# Separability criterion using one observable for special states: Entanglement
detection via quantum quench

Roopayan Ghosh  Department of Physics and Astronomy, University College
London,Gower Street, WC1E6BT, London  Sougato Bose  Department of Physics and
Astronomy, University College London,Gower Street, WC1E6BT, London

###### Abstract

Detecting entanglement in many-body quantum systems is crucial but
challenging, typically requiring multiple measurements. Here, we establish the
class of states where measuring connected correlations in just one basis gives
a necessary condition to detect bipartite separability thus offering a
sufficient condition to detect entanglement, provided the appropriate basis
and observables are chosen. This methodology leverages prior information about
the state, which, although insufficient to reveal the complete state or its
entanglement, enables our one basis approach to be effective. We discuss the
possibility of one observable entanglement detection in a variety of systems,
including those without conserved charges, such as the Transverse Ising model,
reaching the appropriate basis via quantum quench. This provides a much
simpler pathway of detection than previous works. It also shows improved
sensitivity from Pearson Correlation detection techniques.

## Introduction:

Detection of many-body entanglement (bi-partite or multipartite) is an
experimentally difficult task insofar only a handful of experiments [1, 2]
have been able to successfully achieve it. The typical methods involve
creation of a replica of the original many-body system to compute purity[1] or
using a series of randomized measurements to detect Renyi entropy in a trapped
ion quantum simulator[2]. Both of the set-ups were fine tuned and while they
can measure the entanglement of a carefully prepared state in an idealized
setting, measuring or witnessing entanglement of states obtained in many-body
experiments simply and effectively is practically impossible[3].

That being said, entanglement detection in general many-body quantum systems
is crucial for several reasons. Firstly, numerous emergent many-body systems
in solid-state structures, such as multiple electrons confined to nanowires,
can be described by at least one of the Hamiltonians discussed in this work.
Additionally, various other types of Hamiltonians can be realized in
engineered quantum dot arrays. Detecting entanglement between different parts
of such systems is vital for certifying the genuine quantum nature of these
systems. Moreover, entanglement between two components of a many-body system
has been a subject of study for an extended period, with quantifications such
as von Neumann entropy and negativity being employed. The motivation behind
these studies was to comprehend the entanglement structure of such systems,
either to facilitate effective variational descriptions or to assess the
natural availability of resources for quantum communication. However, one
needs to actually verify this entanglement in experiments, and the above
quantifications are difficult to implement experimentally.

Hence it is of paramount importance to connect entanglement detection in
specific classes many-body states with quantities requiring less demanding
experimental resources to measure. To begin, let us look at the most general
definition of an entangled state. An entangled state is defined as one that
cannot be separated into individual states belonging in separate subspaces. In
the simplest scenario of bipartitions, i.e., a system with constituent Hilbert
spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, and the combined space given
by $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, a pure state
$|\psi\rangle\in\mathcal{H}$ is separable iff,

$|\psi\rangle=|\psi_{A}\rangle\otimes|\psi_{B}\rangle$ (1)

where $|\psi_{A[B]}\rangle\in\mathcal{H}_{A[B]}$. On the other hand, a mixed
state represented by density matrix $\rho$ is separable iff

$\rho=\sum_{i}p_{i}\rho_{i}^{A}\otimes\rho_{i}^{B}$ (2)

where $\rho_{i}^{A[B]}\in\mathcal{H}_{A[B]}$ and $\sum_{i}p_{i}=1$.

Entanglement indicates quantum correlations, and measurement of classical
correlations between one set of observables between the corresponding
subsystems is typically not enough to detect entanglement — classical
correlations should necessarily exist in several different observables. The
search for detection of entanglement with least number of measurements have
taken us in various directions in recent years[4, 5, 6, 7, 8, 9, 10, 11, 12,
13], with specific techniques available for spin systems[14, 15, 16, 17, 18,
19], measuring charge fluctuations [20, 21, 22, 23], coherences [24] or number
entropy[25, 26, 27, 28] in systems with locally conserved charge, and
measurements of classical correlations in a set of Mutually Unbiased
Bases(MUBs) or observables[29, 30, 31, 32, 33, 34, 35].

The focus of this work shall be on the efficient detection of bipartite
entanglement utilizing the aforesaid MUBs. Previous studies involving MUBs
have revealed that at least two measurements are necessary to establish a
reliable criterion for detection. However, considering that bipartite
entanglement typically implies classical correlation in many measurements,
naively we expect the existence of at least one measurement that exhibits zero
correlation for separable states, but non-zero correlation for entangled
states. Then by measuring this specific quantity, our problem of entanglement
detection should be resolved. But the challenge lies in the fact that there is
no guarantee of existence of such a quantity that is independent of the state
being examined, thus leading to a wide variety of state-dependent entanglement
witnesses.

