# Channeling quantum criticality

Yijian Zou  Stanford Institute for Theoretical Physics, Stanford University,
Stanford, CA 94305, USA  Shengqi Sang  Perimeter Institute for Theoretical
Physics, Waterloo, Ontario N2L 2Y5, Canada  Timothy H. Hsieh  Perimeter
Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

###### Abstract

We analyze the effect of decoherence, modelled by local quantum channels, on
quantum critical states and we find universal properties of the resulting
mixed state’s entanglement, both between system and environment and within the
system. Renyi entropies exhibit volume law scaling with a subleading constant
governed by a “$g$-function” in conformal field theory (CFT), allowing us to
define a notion of renormalization group (RG) flow (or “phase transitions”)
between quantum channels. We also find that the entropy of a subsystem in the
decohered state has a subleading logarithmic scaling with subsystem size, and
we relate it to correlation functions of boundary condition changing operators
in the CFT. Finally, we find that the subsystem entanglement negativity, a
measure of quantum correlations within mixed states, can exhibit log scaling
or area law based on the RG flow. We illustrate these phenomena for the
critical ground state of the transverse-field Ising model, in which we
identify four RG fixed points of dephasing channels and verify the RG flow
numerically. Our results are relevant to quantum critical states realized on
noisy quantum simulators, in which our predicted entanglement scaling can be
probed via shadow tomography methods.

_Introduction –_ Quantum devices have advanced significantly to the point of
challenging classical computers in sampling and simulation tasks [1, 2, 3, 4,
5]. In the context of quantum many-body physics, they are promising
experimental platforms for realizing long-range entangled quantum phases of
matter including topological order [6, 7] and quantum critical states [8].
Furthermore, their controllability at the single qubit level enables the
exploration of quantum dynamics beyond simple time evolution with a
Hamiltonian, and one prominent example is dynamics involving both unitary
evolution and projective measurements [9, 10, 11, 12, 13, 14, 15].
Nevertheless, quantum devices are noisy and any states prepared are subject to
decoherence. One important question is which properties of quantum matter
survive in a noisy environment, and how to understand any residual order in
the mixed state? Recently, substantial progress has been made in understanding
mixed state versions of gapped quantum systems including symmetry protected
topological phases and topological order [16, 17, 18, 19, 20, 21, 22].

In this work, we study quantum critical states under decoherence, modelled by
local quantum channels. A point of inspiration is prior work [23, 24, 25, 26]
which found that measurements (with outcomes recorded) can have a significant
effect on quantum critical states, in some cases boosting long range
correlation. However, verifying such effects typically requires post-selecting
on monitored pure state trajectories. In contrast, quantum channels can be
viewed as environmental measurement in which the outcomes are averaged over,
and we are interested in properties of the resulting mixed state, thus
avoiding the need for post-selection.

Naively one might expect that local quantum channels cannot induce new
universality classes since they can be represented as finite depth unitaries
acting on an enlarged Hilbert space, and such finite depth unitaries cannot
change the nature of long-distance correlations. While this is true for local,
linear observables of the state, non-linear quantities like entanglement
measures can be significantly affected by local channels.

Indeed, a key finding of our work is that for entanglement measures like Renyi
entropies and entanglement negativity, local quantum channels acting on
quantum critical states can drive phase transitions or more technically,
renormalization group (RG) flows, between different conformal fixed points
labeling different quantum channels. More specifically, we find that in
computing such measures, quantum channels map to boundary conditions for
multiple copies of the original conformal field theory (CFT) describing the
critical state. Under coarse-graining, such boundary conditions can flow to a
variety of conformal fixed points whose classification can be much richer than
that of the single copy CFT.

Concretely, we find that the Renyi entropy of the decohered state, which
quantifies the entanglement between system and environment, is extensive but
has a subleading term dictated by a “g-function” in CFT. This g-function
characterizes the channel with respect to Renyi entropy and dictates the
direction of RG flow between two channels: a channel with a larger $g$ can
flow to a channel with smaller $g$ if weakly perturbed by the latter.
Moreover, near a RG fixed point, the $g$-function has a universal scaling form
determined by critical exponents. One natural consequence of our formalism is
that the Renyi entropy of a subsystem $A$ of the decohered state has a
subleading logarithmic scaling with $|A|$ and its coefficient is given by
scaling dimensions of certain boundary condition changing operators.
Furthermore, Renyi negativity of the subsystem (a measure of entanglement
within the mixed state) can either obey logarithmic scaling or area law based
on the universal properties of the RG fixed point.

We demonstrate all these features for the one dimensional transverse field
Ising critical point and explicitly illustrate the RG flow between different
quantum channels. These universal aspects of entanglement and entropy in
channeled criticality can be probed in quantum simulators using shadow
tomography methods [27, 28].

_Quantum channels–_ We first review useful facts about quantum channels [29].
A quantum channel represents the most general quantum process including
decoherence. Formally, it is a completely positive trace-preserving map
$\rho\rightarrow\mathcal{N}(\rho)$ acting on density matrices. For a quantum
channel, one may define its dual-channel $\mathcal{N}^{*}$, which is a linear
map on operators satisfying
$\textrm{Tr}(\rho\mathcal{N}^{*}(O))=\textrm{Tr}(\mathcal{N}(\rho)O),\ \forall
O$. The relation between $\mathcal{N}$ and $\mathcal{N}^{*}$ is illustrated in
Fig. 1(a).

Here we consider a spatially uncorrelated noise model defined on a qubit
chain. The noise can be modeled as a product of single-qubit channels
$\mathcal{N}=\otimes_{j=1}^{L}\mathcal{N}_{j}$ where each $\mathcal{N}_{j}$ is
a channel acting on the $j^{\text{th}}$-qubit.

One example we focus on is the dephasing channel
$\mathcal{N}_{j}=\mathcal{D}_{p,\vec{v}}$, which represents an environment-
qubit coupling along the
$\sigma_{\vec{v}}:=v_{x}\sigma_{x}+v_{y}\sigma_{y}+v_{z}\sigma_{z}$ direction.
The action of $\mathcal{D}_{p,\vec{v}}$ on a density matrix is

$\mathcal{D}_{p,\vec{v}}(\rho)=\left(1-\frac{p}{2}\right)\rho+\frac{p}{2}\sigma_{\vec{v}}~{}\rho~{}\sigma_{\vec{v}}.$
(1)

The channel is self-dual, i.e.,
$\mathcal{D}^{*}_{p,\vec{v}}=\mathcal{D}_{p,\vec{v}}$. The parameter
$p\in[0,1]$ controls the strength of the noise: when $p=0$,
$\mathcal{D}_{p,\vec{v}}$ is the identity channel and leaves states unchanged;
while if $p=1$, $\mathcal{D}_{p,\vec{v}}$ turns any quantum state into a
classical ensemble.

_General formalism mapping information-theoretic quantities to quantum quench
problem–_ We consider a critical ground state $|\psi\rangle$ under decoherence
of the above type, resulting in density matrix
$\rho=\mathcal{N}(\ket{\psi}\bra{\psi})$. We study information-theoretic
quantities including $n^{\text{th}}$-Renyi entropy and entanglement
negativity, and we map these quantities to quantum quench problems in
$\text{CFT}^{\otimes 2n}$, i.e. a $2n-$copied CFT.

