# Universal structure of measurement-induced information in many-body ground
states

Zihan Cheng  Department of Physics, University of Texas at Austin, Austin, TX
78712, USA  Rui Wen  Department of Physics and Astronomy, and Stewart Blusson
Quantum Matter Institute, University of British Columbia, Vancouver, BC,
Canada V6T 1Z1  Sarang Gopalakrishnan  Department of Electrical and Computer
Engineering, Princeton University, Princeton, NJ 08544, USA  Romain Vasseur
Department of Physics, University of Massachusetts, Amherst, MA 01003, USA
Andrew C. Potter  Department of Physics and Astronomy, and Stewart Blusson
Quantum Matter Institute, University of British Columbia, Vancouver, BC,
Canada V6T 1Z1

###### Abstract

Unlike unitary dynamics, measurements of a subsystem can induce long-range
entanglement via quantum teleportation. The amount of measurement-induced
entanglement or mutual information depends jointly on the measurement basis
and the entanglement structure of the state (before measurement), and has
operational significance for whether the state is a resource for measurement-
based quantum computing, as well as for the computational complexity of
simulating the state using quantum or classical computers. In this work, we
examine entropic measures of measurement-induced entanglement (MIE) and
information (MII) for the ground-states of quantum many-body systems in one-
and two- spatial dimensions. From numerical and analytic analysis of a variety
of models encompassing critical points, quantum Hall states, string-net
topological orders, and Fermi liquids, we identify universal features of the
long-distance structure of MIE and MII that depend only on the underlying
phase or critical universality class of the state. We argue that, whereas in
$1d$ the leading contributions to long-range MIE and MII are universal, in
$2d$, the existence of a teleportation transition for finite-depth circuits
implies that trivial $2d$ states can exhibit long-range MIE, and the universal
features lie in sub-leading corrections. We introduce modified MIE measures
that directly extract these universal contributions. As a corollary, we show
that the leading contributions to strange-correlators, used to numerically
identify topological phases, are in fact non-universal in two or more
dimensions, and explain how our modified constructions enable one to isolate
universal components. We discuss the implications of these results for
classical- and quantum- computational simulation of quantum materials.

##  I Introduction

Entanglement is a fundamentally quantum phenomenon, that serves as the
resource for quantum computation, and offers a powerful lens through which to
classify ground-state and dynamical phases and critical phenomena in quantum
many-body systems. The universal structure of entanglement in quantum many-
body ground-states offers a quantum information theoretic fingerprint of
various quantum orders and critical phenomena. Since local Hamiltonian or
quantum circuit dynamics can generate and propagate entanglement only at a
finite speed, the long distance structure of entanglement is stable under such
dynamics and is therefore useful for characterizing and classifying quantum
phases [1, 2]. By contrast, quantum measurement, an inherently stochastic and
non-unitary operation, can instantly generate non-local entanglement – as
famously illustrated by quantum teleportation. This capability allows short-
depth circuits with measurements to generate certain types of long-range
entangled states [3, 4, 5, 6, 7, 8, 9, 10]. Furthermore, pre-existing
entanglement in many-body states can also serve as a computational resource
[11, 12, 13, 14, 15, 16], and there is a close connection between the
universal phase structure of the pre-measured state and its computational
power [17, 18].

In this work, we investigate the connections between the phase of a various
quantum many-body ground-states in $1d$ and $2d$, and the amount of
measurement-induced long-range entanglement between distant regions,
quantified by two related quantities: measurement-induced entanglement [19]
(MIE) and measurement-induced information (MII), which we define below. The
paper is organized as follows. We begin by reviewing and discussing the
operational significance of MIE and MII and related quantities for various
classical and quantum computational tasks. We then conduct a numerical and
analytic investigation of the universal features of MIE and MII in a variety
of $1d$ and $2d$ phases of matter and critical points through a mixture of
numerical and analytic methods. We focus on states that have an efficient
classical description through exactly solvable models, including free-fermion
descriptions and stabilizer states. Despite their computational simplicity,
this class of states encompasses a wide range of long-range entangled states
including quantum critical points, topologically ordered states, and Fermi
liquids, and provides a rich set of examples.

In $1d$, we show that the leading long-range contribution to MIE and MII have
universal scale-invariant behavior at both conformal and non-conformal
(strongly-disordered) quantum critical points. By contrast, in $2d$ (or
higher-$d$), we argue that the leading contribution to MIE is generally non-
universal, due to the existence of long-range measurement-induced
teleportation properties of short-depth $2d$ circuits, and discuss
implications for other closely-related quantities such as strange correlators
[20]. Despite this, we find that the sub-leading corrections to MIE and MII do
exhibit universal features of topological orders and Fermi surfaces, and
define modified notions of MIE that can directly extract these universal
components.

###  I.1 Definitions

Measurement induced entanglement (MIE) is a tripartite measure of how
measurements affect the entanglement structure of a state, $|\psi\rangle$.
Partition space into three regions: $A$, $B$, and their complement
$M=(AB)^{c}$. Consider measuring $M$ in a fixed basis $b$ with outcome $m$,
which results in the post-measurement state $|\psi_{m}\rangle$ which occurs
with Born probability $p_{m}$. Then, we define the measurement induced
entanglement, ${\rm MIE}_{b}(A)$ of region $A$ to be the entanglement entropy
$S(A)[|\psi_{m}\rangle]$ of the post-measured state averaged over
measurements:

$\displaystyle{\rm
MIE}_{b}(A)[|\psi\rangle]=\sum_{m}p_{m}S(A)[|\psi_{m}\rangle].$ (1)

Here we have included a subscript $b$ to emphasize that the result depends
(often qualitatively) on the choice measurement basis. We will occasionally
drop the measurement-basis subscript where it is clear from context.

Despite its title, MIE, does not necessarily reflect entanglement that is
“induced” by the measurement, but also may capture preexisting entanglement in
the initial state before measurement. To isolate the effects of measurement,
we also define the measurement induced information (MII):

$\displaystyle{\rm
MII}_{b}(A,B)=\sum_{m}p_{m}I(A,B)[|\psi_{m}\rangle]-I(A,B)[|\psi\rangle]$ (2)

where mutual information, $I$, is defined as:

$\displaystyle I(A,B)=S(A)+S(B)-S(AB).$ (3)

Unless otherwise specified, we will consider the case where $ABM$ is in a pure
state, in which case the average mutual information after measurement is
simply related to the MIE as $\sum_{m}p_{m}I(A,B)[|\psi_{m}\rangle]=2{\rm
MIE}(A)$.

MII probes the average amount of mutual information between $A$ and $B$ that
is induced by the measurement, but did not exist before the measurement. Note
that this quantity can be either positive, zero, or negative depending on the
state and measurement basis. Measurements can increase information, resulting
in positive MII, for example, when bipartite information purely between $A$
and $M$ and $M$ and $B$ gets “teleported” by the measurement into entanglement
between $A$ and $B$. A simple example of this arises in a system with four
site spin-1/2 chain, with $A=\\{1\\}$, $M=\\{2,3\\}$ and $B=\\{4\\}$,
$|\psi\rangle$ is a product of singlets on sites 1,2 and 3,4 respectively, and
$M$ is measured in the Bell basis of sites $2,3$. However, measurements can
also reduce or collapse entanglement. A simple example is a three qubit GHZ
state: $(|000\rangle+|111\rangle)/\sqrt{2}$ which has $I=2\log 2$ information
between any pair of qubits before measurement, but collapses to a product
state upon measuring any of the three qubits in the computational ($Z$) basis:
${\rm MII}_{Z}=-2\log 2$ (whereas ${\rm MII}_{X}=0$ for this example).

###  I.2 Significance for quantum and classical computing

In quantum information processing settings, MIE and closely-related quantities
characterize how pre-existing quantum correlations in the state can be used as
a resource for generating entanglement by measurements. MIE and MII also have
implications for the quantum and classical complexity of describing a quantum
state. Here, we briefly review and describe the operational significance of
measures of measurement-induced entanglement for computational tasks.

##### Localizable entanglement:

The localizable entanglement (LE) [21] is defined as the maximum over
measurement bases of the MIE: ${\rm LE}(A,B)=\sup_{b}{\rm MIE}_{b}(A,B)$, for
the special case where regions A and B are single sites. LE upper-bounds
correlation functions, thereby enabling the definition of an entanglement
length scale in many-body systems that can probe non-classical correlations
and has an operational meaning for contexts such as building quantum repeaters
for quantum networks where one wishes to concentrate entanglement of a
multipartite state into two subsystems [21]. The maximization in the
definition of LE makes it difficult to compute, and it is most useful for
establishing bounds.

##### Measurement based quantum computing (MBQC):

MIE also partially characterizes the utility of a state for MBQC [22], where
(adaptive) measurements on an entangled resource state are used to propagate
and process quantum information. Long-range MIE between distant regions $A,B$
of a state is clearly a necessary condition for having a measurement-
propagable computational subspace in MBQC. However, long range MIE is not a
sufficient condition for MBQC as it does not address whether universal
computations can be performed on the propagated information via adaptively
chosen sequence of measurements in region $M$. For example, Haar random states
have long range MIE between any subregions $A,B$, yet are well-known to be
useless for MBQC based on single qubit measurements [23, 24].

