in the case of Euclidean time-dependent
backgrounds [16]. Pseudo-entropy is generically complex-valued as the
transition matrix is not always Hermitian. However, we expect that the real
part of pseudo-entropy, i.e., $\mathrm{Re}[S^{1|2}_{A}]$, can be understood as
the number of Bell pairs averaged over histories evolving from
$|\psi_{1}\rangle$ to $|\psi_{2}\rangle$. For more explanation, see appendix
A. Thus, this quantity is equally sensible in figuring out the underlying
correlation structure as in the case of entanglement entropy. Furthermore, the
mentioned quantity provides a nice quantum order parameter to distinguish
different quantum phases [19, 20]. Refer to [21, 22, 23, 24] for further
recent progress related to pseudo-entropy.

The purpose of this article is to uncover the time evolution of pseudo-entropy
in conformal field theories (CFTs) under postselection, mainly motivated by
the black hole final state proposal, as depicted in the left panel of Fig. 1.
More specifically, we consider an initial state $|\psi_{i}\rangle$ at time
$t=0$, which evolves under the Hamiltonian $H$ until $t=T$ when we perform the
postselection to the final state $\ket{\psi_{f}}$. At time $t$, the initial
state evolves into $|\psi_{1}\rangle=e^{-iHt}\ket{\psi_{i}}$, while the final
state is reverted to $|\psi_{2}\rangle=e^{i(T-t)H}\ket{\psi_{f}}$. In this
setup, sketched in the right panel of Fig. 1, we can define the pseudo-entropy
as in Eq. (2). This is the quantity we will study in this paper. Note that if
we apply the replica method by assuming a cut along the subsystem $A$, the
familiar quantity given by $-{\partial\over\partial
n}\log{Z_{n}\over(Z_{1})^{n}}$, where $Z_{n}$ is the partition function on the
$n$-sheeted geometry, actually coincides with the pseudo-entropy, but not
entanglement entropy for either $|\psi_{1}\rangle$ or $|\psi_{2}\rangle$. This
quantity reduces to the entanglement entropy only when
$|\psi_{1}\rangle=|\psi_{2}\rangle$. Here, we often assume that the CFT under
our consideration has a classical gravity dual in order to obtain analytical
results. Such a CFT, i.e., a so-called holographic CFT, is strongly coupled
and has a large central charge $c$ [25, 26]. For this class of CFTs, we can
evaluate correlation functions by employing the large $c$ factorization such
that they can be computed by Wick contractions.

_2\. Homogeneous postselection and gravity dual.—_ Let us start with the
simplest model where we perform a postselection homogeneously. A tractable
class of postselection can be defined by using the boundary state $\ket{B}$
(or Cardy state [27]) and considering the following two pure states in a given
CFT

$\begin{split}\ket{\psi_{1}}=e^{-iHt}e^{-\delta
H}\ket{B}\,,\,\ket{\psi_{2}}=e^{iH(T-t)}e^{-\delta H}\ket{B}\,,\end{split}$
(3)

where the parameter $\delta$ denotes a UV regularization of the boundary
state. The inner product of the initial state and final state defines a
Euclidean path-integral on the strip with the width $L=2\delta+iT$. We can
describe this strip as $0\leq\text{Im}\,w\leq L$ by taking $(w,\bar{w})$ as a
complex coordinate. Performing the conformal transformation as follows

$z=e^{{\pi w\over L}}\,,$ (4)

one can map this strip into an upper half-plane defined by Im$\,z>0$.

We set the two end points $w_{1},w_{2}$ of the interval $A$ as
$(w_{i},\bar{w}_{i})=(x_{i}+i(\delta+it),x_{i}-i(\delta+it))$ with $i=1,2$.
The $n$-sheeted partition function $Z_{n}$ can be computed by inserting the
twist operator $\sigma_{n}$ at the two end points of $A$ as what is usually
done in the standard computation of entanglement entropy in field theories
[28]. In holographic CFTs, this two-point function has two different saddle
point contributions, namely, (i) the Wick contraction of two twist operators
and (ii) the Wick contraction of each twist operator with their mirror images.
Since the candidates of pseudo-entropy computed from (i) and (ii) are dual to
the length of the connected and disconnected geodesic, we refer to them as the
connected and disconnected contribution, denoted by $S^{\mathrm{con}}_{A}$ and
$S^{\mathrm{dis}}_{A}$, respectively.

More generally, if we consider a conformal map from the original $w$
coordinate to the upper half-plane in $z$ coordinate via a holomorphic map
$z=f(w)$, the two distinct contributions to pseudo-entropy are given by

$\displaystyle S^{\mathrm{con}}_{A}$ $\displaystyle={c\over
6}\log{|f(w_{1})-f(w_{2})|^{2}\over\epsilon^{2}|f^{\prime}(w_{1})||f^{\prime}(w_{2})|}\,,$
(5) $\displaystyle S^{\mathrm{dis}}_{A}$ $\displaystyle={c\over
6}\log{|f(w_{1})-\bar{f}(\bar{w}_{1})||f(w_{2})-\bar{f}(\bar{w}_{2})|\over\epsilon^{2}|f^{\prime}(w_{1})||f^{\prime}(w_{2})|}+2S_{\mathrm{bdy}}\,,$
(6)

where $S_{\mathrm{bdy}}$ is referred to as the boundary entropy [29] and
$\epsilon$ is a UV cutoff corresponding to lattice spacing. This field-
theoretic result perfectly agrees with the holographic entanglement entropy in
AdS/BCFT (anti-de Sitter/boundary CFT) [30, 31, 32].