Figure 1:  (a) Diagrammatic representation of a quantum channel and its dual
channel’s action. (b) Top: graphical representation of tensor $\tau_{n}$
($n=3$) that forward permutes replicas. Each thick leg represents a $n$-copied
original Hilbert space. Bottom: The two expressions of $Z^{(n)}_{A}$ in Eqs.
(2) and (3). (c) Top: path integral representation of Eq. (4), used for
computing Renyi entropy. Note that the denominator is $\langle\psi^{\otimes
n}|\psi^{\otimes n}\rangle=1$. Bottom: path integral representation of the
analogous object for Renyi negativity Eq. (5).

We start with the Renyi entropy of a subsystem $A$:
$S^{(n)}_{A}(\rho):=\frac{1}{1-n}\log\textrm{Tr}(\rho_{A}^{n}),$ where
$\rho_{A}=\textrm{Tr}_{\bar{A}}\rho$. This is a measure of both quantum and
classical correlation between $A$ and rest of the system $\bar{A}$ together
with the environment. It reduces to the von-Neumann entropy after taking the
replica limit $n\rightarrow 1$. The partition function
$Z^{(n)}_{A}:=\textrm{Tr}(\rho_{A}^{n})$ can be rewritten as:

$Z^{(n)}_{A}=\textrm{Tr}(\rho^{\otimes
n}\tau_{n,A})=\textrm{Tr}(\mathcal{N}(|\psi\rangle\langle\psi|)^{\otimes
n}\tau_{n,A}),$ (2)

where $\tau_{n,A}=\prod_{j\in A}\tau_{n,j}$ forward permutes the replicas for
sites in $A$, and acts as identity for sites in $\bar{A}$. Next, we use the
dual channel to rewrite

$Z^{(n)}_{A}=\textrm{Tr}((|\psi\rangle\langle\psi|)^{\otimes
n}B_{\mathcal{N},A}),$ (3)

where $B_{\mathcal{N},A}:=\mathcal{N}^{*\otimes n}(\tau_{n,A})=\otimes_{j\in
A}\mathcal{N}^{*\otimes n}_{j}(\tau_{n,j})$. The two ways to write the
expression are graphically shown in Fig. 1(b). Finally, we use the standard
folding trick for defect CFT [30] to treat each operator $O$ as a state
$|O\rangle\rangle$ in the doubled Hilbert space, whose inner product is
defined as $\langle\langle
O_{1}|O_{2}\rangle\rangle=\textrm{Tr}(O^{\dagger}_{1}O_{2})$. Thus, under the
folding,

$Z^{(n)}_{A}=\langle\langle(\psi\otimes\psi^{*})^{\otimes
n}|B_{\mathcal{N},A}I_{\bar{A}}\rangle\rangle.$ (4)

The bra-state $|(\psi\otimes\psi^{*})^{\otimes n}\rangle\rangle$ is $2n$
copies of the critical state, while the ket-state
$|B_{\mathcal{N}}\rangle\rangle$ is a product state, since the operator
$B_{\mathcal{N},A}$ factorizes among $j$. Thus, we have reformulated the Renyi
entropy as a quantum quench problem from a product state
$|B_{\mathcal{N}}\rangle\rangle$ to a $2n$-copied CFT, whose path integral
representation on the spacetime manifold is shown in Fig. 1(c, top).

Figure 2: $g$-function (universal subleading constant of Renyi entropy) for
complete dephasing channel $\mathcal{D}_{p=1,\vec{v}}$ applied to critical
Ising ground state, where
$\vec{v}=(\sin(\pi\tilde{\theta}/2),0,\cos(\pi\tilde{\theta}/2))$ in the $XZ$
plane. Left: $g(\tilde{\theta},L)$ for different $L$. Center: Data collapse
for $0\leq\tilde{\theta}\leq 0.6$, indicating RG flow from $\mathcal{D}_{z}$
to $\mathcal{D}_{zx}$. Right: Data collapse for $0.95\leq\tilde{\theta}\leq
1$, indicating RG flow from $\mathcal{D}_{x}$ to $\mathcal{D}_{zx}$.

The other quantity that we consider is $n$-th ($n$ is odd) Renyi entanglement
negativity
$N^{(n)}_{A}(\rho):=\frac{1}{1-n}\log\frac{\textrm{Tr}(\\{\rho^{T_{A}}\\}^{n})}{\textrm{Tr}(\rho^{n})}$
for a subsystem $A$, where $(\cdot)^{T_{A}}$ is the partial-transpose
operation that swaps the bra and ket states in $\rho$ for sites within $A$.
$N_{A}^{(n)}$ is a measure of quantum correlation between $A$ and the rest of
the system $\bar{A}$ [31, 32] . A similar derivation shows that

$N^{(n)}_{A}(\rho)=\frac{1}{1-n}\log\frac{\langle\langle(\psi\otimes\psi^{*})^{\otimes
n}|B_{\mathcal{N},A}\tilde{B}_{\mathcal{N},\bar{A}}\rangle\rangle}{\langle\langle(\psi\otimes\psi^{*})^{\otimes
n}|B_{\mathcal{N},A\cup\bar{A}}\rangle\rangle}.$ (5)

where
$\tilde{B}_{\mathcal{N},\bar{A}}=\otimes_{j\in\bar{A}}\mathcal{N}^{*\otimes
n}_{j}(\tau^{-1}_{n,j})$. Thus the calculation of $N_{A}^{(n)}$ is mapped to a
quantum quench from another product initial state to the same $2n$-copied CFT,
see Fig. 1(c, bottom). In such quantum quench problems, it has been shown [33,
34, 35, 36] that a short-range correlated state can be described by a
conformal boundary condition at long distances. We will assume that this is
the case for product states $|B_{\mathcal{N}}\rangle\rangle$ and
$|\tilde{B}_{\mathcal{N}}\rangle\rangle$ 111See appendix for more comments on
the applicability of conformal boundary conditions to quantum quench
problems..

_The $g$ function–_ We now focus on the simplest quantity

$S^{(n)}(\rho)=\frac{1}{1-n}\log Z^{(n)}(L),$ (6)

where $L$ is the total system size. This is the Renyi entropy of the whole
system, or alternatively, the Renyi entanglement entropy between system and
environment. In a boundary CFT, the partition function $Z^{(n)}(L)$ depends on
UV details, but one can define a UV-independent $g$-function [38, 39, 40],

$\log g^{(n)}(L)=\left(1-L\frac{d}{dL}\right)\log Z^{(n)}(L).$ (7)

which plays a similar role for boundary RG flow as the central charge
$c$-function plays for bulk RG flow. It satisfies the following two
properties. First, $\log g^{(n)}(L)$ is a montonically decreasing function of
$L$. Second, for conformal boundary conditions (RG fixed points), $\log
g^{(n)}(L)=\log g^{(n)}$ is independent of $L$, which equals the universal
Affleck-Ludwig boundary entropy [41]. At these fixed points, solving the
differential equation (7) gives $\log Z^{(n)}(L)=(1-n)\alpha^{(n)}L+\log
g^{(n)}$, where $(1-n)\alpha^{(n)}$ is a non-universal integration constant.
One of our main results

$S^{(n)}(\rho)=\alpha^{(n)}L-\frac{1}{n-1}\log g^{(n)}_{\mathcal{N}}$ (8)

then follows from Eq. (6). Given a lattice wavefunction, we may compute
$g^{(n)}_{\mathcal{N}}(L)$ numerically using Eq. (7). We identify a channel
$\mathcal{N}$ as a RG fixed point if $g^{(n)}_{\mathcal{N}}(L)$ becomes a
constant $g^{(n)}_{\mathcal{N}}$ as $L$ exceeds a few lattice spacings. The
identity channel $\mathcal{I}$ is always a RG fixed point with $\log
g^{(n)}_{\mathcal{I}}=0$.