##### Measurement-induced phase transitions:

Measurements can also induce phase-transitions in the post-measured
trajectories, $|\psi_{m}\rangle$ [25, 26, 27, 28, 29, 30, 31, 32, 33]. In
particular, $2d$ random circuits were shown to exhibit a phase transition
between short- and long- range MIE tuned by the circuit depth [34] (the
teleportation fidelity order parameter used in this work is precisely the same
as MIE and MII). This phenomenon was dubbed a teleportation phase transition,
and has since been realized experimentally in superconducting qubit quantum
processors [35]. The existence of long-range MIE in “trivial” states of matter
(related by a finite-depth circuit to an unentangled product state) will play
an important role in our discussion of universality of MIE and MII below.
Analogous to the absence of symmetry-breaking order in low-dimensions, this
teleportation phase transition is believed to be possible only in two or more
spatial dimensions for finite-depth local circuits.

##### Sign problem for Monte Carlo sampling:

Recent work [19] showed that a sign problem for Monte Carlo sampling
amplitudes of a state in the $b$ basis arises when the MIE is larger than the
pre-measurement mutual information: ${\rm MIE}_{b}(A)>I(A,B)[|\psi\rangle]$.
We note that ${\rm MIE}(A,B)-I(A,B)$ differs from ${\rm MII}$ since for a pure
state of AB $I(A,B)=2S(A)=2S(B)$. This observation directly relates the MIE to
the complexity of classical simulations of quantum states.

##### Strange correlators:

MIE also arise in analytic and numerical probes of topology of a state that
arise in the so-called strange correlators [20]:

$\displaystyle\frac{\langle\psi_{m}|O_{A}O_{B}|\psi\rangle}{\langle\psi_{m}|\psi\rangle}$
(4)

where $O_{A,B}$ are (charged) local operators in regions $A,B$ respectively,
and $|\psi_{m}\rangle$ is a product state, for example given by the result of
measuring the system in a given single-site basis. In a path integral
representation, strange correlators for topological states $|\psi\rangle$ are
related to correlators of $O_{A},O_{B}$ at the edge of $|\psi\rangle$, and
hence probe the presence of topological edge modes. Just as ordinary mutual-
information between regions $A$ and $B$ sets an upper bound for correlations
between local operators in these regions, the MIE upper bounds the average
strange correlators of $|\psi_{m}\rangle$ (See Appendix B). Below, we will
show that MIE and conventionally-defined strange-correlators are potentially
dominated by _non-universal_ contributions in two and higher dimensions, and
introduce modified definitions of these that extract the leading universal
components.

##### Complexity of quantum and classical tensor network calculations:

Tensor network states (TNS) provide efficient compressed representations of
low-entangled states, such as the ground states of many local Hamiltonians.
TNS wave-function amplitudes are expressed as a contraction of virtual bond
degrees of freedom: $\langle s_{1},s_{2},\dots
s_{N}|\Psi\rangle=C\left[T^{s_{1}}T^{s_{2}}\dots T^{s_{N}}\right]$, where
$T^{s}_{ijk\dots}$ are tensors with $s$ representing the physical degree of
freedom on each site, $i,j,k,\dots=1\dots\chi$ representing the virtual bond
space, and $C$ denoting summing over virtual indices. As explained below, the
MIE structure of a state relates to measurement-induced phase transitions
(MIPTs) in the bond-space of certain tensor network descriptions of the state,
which informs both the design principles for quantum circuit-based tensor
network calculations on quantum processors, and also the classical complexity
of computing properties of tensor network states.

Classical [36] and quantum-circuit based [37] methods for sampling from tensor
network state (TNS) wave-functions often involve simulating the transfer-
matrix “dynamics” of the virtual (bond) space of a codimension-one subsystem.
For example a standard method to contract $2d$ TNS is to represent the first
row of the TNS as a $1d$ matrix product state (MPS) in bond-space, and
contract the network by evolving this MPS under the action of the row transfer
matrices. For isometric TNS (isoTNS), quantum algorithms for materials
simulation have also been introduced [38] and demonstrated [39], in which
quantum circuit dynamics together incorporating mid-circuit measurements are
used to simulate the non-unitary transfer matrix dynamics. In this context,
the exponential of the MIE between distant co-dimension-one slices, $A,B$ of
the TNS reflects the typical classical memory to sample wave-function
amplitudes $\langle s_{1},s_{2},\dots s_{N}|\Psi\rangle$, with $s_{1}\dots
s_{N}$ representing the measurement outcomes. Similarly, for isoTNS, the MIE
itself represents the average (over values $s_{1},\dots s_{N}$), quantum
resources needed to sample wave-function amplitudes with a quantum computer.

Further, when for TNS with spatial dimension larger than one, the transfer
matrix for calculating $\langle s_{1},s_{2},\dots s_{N}|\Psi\rangle$ involves
dynamics of a many-body system post-selected on measurement outcomes
$s_{1\dots N}$, which may exhibit distinct phases with area- or volume- law
entanglement (with respect to the transfer-matrix dimension), that are
separated by MIPTs [40, 34, 41]. These MIPTs represent a classical
computational complexity phase transition in the difficulty of contracting the
TNS. We note that while such complexity phase transitions may arise for
sampling wave-function amplitudes or related global properties such as
strange-correlators or wave-function overlaps, for many practical purposes one
is interested mainly in correlation functions of local observables that have
recently been argued to not have a complexity phase transition for fixed bond-
dimension [42].

###  I.3 Universal structure of MIE

These connections motivate the need to characterize the universal features of
MIE and MII in quantum many-body states, particularly the ground-states of
local Hamiltonians. For example, for short-range entangled states the MBQC
power is closely connected to the underlying symmetry and topology of the
phase to which the state belongs. Moreover, in quantum simulation algorithms
based on classical or quantum tensor network methods, such relations could
reveal how the properties (symmetry, topology, correlation length, operator
scaling dimensions, etc…) of a state that one wishes to simulate inform the
circuit design principles for its quantum tensor network representation, or
the complexity to perform calculations with classical tensor network methods.

Specifically, we aim to understand: what features of MIE and MII for ground-
states are _universal_ : i.e. which are insensitive to perturbations to the
parent Hamiltonian that do not drive a phase transition, or equivalently,
which cannot be altered by a short-depth quantum circuit, and how are these
measurement-induced entanglement features related to the phase or universality
class of the state in question. To this end, we explore, through analytic and
numerical methods, the universal structure of MIE and MII in a variety of
states with various types of topological orders or criticality. In particular,
we emphasize the difference between the universality in $1d$ and higher
dimensions, where a measurement-induced entanglement phase transition can
occur at a finite circuit depth, while pre-measurement states remain in the
same phase (See Sec.III.1). In the presence of such nonuniversal long-range
behavior, the universal properties only show in the subleading contribution.

The paper is structured as follows. In Sec. II we explore the universal
behavior of MIE and MII in $1d$ ground-states of both gapped, short-range
correlated systems and gapless, critical systems. In Sec. III we discuss the
measurement-induced entanglement phase transition occuring at a finite circuit
depth and potential adjustment of the definition of MIE and MII. Moreover, we
study the universal parts of MIE and MII in topological phases and metals.

##  II MIE in $1d$ systems

We begin by exploring universal properties of MIE and MII in $1d$ ground-
states of both gapped, short-range correlated systems and gapless, critical
systems. We review known structure of MIE for gapped symmetry-protected
topological phases [43, 17], and selected $1+1d$ conformal field theories
(CFTs) were previously explored [19], add additional examples of conformal and
non-conformal critical points, and investigate the stability to irrelevant
perturbations.

###  II.1 Gapped $1d$ states

Absent fine-tuning, typical gapped $1d$ states and generic measurement bases
exhibit short range ${\rm MIE}(A,B)$ that decays exponentially in distance
between $A$ and $B$. This follows from the MPS representation of gapped, $1d$
states: $\langle s_{1},\dots s_{N}|\Psi\rangle={\rm tr}A^{s_{1}}A^{s_{2}}\dots
A^{s_{N}}$, where $A^{s}_{i,j}$ are $\chi\times\chi$ matrices. For unique
gapped ground-states of local Hamiltonians, the matrices $A$ satisfy an
injectivity property [44], which guarantees short-range correlations.

A notable exception arises for the symmetry-protected topological (SPT) phases
with measurements taken in an appropriate symmetry-preserving basis [43],
which exhibit long-range $\lim_{x_{A,B}\rightarrow\infty}{\rm MIE}(A,B)=\log
D$ where $D$ is the dimension of the SPT’s edge states, and $x_{A,B}$ is the
distance between regions $A,B$. This property is directly related to the fact
that $1d$ SPTs can act as good “quantum wires” for MBQC (i.e. are capable of
coherently storing a quDit in the MBQC context). By contrast, as argued in
[43], measurements in generic bases lead to exponentially decaying MIE. Viewed
from the perspective of the MPS transfer matrix dynamics, this short-range MIE
results from mixing between the projective- and ordinary/linear- (a.k.a.
“junk”) symmetry blocks of of the bond-space. We note that, despite this short
range MIE for general measurement bases, it has been shown that generic MBQC
operations can be performed with arbitrary target fidelity by splitting a MBQC
gate into many small operations [17], implemented by a gradually-evolving (and
adaptively-chosen) measurement bases.