Using the conformal map (4) for the present model under a homogeneous
postselection, we can find

$\begin{split}S^{\mathrm{con}}_{A}&={c\over
3}\log\left[{2T\over\pi\epsilon}\sin\left({\pi(x_{2}-x_{1})\over
2T}\right)\right],\\\ S^{\mathrm{dis}}_{A}&={c\over
3}\log\left[{2T\over\pi\epsilon}\sin\left({\pi t\over T}\right)\right]+i\,{\pi
c\over 6}+2S_{\rm bdy}\,,\end{split}$ (7)

by taking the limit $\delta\to 0$. The value of pseudo-entropy is given by the
one which has a smaller real part, viz.,

$\mbox{Re}\,[S^{1|2}_{A}]=\mbox{min}\left[\mbox{Re}\,[S^{\mathrm{con}}_{A}],\mbox{Re}\,[S^{\mathrm{dis}}_{A}]\right]\,.$
(8)

Note that $\mbox{Re}\,[S^{\mathrm{dis}}_{A}]$ vanishes at $t=\epsilon/2$,
which we regard as the initial time with a regularization. Then it increases
logarithmically, reaching the maximum at the middle time $t=T/2$. It again
decreases and vanishes at the final time $t=T-\epsilon/2$ as the boundary
state does not have any real space entanglement [33]. On the other hand,
$\mbox{Re}\,[S^{\mathrm{con}}_{A}]$ is time-independent and is identical to
the entanglement entropy for the vacuum state. Taking into account the
minimization in the definition (8), it is clear that the disconnected
contribution $\mbox{Re}\,[S^{\mathrm{dis}}_{A}]$ dominates at early and late
times. Depending on the value of $S_{\rm bdy}$, there is a chance that
$\mbox{Re}\,[S^{\mathrm{con}}_{A}]$ dominates for a finite period
$t_{\ast}<t<T-t_{\ast}$.

Figure 2: The gravity dual of the Lorentzian BCFT. The blue surface describes
the EOW brane defined in Eq. (11). For a subsystem A (red curve) located on
the asymptotic boundary, the black and green curves denote the corresponding
disconnected geodesics and the connected geodesic, respectively.

The holographic analysis [34] based on AdS/BCFT suggests that a spacelike
boundary in a Lorentzian BCFT has a complex-valued boundary entropy, namely

$S_{\rm bdy}={c\over 6}\log\sqrt{{|{\cal T}|-1\over|{\cal T}|+1}}-i\,{\pi
c\over 12}\,,$ (9)

where ${\cal T}$ is the tension of the end-of-the-world (EOW) brane dual to
the boundary of the BCFT. This takes values in the range ${\cal T}<-1$. The
imaginary part of (9) exactly cancels that of the pseudo-entropy
$S^{\mathrm{dis}}_{A}$.

For our setup associated with (3), we can further construct its gravity dual
as follows. Considering a global AdS3, i.e.,

$ds^{2}=-{T^{2}\over\pi^{2}}\cosh^{2}\rho\,dt^{2}+d\rho^{2}+{T^{2}\over\pi^{2}}\sinh^{2}\rho\,dx^{2}\,,$
(10)

we introduce an EOW brane $Q$ defined by

$\cosh\rho\sin{\pi t\over T}=\cosh\eta_{0}\,,$ (11)

that describes two-dimensional de Sitter spacetime. Here $\eta_{0}$ is related
to the brane tension as ${\cal T}=-\coth\eta_{0}$. Finally, the gravity dual
of the present CFT setup is given by the region surrounded by the AdS
asymptotic boundary $\rho\to\infty$ and the EOW brane $Q$, as illustrated in
Fig. 2. Similar to holographic entanglement entropy [17, 18], the holographic
pseudo-entropy [16] is given by the geodesic length $L_{A}$ in the
corresponding Lorentzian gravity dual. In our case, due to the presence of
$Q$, the geodesic connecting two endpoints of the interval $A$ can end on $Q$,
as shown in Fig. 2. Thus, we can have contributions from disconnected
geodesics, denoted by $L^{\rm dis}_{A}$, in addition to those resulting from a
connected geodesic $L^{\rm con}_{A}$. Accordingly, the real part of
holographic pseudo-entropy is taken to be the minimum as before, i.e.,

$\displaystyle\mbox{Re}\,[S^{1|2}_{A}]=\mbox{min}\left[\mbox{Re}\left[{L^{\rm
con}_{A}\over 4G_{\textrm{\tiny N}}}\right],\mbox{Re}\left[{L^{\rm
dis}_{A}\over 4G_{\textrm{\tiny N}}}\right]\right]\,.$ (12)

Obviously, the connected contribution is the same as the holographic
entanglement entropy in global AdS3 and thus coincides with Eq. (5). The
connected contribution arises only when the connected geodesic does not touch
the EOW brane $Q$, which leads to the following condition

$\sin{\pi t\over T}>\cosh\eta_{0}\cdot\sin{\pi(x_{2}-x_{1})\over 2T}\,.$ (13)

Moreover, a straightforward computation for the sum of two geodesic lengths
reproduces the disconnected contribution as shown in Eqs. (6) and (9). In this
way, we can derive the CFT results from the gravity dual. More details are
presented in appendix B.