_RG flow near a fixed point–_ Let $\mathcal{N}_{p}$ be a family of quantum
channels parameterized by $p$ (e.g, $p$ in the dephasing channel
$\mathcal{D}_{p,\vec{v}}$). We denote the $g$ function of $\mathcal{N}_{p}$ as
$g^{(n)}_{p}(L)$. Let $p=p_{c}$ be a RG fixed point with the $g$-function
$g_{c}$. Near $p=p_{c}$, the RG flow equation can be linearized:
$\frac{dp}{d\log L}=(p-p_{c})/\nu$, which implies $|p(L)-p_{c}|\sim L^{1/\nu}$
for the effective $p(L)$ at length scale $L$. The $g$-function only depends on
the latter, and thus

$g^{(n)}_{p}(L)=\bar{g}^{(n)}(|p-p_{c}|L^{1/\nu}).$ (9)

$\nu$ is the analog of the “correlation length exponent” in critical
phenomena. If $\nu>0$, then the fixed point $\mathcal{N}_{p_{c}}$ is unstable,
and it flows into another fixed point with the $g$-function
$g^{\prime}<g_{c}$. In such cases, the scaling function $\bar{g}^{(n)}$ is a
monotonically decreasing function with $\bar{g}^{(n)}(0)=g_{c}$ and
$\bar{g}^{(n)}(\infty)=g^{\prime}$.

Figure 3: Fixed-point channels for Ising model and their universal properties.
Left: $g$-function of the four fixed points among dephasing channels and their
RG flow. The dashed line indicates a continuous set of fixed points and the
solid lines indicate direction of the RG flow. Center: $I^{(2)}(A,\bar{A})$ of
a single interval at the four fixed points. All four slopes are estimated to
be close to $4\Delta^{(2)}_{\mathcal{I}\mathcal{N}}\approx 1/4$. Right:
$I^{(2)}(A,B)$ of a two disjoint intervals at the four fixed points. The
scaling dimension $\Delta^{(2)}_{O}$ is close to $1/4,1,1/8,1/4$ for the four
fixed points $\mathcal{I},\mathcal{D}_{z},\mathcal{D}_{zx},\mathcal{D}_{x}$,
respectively.

_Subsystem entropy–_ Now we consider the entropy of a subsystem, whose
correponding quantum quench problem is from a spatially inhomogeneous initial
state, see Fig. 1(c, top). Such initial state can be described in CFT by
inserting boundary condition changing operator
$\phi^{(n)}_{\mathcal{I}\mathcal{N}}$ and
$\phi^{(n)}_{\mathcal{N}\mathcal{I}}=\phi^{(n)*}_{\mathcal{I}\mathcal{N}}$ at
the intersections [42, 43, 44]. There are $2m$ insertions if $A$ contains $m$
disjoint intervals. In particular, if $A$ is a single interval with length
$L_{A}$, then the partition function $Z^{(n)}(L)$ is

$\langle\phi^{(n)}_{\mathcal{I}\mathcal{N}}(0)\phi^{(n)*}_{\mathcal{I}\mathcal{N}}(L_{A})\rangle=C\left(\frac{L}{\pi}\sin\frac{\pi
L_{A}}{L}\right)^{-2\Delta^{(n)}_{\mathcal{I}\mathcal{N}}},$ (10)

where $\Delta^{(n)}_{\mathcal{I}\mathcal{N}}$ is the scaling dimension of
$\phi^{(n)}_{\mathcal{I}\mathcal{N}}$ and $C$ is a normalization constant. Eq.
(10) together with Eq. (6) gives another main result

$S^{(n)}(\rho_{A})=\alpha^{(n)}L_{A}+\frac{2\Delta^{(n)}_{\mathcal{I}\mathcal{N}}}{n-1}\log\left(\frac{L}{\pi}\sin\left(\frac{\pi
L_{A}}{L}\right)\right)+O(1).$ (11)

In order to numerically verify the formula, it is useful to consider the Renyi
mutual information
$I^{(n)}(A,B)=S^{(n)}(\rho_{A})+S^{(n)}(\rho_{B})-S^{(n)}(\rho_{AB})$.
Choosing $B=\bar{A}$ to be the complement of $A$, we obtain

$I^{(n)}(A,\bar{A})=\frac{4\Delta^{(n)}_{\mathcal{I}\mathcal{N}}}{n-1}\log\left(\frac{L}{\pi}\sin\left(\frac{\pi
L_{A}}{L}\right)\right)+O(1),$ (12)

in which the volume law pieces have cancelled each other. Similar results hold
for Renyi negativity $N^{(n)}_{A}(\rho)$, except that the boundary condition
changing operator is different.

For two disjoint intervals $A=[x_{1},x_{2}],B=[x_{3},x_{4}]$, we can show that
$I^{(n)}(A,B)$ is only a function of the cross ratio
$\eta=(X_{12}X_{34})/(X_{13}X_{24})$, where $X_{ij}=\sin(\pi|x_{i}-x_{j}|/L)$.
At small $\eta$, we may fuse the two boundary condition changing operators
$\phi^{(n)}_{\mathcal{I}\mathcal{N}}\times\phi^{(n)*}_{\mathcal{I}\mathcal{N}}=I+O^{(n)}+\cdots$,
where $O^{(n)}$ is the second lowest operator in the operator product
expansion. This implies the scaling

$I^{(n)}(A,B)=\text{const.}\times\eta^{\Delta^{(n)}_{O}}~{}(\eta\ll 1).$ (13)

Note that $O^{(n)}$ is a boundary operator at the boundary condition
$|S_{\mathcal{I}}\rangle\rangle$, which we may unfold to get a bulk local
operator in the $n$-copied CFT. Thus, $\Delta^{(n)}_{O}$ must be a sum of $n$
scaling dimensions in the original CFT.

_Critical Ising model under dephasing noise–_ As an example, we study the
effect of dephasing noise on the ground state of the 1d transverse field Ising
model

$H=-\sum_{i=1}^{L}\sigma_{x,i}\sigma_{x,i+1}-\sum_{i=1}^{L}\sigma_{z,i}.$ (14)

For the dephasing noise $\mathcal{D}_{p,\vec{v}}$, we restrict the direction
to the $XZ$ plane, i.e.,
$\vec{v}=(\sin(\pi\tilde{\theta}/2),0,\cos(\pi\tilde{\theta}/2))$, where
$0\leq\tilde{\theta}\leq 1$. For Renyi index $n=2$, we find four RG fixed
points: (1) identity channel $\mathcal{I}=\mathcal{D}_{0,\vec{v}}$ with $\log
g_{\mathcal{I}}=0$, (2) complete dephasing in $Z$ direction $\mathcal{D}_{z}$
with $\log g_{z}=0$, (3) complete dephasing in $X$ direction $\mathcal{D}_{x}$
with $\log g_{x}=-\log 2$ and (4) complete dephasing $\mathcal{D}_{zx}$ at
$\tilde{\theta}\approx 0.8$, with $\log g_{zx}=-2\log 2$. For complete
dephasing, the entropy reduces to the “classical Shannon entropy” of Ref.
[45], which considered cases (2) and (3).