###  II.2 Gapless or Critical $1d$ states

We next turn to gapless critical states with emergent conformal invariance,
i.e. which are described in the continuum limit by a conformal field theory
(CFT). Previous numerical investigations on select critical states of Ising-
like spin chains using density matrix renormalization group [19] demonstrated
that MIE of regions $A=[x_{1},x_{2}]$, $B=[x_{3},x_{4}]$, depend only on the
cross-ratio:

$\displaystyle\eta=x_{12}x_{34}/x_{13}x_{24},$ (5)

and decayed as $\eta^{\alpha}$ for small $\eta$ (i.e. distance between
intervals much larger than interval size).

While suggestive power-law decaying behavior does not necessarily imply
universality. For example, there are other closely-related setups in which
_non-universal_ power-law behavior can arise. Namely, Ref. [45], observed
power-law dependence in Loschmidt echo of a CFT with a random time-dependent
noise applied to a boundary, where the power was continuously tuned by non-
universal parameters of the noise. On its face, this problem appears very
different, yet, it is described by a very-similar field theory construction as
MIE. Specifically, a path-integral description Loschmidt echo for the
stochastically-driven boundary CFT is related by a conformal mapping to the
path integral setup for calculating a second Renyi entropy version a forced-
MIE in which the measurement outcomes are “forced” to be equal to the
stochastic boundary drive. A potentially important difference between the
average MIE and this stochastic boundary drive, is that, in the MIE, the
averaging is weighted by the Born probability of the measurement outcomes,
whereas in the stochastic drive setting, it is externally imposed by the
drive. Does this weighting by Born probabilities restore universality? Or
could it be that the observed power-law decays of MIE for CFT states is more
akin to the non-universal behavior observed in the driven boundary CFT?

Figure 1: Data collapse of MIE in XX chain versus cross ratio
$\eta=x_{12}x_{34}/x_{13}x_{24}$ with fill factor (a) $n_{f}=0.5$ and (b)
$n_{f}=0.375$. Inserted figure in (a) shows the geometry of MIE regions where
measured region is in black. Data are averaged over $10^{6}$ samples.

#### II.II.2.II.2.1) XX chain

To investigate this question, we numerically study the MIE of an XX spin chain
with periodic boundary condition:

$\displaystyle H_{\rm
XX}=\sum_{i}J(\sigma^{x}_{i}\sigma^{x}_{i+1}+\sigma^{y}_{i}\sigma^{y}_{i+1})$
(6)

which can be mapped, by a Jordan-Wigner (JW) transformation, to a free fermion
chain with nearest-neighbor hopping $J$ at half-filling, and exhibits
Luttinger liquid ground-state described by a free-boson CFT. Specifically we
consider the ferromagnetic interaction $J=-\frac{1}{2}$, and then the
Hamiltonian in free fermion language takes the form

$\displaystyle
H=-\sum_{i}\left(c^{\dagger}_{i}c_{i+1}+c^{\dagger}_{i+1}c_{i}\right)+{\rm
const.}$ (7)

For the free fermion systems, the single orbital measurements can be
implemented based on correlation matrix $C_{ij}=\langle
c^{\dagger}_{i}c_{j}\rangle$ (See Appendix A). The correlation matrix for the
ground state of Hamiltonian in Eq. (7) with periodic boundary condition is
given by:

$\displaystyle
C_{ij}=\frac{1}{L}\left(1+e^{inx}+2\frac{\sin\frac{(n-1)x}{2}}{\sin\frac{x}{2}}\cos\frac{nx}{2}\right)$
(8)

where $x=\frac{2\pi}{L}\left(i-j\right)$ and
$n=\left\lfloor\frac{Ln_{f}}{2}\right\rfloor$. $n_{f}$ is the filling factor,
which is $0.5$ when there is no external field. In Fig. 1 (a) we show the data
collapse of MIE for $L=128$ and $L=256$ with $\eta$ over several orders of
magnitude, giving $\alpha\approx 0.3$, which is much smaller the $\alpha=2.$
in the forced measurement case [46]. Note that in the cross ratio, the finite-
size effect has been taken into account by replacing $x_{ij}=|x_{j}-x_{i}|$
with the chord length
$\frac{L}{\pi}\sin\left(\frac{\pi|x_{j}-x_{i}|}{L}\right)$.

To test the universality of power-law relation observed in MIE, we perturb the
idealized XX model with chemical potential $\mu\sum_{i}\sigma_{i}^{z}$, which
modifies the filling factor to $n_{f}=\frac{\arccos\mu}{\pi}$ and Eq. (8)
still holds. In Fig. 1 (b) the data collapse of MIE with similar power-law
exponent indicates the universality of MIE.

On the other hand, the mutual information from $1+1d$ CFT calculation scales
as $\sim\eta^{\frac{1}{2}}$ at the small $\eta$ limit [47], thus MII is
positive and dominated by the measurement-induced part at the large distance
limit.

Figure 2: Number of measurement-induced Bell pairs in a random singlet state
between $A$ and $B$ with $x_{23}=x_{14}=r$ for (a) odd $|A|=|B|$ and (b) even
$|A|=|B|$. Data are averaged over $10^{6}$ random singlet configurations.

#### II.II.2.II.2.2) Random singlet phase

Phase transitions and critical phenomena in $1d$ disordered systems often
exhibit a flow to so-called infinite randomness fixed points in which the
long-distance low-energy behavior are governed by rare-region effects leading
to slow-glassy dynamics, strong differences between average and typical
correlation functions [48, 49]. A classic example arises in a $1d$ random
antiferromagnetic Heisenberg spin-$\frac{1}{2}$ chain:

$\displaystyle H=\sum_{i}J_{i}\boldsymbol{S}_{i}\cdot\boldsymbol{S}_{i+1},$
(9)

where $J_{i}\in\left[0,J\right]$ are random, and identically and independently
distributed for each bond. Up to RG-irrelevant local dressing from quantum
fluctuations, the ground-state of this model is a random singlet (RS) state in
which each spin is locked into a singlet state with an another partner. The
distance $r$ between a singlet pair satisfies the distribution $\sim 1/r^{2}$.
The entanglement in these long range singlets produce a log-violation of area-
law with single-region entanglement is proportional to the number of singlets
starting from an interval $A$ of length $l$ and terminating elsewhere [50]:

$\displaystyle S(A)=\frac{\log 2}{3}\log l+k$ (10)

where $k$ is some non-universal constant. Similarly, the mutual information
between disjoint intervals $A$ and $B$ is proportional to the number of
singlets spanning between $A$ and $B$ [51]:

$\displaystyle I(A,B)=-\frac{\log 2}{3}\log(1-\eta),$ (11)

where $\eta$ is the cross ratio defined in Eq. (5). For the case small
intervals $A$ and $B$ separated by a large distance $r$, $I(A,B)$ scales as
$\sim 1/r^{2}$.

Now we consider the MIE and MII for such a RS state between two pre-measured
disjoint intervals $A$ and $B$. Measurement in any product-state basis (e.g.
the $S^{z}$ basis) collapses the entanglement from all singlets between $A/B$
and $M$. The only surviving entanglement arises from singlets that directly
connected $A$ and $B$ in the pre-measured state, resulting in
$\mathrm{MIE}_{S^{z}}(A)=I(A,B)\left[|\psi\rangle\right]/2$ and
$\mathrm{MII}_{S^{z}}(A,B)=0$.

On the other-hand, measurement in the Bell basis has the effect of
“teleporting” singlets between $A/B$ and $M$ into Bell-pairs in $A\cup B$,
leaving local Bell-pairs in $M$. Specifically we consider the Bell
measurements on nearest-neighbor spin pairs in a measurement region containing
an even number of spins. For the simplest case, i.e. the un-measured regions
$A$ and $B$ only contain single sites. In this case: $n_{AB,M}$ has to be $1$
since all other sites are paired after Bell measurements and no existing pair
can be eliminated. In result, $\mathrm{MIE}_{\rm Bell}=\log 2$ and
$\lim_{L\rightarrow\infty}\mathrm{MII}_{\rm Bell}=2\log 2$. The similar long
range behavior is expected for odd number of $|A|$ and $|B|$ (for random
singlets, we have assumed total system size is even), which is numerically
verified in Fig. 2(a). However, for the cases $A$ and $B$ containing even
number of site, there is no guarantee for the formation of long-range Bell
pairs. Instead, measurements can either teleport an $A/M$ singlet into an
$A/B$ singlet contributing to MII, or they might teleport the $A/M$ singlet
into a local singlet with both spins in $A$, which does not contribute to MII.
The ratio of these two possibilities depends in a complicated fashion on the
disorder configuration and geometry of the regions, but nevertheless yields a
universal scaling form. To investigate it, we perform a numerical calculation
using the strong-disorder renormalization group method. The results shown in
Fig. 2(b) demonstrate for even number of $|A|$ and $|B|$ the
$\mathrm{MII}_{\rm Bell}(A,B)$ features a power-law decaying $\sim r^{-0.34}$.

Figure 3: (Left) Schematic figure for MERA tensor networks; (Right) Mapped
statistical mechanics model after Haar random average. Mini-cut domain wall is
indicated by the orange wave line.

#### II.II.2.II.2.3) Random MERA states

Another tractable model for computing MIE are random multi-scale entanglement
renormalization (MERA) tensor networks [52, 53]. Entanglement features of
MERAs with either non-unitary Gaussian random tensors or Haar random unitaries
and isometries can be computed by mapping them, via a replica trick, to a
classical statistical mechanics model [54, 55, 56, 57, 58, 59]. In the limit
of large bond dimension, $D\rightarrow\infty$, the statistical mechanics model
calculations become tractable and reduced to pure geometric quantities that
are identical to the Ryu-Takanagi principle for computing entanglement via
holographic field theory/gravity duality [54].