In this work we find a common observable which can efficiently detect
bipartite entanglement for a wide class of states. We demonstrate that, (i) it
is in fact possible to detect entanglement of any arbitrary mixed state via
measuring correlations of one observable independently of the measured state,
if they are separable to purely real states or states with the density matrix
having purely imaginary off diagonal elements. (ii) We also show that in
presence of a locally conserved charge such as particle number/magnetization,
detection of entanglement in any generic state (i.e. need not be separable to
purely real or imaginary) is possible by this technique. 111There are several
methods already available for entanglement detection in this case,but mostly
focusing on pure states. [20, 21, 22, 23, 24, 25, 26, 27, 28]. However this
method provides a new correlation witness from the point of view of MUBs for
all kinds of states. (iii) We then test the efficacy of the observable to
detect entanglement in mixed states generated by different Hamiltonians via
exact numerical simulations.

In what follows, since our proposed criterion is based on MUB, we first define
MUBs and discuss a specific entanglement witness which utilizes them. Then we
elaborate our criterion and provide examples comparing efficiency of both
approaches.

## Mutually Unbiased Basis(MUB):

A set of bases $\\{\bm{\alpha}\\}=\\{\ket{\alpha_{i}}\\}$ and
$\\{\bm{\alpha^{\prime}}\\}=\\{\ket{\alpha^{\prime}_{j}}\\}$ where $i,j\in
1,\ldots,d$, in a Hilbert space $\mathcal{H}=\mathbb{C}^{d}$ is mutually
unbiased iff,

$|\langle\alpha_{i}|\alpha^{\prime}_{j}\rangle|^{2}=1/d\hskip
36.135pt\forall\hskip 14.45377pti,j$ (3)

## Entanglement witnessing using MUBs

Entanglement detection using MUBs requires measurement of correlations. Among
the several different quantifiers which have been addressed in previous works
[29, 30, 31, 32, 33, 34], we focus on the Pearson correlation measurement, as
it was shown to be among the more efficient measures.[31]

Consider a system $S$, subdivided into two subsystems $A$ and $B$. Let us
denote an observable measured in subsystem $A$ as $\mathcal{O}_{A}$ and for
$B$ as $\mathcal{O}_{B}$. Then the Pearson correlation between the observables
is given by the normalized connected correlation,

$\mathcal{P}_{\mathcal{O}}=\frac{\langle\mathcal{O}_{A}\otimes\mathcal{O}_{B}\rangle-\langle\mathcal{O}_{A}\otimes\mathbb{I}\rangle\langle\mathbb{I}\otimes\mathcal{O}_{B}\rangle}{\sqrt{\langle\mathcal{O}_{A}^{2}\otimes\mathbb{I}\rangle-\langle\mathcal{O}_{A}\otimes\mathbb{I}\rangle^{2}}\sqrt{\langle\mathbb{I}\otimes\mathcal{O}_{B}^{2}\rangle-\langle\mathbb{I}\otimes\mathcal{O}_{B}\rangle^{2}}},$
(4)

where the expectation value is taken with respect to the state of the system
$S$. For MUB measurement, if we assume that the set of eigenvectors for
$\mathcal{O}_{A}$ is $\\{\ket{\alpha}\\}$, then we can find another observable
$\mathcal{O}^{\prime}_{A}$ with eigenvectors $\\{\ket{\alpha^{\prime}}\\}$
fulfilling Eq. 3. A similar construction can be done for subsystem $B$. Then

$|\mathcal{P}_{\mathcal{O}}|+|\mathcal{P}_{\mathcal{O}^{\prime}}|>1$ (5)

provides a sufficient but not necessary condition for the two halves of the
system to be entangled [31]. Further improvements on the bound were seen if
correlations were measured in more than two MUBs.

Hereafter, we shall focus in the opposite direction, namely, we shall
establish the case where measurements in two basis are not needed, i.e., non-
zero connected correlations in one basis immediately indicates entanglement.