_3\. Inhomogeneous postselection and pseudo-entropy.—_ Next, we focus on an
example of inhomogeneous postselection models motivated by the black hole
final state projection. Namely, at $t=0$ we project the left part $x<0$ and
right part $x>0$ to a boundary state $\ket{B}$ and the CFT vacuum $\ket{0}$,
respectively. This is realized by the (Euclidean) path-integral on the
$w$-sheet as depicted in the left of Fig. 3. We regard the projection on $x<0$
as the black hole final state 111One may wonder whether the boundary state
$\ket{B}$ imposed on $x<0$ can mimic the black hole final state since the
latter is expected to be very random [3]. However, it is worth emphasizing
that, as commented in [7], the state which is expected to be random should be
a time-reverted state of the black hole final state, but not itself.
Therefore, there is no apparent contradiction that a boundary state can mimic
the black hole final state.. Although the boundary state is based on a local
boundary condition and does not have any real space entanglement [33], we
expect that the time evolution $e^{i(T-t)H}$ may lead the state to a random
chaotic one. We also qualitatively mimic the creation of entangled pairs due
to Hawking radiation during the time evolution of the initial boundary state
as similar to quantum quenches [11].

Figure 3: A sketch of Euclidean path-integral on the $w$-sheet for
inhomogenous postselection and its conformal transformation to the upper half-
plane in terms of the $z$ coordinate. For $x<0$, the path-integral is
terminated at $\text{Im}\,w=0$ describing the projection to $\ket{B}_{x<0}$,
while for $x>0$ it extends to $\text{Im}\,w=\infty$, corresponding to the
projection to the vacuum state $\ket{0}_{x>0}$ on the right half.

Taking the initial state as the quantum quench state $e^{-\delta H}\ket{B}$,
and fixing the final state as $e^{-\delta
H}\ket{B}_{x<0}\otimes\ket{0}_{x>0}$, we can calculate the pseudo-entropy by
using the general formulae shown in Eqs. (5) and (6). We choose the subsystem
$A$ as an interval whose two endpoints are denoted by $(w_{1},\bar{w}_{1})$
and $(w_{2},\bar{w}_{2})$. More explicitly, one can find ($i=1,2$)

$w_{i}=x_{i}+i\left(-\delta-i(T-t)\right)\,,\bar{w}_{i}=x_{i}-i\left(-\delta-i(T-t)\right)\,.$
(14)

We can also map the $w$-sheet to an upper half-plane ($z$-plane) via the
following conformal transformation

$\begin{split}w&=\sqrt{z}-{a\over 2}\log\left[{1+{\sqrt{z}\over
a}\over{\sqrt{z}\over a}-1}\right]-{a\over 2}\pi i\,,\\\
\bar{w}&=\sqrt{\bar{z}}-{a\over 2}\log\left[{1+{\sqrt{\bar{z}}\over
a}\over{\sqrt{\bar{z}}\over a}-1}\right]+{a\over 2}\pi i\,,\\\ \end{split}$
(15)

by choosing $a={2\over\pi}\left(2\delta+iT\right)$.

For simplicity, we first examine the simple cases with $|x|\gg T$ to obtain
some analytical results. To wit, one can get $z\simeq(x-t)^{2},\
\bar{z}\simeq(x+t)^{2}\,,$ and $z\simeq a^{2}\\!+\\!4a^{2}e^{{2\over
a}(x-t)}\,,\bar{z}\simeq a^{2}\\!+\\!4a^{2}e^{{2\over a}(x+t)}\,,$ by taking
the limit $x\gg T$ and $x\ll-T$, respectively. It is then straightforward to
evaluate the pseudo-entropy in the following three cases: (a) $x_{1},x_{2}\gg
T$, (b) $x_{2}\gg T,x_{1}\ll-T$, and (c) $x_{1},x_{2}\ll-T$ by setting
$\delta=0$ and noting that $t<T\ll|x_{1,2}|$.

In the case (a), $x_{1},x_{2}\gg T$, we obtain

$\begin{split}\mbox{Re}\,[S^{\mathrm{con}}_{A}]&\simeq{c\over
6}\log\left[{(x_{1}-x_{2})^{2}(x_{1}+x_{2})^{2}\over
4x_{1}x_{2}\epsilon^{2}}\right]\,,\\\
\mbox{Re}\,[S^{\mathrm{dis}}_{A}]&\simeq{c\over
3}\log\left[{2t\over\epsilon}\right]+2\tilde{S}_{\rm bdy}\,,\end{split}$ (16)

where $\tilde{S}_{\rm bdy}$ is the real part of the boundary entropy $S_{\rm
bdy}$ for space-like surface. At $t=0$, the disconnected one is favored and we
have $\mbox{Re}\,[S^{\mathrm{dis}}_{A}]=0$. As time evolves, it grows
logarithmically as $\mbox{Re}\,[S^{\mathrm{dis}}_{A}]\simeq{c\over
3}\log{2t\over\epsilon}$ and eventually $\mbox{Re}\,[S^{\mathrm{con}}_{A}]$
becomes dominant. This transition behavior is schematically described by

$\mbox{Re}\,[S_{A}]\simeq\min\left[{c\over
3}\log{|x_{1}-x_{2}|\over\epsilon},{c\over
3}\log{2t\over\epsilon}+2\tilde{S}_{\rm bdy}\right]\,.$ (17)

We can intuitively understand this result as follows. The state
$\ket{\psi_{i}}=e^{-itH}e^{-\delta H}\ket{B}$ has quantum entanglement for the
length scale $l<2t$ due to causal propagation. Thus, if $|x_{1}-x_{2}|<2t$,
both the initial state and final states have the corresponding quantum
entanglement. However, if $|x_{1}-x_{2}|>2t$, the initial state does not have
the entanglement at the length scale $|x_{1}-x_{2}|$. These arguments
qualitatively explain the behavior of Eq. (17). It is still intriguing that
the growth is given in terms of the logarithmic one as opposed to the linear
growth of entanglement entropy in global quenches.