In the stat-mech mapping [60], the entanglement entropy of a boundary region
$A$ maps to the free energy in a classical statistical mechanics model of
generalized “spins” sitting on the vertices of the tensor network, with
ferromagnetic interactions of strength $J=\log D$ between spins connected by
edges of the network, and boundary “magnetic” fields of strength $h=\log D$
[54, 55, 58], which explicitly break the replica permutation symmetry. For the
simplicity, we describe the two-replica case where the model reduces to an
Ising model with two spin configurations $\uparrow,\downarrow$ (for general
number, $Q$, of replicas, the resulting model is Potts-like with $Q$ spin-
flavors, but the universal features inside the ordered phase of the model at
large-$D$ are expected to be independent of $Q$). The phase diagram of Ising
models on graphs with hyperbolic geometry has been studied both analytically
[61] and numerically[62]. Unlike for Euclidean geometries, where boundary
conditions do not effect bulk critical properties, on hyperbolic graphs, the
boundary contains an extensive fraction of the total number of sites, and the
resulting phase diagram is sensitive to boundary conditions. For fixed
boundary conditions, with an explicitly symmetry-breaking field at the
boundary, there is a single bulk order-to-disorder transition. By contrast,
for free-boundary conditions, the “ordered” phase splits into two phases: a
low temperature uniformly-ordered phase with a single spontaneously-chosen
magnetization, and a moderate temperature phase with a finite fraction of
disordered spins.

In the present context, the single-region entanglement $S(A)$ is given by the
free energy cost having $\uparrow$ boundary fields outside of $A$ and
$\downarrow$ boundary fields in its counterpart $\bar{A}$, which forces a
domain wall (DW) into the system. Therefore, $S(A)$ corresponds to the fixed-
boundary condition hyperbolic-Ising model, and exhibits a single (dis)ordering
phase transition at a critical $D_{c}$. Below this critical bond-dimension,
there is only a local cost to inserting a boundary domain wall resulting in
area-law entanglement ($S(A)\sim$ constant). Above this critical bond-
dimension, the bulk is ordered, and the DW tension (energy cost per unit
length) is proportional to $\log D/D_{c}$. In the following we consider the
limit of large $D$, where the effective spin model will be in an ordered,
ferromagnetic phase. Here, fluctuations in the DW shape are strongly penalized
and the DW follows a minimal cut of the interaction edges. For a MERA tensor
network with a $1d$ boundary, the minimal cut connects the ends of region $A$
interval dives down into the bulk as shown in Fig. 3 and has length
$\sim\log|A|$. which gives $S(A)\sim\log D\log|A|$. The mutual information for
two disjoint intervals $A=\left[x_{1},x_{2}\right]$ and
$B=\left[x_{3},x_{4}\right]$ is given by the free energy of two competing cut
configurations for $S(A\cup B)$

$\displaystyle
I\left(A,B\right)=F_{A}+F_{B}+\log\left(e^{-F_{A}-F_{B}}+e^{-F_{AB}}\right),$
(12)

where $F_{A}+F_{B}\sim\log D\log x_{12}x_{34}$ (Fig. 4(a)) and $F_{AB}\sim\log
D\log x_{14}x_{23}$ (Fig. 4(b)). In the limit of $D\rightarrow\infty$,
$I\left(A,B\right)$ shows an abrupt jump as a function of
$\tilde{\eta}=x_{12}x_{34}/x_{14}x_{23}$, which relates to the cross ratio
defined in Eq. (5) as $1/\eta=1/\tilde{\eta}+1$,

$\displaystyle I\left(A,B\right)\sim\begin{cases}\log
D\log\tilde{\eta}&\tilde{\eta}>1\\\ \tilde{\eta}^{\log
D}&\tilde{\eta}<1\end{cases}.$ (13)

Figure 4: (a), (b) two possible minimal cuts for $S(A\cup B)$; (c) two
possible minimal cuts for ${\rm MIE}(A)$.

We now turn to average MIE and MII for this random MERA. We note that, the
random nature of the tensors means that the average MIE does not depend on
measurement basis. The principal difference in the stat-mech mapping is that
the measured region $M$ has free-boundary condition, and thus exhibits a
_distinct, separate phase transition_ from that of the (fixed-boundary
condition) stat-mech model for entanglement (without measurement). As a result
of the free-boundary conditions, the DW ends are no longer linearly-confined
to the ends of the boundary of entanglement region, but can fluctuate into
measured regions. The cost to change the size of the minority spin domain
encapsulating $A$ from the minimum domain, to $x>|A|$, scales as $\log D\log
x/|A|$. This competes with an entropic gain $\sim\log x$ from the
fluctuations. The competition of energetic vs. entropic logarithmic factors is
familiar from Kosterlitz-Thouless (KT) transitions arising in $1d$ XY models
and discrete $1d$ spin chains with long range interactions decaying as $\sim
1/r^{2}$ with distance $r$. By analogy, there should be a critical bond-
dimension $D_{c}^{\prime}>D_{c}$ at which the stat-mech model for MIE changes
from an ordered phase $(D_{c}<D<D_{c}^{\prime})$ with short-range MIE, to an
ordered phase $(D>D_{c}^{\prime})$ with $\mathrm{MIE}(A)\sim\log|A|$ as
discussed above. This phase transition is analogous to the “finite-time”
teleportation transition in $1d$ quantum circuits with power-law range gates
discussed in [34]. We will discuss the implications to MIE for a related
teleportation transition in finite-depth $2d$ circuits below. There we will
argue that the leading contribution to MIE in the teleporting phase is long-
range and non-universal.

In the large-$D$ limit, such fluctuations are strongly suppressed, and thus
the leading order of MIE is given by

$\displaystyle\mathrm{MIE}(A)\sim\log
D\log\left(\min\\{x_{12},x_{34}\\}\right),$ (14)

which is equal to the minimum pre-measurement of entanglement entropy in $A$
and $B$ (min-cuts are given in Fig. 4(c)). We note that this result holds even
when the distance between $A$ and $B$ becomes very large such that the pre-
measurement $I(A,B)\approx 0$. In such case one has

$\displaystyle\mathrm{MII}(A,B)\approx 2\mathrm{MIE}(A)\sim 2\log
D\log\left(\min\\{x_{12},x_{34}\\}\right).$ (15)

This behavior of MIE qualitatively differs from that found in the free fermion
(XX-chain) studied numerically in the previous section. This deviation is a
direct consequence of the different phase diagrams for ferromagnetic spin
models on hyperbolic geometries with fixed- or free- boundary conditions. For
the CFT, MIE followed the behavior of a four-point function, depending only on
the cross-ratio: $x_{12}x_{34}/x_{13}x_{24}$. This suggests a possible
difference in the structure of MIE between holographic tensor network states
and (minimal-model) CFTs. This behavior contrasts that for the pre-measurement
mutual-information, $I(A,B)$, which behaved like a four-point function for
both models.

##  III MIE in $2d$

We next turn to investigating the structure of measurement-induced
entanglement in $2d$ states. Here, the existence of teleportation phase
transitions in finite-depth $2d$ circuits dramatically alters the structure
and universality of MIE [40, 34]. Nevertheless, we find that universal
signatures of topology and non-local entanglement associated with Fermi
surfaces still arises as subleading corrections to MIE, and discuss how to
directly extract these universal signatures.

Figure 5: Schematic figure for the traced-MII: mutual information between $A$
and $B$ with green regions $T$ traced out and grey regions $M$ measured out,

###  III.1 Trivial Gapped States

A trivial gapped state is one that can be produced from a product state by a
finite depth local circuit. Unlike $1d$ systems, measurements in $2d$ (or
higher dimensions) can induce nontrivial entanglement phase transition with
finite-depth random circuits [40, 34], i.e. after applying $t>t_{c}$ layers of
random circuits, arbitrary initial state can feature extensive measurement-
induced entanglement $\mathrm{MIE}\left(A\right)=O\left(|A|\right)$.
Therefore, the leading order of $\mathrm{MIE}$ in $2d$ is some non-universally
constant and on the other hand, universal behaviors, such power-law decaying
will be overshadowed.