## One measurement entanglement detection:

For completeness, we shall discuss the cases of both pure and mixed states,

  1. (I)

Pure states: In the study of many-body physics we often work with pure states,
when we isolate the experimental set-up from interactions with the
environment. In this case,

$\mathcal{C}_{1}=\langle\mathcal{O}_{A}\otimes\mathcal{O}_{B}\rangle-\langle\mathcal{O}_{A}\otimes\mathbb{I}\rangle\langle\mathbb{I}\otimes\mathcal{O}_{B}\rangle$
(6)

$=0$, is a necessary but not sufficient condition for zero entanglement
between the partitions. This indicates a non zero value of $C_{1}$ is a
sufficient condition for entanglement.

The proof of necessity directly follows from Eq. (1). The lack of sufficiency
condition for separability can be seen from special choices of pure states.
For example, if we take the entangled singlet state
$\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})$ and
take $\mathcal{O}_{A}=\sigma_{z}$ and $\mathcal{O}_{B}=\sigma_{x}$, we shall
get no correlation.

  2. (II)

Mixed states: For generic mixed states, $\mathcal{C}_{1}=0$ is neither a
necessary nor a sufficient for separability. To illustrate this, consider a
separable state $\rho_{CC}$ given by,

$\rho_{CC}=\frac{1}{d}\sum_{i=1}^{d}\ket{\alpha_{i}}\bra{\alpha_{i}}\otimes\ket{\beta_{i}}\bra{\beta_{i}}$
(7)

where $\\{|\beta\rangle\\}$ is the eigenbasis of $\mathcal{O}_{B}$. In this
case, $\mathcal{C}_{1}$ can show high correlations, but

$\mathcal{C}_{2}=\langle\mathcal{O}^{\prime}_{A}\otimes\mathcal{O}^{\prime}_{B}\rangle-\langle\mathcal{O}^{\prime}_{A}\otimes\mathbb{I}\rangle\langle\mathbb{I}\otimes\mathcal{O}^{\prime}_{B}\rangle$
(8)

will be equal to $0$.

For the generic separable mixed state of Eq. (2) correlations may exist in
multiple bases, but there is no entanglement present. This can be seen by
choosing appropriate $\mathcal{O}_{A}$ and $\mathcal{O}_{B}$ to compute,

$\displaystyle\langle\mathcal{O}_{A}\otimes\mathcal{O}_{B}\rangle$
$\displaystyle=\sum_{i}p_{i}{\rm Tr}[\rho_{i}^{A}\mathcal{O}_{A}]{\rm
Tr}[\rho_{i}^{b}\mathcal{O}_{B}]$
$\displaystyle\langle\mathcal{O}_{A}\otimes\mathbb{I}\rangle\langle\mathbb{I}\otimes\mathcal{O}_{B}\rangle$
$\displaystyle=\sum_{i}p_{i}{\rm
Tr}[\rho_{i}^{A}\mathcal{O}_{A}]\sum_{j}p_{j}{\rm
Tr}[\rho_{j}^{b}\mathcal{O}_{B}]$ (9)

If $p_{i}=1$, i.e., the system is separable to $\rho=\rho^{A}\otimes\rho^{B}$,
then $\mathcal{C}_{1}=\mathcal{C}_{2}=0$, and the condition becomes necessary.
For all other typical cases it is not. Thus, entanglement witness bounds
usually involve computation of correlations in several measurements.

To overcome this difficulty, we attempt to construct a common observable for
all density matrices whose measurement shall provide the necessary criterion.
As mentioned before, a common observable is not guaranteed to exist for
arbitrary density matrices, so we search for classes of states which allow us
to do so.

## Construction of the observable:

Let us consider the case of L-qubits, i.e. Hilbert spaces of dimension
$2^{L}$. The generalization to qudits is provided in Appendix B. From the
previous discussion of $\rho_{CC}$, choosing an observable whose eigenbasis is
mutually unbiased to $\\{|\alpha\rangle,|\beta\rangle\\}$ is clearly necessary
(For simplicity we hereafter take $\\{|\alpha\rangle,|\beta\rangle\\}$ as the
computational $\sigma_{z}$ basis of the respective systems). Additionally, for
separable states not of the form of $\rho_{CC}$, any arbitrary mutually
unbiased observable does not yield $\mathcal{C}_{2}=0$. Ideally since we also
want an observable easily implementable in experiments, the special observable
should to be realized by very local (involving on-site or nearest neighbour)
transformations. While this is not easily realizable for a collection of
qudits, in the L-qubit scenario, a special observable whose eigenbasis is
mutually unbiased to the computational basis can be created by rotating
$\mathcal{O}_{A[B]}$ via

$U_{A[B]}=(e^{-i\sigma^{x}\pi/4})^{\otimes L_{A[B]}},$ (10)

where $L_{A[B]}$ is the length of subsystem $A[B],L_{A}+L_{B}=L$.