We also observe the RG flow between these four fixed points, which manifests
the monotonicity of the $g$-function. Near the $\mathcal{D}_{z}$ fixed point,
we can turn on a small perturbation to $\tilde{\theta}$ and observe the RG
flow to the $\mathcal{D}_{zx}$ fixed point. The universal data collapse
indicates the critical exponent is $\nu\approx 2.0$. Analogously, near the
$\mathcal{D}_{x}$ fixed point, we can turn on a perturbation to
$\tilde{\theta}$ and observe the RG flow to the $\mathcal{D}_{zx}$ fixed
point. The universal data collapse shows that $\nu\approx 1.0$. Another
example is the RG flow from $\mathcal{I}$ fixed point to complete dephasing
$\mathcal{D}_{zx}$ and $\mathcal{D}_{x}$ by turning on a small $p$, see
appendix. Finally, all dephasing channels $\mathcal{D}_{p,z}$ have the same
$\log g=0$, which suggests that they constitute a continuous family of fixed
points connected by marginal deformations. These RG flows are summarized in
Fig. 3. We have also checked that the RG flows in this example are the same
for higher Renyi indices such as $n=3$, although consistency betwen Renyi
indices may not hold generally 222In fact, we have numerically observed that
the RG flow for dephasing in $Y$ direction is not consistent across Renyi
indices. For $n=2$, a slight dephasing in $Y$ is a relevant perturbation while
for $n=3$ it appears to be an irrelevant perturbation..

The entropy of a subsystem at the four RG fixed points is shown in Fig. 3. For
a single interval, we find that they all satisfy Eq. (11) with the same
$\Delta^{(2)}_{\mathcal{I}\mathcal{N}}=1/16$. For the identity channel, the
operator $\phi^{(2)}_{\mathcal{I}\mathcal{N}}$ is the branch-point twist
operator, with the scaling dimension $c/8=1/16$. The fact that the scaling
dimensions coincide for dephasing channel and identity channel was also
observed numerically in Ref. [47]. In order to distinguish the channels, we
compute the Renyi mutual information $I^{(n)}(A,B)$ of two disjoint intervals.
We see that the scaling dimensions of $\Delta^{(2)}_{O}$ vary among the four
fixed points. These dimensions can be understood analytically in terms of
local operators in the two-copied Ising model, see appendix for details.

Finally, we study the Renyi negativity $N^{(3)}_{A}$ of a subsystem $A$ under
dephasing noise, see Fig. 4. For dephasing in $Z$ direction, we find a
logarithmic scaling with continuously changing coefficients, which is an
indication of a continuous set of RG fixed points. For dephasing in $X$
direction, even with a small strength $p$, the Renyi negativity obeys an area
law, in agreement with the previous observation that the channel flows to
complete dephasing.

Figure 4: Renyi negativity $N^{(3)}_{A}$ of a subsystem $A$ for the ground
state of critical Ising model ($L=64$) under various choices of dephasing
noise.

_Discussion–_ In this work we have established a notion of RG flow and
transitions between quantum channels acting on a critical wavefunction. We
have shown that a $g$-function gives the volume-independent part of the Renyi
entropy at the RG fixed points and established its monotonicity under the RG
flow. We have demonstrated the RG flow between quantum channels using the
example of the transverse field Ising model and dephasing noise, although
generalization to other noise models, such as depolarizing noise, is
straightforward and also yields interesting phenomena (see appendix for
details).

Our notion of RG flow can be naturally applied to measurement-induced
quantities of a critical wavefunction, such as those considered in Refs. [24,
25]. Another future direction is to classify the conformal boundary conditions
for the fixed-point quantum channels in the CFT and take the replica limit. It
is not yet known how this could be done for a generic CFT. Even within minimal
models only a few examples have been addressed [48]. Regarding the holographic
correspondence between a CFT on the boundary of a spacetime and a
gravitational theory in the bulk, it would be interesting to explore the
holographic dual of quantum channels acting on the boundary; many recent works
have considered coupling the boundary CFT to a bath, and perhaps our formalism
could be related to replica wormholes [49, 50, 51]. In another direction,
while this work only considered one layer of local quantum channels, it may
also be interesting to explore the entanglement dynamics of random local
channels [52] on quantum critical (and other long-range entangled) states.
Finally, we comment that experimental observation of our main results
involving Renyi entropy and entanglement negativity can be achieved on quantum
computers and simulators using recent techniques of shadow tomography [27,
28].

_Acknowledgements–_ We thank Runze Chi, Paolo Glorioso, Yaodong Li, Yuhan Liu,
Peter Lu , Alex May, Xiao-Liang Qi, Guifre Vidal, and Fei Yan for helpful
discussions. Y.Z. particularly thanks Ehud Altman, Samuel Garatt and Zack
Weinstein for explanation of their work Ref. [25]. We thank Jacob Lin and
Weicheng Ye for collaboration on a related project. S.S. and T.H. are
supported by Perimeter Institute and NSERC. Research at Perimeter Institute is
supported in part by the Government of Canada through the Department of
Innovation, Science and Economic Development Canada and by the Province of
Ontario through the Ministry of Colleges and Universities. Y. Z. is supported
by the Q-FARM fellowship at Stanford University.

_Note added:_ While completing this manuscript, we noticed a related
independent work [53].

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## Ising model with finite dephasing and depolarizing noise

In this appendix, we consider channels beyond fixed point channels. In
particular, we consider finite dephasing and depoloarizing channels and their
flow to the fixed points. The dephasing channel is defined by

$\mathcal{D}_{p,\vec{v}}(\rho)=\left(1-\frac{p}{2}\right)\rho+\frac{p}{2}\sigma_{\vec{v}}~{}\rho~{}\sigma_{\vec{v}}.$
(15)

We fix $\vec{v}$ and continuously change $p$ away from the fixed point
$p_{c}=0$. We compute the $g$-function for the second Renyi entropy. As shown
in Fig. 5, the $g$-function has a universal form

$g^{(2)}(p,L)=g^{(2)}(pL^{1/\nu}),$ (16)

where $\nu$ depends on the direction $\vec{v}$, indicating flow from
$\mathcal{I}$ to different IR fixed points. For flow to $\mathcal{D}_{zx}$, we
have $1/\nu\approx 0.8$, and for flow to $\mathcal{D}_{x}$, we have
$1/\nu\approx 0.7$.

Figure 5: $g$-function of dephasing channel $\mathcal{D}_{p,\vec{v}}$ where
$0\leq p\leq 0.3$. Left: $\tilde{\theta}=0.6$ in the $ZX$ direction,
indicating RG flow from $\mathcal{I}$ to $\mathcal{D}_{zx}$ Right:
$\tilde{\theta}=1$, indicating RG flow from $\mathcal{I}$ to
$\mathcal{D}_{x}$.