#### III.III.1.III.1.1) Leading contributions to strange correlators are non-
universal

The existence of extensive term in MIE also implies a non-universal constant
part in strange correlators in $2d$, that there exists two trivial states
$|\psi\rangle$ and $|\psi_{m}\rangle$ giving long-ranged strange correlator in
Eq. (4) and thus overshadows the diagnosis of nontrivial topological states,
which contributes a power-law decaying [20]. To show this explicitly, we
consider a $2d$ trivial initial state $|\psi\rangle$ with application of
$t>t_{c}$ layers of random circuits which doesn’t have pre-measurement long-
range correlation but is in the “teleportation phase” [34]:
$\mathrm{MIE}(A)=O\left(|A|\right)$. Specifically, we consider local Hilbert
dimension $D=2$, $|A|=|B|=1$ separated by distance $r$ and the rest degrees of
freedom are measured out. Then the post-measurement state can be written as

$\displaystyle|\psi\rangle_{AB}=\lambda_{0}|a_{0}\rangle|b_{0}\rangle+\lambda_{1}|a_{1}\rangle|b_{1}\rangle,$
(16)

where $|a_{0/1}\rangle$ and $|b_{0/1}\rangle$ are orthonormal local basis
supported by $A$ and $B$. In the “teleportation phase”
$\mathrm{MIE}(A)=-\left(\lambda_{0}^{2}\log\lambda_{0}^{2}+\lambda_{1}^{2}\log\lambda_{1}^{2}\right)=O(1)$,
which indicates $\lambda_{0/1}$ are both finite non-zero numbers. Then by
choosing $O_{A}=|a_{0}\rangle\langle a_{1}|$, $O_{B}=|b_{0}\rangle\langle
b_{1}|$ and $|\psi_{m}\rangle_{AB}=|a_{0}\rangle|b_{0}\rangle$, one can verify
that the (connected) strange correlator between $|\psi\rangle$ and
$|\psi_{m}\rangle=|\psi_{m}\rangle_{AB}|\psi_{m}\rangle_{M}$

$\displaystyle\frac{\langle\psi_{m}|O_{A}O_{B}|\psi\rangle}{\langle\psi_{m}|\psi\rangle}-\frac{\langle\psi_{m}|O_{A}|\psi\rangle}{\langle\psi_{m}|\psi\rangle}\frac{\langle\psi_{m}|O_{B}|\psi\rangle}{\langle\psi_{m}|\psi\rangle}=\frac{\lambda_{1}}{\lambda_{0}},$
(17)

which is generally a non-vanishing number for finite $\mathrm{MIE}(A)$
independent of the distance $r$. Note that the previous choice of $O_{A}$,
$O_{B}$ and $|\psi_{m}\rangle_{AB}$ can be in arbitrary local orthonormal
basis if we consider the typical case under random circuits, where each
projector $|a_{0}/b_{0}\rangle\langle a_{1}/b_{1}|$ will typically contribute
with a prefactor $O\left(\frac{1}{4}\right)$, and then the (connected) strange
correlator will still have a finite long-range behavior.

###  III.2 Isolating the universal features of MIE and strange correlators

In this section we introduce two modified versions of MIE (which can be
readily adapted to strange correlators), that isolate the universal
contributions of MIE that depend only on the phase of matter for the pre-
measured state, from the non-universal ones arising from the teleportation
transition.

##### Scaled MIE

To probe the universal behavior of $\mathrm{MIE}$ or strange correlator, one
has to go to the subleading term with power-law decaying. Generally, the
measurement-induced non-universal effect is proportional to the size of post-
measurement region $A$, thus to get rid of the non-universal extensive term of
MIE, one can redefine MIE as

$\displaystyle\widetilde{\mathrm{MIE}}(A)=2\mathrm{MIE}_{1/2}(A/2)-\mathrm{MIE}(A),$
(18)

where $\mathrm{MIE}_{1/2}(A/2)$ represents the MIE for a system scaled by
$1/2$ in the particular direction of interest, not only for the region $A$.
Within this definition, the extensive terms in MIE are canceled, while the
sub-leading power-law relation is retained. A non-ideal aspect of this
difference between scaled MIE is that it assumes a specific form of the
subleading universal corrections. In the next section, we introduce an
alternative means to isolate the universal aspects of MIE from the non-
universal teleportation transition ones, that is agnostic to the precise
scaling structure of the subleading universal terms.

##### Partially-traced MII

Another possible solution is to consider the quantity that is not affected by
the measurement-induced phase transition but still features the universal
behaviors. One candidate, shown in Fig. 5, is the partialy-traced MII, where
instead of considering the total mutual information between two post-measured
regions $A$ and $B$, we consider the mutual information between two subregions
$A_{0}$, $B_{0}$ of them. In this case, the left region $A/A_{0}\cup B/B_{0}$
is traced over (see Fig. 5) and thus imposes a definite spin configuration in
the statistical mechanics picture which makes this quantity always vanishing
in the long-range limit for trivial initial states.

For the models considered in the remainder of the paper, we find that the
ground-states naturally lie in the non-teleporting phase. For this reason, we
do not need to explicitly modify MIE to subtract a non-universal extensive
piece.

###  III.3 Chern insulator

Figure 6: (a) Two band model on a square lattice, where nearest-neighbor
hopping amplitudes are $t_{1}e^{i\pi/4}$ along the arrow direction and next-
nearest-neighbor hopping amplitudes are $t_{2}$ and $-t_{2}$ along the dashed
and dotted diagonals. (b) MIE on a cylinder geometry where grey region is
measured out.  Figure 7: MIE in the two-band model with cylinder geometry for
(a) $V=0$ and (b) $V=0.8$, where the result is averaged over $120$ measurement
realizations.

In Chern insulators, the strange correlator decays as a power law (in
particular, for the free-fermion case $\sim r^{-1}$) [20, 63, 64], which can
be understood by mapping to the standard correlator in the one-dimensional
CFT. Since in the CFT, MIE is lower-bounded by the averaged square of strange
correlators (see Appendix. B), we expect the universal part of MIE in Chern
insulators also features a power-law behavior.

To investigate this question, we numerically study the MIE in a two band model
on a square lattice introduced by Ref. [65], which holds non-trivial Chern
number. As shown in Fig. 6(a), the model consists of two sublattices $a$ and
$b$, where nearest-neighbor hopping amplitudes are $t_{1}e^{i\pi/4}$ along the
arrow direction and next-nearest-neighbor hopping amplitudes are $t_{2}$ and
$-t_{2}$ along the dashed and dotted diagonals. To compare the results with
trivial insulator, we add a staggered one-site potential, which is $V$ for $a$
sublattice and $-V$ for $b$ sublattice. For $-4t_{2}/t_{1}<V<4t_{2}/t_{1}$,
the band features Chern number $C=1$, while for $V>4t_{2}/t_{1}$ or
$V<-4t_{2}/t_{1}$, the band is trivial.

Specifically, we let $t_{1}=1.$ and $t_{2}=0.1$ and consider the MIE between
two rings with width $2$ separated by distance $r$ on an $L\times L$ lattice
with cylinder geometry (Fig. 6(b)). As shown in Fig. 7(a), the MIE for
nontrivial Chern number ($V=0$) features a power-law decay $\sim r^{-0.9}$
satisfying the lower bound given by the strange correlator. As a comparison,
in a trivial insulator ($V=0.8$) the MIE decays exponentially (Fig. 7(b)).
Since the ground state of Chern insulators has short-ranged pre-measurement
entanglement, the MII also features power-law decaying.

Figure 8: MIE in the $2d$ free fermion model on the square lattice with
cylinder geometry, where the result is averaged over $120$ measurement
realizations.

###  III.4 Metals

As a final numerical example, we consider the MIE of a non-interacting metal,
with measurements taken in the basis of occupation number on each site. Due to
the presence of a Fermi-surface of gapless excitations, the metallic state has
a large entanglement before measurement, with entanglement of a region of size
$L$ scaling as $S\sim k_{F}L\log L$ where $k_{F}$ is the Fermi wave-vector. We
consider MIE in a free fermion tight-binding model

$\displaystyle H=-\sum_{\langle
i,j\rangle}\left(c^{\dagger}_{i}c_{j}+c^{\dagger}_{j}c_{i}\right),$ (19)

on a two-dimensional square lattice, where the ground state is a metal.

Similar to the Chern insulator case, we consider the MIE between two rings
with width $2$ separated by distance $r$ on an $L\times L$ lattice with
cylinder geometry (Fig. 6(b)). As shown in Fig. 8, the MIE features a power-
law decaying $\sim r^{-0.23}$, where corresponds to $\alpha\approx 0.115$
(considering $\eta\sim r^{-2}$ in the large distance limit), which smaller
than $\alpha\approx 0.3$ of the $1d$ case. Since the pre-measurement $2d$ free
fermion systems can be decoupled into $1d$ chains, the mutual information
scales as $\sim\eta$ [66] 111Note that although JW transformation can map the
XX model to $1d$ free fermion chain, they actually have different mutual
information behavior due to the implicit non-local structure from JW
transformation [67]).. By contrast, the random measurement outcomes break the
translation invariance, such that the MIE of the metal does _not_ reduce to
decoupled copies of $1d$ free-fermion systems for each momentum parallel to
the measurement-region boundary. In particular, the MIE of the $2d$ metal
decays with a different power of $r$ than for the $1d$ free fermion system.
Since this pre-measurement mutual information decays much more rapidly than
the MIE, the MII in large distance limit is dominated by the measurement-
induced part and scales as $r^{-0.23}$.

###  III.5 Topological orders (String-net liquids)

While leading order of MIE in a trivial gapped state shows no universal
behavior, MIE of a topologically ordered state contains a long-range
(constant) term that depends only on the topological information of the
measurement scheme, namely it depends only on the homological class of the
measurement region and is invariant upon deformation of the measurement
region.

Measurements of a sub-region of a topologically-ordered state can be
interpreted as imposing boundary conditions on the unmeasured region. To see
this, observe that after performing a local measurement in a region $M$, the
state in $M$ is in a tensor product state that has no topological order while
the unmeasured region remains topologically ordered. Therefore the interface
between the two regions is an interface between topological order and trivial
state. Boundaries of $2d$ topological orders are well-understood–they are in
1-1 correspondence with different ways of condensing anyons of the topological
order. This classification and characterization of boundary conditions allow
us to analyze the post-measurement state of a topologically order state
systematically.