By explicit construction , we see $U$ has the following properties.

  1. (i)

$(U_{A[B]})_{p,q}=\frac{1}{\sqrt{d_{A[B]}}}e^{-i\pi(\phi_{p,q})/2}$, where
$\phi_{p,q}\in[0,L_{A[B]}]\cap\mathbb{Z}$

  2. (ii)

$U_{A[B]}^{T}=U_{A[B]}\implies\phi_{p,q}=\phi_{q,p}$.

  3. (iii)

$(U_{A[B]})_{p,q}(U_{A[B]})_{d-p+1,q}=\frac{1}{d_{A[B]}}e^{-i\pi
L_{A[B]}/2}\implies\phi_{p,q}+\phi_{d-p+1,q}=L_{A[B]}$

where $d_{A[B]}=2^{L_{A[B]}}$, the dimensions of the subsystem. These
properties can be extracted from the tensor product structure of $U_{A}$ for
any $L_{A}$, by building from the $L_{A}=1$ case, which in the
$\\{\ket{0},\ket{1}\\}$ basis is,

$U_{A}=\left(\begin{array}[]{cc}\frac{1}{\sqrt{2}}&-\frac{i}{\sqrt{2}}\\\
-\frac{i}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\ \end{array}\right)$ (11)

A detailed explanation is provided in Appendix. A

Moving to the Schroedinger picture we obtain the density matrix after the
evolution,

$\rho^{\prime}_{i_{A},i_{B},j_{A},j_{B}}=(U_{A}\otimes
U_{B})_{i_{A},i_{B},k_{A},k_{B}}\rho_{k_{A},k_{B},l_{A},l_{B}}(U_{A}\otimes
U_{B})^{\dagger}_{l_{A},l_{B},j_{A},j_{B}}$ (12)

We see that the connected correlator with respect to $\rho^{\prime}$ is
identically $0$, when the following conditions are fulfilled,

Condition (i)

  1. 1.

$\rho$ is separable (i.e. given by Eq. (2)) to purely real [ or with purely
imaginary off-diagonal elements] density matrices, i.e.
$\rho_{i}^{A[B]}\in\mathbb{R}^{d_{A[B]}\times d_{A[B]}}$ or
$(\rho^{A[B]}_{i})_{k,l,k\neq l}\in i\mathbb{R}^{d_{A[B]}\times d_{A[B]}}$.

  2. 2.

The observable $\mathcal{O}$ is diagonal in the computational basis222This
means that if we want to measure bipartite entanglement not in real space but
any other space, for example in momentum space, we have to choose the
observable to be diagonal in the computational basis of that space. with
eigenvalues $E_{j}=\\{f(j)+c_{1}\ni
f(j)=-[+]f(d-j+1),j\in[1,d]\cap\mathbb{Z},c_{1}\in\mathbb{R}\\}$. Examples of
such observables include total particle number or total magnetization in many-
body systems, or the $S_{z}$ index in large spin systems, all typically
measurable by local measurements. The ‘$[+]$’ is required for imaginary case
of the previous condition.

OR

Condition (ii)
$\rho$ is separable with a locally conserved charge, such as magnetization or
particle number, and we perform subsystem measurements of the conserved
charge.

## Proof of condition (i):

We first show our conjecture is valid when the first two statements are
fulfilled. Dropping the indices $i$ from Eq. (2) and subscript $A$ from
$d_{A}$ for brevity, we see that,

$\displaystyle\begin{split}&{\rm Tr}[\rho^{\prime
A}\mathcal{O}_{A}]=\sum_{p=1}^{d/2}\sum_{q,r}^{d}(f(p)+c_{1})(U_{A})_{p,q}\rho_{qr}(U_{A})^{*}_{r,p}\\\
&+\sum_{p=1}^{d/2}\sum_{q,r}^{d}(-f(p)+c_{1})(U_{A})_{d-p+1,q}\rho_{q,r}(U_{A})^{*}_{r,d-p+1}\end{split}$
(13)