Finally, we consider a different class of noise channels, the depolarizing
noise, whose action on a single qubit is

$\Delta_{p}(\rho)=\left(1-p\right)\rho+\frac{p}{2}I.$ (17)

At $p=0$, the channel is the $\mathcal{I}$ channel, which is a fixed point
with $\log g=0$. At $p=1$, the channel turns any state into a maximally mixed
state, thus it is also a fixed point with $\log g=0$. We denote the latter
fixed point by $\Delta_{1}$. Surprisingly, we find another fixed point at
$p\approx 0.7$, which has $\log g^{(2)}\approx-\log 2$, see Fig. 6. We denote
this fixed point as $\Delta$. By turning on a small perturbation to $p$ at
either the $\mathcal{I}$ ($p=0$) or $\Delta_{1}$ ($p=1$) fixed point, the
channel flows to the new fixed point $\Delta$. The data collapse shows that
the critical exponent is $1/\nu\approx 0,7$ and $1/\nu\approx 0.9$,
respectively.

We suspect that the finite-depolarization fixed point $\Delta$ coincides with
the complete dephasing $\mathcal{D}_{x}$ fixed point for the ground state of
transverse-field Ising model. They both have $\log g^{(2)}\approx-\log 2$, and
the scaling dimension of the boundary condition changing operator
$\phi^{(2)}_{\mathcal{I}\mathcal{N}}$ are both approximately $1/16$.
Furthermore, the RG flow from $\mathcal{I}$ to both fixed point has the same
critical exponent $1/\nu\approx 0.7$, and the subsystem entropy both shows a
transient scaling with close transient scaling dimensions (see the section
below). We leave it as future work to determine whether $\Delta$ is the same
fixed point as $\mathcal{D}_{x}$ from the boundary CFT perspective.

Figure 6: $g$-function of depolarization channel $\Delta_{p}$. Left: $g(p,L)$
for different $L$. Center: Data collapse for $0\leq p\leq 0.2$, indicating RG
flow from $\mathcal{I}$ to $\Delta$. Right: Data collapse for $0.85\leq p\leq
1$, indicating RG flow from $\Delta_{1}$ to $\Delta$.

## Transient behavior of subsystem entropy

We have shown that at the fixed points the subsystem entropy satisfies a
simple formula

$I^{(n)}(A,\bar{A})=\frac{4\Delta^{(n)}_{\mathcal{I}\mathcal{N}}}{n-1}\log\left(\frac{L}{\pi}\sin\left(\frac{\pi
L_{A}}{L}\right)\right)+O(1).$ (18)

Away from the fixed points, the formula is expected to hold in the limit of
$L_{A}\rightarrow\infty,L\rightarrow\infty$ while keeping $L_{A}/L$ a $O(1)$
constant. However, when the channel is in the middle of a RG flow between
fixed points, interesting transient behaviors occur at finite sizes. We show
numerically that Eq. (18) still approximately holds, but with scaling
dimensions $\Delta^{(n)}_{\mathcal{I}\mathcal{N}}$ drifting with the system
sizes. We study two examples of RG flow in more detail below.

###  .1 RG flow from $\mathcal{D}_{x}$ to $\mathcal{D}_{zx}$

As a first example, we study the complete dephasing channel
$\mathcal{D}_{p,\vec{v}}$ with $p=1$ and
$\vec{v}=(\sin(\pi\tilde{\theta}/2),0,\cos(\pi\tilde{\theta}/2))$ acting on
the ground state of the critical Ising model. We choose $\tilde{\theta}$ close
to $1$ to observe a RG flow from the UV fixed point $\mathcal{D}_{x}$ to IR
fixed point $\mathcal{D}_{zx}$. We plot the Renyi mutual information
$I^{(2)}(A,\bar{A})$ in Fig. 7.

Figure 7: Renyi mutual information $I^{(2)}(A,\bar{A})$ for the complete
dephasing channel acting on the ground state of Ising model.

We see that at large $L_{A}$ and $L$ the subsystem entropy agrees with the IR
fixed point with $4\Delta^{(2)}_{\mathcal{I}\mathcal{N}}=1/4$. At smaller
sizes we see a transient scaling with
$4\Delta^{(2)}_{\mathcal{I}\mathcal{N}}=1/2$. The crossover between the two
scaling can be tuned by varying $\tilde{\theta}$. The length scales where we
observe the transient scaling become broader if we tune $\tilde{\theta}$
closer to the UV fixed point $\tilde{\theta}=1$, as the RG flow becomes
longer. We conjecture that the operator with
$4\Delta^{(2)}_{\mathcal{I}\mathcal{N}}=1/2$ is the subleading boundary
condition changing operator from $|B_{\mathcal{I}}\rangle\rangle$ to
$|B_{\mathcal{N}}\rangle\rangle$ at the IR fixed point, where
$\mathcal{N}=\mathcal{D}_{zx}$. We leave it for future work to identify these
operators in the replicated CFT.

###  .2 RG flow from $\mathcal{I}$ to $\Delta$

Next, we consider the RG flow from identity channel $\mathcal{I}$ to finite
depolarization $\Delta$ fixed point. We consider depolarization channel
$\Delta_{p}$ with $p=0.1,0.2,0.3$ and plot the Renyi mutual information
$I(A,\bar{A})$ in Fig. 8. Again, we see that there is a transient behavior at
small sizes near the UV fixed point. At $p=0.1$, Eq. (18) holds with
$4\Delta^{(2)}_{I\mathcal{N}}\approx 0.18$ for all sizes $L\leq 256$. At
larger $p$ and larger sizes, the IR scaling with
$4\Delta^{(2)}_{I\mathcal{N}}\approx 1/4$ starts to emerge. The appearance of
drifting scaling dimension at small sizes is similar to the previous example.

Figure 8: Renyi mutual information $I^{(2)}(A,\bar{A})$ for the depolarization
channel acting on the ground state of Ising model.

We note that the same transient behavior is observed for finite dephasing
channel $\mathcal{D}_{p,x}$. At small $p$ and small system sizes, e.g.,
$p=0.1$ and $L\leq 256$, we also observe that Eq. (18) holds with
$4\Delta^{(2)}_{I\mathcal{N}}\approx 0.18$. This provide additional evidence
that $\Delta$ and $\mathcal{D}_{x}$ may be the same fixed point, although a
more detailed field-theoretic computation must be done in order to show it.

## Free fermion with amplitude damping noise

In this appendix, we consider the free fermion CFT and a special instance of
Gaussian channel. The covariance matrix formalism makes it easy to access all
Renyi entropy and negativity. In particular, we can directly compute the Von
Neumann entanglement entropy and negativity, which is hard for a tensor
network state.

We consider the free Majorana fermion with Neveu-Schwarz boundary condition

$H=-2i\sum_{j=1}^{2L}\psi_{j}\psi_{j+1},$ (19)

where $\psi_{j}$ is the Majorana operator that satisfies
$\\{\psi_{j},\psi_{l}\\}=\delta_{jl}$, and $\psi_{2N+1}=-\psi_{1}$. The
Hamiltonian is equivalent to the transverse field Ising model under the
Jordan-Wigner transformation. Yet, a local channel in the spin representation
may not be a local channel in the fermion representation.