In the case (b), $x_{2}\gg T$ and $x_{1}\ll-T$, we obtain

$\begin{split}\mbox{Re}\,[S^{\mathrm{con}}_{A}]&\simeq{c\over
6}\log\left[{\pi^{2}x_{2}^{3}\over 32T\epsilon^{2}}\right]\,,\\\
\mbox{Re}\,[S^{\mathrm{dis}}_{A}]&\simeq{c\over 6}\log\left[{4Tt\sin\left({\pi
t\over T}\right)\over\pi\epsilon^{2}}\right]+2\tilde{S}_{\rm
bdy}\,.\end{split}$ (18)

Since we assume $x_{2}\gg T>t$, the disconnected one is favored at any time.
It starts with $\mbox{Re}\,[S^{\mathrm{dis}}_{A}]=0$ as in the previous case
and grows logarithmically as $\mbox{Re}\,[S^{\mathrm{dis}}_{A}]\simeq{c\over
3}\log{2t\over\epsilon}$ initially. Then it reaches a maximum and starts
decreasing. At the final time $t\simeq T$, it is reduced to
$\mbox{Re}\,[S^{\mathrm{dis}}_{A}]\simeq{c\over 6}\log{2T\over\epsilon}$. This
result can be explained by noting that the entanglement in the part
$x_{1}<x<0$ of region $A$ becomes trivial at the final time due to
postselection.

In the case (c), $x_{1},x_{2}\ll-T$, we simply reproduce our previous results
(5) and (6). This can be easily understood if we note that for $x\ll-T$, the
space looks like a strip having width $T$, identical to the homogeneous
postselection model (3).

Figure 4: The plot for the real part of pseudo-entropy Re$[S_{A}]$ for
subsystems $A=[0,\infty]$ (left) and $A=[0,5]$ (right) as a function of time
$t$. We choose $T=5$, $\delta=\epsilon=0.01$ and $\tilde{S}_{\rm bdy}=0$.

Next, we shall choose the subsystem to be $A=[0,y]$ with $y>0$ to model the
radiation subsystem for the evaporating black hole, where the black hole final
state is imposed on $x<0$ at $t=T$. We numerically plot the real part of
pseudo-entropy in Fig. 4. In particular, when we choose $y=\infty$, where only
the disconnected geodesic is available, it looks like a Page curve, i.e.,
starting from $\mbox{Re}\,[S_{A}]=0$ and ending up with
$\mbox{Re}\,[S_{A}]=0$. Indeed, since we have $w\simeq-{z^{3/2}\over 3a^{2}}$
when $z\simeq 0$, we can estimate the value at $t\simeq T$ as follows

$\displaystyle\mbox{Re}\,[S^{\mathrm{dis}}_{A}]\sim{c\over
12}\log{|w|^{2}\over\epsilon^{2}}\simeq{c\over
6}\log{(T-t)^{2}+\delta^{2}\over\epsilon^{2}}\,.$ (19)

At $t=T$, this is of the same order as at the initial time $t=0$. Therefore,
we can conclude that the pseudo-entropy vanishes at $t=T$ by choosing
$\delta=O(\epsilon)$. More generally, this causes the phenomenon that the real
part of pseudo-entropy is highly reduced at $|x|=T-t$. We may interpret that
this is due to the negative energy flux emitted at the end point of the
projection region, $x=0$, at $t=T$.

Figure 5: A sketch of the path-integral description of partial postselection
(left) and its conformal map to the upper half-plane (right).

_4\. Partial postselection and pseudo-entropy.—_ Although the inhomogeneous
postselection model studied before mimics a black hole spacetime with a
spacelike singularity, there is a crucial difference with the black hole final
state proposal [3]. In the former, the postselection is only imposed on the
singularity, while no operation is performed to the compliment part.

To model this feature, we shall consider partial postselection in a two-
dimensional CFT, where we make a projection only for the left half $x<0$. We
again choose a global quench state $e^{-\delta H}|B\rangle$ at $t=0$ as the
initial state and consider its time evolution until the partial postselection
is imposed at $t=T$. We take the postselected state as a boundary state
$|B\rangle_{x<0}$, while keeping the right half $x>0$ free. Therefore, right
after the postselection at $t=T$, the quantum state associated with the whole
system turns out to be

$\displaystyle\,|B\rangle_{x<0}\otimes\left[\langle
B|_{x<0}~{}\left(e^{-iTH}e^{-\delta H}|B\rangle\right)\right]_{x>0}$
$\displaystyle=$ $\displaystyle\left(|B\rangle\langle B|_{x<0}\otimes
I_{x>0}\right)\cdot e^{-iTH}e^{-\delta H}|B\rangle\,,$ (20)

where $I_{x<0}$ is the identity matrix for the left half.

Next, we focus on the pseudo-entropy at immediate time $0<t<T$. In this case,
the initial state and the final state are given by

$\begin{split}|\psi_{1}\rangle&=e^{-itH}e^{-\delta H}|B\rangle\,,\\\
|\psi_{2}\rangle&=e^{-i(t-T)H}\left(|B\rangle\langle B|_{x<0}\otimes
I_{x>0}\right)\cdot e^{-iTH}e^{-\delta H}\ket{B}\,.\end{split}$ (21)

Taking subsystem $A$ to be an interval, the pseudo-entropy (2) in this setup
can be computed via the Euclidean path-integral shown in the left panel of
Fig. 5. Using the conformal map

$w(z)=f^{-1}(z)=\alpha\log(1-z)+\alpha\beta\log\left({z+\beta\over\beta}\right)\,,$
(22)

we can map the strip geometry with a slit to the upper half-plane as shown in
the right of Fig. 5. The parameters $\alpha$ and $\beta$ are fixed by
$\pi\alpha=\delta+iT$ and $\pi\alpha\beta=\delta-iT$, respectively. For the
subsystem $A=[0,y]$, with $y>0$, we plot the real part of pseudo-entropy in
Fig. 6. This again shows a Page curve-like behavior.