#### III.III.5.III.5.1) Toric code on a torus

As a warmup example let us consider the toric code, $D(\mathbb{Z}_{2})$, on a
torus, $T^{2}$, modeled is a spin $\frac{1}{2}$ degree of freedom at each link
of a square lattice, with Hamiltonian:

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where $Z,X$ are Pauli operators. If the measurement region is contractable,
then due to the topological nature of the system, there is no long-range
measurement-induced entanglement. Instead we consider measuring a non-
contractable region $M$, that separates two unmeasured regions $A,B$ as shown
in Fig. 9. The boundaries between measured and unmeasured region are on non-
contractible loops which we call $l_{1}$. The other non-contractible loop
intersects $l_{1}$ once and is called $l_{2}$.

Figure 9: MIE of toric code on a torus. The torus is divided into regions
$A,B,M$. The $M$ region is then measured out and we are interested in the MIE
between $A$ and $B$. Two non-contractible loops $l_{1},l_{2}$ are shown in
blue and red.

The ground space of the toric code Hamiltonian on the torus is 4-fold
degenerated and is invariant under the action of string (loop) operators on
$l_{1},l_{2}$. The ground state is invariant under application of string
operators that are labelled by anyon types, $\\{e,m,f=e\times m\\}$. A
generating set of string operators can be chosen as $e$-strings and
$m$-strings. An $e$-string is a product of $Z$ operators along a path of the
lattice while an $m$-string is a product of $X$ operators along a path of the
dual lattice. We denote loop operators of type $a=e,m$ on loops $l_{i=1,2}$ as
$W^{a}_{i}$. Then these operators preserve the ground space and their
eigenvalues can be used to label states in the ground space. In the language
of string-net condensation the ground states are equal-weight superpositions
of closed $e$-loop configurations, where we define links with $X=-1$ as having
an $e$-loop segment. Then the 4-fold degenerated states can be characterized
by the parity of number of loops along non-contractible directions
$l_{1},l_{2}$. This way of labelling provides us with a basis of the ground
space which we denote as $|\alpha,\beta\rangle_{SN}$ and call string-net
basis, $\alpha,\beta=0,1$ are the parities of number of loops along
$l_{1},l_{2}$ respectively.

Now let us consider starting with a state with even parity of loops long
$l_{1},l_{2}$: $|0,0\rangle_{SN}$ and measuing in $X$-basis in region $M$. The
measurement will then project onto some fixed loop configuration in $M$, while
the unmeasured region remains a equal weight superposition of various loop
configurations. From the measurement outcomes we can infer the parity of loops
along $l_{1}$ in $M$. If the measurements yield a result such that the number
of loops along $l_{1}$ in $M$ is even, then we know that the total number of
loops along $l_{1}$ in $A,B$ must be even since the initial state has even
parity of loops along $l_{1}$. This condition entangles the string-net
configurations in $A,B$: they can have either both even or both odd parity
along $l_{1}$. Apart from this condition the states in $A,B$ are still equal
weight superpositions of all possible loop configurations. The post-
measurement state in region $A,B$ is thus a Bell pair with entanglement
entropy $\log 2$. On the other hand if the measurement projects onto a state
in region $M$ with odd number of loops along $l_{1}$, then the postmeasurement
state must have opposite parities of loops along $l_{1}$ in $A$ and $B$. In
this case the post-measurement state is again a Bell-pair and has entanglement
entropy $\log 2$. Therefore after averaging over measurement outcomes with
Born probability we have ${\rm MIE}(A,B)=\log 2$. One can verify that starting
with any of the 4 states $|\alpha,\beta\rangle_{SN}$ will yield ${\rm
MIE}(A,B)=\log 2$ if one measures in $X$-basis.

A scheme dual to the above measurement scheme is to start with a state with
definite parities of dual loops along $l_{1},l_{2}$222One says there is a a
dual string at a link if $Z=-1$ at that link, and a dual string lives on a
path of the dual lattice. and make measurements in $Z$-basis, which will also
give ${\rm MIE}(A,B)=\log 2$.

The above example illustrates that the ${\rm MIE}$ of a topological order is a
constant that is invariant under deformations of the measurement region for
certain measurement basis. But several question remains. How exactly is the
${\rm MIE}$ related to the data of the topological order such as quantum
dimension of anyons? Can the result for toric code be generalized to other
topological orders? We address these questions by reformulating the above
calculation in a way that makes it suitable for generalizaiton, along the way
these questions will be resolved automatically.

#### III.III.5.III.5.2) Minimally entangled states

Apart from the string-net basis $|\alpha,\beta\rangle_{SN}$ discussed before,
there are other basis for the ground space of a topological order on a torus.
An important basis consists of the minimally entangled states (MES) [68, 69].
MES will be crucial in obatining the MIE for a general ground state of abelian
quantum double models. Here we briefly summarize their construction. Consider
all loop operators on a non-contractible loop $l_{1}$, these are generated by
$W^{e}_{1},W^{m}_{1}$ as an algebra. Since these two operators commute, we can
diagonalize them simultaneously within the ground space. The common
eigenstates of the loop operators on $l_{1}$ can be labelled by anyon types:
$|a\rangle,a=1,e,m,f$, with the property that the eigenvalue under action by
$W^{a}_{1}$ is given by the braiding statistics of anyons:
$W^{a}_{1}|b\rangle=e^{i\theta(a,b)}|b\rangle$. Here $\theta(a,b)$ is the
braiding phase between anyon $a$ and $b$. The state $|a\rangle$ can be viewed
as having an anyon $a$ threading through the center of the torus333In other
words the state $|a\rangle$ can be viewed as prepared by a path integral in
the solid torus, with an anyon string $a$ inserted in the center $S^{1}$.,
then the action of $W^{b}_{1}$ is performing a braiding between $a$ and $b$ in
spacetime. The string-net basis is related to MES via a linear transformation.
To derive the transformation, notice the effect of the operators $W^{e,m}_{1}$
on the string-net configuration: $W^{e}_{1}$ creates a loop along $l_{1}$
while $W^{m}_{1}$ measures the parity of loops along $l_{2}$. We can then
deduce the action of $W^{a}_{1}$ on the string-net basis:

$\displaystyle
W^{e}_{1}|0,0\rangle_{SN}=|1,0\rangle_{SN},~{}W^{m}_{1}|0,0\rangle_{SN}=|0,0\rangle,$
(21) $\displaystyle
W^{e}_{1}|1,0\rangle_{SN}=|0,0\rangle_{SN},~{}W^{m}_{1}|1,0\rangle_{SN}=|1,0\rangle_{SN},$
(22) $\displaystyle
W^{e}_{1}|0,1\rangle_{SN}=|1,1\rangle_{SN},~{}W^{m}_{1}|0,1\rangle_{SN}=-|0,1\rangle,$
(23) $\displaystyle
W^{e}_{1}|1,1\rangle_{SN}=|0,1\rangle_{SN},~{}W^{m}_{1}|1,1\rangle_{SN}=-|1,1\rangle_{SN},$
(24)

from which we can form superpositions to obtain (unnormalized) eigenstates of
loop operators $W^{a}_{1}$: $|1\rangle=|0,0\rangle_{SN}+|1,0\rangle_{SN}$,
$|m\rangle=|0,0\rangle_{SN}-|1,0\rangle_{SN}$,
$|e\rangle=|1,1\rangle_{SN}+|0,1\rangle_{SN}$,
$|f\rangle=|1,1\rangle_{SN}-|0,1\rangle_{SN}$. The inverse transformation is:
$|0,0\rangle_{SN}=|1\rangle+|m\rangle$,
$|1,0\rangle_{SN}=|1\rangle-|m\rangle$,
$|0,1\rangle_{SN}=|e\rangle-|f\rangle$,
$|1,1\rangle_{SN}=|e\rangle+|f\rangle$.

Now consider taking an MES $|a\rangle$ as our initial state and denote the
post-measurement state as $P_{m}|a\rangle$ where $P_{m}$ is the projection
operator for a measurement outcome $m$. The state $P_{m}|a\rangle$ remains
topologically ordered in the unmeasured region $A,B$, since the stabilizers
$A_{v},B_{f}$ are not altered by the measurements for $v,f$ in $A\cup B$. A
crucial observation is that the post-measurement state $P_{m}|a\rangle$
remains a common eigenstate of loop operators $W^{a}_{1}$ in region $A,B$.
This is due to the fact that the loop operators $W^{a}_{1}$ are topological
and can be freely deformed to be supported entirly in region $A$ or $B$
without changing their action on the states prior to measurement. Therefore
since region $A,B$ are not affected by the measurements, the operators
$W^{a}_{1}$ commute with the projector $P$, and $P_{m}|a\rangle$ remains an
eigenstate of $W^{a}_{1}$ with the same eigenvalues as the initial state
$|a\rangle$, that is, $W^{b}_{1}P_{m}|a\rangle=e^{i\theta(a,b)}P_{m}|a\rangle$

After measurement, the quantum state in $M$ becomes a trivial product state,
therefore the $A,B$ to $M$ interfaces become gapped boundaries of the
unmeasured region $A,B$. Region $A,B$ now host topological order on cylinder
with gapped boundaries, whose ground space is finitely degenerated and can be
characterized by the action of $W^{a}_{1}$s. Similar to the situation on a
torus we can label the states in region $A,B$ by anyon types and they satisfy
the relation $W^{1}_{b}|a\rangle=e^{i\theta(a,b)}|a\rangle$. One can again
picture the state as having an anyon $b$ threading through the center of the
cylinder. The fact that the post-measurement state $P_{m}|a\rangle$ remains an
eigenstates of $W^{b}_{1}$ with eigenvalues $e^{i\theta(a,b)}$ fixes the
states in region $A,B$ to be $|a\rangle_{A},|a\rangle_{B}$. Therefore we
conclude for an MES,

$\displaystyle\frac{P|a\rangle}{\sqrt{\langle
a|P|a\rangle}}=|a\rangle_{A}\otimes|a\rangle_{B}\otimes|\phi\rangle_{M}$ (25)

This shows for MES there is no MIE for any measurement basis. The
decomposition of post-measurement state Eq. (25) allows us to directly
calculate ${\rm MIE}$ for a generic initial state by expanding the initial
state in the MES basis.