Here we use the covariance matrix technique to simulate the amplitude damping
channel, which is a Guassian channel. The covariance matrix, defined by

$M_{jl}=-i\textrm{Tr}((\psi_{j}\psi_{l}-\delta_{jl})\rho),$ (20)

fully characterizes a Guassian state $\rho$. One may introduce complex fermion
operators
$c_{j}=(\psi_{2j}-i\psi_{2j+1})/\sqrt{2},c^{\dagger}_{j}=(\psi_{2j}+i\psi_{2j+1})/\sqrt{2}$
and the number operator $c^{\dagger}_{j}c_{j}$. The amplitude damping channel
$\mathcal{A}_{p}$ for a single fermion can be realized by

$\mathcal{A}_{p}(\rho_{S})=\textrm{Tr}_{A}[U_{p,SA}(\rho_{S}\otimes|0\rangle_{A}\langle
0|)U^{\dagger}_{p,SA}],$ (21)

where $|0\rangle_{A}$ is the ground state of the number operator
$c^{\dagger}_{A}c_{A}$ of the ancilla, and

$U_{p,SA}=e^{\theta(c^{\dagger}_{S}c_{A}-c^{\dagger}_{A}c_{S})},~{}~{}p=\sin^{2}\theta.$
(22)

Such representation makes it clear that the amplitude damping channel is a
Gaussian channel. The channel on the total system is a product of single-
fermion channels, $\mathcal{N}_{p}=\otimes_{j}\mathcal{A}_{p,j}$. The
covariance matrix $M_{p}$ of the decohered state $\mathcal{N}_{p}(\rho)$ is

$M_{p}=(1-p)M+pM^{(0)},$ (23)

where $M$ is the covariance matrix of $\rho$ and $M^{(0)}$ is the covariance
matrix of $|0\rangle^{\otimes L}$ with nonzero entries
$M^{(0)}_{2i-1,2i}=-M^{(0)}_{2i,2i-1}=1$. Using the covariance matrix, one may
compute all the Renyi mutual information $I^{(n)}(A,B)$ and Renyi negativity
$\mathcal{E}^{(n)}(A,B)$ of any two subsystems. One can directly take the
replica limit to obtain the Von-Neumann mutual information $I(A,B)$ and the
logarithmic negativity $\mathcal{E}(A,B)$ of the decohered state. We note that
here the negativity is defined through partial time reversal rather than
partial transpose, following the computation of Refs. [54, 55].

As shown in the main text, we find that the amplitude damping channel forms a
continuous set of conformal fixed points with $\log g^{(n)}=0$ for any $n$.
The scaling dimension of the boundary condition changing operator
$\phi^{(n)}_{\mathcal{I}\mathcal{N}}$ also continuously changes with the
damping strength $p$, as is evident by the (Renyi) mutual information
$I^{(n)}(A,\bar{A})$. Likewise, the (Renyi) negativity
$\mathcal{E}^{(n)}(A,\bar{A})$ shows a similar logarithmic form with
continuously changing coefficients.

Figure 9: Mutual information (left) and negativity (right) for free fermion
critical state subject to amplitude damping channel of strength $p$.

## Analytical treatment of Renyi mutual information

Here we present analytical result of Renyi mutual information $I^{(n)}(A:B)$,
where $A$ and $B$ are of $O(1)$ size. We show that $I^{(n)}(A:B)$ decays as a
power law of the distance and the power is determined by the quantum channel.
For the case of dephasing channel and $|A|,|B|=1$, we further take the replica
limit $n\rightarrow 1$ and consider the Von-Neumann mutual information. Our
results provide a concrete example where any Renyi mutual information with
$n>1$ is not monotonic under local quantum channels, in sharp constrast with
the Von-Neumann mutual information.

###  .3 Renyi mutual information of two sites

Let $|A|=|B|=1$, then

$I^{(n)}(A:B)=\frac{1}{n-1}\log\frac{\langle\mathcal{N}^{*}(\tau_{n,A})\mathcal{N}^{*}(\tau_{n,B})\rangle}{\langle\mathcal{N}^{*}(\tau_{n,A})\rangle\langle\mathcal{N}^{*}(\tau_{n,B})\rangle},$
(24)

where the expectation value is taken on the ground state of $n$-copied CFT.
Let the distance of $A$ and $B$ be $l$ and let

$d=\frac{l}{\pi}\sin\frac{\pi l}{L}$ (25)

be the chord distance. We can expand the lattice operator
$B^{(n)}_{\mathcal{N}}=\mathcal{N}^{*}(\tau_{n})$ into a sum of CFT operators
in $n$-copied CFT,

$B^{(n)}_{\mathcal{N}}=a_{0}I+a_{1}O^{(n)}+\cdots,$ (26)

where the $\cdots$ represents operators with higher scaling dimensions. The
correlation function $\langle O^{(n)}(x)O^{(n)}(y)\rangle\sim
d^{-2\Delta^{(n)}_{O}}$, thus

$I^{(n)}(A:B)\sim d^{-2\Delta^{(n)}_{O}}.$ (27)

###  .4 Renyi index $n=2$

For complete dephasing, we compute the lattice operator and find their CFT
counterpart using the dictionary in Ref. [56]. Below we list results for
various choices of $\vec{v}$ in the dephasing channel. For the fixed points
$\mathcal{D}_{z}$, $\mathcal{D}_{x}$, $\mathcal{D}_{zx}$, we find that the
leading CFT operator to $O^{(2)}$ is $\varepsilon\otimes
I+I\otimes\varepsilon$, $\sigma\otimes\sigma$ and $\sigma\otimes
I+I\otimes\sigma$, with scaling dimension $1,1/4,1/8$ respectively. This
agrees with the numerical result shown in the main text,
$I^{(2)}(A,B)\sim\eta^{\Delta^{(n)}_{O}}$. However, we note an important
difference: in the main text we consider $|A|$ and $|B|$ to be much larger
than the lattice spacing rather than $|A|=|B|=1$. It is not a priori the case
that the scaling operators $O^{(n)}$ are the same for the two cases. Yet for
the dephasing fixed points we observe that they coincide.

We also note that the exponent $1/8$ for $\mathcal{D}_{zx}$ is smaller than
that of the identity channel, meaning that the channel increases the Renyi
mutual information. As we show shortly, this happens for all Renyi indices
$n>1$, but ceases to hold at $n=1$.

Dephasing direction $\vec{v}$ |  $O^{(2)}$ (leading) | CFT operator | $\Delta^{(2)}_{O}$
---|---|---|---
$\vec{x}$ | $X\otimes X$ | $\sigma\otimes\sigma$ | $1/4$
$\vec{y}$ | $Y\otimes Y$ | $\partial_{\tau}\sigma\otimes\partial_{\tau}\sigma$ | $9/4$
$\vec{z}$ | $Z\otimes Z$ | $\varepsilon\otimes I+I\otimes\varepsilon$ | $1$
Generic $(v_{x},v_{y},0)$ | $X\otimes X$ | $\sigma\otimes\sigma$ | $1/4$
Generic $(v_{x},0,v_{z})$ | $X\otimes Z+Z\otimes X$ | $\sigma\otimes I+I\otimes\sigma$ | $1/8$
Generic $(0,v_{y},v_{z})$ | $Z\otimes Z$ | $\varepsilon\otimes I+I\otimes\varepsilon$ | $1$
Generic $(v_{x},v_{y},v_{z})$ | $X\otimes Z+Z\otimes X$ | $\sigma\otimes I+I\otimes\sigma$ | $1/8$
Table 1: Scaling dimension of $O^{(2)}$ under complete dephasing ($p=1$) in
different directions. (“Generic” means all components are nonzero.)