Figure 6: The plot for the real part of pseudo-entropy Re$[S_{A}]$ for
subsystems $A=[0,\infty]$ (left) and $A=[0,5\pi]$ (right) as a function of
time $t$. We choose the parameters $T=2\pi$, $\delta=\varepsilon=0.05\pi$.

_5\. Discussion.—_ We have studied the time evolution of pseudo-entropy under
postselection, whose real part provides an estimation of the amount of quantum
entanglement, i.e., the number of Bell pairs averaged over histories between
the initial state and the postselected final state. In a two-dimensional CFT
setup, where postselection applies to the left region, $x<0$, the mentioned
entanglement measure associated with the region $x>0$ possesses a Page curve-
like behavior. It grows by starting from zero, reaches its maximum, decreases,
and eventually vanishes at the time of the partial postselection. Our setups
model black hole evaporation according to the final state projection scenario.
One may wonder how these models are related to the island picture [36, 37, 38]
resulting in the Page curve. Notably, in our analysis, the unitarity-caused
deviation from monotonical increase arises due to the past evolution of the
postselected final state. This qualitatively looks similar to the island
scenario suggesting a modification of the Hilbert space structure inside the
black hole. However, state projection seems to affect correlations already in
the initial stages of evaporation. There is no sudden appearance of a disjoint
entanglement region on a Cauchy slice residing in the interior right after the
Page time.

Considerations that seem to indicate a tension between the final state
proposal and the presence of a smooth horizon have been discussed in [39, 40],
where a resolution to the puzzle raised in [40] has been proposed in [41].
Nevertheless, starting from the arguments in [42], we shall emphasize that
indications of a drama for infalling observers are merely an artifact of the
factorizable Hilbert space within the improper semiclassical treatment. On the
other hand, the notion of a classical singularity may already have to be given
up when the black hole is young, following the arguments by Page [43]. This
motivates considering sequential projective measurements scanning through a
smeared interior patch. It might be possible that associated correlations
responsible for the Page curve make it to the exterior near-horizon region,
namely in a nontrivially protected form [44]. A correct treatment of the
singularity might thus have imprints not only behind the horizon. We want to
return to some of these aspects in the near future. Another interesting
direction is studying state projection in moving mirrors [45] by employing
their holographic formulation [46, 47].

###### Acknowledgements.

_Acknowledgments.—_ We are grateful to Tokiro Numasawa and Kotaro Tamaoka for
useful discussions and especially to Ahmed Almheiri for valuable comments on a
draft of this paper. We would also like to thank the YITP workshop: ”KIAS-YITP
2021: String Theory and Quantum Gravity ” (YITP-W-21-18), hosted by YITP,
Kyoto U, for stimulating discussions and comments from participants, where
this work was presented by T. K. This work is supported by MEXT-JSPS Grant-in-
Aid for Transformative Research Areas (A) ”Extreme Universe”, No. 21H05187. T.
T. is also supported by JSPS Grant-in-Aid for Scientific Research (A) No.
21H04469. S-M. R. and T. T. are supported by the Simons Foundation through the
“It from Qubit” collaboration. T. T. is also supported by Inamori Research
Institute for Science and World Premier International Research Center
Initiative (WPI Initiative) from MEXT. Z. W. is supported by Grant-in-Aid for
JSPS Fellows No. 20J23116.