Let us reproduce the string-net states ${\rm MIE}$ using the MES formalism.
The state $|0,0\rangle_{SN}$ can be expanded in MES as
$|0,0\rangle_{SN}=\frac{1}{\sqrt{2}}(|1\rangle+|m\rangle)$. Apply the
projection operator to the initial state and use the decomposition Eq. (25),
we conclude 444Here we also used the fact that the norm square of the post
measurement state $\langle a|P|a\rangle$ is the same for $a=1,m$. This can be
justified by noticing that $|a\rangle$s are related by the action of
$W^{a}_{2}$s: $|a\rangle=W^{a}_{2}|1\rangle$, therefore $\langle
m|P|m\rangle=\langle 1|W^{m}_{2}PW^{m}_{2}|1\rangle$. Since $W^{m}_{2}$ is a
string of $X$ operators that commutes with the measurements, we have $\langle
m|P|m\rangle=\langle 1|P|1\rangle$.

$\displaystyle
P|0,0\rangle_{SN}=\frac{1}{\sqrt{2}}(|1\rangle_{A}|1\rangle_{B}+|m\rangle_{A}|m\rangle_{B})|\phi\rangle_{M}$
(26)

which is a Bell-pair with entanglement entropy $\log 2$ regardless of
measurement outcome. Therefore MIE for $|0,0\rangle_{SN}$ is $\log 2$, in
agreement with the result obtained earlier using the string-net picture of the
ground state wavefunction.

####  III.III.5.III.5.3) MIE in Abelian quantum double models

The toric code analysis can be readily generalized to general Abelian
topological orders. From the decomposition Eq. (25) we know the MES has zero
MIE. Therefore a non-zero MIE can only be obtained if one starts with a non-
MES state, such as the string-net states $|SN,\alpha,\beta\rangle$ of the
toric code. We show that the string-net states of toric code can be
generalized to generic abelian quantum double models $D(G)$, and their MIE
contains information about the order of the gauge group $G$.

Figure 10: Definition of vertex and face operators in quantum double models.

The abelian quantum double model $D(G)$ with abelian $G$, can be defined on
the square lattice as follows [70, 71]. There is a $|G|$-dimensional Hilbert
space on every link of the lattice, with a natural basis $|g\rangle,g\in G$.
The lattice is endowed with an orientation of edges and the Hamiltonian are
built with two types of operators $A^{g}_{v},B^{h}_{f}$, defined on vertices
and faces of the lattice respectively. $A^{g}_{v}$ changes the states on the 4
edges adjacent to $v$ from $|h\rangle$ to $|gh\rangle$ or $|g^{-1}h\rangle$,
according to whether the edge is pointing towards or away from the vertex $v$.
$B^{h}_{f}$ forces the “flux” through the face $f$ to be $h$. These operators
are summarized in Fig. 10. The Hamiltonian is then built from these two types
of operators as follows:

$\displaystyle H=-\sum_{v}A_{v}-\sum_{f}B_{f},$ (27) $\displaystyle
A_{v}:=\frac{1}{|G|}\sum_{g}A^{g}_{v},~{}B_{f}:=B^{e}_{f}$ (28)

$A_{v}$ can be viewed of as the Gauss’s law constraint and $B_{f}$ can be
viewed as the zero-flux condition. The ground space of the quantum double can
also be described by the picture of string-net condensation. However one needs
to use a basis dual to $|g\rangle$ to express the ground state as a string-net
liquid on the direct lattice.555Alternatively one can write the ground state
of the quantum double as a string-net liquid in the $|g\rangle$-basis on the
dual lattice.  For any character of the group $G$:
$\chi\in\widehat{G}=\hom[G,U(1)]$, define a dual basis on an edge as
$|\chi\rangle:=\frac{1}{\sqrt{|G|}}\sum_{g}\chi(g)|g\rangle$. This can be
viewed as a group Fourier transformation of the basis $|g\rangle$, with
inverse transformation
$|g\rangle=\frac{1}{\sqrt{|G|}}\sum_{\chi\in\widehat{G}}\chi^{*}(g)|\chi\rangle$.
Then acting on dual basis $|\chi_{i}\rangle$, where $i=1,2,3,4$ labels 4 edges
adjacent to $v$, we have

$\displaystyle
A_{v}\sim\frac{1}{|G|}\sum_{g}\chi_{1}^{s_{1}}(g)\chi^{s_{2}}_{2}(g)\chi^{s_{3}}_{3}(g)\chi^{s_{4}}_{4}(g),$
(29)

where $s_{i}$ is the orientation of edge $i$ with respect to $v$. From
properties of group Fourier transformation we have

$\displaystyle\frac{1}{|G|}\sum_{g}\chi_{1}^{s_{1}}(g)\chi^{s_{2}}_{2}(g)\chi^{s_{3}}_{3}(g)\chi^{s_{4}}_{4}(g)=\delta_{\prod_{i}\chi_{i}^{s_{i}},1}.$
(30)

Hence, the vertex term $A_{v}$ enforces the product of characters on edges
adjacent to a vertex to be the trivial character. On the other hand the face
term can be written in the $|g_{i}\rangle$ basis as
$B_{f}\sim\delta_{g_{1}g_{2}g_{3}g_{4},e}=\frac{1}{|G|}\sum_{\chi}\chi(g_{1}g_{2}g_{3}g_{4})$.
Therefore in the dual basis we have
$B_{f}=\frac{1}{|G|}\sum_{\chi}B^{\chi}_{f}$, where $B^{\chi}_{f}$ changes the
states on the the edges of the face $f$ from $|\chi_{i}\rangle$ to
$|\chi\chi_{i}\rangle$. We see that in the dual basis the roles of $A_{v}$ and
$B_{f}$ are interchanged. Then one can now view $A_{v}$ as enforcing the
condition that in a ground state the characters flowing through any vertex is
1. $B_{f}$ terms then form a superposition of all possible such
configurations. We can now say that the ground state of the Hamiltonian Eq.
(27) is a string-net liquid on the direct lattice, with string types labelled
by characters of $G$.

We are now in place to define a string-net basis for a general abelian quantum
double model defined on a torus. In a given string-net configuration, the
string type along a non-contractible direction $l_{1,2}$ is $\chi_{1,2}$. Then
the ground states with definite $\chi_{1},\chi_{2}$ are called string-net
states and are denoted by $|\chi_{1},\chi_{2}\rangle_{SN}$. We now make
connection to the MES, defined for any topological order on a torus. The MES
are labelled by anyon types. For abelian quantum double models, the anyon
types are labelled by pairs $(g,\chi),g\in G,\chi\in\widehat{G}$, with self
and mutual statistics given by
$\theta(g,\chi)=\chi(g),\theta((g,\chi),(g^{\prime},\chi^{\prime}))=\chi(g^{\prime})\chi^{\prime}(g)$
[70, 71]. $\chi$-type anyon is associated with violation of the vertex terms
$A_{v}$ of the Hamiltonian, $g$-type anyon is associated with violation of
face terms $B_{f}$. To find the transformation between MES $|g,\chi\rangle$
and string-net states $|\chi_{1},\chi_{2}\rangle$, we analyze the effect of
loop operators $W^{1}_{g,\chi}$ on the string-net states. The loop operator
$W^{1}_{g,\chi}$ moves an anyon $(g,\chi)$ around $l_{1}$. It is an ribbon
operator with support on a direct path and a dual path, both along $l_{1}$. It
changes the states on the direct path from $|\chi_{i}\rangle$ to
$|\chi\chi_{i}\rangle$, and the states on the dual path from $|g_{i}\rangle$
to $|gg_{i}\rangle$. Therefore it acts on the string-net states as
$W^{1}_{g,\chi}|\chi_{1},\chi_{2}\rangle_{SN}=\chi_{2}(g)|\chi\chi_{1},\chi_{2}\rangle_{SN}$.
To obtain MES, which are eigenstates of $W^{1}_{g,\chi}$, we simply perform
Fourier transform to the first label of string-net states and define:
$|g,\chi\rangle=\frac{1}{\sqrt{|G|}}\sum_{\chi_{1}}\chi^{*}_{1}(g)|\chi_{1},\chi_{2}\rangle_{SN}$.
One can verify the states $|g,\chi\rangle$ satisfies
$W^{1}_{g^{\prime},\chi^{\prime}}|g,\chi\rangle=\chi_{(}g^{\prime})\chi^{\prime}(g)|g,\chi\rangle$.
The phase $\chi(g^{\prime})\chi^{\prime}(g)$ is exactly the braiding between
two anyons $(g,\chi)$ and $(g^{\prime},\chi^{\prime})$.Thus the states
$|g,\chi\rangle$ are indeed the MES. The inverse transformation is given by
the inverse Fourier transformation:
$|\chi_{1},\chi_{2}\rangle_{SN}=\frac{1}{\sqrt{|G|}}\sum_{g}\chi_{1}(g)|g,\chi_{2}\rangle$.