Then using the properties of $U_{A}$ we can simplify Eq. 13 by realizing that,

$\displaystyle\sum_{p=1}^{d/2}\sum_{q,r}^{d}f(p)(U_{A})_{d-p+1,q}\rho_{q,r}(U_{A}^{*})_{r,d-p+1}$
$\displaystyle=\frac{1}{d}\sum_{p=1}^{d/2}\sum_{q,r}^{d}f(p)e^{-i\pi/2[(L_{A}-\phi_{p,q})-(L_{A}-\phi_{r,p})]}\rho_{q,r}$
$\displaystyle=\frac{1}{d}\sum_{p=1}^{d/2}\sum_{q,r}^{d}f(p)e^{-i\pi/2(\phi_{p,q}-\phi_{r,p})}\rho_{q,r}$
$\displaystyle=\sum_{p=1}^{d/2}\sum_{q,r}^{d}f(p)(U_{A})_{p,q}\rho_{q,r}(U_{A}^{*})_{r,p}$
(14)

where in the third step, we have used $U_{A}^{T}=U_{A}$ and
$\rho_{q,r}=\rho_{r,q}$, for purely real $\rho_{A}$. For purely imaginary off-
diagonal elements of $\rho_{A}$, we shall have $\rho_{q,r}=-\rho_{r,q}$, hence
we need to choose $f(j)=f(d-j+1)$. Then Eq. 13 immediately simplifies to yield

$\displaystyle{\rm Tr}[\rho^{\prime A}\mathcal{O}_{A}]$ $\displaystyle=$
$\displaystyle\sum_{p=1}^{d}\sum_{q,r}^{d}c_{1}(U_{A})_{p,q}\rho_{q,r}(U_{A})^{*}_{r,p}$
(15) $\displaystyle=c_{1}{\rm Tr}[\rho^{\prime A}]=c_{1}$

Thus we have proven that for any real [imaginary] density matrix
$\langle\mathcal{O}^{\prime}_{A}\otimes\mathbb{I}\rangle$ gives a constant
value, $c_{1}$ depending on parameters of the operator chosen. If we perform
the same unitary evolution on partition $B$ we will obtain a similar result
for $\langle\mathbb{I}\otimes\mathcal{O}^{\prime}_{B}\rangle$, say $c_{2}$.
Then invoking the separability condition,
$\rho=\sum_{i}p_{i}\rho_{i}^{A}\otimes\rho_{i}^{B}$ we can compute
$\langle\mathcal{O}^{\prime}_{A}\otimes\mathcal{O}^{\prime}_{B}\rangle$ to
obtain $c_{1}c_{2}$. Note that since the expectation values now depends on the
operator being used and not the density matrix, for all values of index $i$ we
shall obtain the same values.

Hence the connected correlator becomes

$\displaystyle\langle\mathcal{O}^{\prime}_{A}\otimes\mathcal{O}^{\prime}_{B}\rangle-\langle\mathcal{O}^{\prime}_{A}\otimes\mathbb{I}\rangle\langle\mathbb{I}\otimes\mathcal{O}^{\prime}_{B}\rangle$
(16) $\displaystyle=$
$\displaystyle\sum_{i=1}^{d}p_{i}c_{1}c_{2}-(\sum_{i=1}^{d}p_{i}c_{1})(\sum_{j=1}^{d}p_{j}c_{2})$
$\displaystyle=$ $\displaystyle 0$

Eq. 16 thus defines a necessary condition for separability when the two
conditions are fulfilled. Note that practically we can to choose
$\mathcal{O}_{A}=\mathcal{O}_{B}$ to ensure that we measure only one
observable. Hence,
For a class of states, which, if separable, are only separable into purely
real states [or with purely imaginary off-diagonal elements in $\rho$], a
single measurement of the connected correlation in special mutually unbiased
bases (MUBs) to the eigenbasis of the bipartite system is sufficient to detect
entanglement.

Thus with no prior knowledge of the state, a non-zero value in our test would
indicate it is not separable to completely real (or having completely
imaginary off-diagonal elements) states. This condition then immediately
becomes useful to detect entanglement of rebits[38, 39, 40] which involves
qubits in real Hilbert space. Previous experimental detection of entanglement
of rebits[41] needed to take measures to ensure that the detection was truly
in real space, but our method automatically ensures that entanglement, if
detected, is true real space entanglement and thus simplifies the experimental
setup considerably. In fact, it has been seen before that for classes of
states bipartite entanglement and quantum simulation in qubit systems can also
be achieved via system of rebits with an additional ancilla rebit[42, 43, 44].
Here our method will be quite useful, but a complete mapping of our results
for mixed states with the ancilla rebit method is beyond the scope of this
work.

However, we can explicitly construct classes of density matrices (states) for
which our result will hold for two qubits as follows.