## References

  * Skinner  _et al._ [2019] B. Skinner, J. Ruhman, and A. Nahum, Measurement-Induced Phase Transitions in the Dynamics of Entanglement, Phys. Rev. X 9, 031009 (2019), arXiv:1808.05953 [cond-mat.stat-mech] .
  * Li  _et al._ [2018] Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018), arXiv:1808.06134 [quant-ph] .
  * Horowitz and Maldacena [2004] G. T. Horowitz and J. M. Maldacena, The Black hole final state, JHEP 02, 008, arXiv:hep-th/0310281 .
  * Hawking [1974] S. W. Hawking, Black hole explosions, Nature 248, 30 (1974).
  * Hawking [1975] S. W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43, 199 (1975), [Erratum: Commun.Math.Phys. 46, 206 (1976)].
  * Hawking [1976] S. W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys. Rev. D 14, 2460 (1976).
  * Gottesman and Preskill [2004] D. Gottesman and J. Preskill, Comment on ‘The Black hole final state’, JHEP 03, 026, arXiv:hep-th/0311269 .
  * Page [1993a] D. N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71, 1291 (1993a), arXiv:gr-qc/9305007 .
  * Page [1993b] D. N. Page, Information in black hole radiation, Phys. Rev. Lett. 71, 3743 (1993b), arXiv:hep-th/9306083 .
  * Bekenstein [1973] J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973).
  * Calabrese and Cardy [2005] P. Calabrese and J. L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504, P04010 (2005), arXiv:cond-mat/0503393 .
  * Rajabpour [2015a] M. A. Rajabpour, Post measurement bipartite entanglement entropy in conformal field theories, Phys. Rev. B 92, 075108 (2015a), arXiv:1501.07831 [cond-mat.stat-mech] .
  * Rajabpour [2015b] M. Rajabpour, Fate of the area-law after partial measurement in quantum field theories, arXiv preprint arXiv:1503.07771 (2015b).
  * Rajabpour [2016] M. A. Rajabpour, Entanglement entropy after a partial projective measurement in $1+1$ dimensional conformal field theories: exact results, J. Stat. Mech. 1606, 063109 (2016), arXiv:1512.03940 [hep-th] .
  * Numasawa  _et al._ [2016] T. Numasawa, N. Shiba, T. Takayanagi, and K. Watanabe, EPR Pairs, Local Projections and Quantum Teleportation in Holography, JHEP 08, 077, arXiv:1604.01772 [hep-th] .
  * Nakata  _et al._ [2021] Y. Nakata, T. Takayanagi, Y. Taki, K. Tamaoka, and Z. Wei, New holographic generalization of entanglement entropy, Phys. Rev. D 103, 026005 (2021), arXiv:2005.13801 [hep-th] .
  * Ryu and Takayanagi [2006] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006), arXiv:hep-th/0603001 .
  * Hubeny  _et al._ [2007] V. E. Hubeny, M. Rangamani, and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07, 062, arXiv:0705.0016 [hep-th] .
  * Mollabashi  _et al._ [2021a] A. Mollabashi, N. Shiba, T. Takayanagi, K. Tamaoka, and Z. Wei, Pseudo Entropy in Free Quantum Field Theories, Phys. Rev. Lett. 126, 081601 (2021a), arXiv:2011.09648 [hep-th] .
  * Mollabashi  _et al._ [2021b] A. Mollabashi, N. Shiba, T. Takayanagi, K. Tamaoka, and Z. Wei, Aspects of pseudoentropy in field theories, Phys. Rev. Res. 3, 033254 (2021b), arXiv:2106.03118 [hep-th] .
  * Camilo and Prudenziati [2021] G. Camilo and A. Prudenziati, Twist operators and pseudo entropies in two-dimensional momentum space (2021), arXiv:2101.02093 [hep-th] .
  * Nishioka  _et al._ [2021] T. Nishioka, T. Takayanagi, and Y. Taki, Topological pseudo entropy, JHEP 09, 015, arXiv:2107.01797 [hep-th] .
  * Goto  _et al._ [2021] K. Goto, M. Nozaki, and K. Tamaoka, Subregion Spectrum Form Factor via Pseudo Entropy (2021), arXiv:2109.00372 [hep-th] .
  * Miyaji [2021] M. Miyaji, Island for Gravitationally Prepared State and Pseudo Entanglement Wedge (2021), arXiv:2109.03830 [hep-th] .
  * Heemskerk  _et al._ [2009] I. Heemskerk, J. Penedones, J. Polchinski, and J. Sully, Holography from Conformal Field Theory, JHEP 10, 079, arXiv:0907.0151 [hep-th] .
  * Hartman  _et al._ [2014] T. Hartman, C. A. Keller, and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP 09, 118, arXiv:1405.5137 [hep-th] .
  * Cardy [1989] J. L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324, 581 (1989).
  * Calabrese and Cardy [2004] P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406, P06002 (2004), arXiv:hep-th/0405152 .
  * Affleck and Ludwig [1991] I. Affleck and A. W. W. Ludwig, Universal noninteger ’ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67, 161 (1991).
  * Takayanagi [2011] T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett. 107, 101602 (2011), arXiv:1105.5165 [hep-th] .
  * Fujita  _et al._ [2011] M. Fujita, T. Takayanagi, and E. Tonni, Aspects of AdS/BCFT, JHEP 11, 043, arXiv:1108.5152 [hep-th] .
  * Karch and Randall [2001] A. Karch and L. Randall, Open and closed string interpretation of SUSY CFT’s on branes with boundaries, JHEP 06, 063, arXiv:hep-th/0105132 .
  * Miyaji  _et al._ [2015] M. Miyaji, S. Ryu, T. Takayanagi, and X. Wen, Boundary States as Holographic Duals of Trivial Spacetimes, JHEP 05, 152, arXiv:1412.6226 [hep-th] .
  * Akal  _et al._ [2020] I. Akal, Y. Kusuki, T. Takayanagi, and Z. Wei, Codimension two holography for wedges, Phys. Rev. D 102, 126007 (2020), arXiv:2007.06800 [hep-th] .
  * Note [1] One may wonder whether the boundary state $\mathinner{|{B}\rangle}$ imposed on $x<0$ can mimic the black hole final state since the latter is expected to be very random [3]. However, it is worth emphasizing that, as commented in [7], the state which is expected to be random should be a time-reverted state of the black hole final state, but not itself. Therefore, there is no apparent contradiction that a boundary state can mimic the black hole final state.
  * Penington [2020] G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09, 002, arXiv:1905.08255 [hep-th] .
  * Almheiri  _et al._ [2019] A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12, 063, arXiv:1905.08762 [hep-th] .
  * Almheiri  _et al._ [2020] A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03, 149, arXiv:1908.10996 [hep-th] .
  * Lloyd and Preskill [2014] S. Lloyd and J. Preskill, Unitarity of black hole evaporation in final-state projection models, JHEP 08, 126, arXiv:1308.4209 [hep-th] .
  * Bousso and Stanford [2014] R. Bousso and D. Stanford, Measurements without Probabilities in the Final State Proposal, Phys. Rev. D 89, 044038 (2014), arXiv:1310.7457 [hep-th] .
  * [41] Talk by A. Almheiri, Comments on the black hole final state proposal and the gravitational path integral, at YITP workshop ”Recent progress in theoretical physics based on quantum information theory”, Mar. 2021, http://www2.yukawa.kyoto-u.ac.jp/ qith2021/Almheiri.pdf.
  * Almheiri  _et al._ [2013] A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02, 062, arXiv:1207.3123 [hep-th] .
  * Akal [2021] I. Akal, Information storage and near horizon quantum correlations (2021), arXiv:2109.01639 [hep-th] .
  * Akal [2020] I. Akal, Universality, intertwiners and black hole information (2020), arXiv:2010.12565 [hep-th] .
  * [45] I. Akal, T. Kawamoto, S.-M. Ruan, T. Takayanagi, and Z. Wei, in preparation.
  * Akal  _et al._ [2021a] I. Akal, Y. Kusuki, N. Shiba, T. Takayanagi, and Z. Wei, Entanglement Entropy in a Holographic Moving Mirror and the Page Curve, Phys. Rev. Lett. 126, 061604 (2021a), arXiv:2011.12005 [hep-th] .
  * Akal  _et al._ [2021b] I. Akal, Y. Kusuki, N. Shiba, T. Takayanagi, and Z. Wei, Holographic moving mirrors, Class. Quant. Grav. 38, 224001 (2021b), arXiv:2106.11179 [hep-th] .
  * Gell-Mann and Hartle [2018] M. Gell-Mann and J. B. Hartle, Quantum mechanics in the light of quantum cosmology (2018), arXiv:1803.04605 [gr-qc] .