###  .5 Higher Renyi index

For general replica number $n\geq 2$, we first consider the $\vec{z}$
direction dephasing,

$B^{(n)}_{\mathcal{N}}:=\mathcal{D}^{*\otimes
n}_{p=1,\vec{z}}(\tau_{n})=\sum_{i=0}^{1}|i^{\otimes n}\rangle\langle
i^{\otimes n}|$ (28)

In Pauli basis, this is

$\mathcal{D}^{*\otimes n}_{p=1,\vec{z}}(\tau_{n})=\frac{1}{2^{n-1}}\sum_{s\in
S_{\vec{z}}}s,$ (29)

where $S_{\vec{z}}$ is the stabilizer group consisting of even number of
product of Pauli $Z$’s. Note that this expression also holds if we change $z$
to any $\vec{v}$ and correspondingly the stabilizer group $S_{\vec{z}}$ to
$S_{\vec{v}}$ (which is isomorphic to $S_{\vec{z}}$ by changing $Z$ to
$\vec{v}\cdot\vec{\sigma}$). Now we consider the correlation function of this
operator on $A$ and $B$. Taking into account that the correlation function of
a single Pauli string factorizes into product of correlation functions of one
replica, we obtain

$\displaystyle\langle\psi^{\otimes
n}|B^{(n)}_{\mathcal{N}}(x)B^{(n)}_{\mathcal{N}}(y)|\psi^{\otimes n}\rangle$
$\displaystyle=$
$\displaystyle((1+x)^{2}+C)^{n}+((1-x)^{2}+C)^{n}+2(1-x^{2}-C)^{n}$ (30)
$\displaystyle\langle\psi^{\otimes n}|B^{(n)}_{\mathcal{N}}(x)|\psi^{\otimes
n}\rangle\langle\psi^{\otimes n}|B^{(n)}_{\mathcal{N}}(y)|\psi^{\otimes
n}\rangle$ $\displaystyle=$
$\displaystyle(1+x)^{2n}+(1-x)^{2n}+2(1-x^{2})^{n}$ (31)

where
$x=\langle\sigma_{\vec{v}}\rangle,C=\langle\sigma_{\vec{v}}(x)\sigma_{\vec{v}}(y)\rangle_{c}$
is the connected correlation function of the ground state, and
$\sigma_{\vec{v}}\equiv\vec{v}\cdot\vec{\sigma}$. The Renyi mutual information
is given by

$I^{(n)}(A,B)=\frac{1}{n-1}\log\frac{\langle\psi^{\otimes
n}|B^{(n)}_{\mathcal{N}}(x)B^{(n)}_{\mathcal{N}}(y)|\psi^{\otimes
n}\rangle}{\langle\psi^{\otimes n}|B^{(n)}_{\mathcal{N}}(x)|\psi^{\otimes
n}\rangle\langle\psi^{\otimes n}|B^{(n)}_{\mathcal{N}}(y)|\psi^{\otimes
n}\rangle}.$ (32)

In the case of $n=2$, this reduces to

$I^{(2)}(A,B)=\log\left(1+\frac{x^{2}C+C^{2}}{(1+x^{2})^{2}}\right)$ (33)

In the limit where $C\ll 1$, we obtain

$I^{(2)}(A,B)=\frac{x^{2}C+C^{2}}{(1+x^{2})^{2}}~{}~{}(C\ll 1)$ (34)

This is in agreement with our previous subsections. If $x=0$, that is, if the
dephasing direction has $v_{x}=0$, then $I^{(2)}$ is proportional to the
square of correlation function. However, if $x\neq 0$, then $I^{(2)}$ gets
promoted to linear in the correlation function. This means that dephasing can
actually increase the second Renyi mutual information (e.g., for
$\mathcal{D}_{zx}$). More generally, if $n\geq 2$, we have

$I^{(n)}(A,B)=\frac{1}{n-1}\log\left(1+\frac{n((1+x)^{n-1}-(1-x)^{n-1})^{2}C+O(C^{2})}{((1+x)^{n}+(1-x)^{n})^{2}}\right),$
(35)

which reduces to

$I^{(n)}(A,B)=\frac{n}{n-1}\frac{((1+x)^{n-1}-(1-x)^{n-1})^{2}}{((1+x)^{n}+(1-x)^{n})^{2}}C+O(C^{2}),~{}~{}~{}(n\geq
2,C\ll 1)$ (36)

Thus, for all Renyi mutual information $I^{(n)}(A,B)$ with $n>1$, it can be
promoted to linear in correlation function by acting with a dephasing channel
$\mathcal{D}_{p=1,\vec{v}}$ if $C:=\langle\sigma_{\vec{v}}\rangle\neq 0$.

###  .6 Replica limit $n\rightarrow 1$

One important point is that the Von Neumann mutual information cannot increase
under noise channels. To see this, we use the data processing inequality in
quantum information theory

$S(\mathcal{N}(\rho)|\mathcal{N(\sigma)})\leq S(\rho|\sigma)$ (37)

Using the fact that $I(A,B)=S(\rho_{AB}|\rho_{A}\otimes\rho_{B})$ and the fact
that $\mathcal{N}=\mathcal{N}_{A}\otimes\mathcal{N}_{B}$, we obtain

$I(A,B)_{\mathcal{N}(\rho)}\leq I(A,B)_{\rho}.$ (38)

The data processing inequality does not apply to the Renyi entropies, so the
Renyi mutual information is not monotonic under product channels.

The information-theoretic argument dictates that the Von Neumann mutual
information after applying the noise channel should be at least quadratic in
the original correlation functions. In order to see this, we take the replica
limit $n\rightarrow 1$. The result is

$I(A,B)=\left(H\left(\frac{(1+x)^{2}+C}{4}\right)+H\left(\frac{(1-x)^{2}+C}{4}\right)+2H\left(\frac{1-x^{2}-C}{4}\right)\right)-(C\leftarrow
0)$ (39)

where $H(x)=x\log x$. Note that the approximation with $C\ll 1$ must come
after taking the replica limit. Using the Taylor expansion around $C=0$, and
the derivatives $H^{\prime}(x)=1+\log x,~{}H^{\prime\prime}(x)=1/x$, we obtain

$I(A,B)=\frac{C^{2}}{2(1-x^{2})^{2}}~{}~{}(C\ll 1).$ (40)

Note that the linear term in $C$ exactly cancels. Thus we expect that

$I(A,B)=O(d^{-4\Delta_{\vec{v}\cdot\vec{\sigma}}})$ (41)

under complete dephasing. See the table below for a list of scaling dimensions
for different dephasing channels. We no longer observe an increase in the Von-
Neumann mutual information, as required by the data processing inequality.