Now using the decomposition of MES after measurement Eq. (25), we have for
string-net states

$\displaystyle
P|\chi_{1},\chi_{2}\rangle_{SN}=\frac{1}{\sqrt{|G|}}\sum_{g}\chi_{1}(g)|g,\chi_{2}\rangle_{A}\otimes|g,\chi_{2}\rangle\otimes|\phi\rangle_{M}$
(31)

The entanglement entropy between $A$ and $B$ is therefore $\sum_{g\in
G}\frac{1}{|G|}\log|G|=\log|G|$ regardless of measurement outcome. We conclude
that for string-net states we have ${\rm MIE}=\log G=\log D$. We used the fact
that the quantum double $D(G)$ has total quantum dimension $D=|G|$.

##  IV Discussion

This exploration of measurement-induced entanglement (MIE) and information
(MII) in ground-states of various systems reveals the presence of universal
features depending on the topology or universality class for critical or
gapless systems. This universal structure provides distinct, complementary
information from that contained in the single-interval entanglement entropy of
the state. These quantities are relevant for assessing the computational power
of the state for measurement-based quantum computing, and have operational
significance for the classical and quantum complexity of simulating that
state. With the exception of gapped topological phases, our present
understanding of MIE and MII comes entirely from numerical simulations of
piecemeal examples. To obtain a systematic understanding of the structure of
MIE and MII, analytic methods for computing these from field theory
descriptions would be highly valuable, and are an important challenge for
future work. For example, while numerical simulations of the XX spin chain
show that the MIE behaves like a four-point correlation function, although the
scaling exponent does not appear to be simply related to known bulk- or
boundary- scaling dimensions. This suggests that the MIE for $1+1d$ CFTs
could, perhaps, be governed by new classes of scaling operators that have not
been previously considered. An analytic understanding would be especially
valuable for studying higher-dimensional gapless systems, such as non-Fermi
liquids, and gapless quantum spin liquids, where purely (classical) numerical
methods are challenging to implement at large scale.

Acknowledgements –  We thank Tim Hsieh and Ehud Altman for stimulating
conversations. This work was supported by the US Department of Energy under
awards DOE DE-SC0022102 (A.C.P.), the Air Force Office of Scientific Research
under Grant No. FA9550-21-1-0123 (R.V.) and by a Sloan Research Fellowship
(A.C.P.). We thank the Kavli Institute of Theoretical Physics (KITP) and the
Aspen Center for Physics where part of this work were completed. KITP is
supported in part by the National Science Foundation under Grant No. NSF
PHY-1748958.

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##  Appendix A Measurements in Gaussian fermion states

A Gaussian fermion state can be entirely captured by its single particle
correlation function $C_{ij}\equiv\langle c^{\dagger}_{i}c_{j}\rangle$ [72].
Here we consider the effect of single orbital measurements on Gaussian fermion
states, where the on-site projectors are given by

$\displaystyle P_{1}=c_{a}^{\dagger}c_{a},\quad
P_{0}=c_{a}c_{a}^{\dagger}=1-c_{a}^{\dagger}c_{a},$ (32)

with probabilities

$\displaystyle p_{1}=C_{aa},\quad p_{0}=1-C_{aa}.$ (33)

where $a$ is the measured orbital.

When $P_{1}$ is applied, the updated correlation matrix is given by

$\displaystyle C^{\prime}_{ij}=\frac{\langle
c^{\dagger}_{a}c_{a}c^{\dagger}_{i}c_{i}c^{\dagger}_{a}c_{a}\rangle}{C_{aa}}=\left\\{\begin{aligned}
\ &1&\mathrm{if}\ i=j=a\\\ \ &C_{ij}-\frac{C_{ia}C_{aj}}{C_{aa}}&\mathrm{if}\
i\neq a,\ j\neq a\\\ \ &0&\mathrm{otherwise}\end{aligned}\right.$ (34)

and when $P_{0}$ is applied, the updated correlation matrix is given by

$\displaystyle C^{\prime}_{ij}=\frac{\langle
c_{a}c^{\dagger}_{a}c^{\dagger}_{i}c_{i}c_{a}c^{\dagger}_{a}\rangle}{1-C_{aa}}=\left\\{\begin{aligned}
\ &0&\mathrm{if}\ i=j=a\\\ \
&C_{ij}+\frac{C_{ia}C_{aj}}{1-C_{aa}}&\mathrm{if}\ i\neq a,\ j\neq a\\\ \
&0&\mathrm{otherwise}\end{aligned}\right.$ (35)

where multi particle correlators can be decomposed to single particle
correlators by Wick’s theorem. With the update rule, one can easily obtain the
postmeasurement correlation functions satisfying Born probability.

##  Appendix B MIE upper-bounds strange correlators

Mutual information between regions $A$ and $B$ upper bounds connected
correlators between operators supported in these regions. In this section, we
show that the MIE similarly upper bounds _strange_ correlators. Consider a
state $|\psi\rangle$ which is partitioned into three regions: $A$, $B$, and
$C$:

$\displaystyle|\psi\rangle=\sum_{a,b,c}\phi_{abc}|abc\rangle,$ (36)

where $a$, $b$, and $c$ label the local degree of freedom in $A$, $B$, and
$C$. We apply measurements on region $C$ with outcome $|m_{c}\rangle$, and
then the resulting wavefunction is

$\displaystyle|\psi_{c}\rangle=\frac{\langle
m_{c}|\psi\rangle}{\sqrt{p_{m_{c}}}}=\sum_{a,b}\frac{\phi_{abm_{c}}}{\sqrt{p_{m_{c}}}}|ab\rangle,$
(37)

with probability $p_{m_{c}}=\sum_{a,b}|\phi_{abm_{c}}|^{2}$. The averaged MIE
in $A$ is defined as

$\displaystyle\overline{{\rm
MIE}}(A)=\sum_{m_{c}}p_{m_{c}}S_{A}\left(|\psi_{c}\rangle\right).$ (38)

In the same setting, we can also define a strange correlator between a post-
measurement product state $|m\rangle=|m_{a}m_{b}m_{c}\rangle$ and the
interested state $|\psi\rangle$

$\displaystyle\mathrm{SC}\left(O_{A},O_{B}\right)=\frac{\langle
m|O_{A}O_{B}|\psi\rangle}{\langle m|\psi\rangle}$ (39)

where $m_{a}$ and $m_{b}$ are meaurement outcomes in regions $A$ and $B$,
which are not measured in the MIE setting, and $O_{A}$ and $O_{B}$ are charged
local operators in regions $A$ and $B$.

To show the connection between strange correlators and MIE, we first rewrite
the MIE as the mutual information between $A$ and $B$ in the post-measurement
state $|\psi_{c}\rangle$

$\displaystyle{\rm
MIE}(A)=\frac{1}{2}\sum_{m_{c}}p_{m_{c}}I(A,B)\left[|\psi_{c}\rangle\right].$
(40)

Then we can follow Ref. [73] to express the mutual information as a relative
entropy

$\displaystyle
I(A,B)\left[|\psi_{c}\rangle\right]=S\left(\rho^{c}_{AB}|\rho^{c}_{A}\otimes\rho^{c}_{B}\right),$
(41)

where $\rho_{AB}^{c}=|\psi_{c}\rangle\langle\psi_{c}|$ and
$\rho_{A/B}^{c}=\text{tr}_{B/A}\left(|\psi_{c}\rangle\langle\psi_{c}|\right)$.
Then using the norm bound [74]
$S(\rho|\sigma)\geq\frac{1}{2}||\rho-\sigma||^{2}_{1}$ and the trace
inequality $||X||_{1}\geq\text{tr}\left(XY\right)/||Y||$, we can obtain

$\displaystyle
I(A,B)\left[|\psi_{c}\rangle\right]\geq\frac{1}{2}\frac{\left(\text{tr}\left(\rho_{AB}^{c}Y\right)-\text{tr}\left(\rho_{A}^{c}\otimes\rho_{B}^{c}Y\right)\right)^{2}}{||Y||^{2}}.$
(42)

Let $Y=|m_{a}m_{b}\rangle\langle m_{a}m_{b}|O_{A}O_{B}$, then the right hand
side of the inequality becomes

$\displaystyle~{}\frac{p_{m_{a}m_{b}m_{c}}^{2}}{2\overline{O}^{2}_{A}\overline{O}^{2}_{B}p_{m_{c}}^{2}}\left[\mathrm{SC}(O_{A},O_{B})\right]^{2}$
(43)

where $\overline{O}_{A/B}=\langle m_{A/B}|O_{A/B}|m_{A/B}\rangle$ and we have
used the fact that each changed local operator has zero expectation value.

Combining Eq. (40)– Eq. (43), we can obtain

$\displaystyle{\rm MIE}(A)\geq
c_{0}\sum_{m_{c}}\frac{p^{2}_{m_{a}m_{b}m_{c}}}{p_{m_{c}}}\left[\mathrm{SC}(O_{A},O_{B})\right]^{2}$
(44)

where $c_{0}$ is some non-universal constant depending on the choice of
$|m_{a/b}\rangle$ and $O_{A/B}$. Therefore, MIE is lower-bounded by the
average of square of strange correlators weighted by some nonuniversal joint
measurement probability.