##  I Appendix A: Pseudo-entropy and postselection

In the following, we present an argument suggesting that the real part of
pseudo entropy can measure quantum entanglement when we take an average over
the evolution from the initial state to the final state under postselection.

Let us consider the decoherence function [48] for the projective measurement
$\sum_{k}\Pi_{k}=1$ defined by

$\displaystyle
D_{k,l}={1\over\mbox{Tr}[\rho_{1}\rho_{2}]}\mbox{Tr}[\rho_{2}\Pi_{k}\rho_{1}\Pi_{l}]\,,$
(23)

where $\rho_{1}$ and $\rho_{2}$ denote the initial and final density matrix,
respectively. This quantity measures the inference of probability of $k$-event
and $l$-event. When the decoherence function is diagonal, the histories
completely decohere. If we set $\rho_{2}=I$, the above quantity is reduced to
the ordinary probability distribution

$D_{k,l}=\mbox{Tr}\,[\Pi_{k}\rho_{i}\Pi_{l}]\,.$ (24)

When both the initial and final state are pure, we have

$D_{k,l}={\langle\psi_{f}|\Pi_{k}|\psi_{i}\rangle\langle\psi_{i}|\Pi_{l}|\psi_{f}\rangle\over|\langle\psi_{f}|\psi_{i}\rangle|^{2}}\,.$
(25)

As in [16], let us consider two-qubit states $|\psi_{1}\rangle$ and
$|\psi_{2}\rangle$ of the following form

$\begin{split}|\psi_{1}\rangle&=c_{1}|00\rangle_{AB}+s_{1}|11\rangle_{AB}\,,\\\
|\psi_{2}\rangle&=c_{2}|00\rangle_{AB}+s_{2}|11\rangle_{AB}\,,\end{split}$
(26)

where $c_{1,2}$ and $s_{1,2}$ can take any complex values with the constraints
$|c_{1}|^{2}+|s_{1}|^{2}=1$ and $|c_{2}|^{2}+|s_{2}|^{2}=1$ imposed. To have a
better counting of Bell pairs, we take the asymptotic limit $M\to\infty$, by
considering $M$ copies of the original states:
$|\psi_{i}\rangle=(|\psi_{1}\rangle)^{\otimes M}$ and
$|\psi_{f}\rangle=(|\psi_{2}\rangle)^{\otimes M}$. The total Hilbert space now
consists of $2M$ qubits, i.e., $M$ copies of the $A$ spin and $M$ copies of
the $B$ spin. We call the former $\tilde{A}$ and the latter $\tilde{B}$. We
choose $\Pi_{k}$ to be the projection which acts only on the $M$ spins in
$\tilde{A}$ such that the states with $k$ up spins (i.e., $|1\rangle$) and
$M-k$ down spins (i.e., $|0\rangle$) for $M$-qubit states are selected.
$\Pi_{k}$ acts on $\tilde{B}$ as an identity. After the projection by
$\Pi_{k}$, we obtain a state with maximal entanglement between $\tilde{A}$ and
$\tilde{B}$. Due to this projection there remain ${}_{M}C_{k}\equiv{M!\over
k!(M-k)!}$ states. This is the same procedure as in [16], where an operational
interpretation of pseudo-entropy was presented for this special class of
states.

We naturally define the averaged value $\bar{N}$ of Bell pairs when we fix
both the initial and final state in the asymptotic limit as follows

$\bar{N}=\lim_{M\to\infty}{N_{\rm max}\over M}\,,$ (27)

where

$\displaystyle N_{\rm
max}=\sum_{k,l}D_{k,l}\left({\log({}_{M}C_{k})+\log({}_{M}C_{l})\over
2}\right)\,.$ (28)

Here, $N_{max}$ estimates the maximal number of Bell pairs, which can be
distilled by local operation and classical communication (LOCC) in the
asymptotic limit $M\to\infty$ when we take the average over histories from the
initial state to the final state. We can explicitly write

$\displaystyle D_{k,l}=p_{k}p^{*}_{l}\,,$ (29)

with

$\displaystyle
p_{k}={\langle\psi_{f}|\Pi_{k}|\psi_{i}\rangle\over\langle\psi_{f}|\psi_{i}\rangle}\,.$
(30)