Dephasing direction $\vec{v}$ |  $\vec{v}\cdot\vec{\sigma}$ (leading) | CFT operator | $4\Delta_{\vec{v}\cdot\vec{\sigma}}$
---|---|---|---
$\vec{x}$ | $X$ | $\sigma$ | $1/2$
$\vec{y}$ | $Y$ | $\partial_{\tau}\sigma$ | $9/2$
$\vec{z}$ | $Z$ | $\varepsilon$ | $4$
Generic $(v_{x},v_{y},0)$ | $X$ | $\sigma$ | $1/2$
Generic $(v_{x},0,v_{z})$ | $X$ | $\sigma$ | $1/2$
Generic $(0,v_{y},v_{z})$ | $Z$ | $\varepsilon$ | $4$
Generic $(v_{x},v_{y},v_{z})$ | $X$ | $\sigma$ | $1/2$
Table 2: Mutual information scaling under complete dephasing ($p=1$) in
different directions.

## Applicability of conformal boundary conditions

In a global quantum quench problem to a critical Hamiltonian, not all short-
range entangled initial state flows to a conformal boundary state. Yet, it has
been argued by Cardy [36] that the ground state of the CFT Hamiltonian
perturbed by a relevant deformation flows to a Cardy state, a conformal
boundary state subject to physical constraints.

In this appendix, we show that the state $|B_{\mathcal{N}}\rangle\rangle$ for
a dephasing channel considered in the main text can be associated with such
relevant deformations to the $2n$-copied Ising model. Thus, if the above
Cardy’s conjecture holds, we can argue that $|B_{\mathcal{N}}\rangle\rangle$
in the main text all flows to conformal boundary states. We also provide an
example of $\mathcal{N}$ such that the argument fails and such that
$|B_{\mathcal{N}}\rangle\rangle$ does not flow to a conformal boundary state.
We will stay within replica index $n=2$ in this appendix.

To start with, we consider the case where $\mathcal{N}=\mathcal{D}_{p=1,x}$.
In this case the operator

$B^{(2)}_{\mathcal{N}}:=\mathcal{N}^{*\otimes
2}(\tau_{2})=\frac{1}{2}(I+X\otimes X)$ (42)

Under the folding, the operator becomes a state in 4 replicas, which is

$|B^{(2)}_{\mathcal{N}}\rangle\rangle=\frac{1}{2}(|++++\rangle+|----\rangle),$
(43)

where $|+\rangle$ or $|-\rangle$ are the eigenstates of $X$ in one replica.
One parent Hamiltonian of $|B^{(2)}_{\mathcal{N}}\rangle\rangle$ is

$H^{(2)}_{int,\mathcal{N}}=X\otimes X\otimes I\otimes I+I\otimes X\otimes
X\otimes I+I\otimes I\otimes X\otimes X.$ (44)

The state $|B^{(2)}_{\mathcal{N}}\rangle\rangle$ is its unique ground state
upon fixing the total parity. Note that this Hamiltonian is defined on one
lattice site across the four replicas. Now we consider the replicated CFT
Hamiltonian

$H^{(2)}_{C}=H_{C}\otimes I\otimes I\otimes I+I\otimes H_{C}\otimes I\otimes
I+I\otimes I\otimes H_{C}\otimes I+I\otimes I\otimes I\otimes H_{C},$ (45)

where $H_{C}$ is the Hamiltonian for one copy of critical Ising model. Now we
consider the perturbation of $H^{(2)}_{C}$ by the parent Hamiltonian,

$H^{(2)}=H^{(2)}_{C}+\lambda\sum_{j=1}^{L}H^{(2)}_{int,\mathcal{N},j}$ (46)

The interaction term $H^{(2)}_{int,\mathcal{N}}$ is a relevant deformation
since its scaling dimension is $2\Delta_{\sigma}=1/4<2$. At long wavelengths,
the coupling $\lambda$ flows to infinity, and the ground state of $H^{(2)}$ is
qualitatively the same as the ground state of $H^{(2)}_{int,\mathcal{N}}$,
which is $|B^{(2)}_{\mathcal{N}}\rangle\rangle$. Thus, if the aforementioned
Cardy’s conjecture holds, then $|B^{(2)}_{\mathcal{N}}\rangle\rangle$ can be
described by a conformal boundary state.

In order to generalize the argument to other dephasing channels, we just have
to replace $X$ in the parent Hamiltonian $H^{(2)}_{int,\mathcal{N}}$ by
$\sigma_{\vec{v}}=\vec{v}\cdot\vec{\sigma}$. If $\vec{v}$ is in the $XZ$
plane, then all terms in the parent Hamiltonian are relevant (they are either
$\sigma\otimes\sigma$, $\sigma\otimes I$ or $\sigma\otimes\varepsilon$), and
the argument above goes through. We note in passing that for all Renyi index
$n\geq 2$ the argument also holds. However, if $\vec{v}$ is in the $YZ$ plane,
then not all terms are relevant. For example, the term $Y\otimes Y\otimes
I\otimes I$ has a scaling dimension $2\Delta_{\partial\sigma}=9/4>2$. Then it
is no longer guaranteed that the boundary state
$|B^{(2)}_{\mathcal{N}}\rangle\rangle$ flows to a conformal boundary
condition.

The applicability of conformal boundary condition is crucial to all the main
results. The main results could fail, if the boundary state
$B^{(n)}_{\mathcal{N}}$ does not flow to a conformal boundary state. Indeed,
if we choose $\vec{v}=(0,\sin\theta,\cos\theta)$ in the $YZ$ plane, then we
observe that the $g$-function is monotonically increasing rather than
decreasing for a range of $\theta$, e.g., $0<\theta<\pi/4$. Yet, the
$g$-function still appears to converge at large $L$, indicating scale
invariance. These boundary conditions may be examples of scale invariant but
not conformal boundary conditions in the replicated CFT. We leave it as future
work to explore these boundary conditions.

## Numerical techniques

In this appendix, we describe matrix product state techniques to evaluate
$S^{(n)}_{A}(\mathcal{N}(\ket{\psi}\bra{\psi}))$. For the sake of
presentation, all the tensor diagrams were drawn for the special case that
$n=2$, $L=4$ and $A=\\{1,2\\}$, however the generalization to generic cases is
straightforward.

By performing density matrix renormalization group (DMRG) simulation, one is
able to obtain the matrix product state (MPS) representation of the critical
ground state $\ket{\psi}$ of interest:

$\ket{\psi}=\vbox{\hbox{\includegraphics[height=28.45274pt]{fig_mps.png}}}$
(47)

Accordingly, the partition function $Z^{(n)}_{A}$ can be diagrammatically
represented as:

$Z^{(n=2)}_{A=\\{1,2\\}}=\vbox{\hbox{\includegraphics[height=142.26378pt]{fig_mps_renyiS.png}}}$
(48)

By contracting from left to right, the diagram can be contracted with a time
complexity $O(L\chi^{2n+1})$, where $\chi$ is the maximum bond dimension of
the MPS. Using the above techniques, we can evaluate $Z^{(2)}$ for $\chi$ up
to $100$ and $L$ up to $256$.

By replacing $I$ with $\tau^{-1}_{n}$, the same technique can also be used for
evaluating Renyi entanglement negativity.