Note that $p_{k}$ take complex values in general since we allow $c_{1,2}$ and
$s_{1,2}$ to take complex values. Due to the identity $\sum_{k}p_{k}=1$, we
can rewrite $N_{\rm max}$ as

$N_{\rm max}=\mbox{Re}\left[\sum_{k}p_{k}\log({}_{M}C_{k})\right]\,.$ (31)

As shown in [16], the quantity $\sum_{k}p_{k}\log({}_{M}C_{k})$ coincides with
the pseudo-entropy (2) in the limit $M\to\infty$. Therefore, it implies that
the real part of pseudo-entropy can be interpreted as the number of
distillable Bell pairs averaged over the history between the initial state and
final state. In this argument, we have assumed the specific class of initial
and final states given by (26). They share the same basis of spins, i.e.,
$|00\rangle_{AB}$ and $|11\rangle_{AB}$, and we can easily fix the form of the
projection $\Pi_{k}$. To extend this analysis to general states, we need to
pick up an appropriate projection $\Pi_{k}$ to extract Bell pairs, which is
not obvious for generic choices of $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$. We
will leave this general argument to future work.

##  II Appendix B: Analysis of end-of-the-world branes in $\text{AdS}_{3}$

We focus on the end-of-the-world (EOW) brane in global AdS3 (10) defined by
(11). This surface is described by its world-sheet coordinates $(\tau,x)$
introduced as follows

$\begin{split}X_{0}&=\cosh\rho\cos{\pi t\over T}=\sinh\tau\sinh\eta_{0}\,,\\\
X_{1}&=\sinh\rho\sin{\pi x\over T}=\cosh\tau\sinh\eta_{0}\sin{\pi x\over
T}\,,\\\ X_{2}&=\sinh\rho\cos{\pi x\over T}=\cosh\tau\sinh\eta_{0}\cos{\pi
x\over T}\,,\\\ X_{3}&=\cosh\rho\sin{\pi t\over T}=\cosh\eta_{0}\,,\\\
\end{split}$ (32)

where the original AdS3 defined by $X_{0}^{2}+X_{3}^{2}=X_{1}^{2}+X_{2}^{2}+1$
in the spacetime with line element
$ds^{2}=-(dX_{0})^{2}-(dX_{3})^{2}+(dX_{1})^{2}+(dX_{2})^{2}$. The induced
metric of the brane $Q$ is derived as

$\displaystyle ds^{2}|_{Q}=\sinh^{2}\eta_{0}\left(-d\tau^{2}+{\pi^{2}\over
T^{2}}\cosh^{2}\tau dx^{2}\right),$ (33)

which is nothing but a two-dimensional de Sitter space.

To evaluate the geodesic length, it is useful to rewrite the global AdS3 in
terms of Poincare coordinates, i.e.,

$\begin{split}X_{0}&={1+z_{\textrm{\tiny P}}^{2}+x_{\textrm{\tiny
P}}^{2}-t_{\textrm{\tiny P}}^{2}\over 2z_{\textrm{\tiny P}}}\,,\\\
X_{3}&={t_{\textrm{\tiny P}}\over z_{\textrm{\tiny P}}}\,,\\\
X_{1}&={x_{\textrm{\tiny P}}\over z_{\textrm{\tiny P}}}\,,\\\
X_{2}&={1-z_{\rm{P}}^{2}-x_{\textrm{\tiny P}}^{2}+t_{\textrm{\tiny
P}}^{2}\over 2z_{\textrm{\tiny P}}}\,,\\\ \end{split}$ (34)

with the metric $ds^{2}={dz_{\textrm{\tiny P}}^{2}-dt_{\textrm{\tiny
P}}^{2}+dx_{\textrm{\tiny P}}^{2}\over z_{\textrm{\tiny P}}^{2}}$. The surface
$Q$ mapped to the plane is thus located at

$\displaystyle{t_{\textrm{\tiny P}}\over z_{\textrm{\tiny
P}}}=\cosh\eta_{0}\,.$ (35)

It is also useful to note that we have $x_{\textrm{\tiny P}}={\sin{\pi x\over
T}\over\cos{\pi t\over T}+\cos{\pi x\over T}}$ and $t_{\textrm{\tiny
P}}={\sin{\pi t\over T}\over\cos{\pi t\over T}+\cos{\pi x\over T}}$ at the AdS
boundary .

In Poincare coordinates, the geodesic length between a boundary point at
$t_{\textrm{\tiny P}}$ and the surface $Q$ is given by
$\log\left[{2t_{P}\over\epsilon}e^{\eta_{0}}\right]$ (refer to e.g., [34]). As
a result, we can reproduce Eq. (6). The condition that the connected geodesic
connecting two boundary points $(t_{\textrm{\tiny P}},x^{(1)}_{\textrm{\tiny
P}})$ and $(t_{\textrm{\tiny P}},x^{(2)}_{\textrm{\tiny P}})$ does not touch
the surface $Q$ is given by

$|x^{(2)}_{\textrm{\tiny P}}-x^{(1)}_{\textrm{\tiny P}}|<2z^{*}_{\textrm{\tiny
P}}\,,$ (36)

where $z^{*}_{\textrm{\tiny P}}$ denotes the value of $z_{\textrm{\tiny P}}$
at the time $t_{\textrm{\tiny P}}$ on the surface $Q$, i.e.,
$z^{*}_{\textrm{\tiny P}}={t_{\textrm{\tiny P}}\over\cosh\eta_{0}}$. This
explains the condition given in Eq. (13).