The density matrix for a single qubit can be written in the most general form
as $\frac{1}{2}(\mathbb{I}+v_{j}\sigma^{j})$ where the index $j=1,2,3$,
$\sigma^{j}$ denotes the Pauli matrices and $\vec{v}=v_{j}$ denotes a vector
on the Bloch sphere. To ensure reality we need to restrict ourselves to the
$XZ$ plane , i.e. choose $v_{2}=0$. This then allows to express one class of
real separable two qubit density matrices as,

$\displaystyle\rho_{S}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}p_{i}[\begin{pmatrix}\frac{1}{2}+\frac{\sqrt{1-r_{i}^{2}}}{2}&\frac{r_{i}}{2}\\\
\frac{r_{i}}{2}&\frac{1}{2}+\frac{\sqrt{1-r_{i}^{2}}}{2}\\\ \end{pmatrix}$
$\displaystyle\otimes$
$\displaystyle\begin{pmatrix}\frac{1}{2}+\frac{\sqrt{1-s_{i}^{2}}}{2}&\frac{s_{i}}{2}\\\
\frac{s_{i}}{2}&\frac{1}{2}+\frac{\sqrt{1-s_{i}^{2}}}{2}\\\ \end{pmatrix}]$
(17)

where ${r_{i},s_{i}}\in[-1,1]$ and $\sum_{i=1}^{m}p_{i}=1$. $r_{i}$ and
$s_{i}$ correspond to the $v_{1}$ element of the $i^{{\rm th}}$ density matrix
in subsystems $A$ and $B$ respectively. For simplicity let us choose $m=2$. We
can then create a new class of states by mixing in entangling states with
strength $\epsilon$ in the $\\{\ket{00},\ket{11}\\}$ plane of the form
$\ket{\psi}=c\ket{00}+\sqrt{1-c^{2}}\ket{11}$. Subsequently, entanglement can
be detected via $\mathcal{C}_{2}$ for the class of two qubit density matrices
$\rho=(1-\epsilon)\rho_{S}+\epsilon\ket{\psi}\bra{\psi}$ 333We have chosen a
simple form of the mixing state, more complex ones such as linear combinations
of different such states are also valid parameterizations. generated in this
manner with the elements,

$\begin{split}\rho_{11}&=\frac{1-\epsilon}{8}(a_{1}^{+}b_{1}^{+}+a_{2}^{+}b_{2}^{+})+c^{2}\epsilon\\\
\rho_{22}&=\frac{1-\epsilon}{8}(a_{1}^{+}b_{1}^{-}+a_{2}^{+}b_{2}^{-})\\\
\rho_{33}&=\frac{1-\epsilon}{8}(a_{1}^{-}b_{1}^{+}+a_{2}^{-}b_{2}^{+})\\\
\rho_{44}&=\frac{1-\epsilon}{8}(a_{1}^{-}b_{1}^{-}+a_{2}^{-}b_{2}^{-})+(1-c^{2})\epsilon\\\
\rho_{12}&=\rho_{21}=\frac{1-\epsilon}{8}\sum_{i=1,2}s_{i}a_{i}^{+}\\\
\rho_{13}&=\rho_{31}=\frac{1-\epsilon}{8}\sum_{i=1,2}r_{i}b_{i}^{+}\\\
\rho_{14}&=\rho_{41}=\frac{1-\epsilon}{8}\sum_{i=1,2}r_{i}s_{i}+c\sqrt{1-c^{2}}\epsilon\\\
\rho_{23}&=\rho_{32}=\frac{1-\epsilon}{8}\sum_{i=1,2}r_{i}s_{i}\\\
\rho_{24}&=\rho_{42}=\frac{1-\epsilon}{8}\sum_{i=1,2}r_{i}b_{i}^{-}\\\
\rho_{34}&=\rho_{43}=\frac{1-\epsilon}{8}\sum_{i=1,2}s_{i}a_{i}^{-}\\\
\end{split}$ (18)

where $a_{i}^{\pm}=(1\pm\sqrt{1-r_{i}^{2}})$,
$b_{i}^{\pm}=(1\pm\sqrt{1-s_{i}^{2}})$ and $\epsilon,|c|\in[0,1]$ and
$c\in\mathbb{R}$. One can verify this using exact diagonalization, by
comparing with the known entanglement measure obtained from the positive
partial transpose criterion, negativity $\mathcal{N}$, defined as

$\mathcal{N}=\frac{||\rho^{\Gamma_{A}}||_{1}-1}{2},$ (19)

which returns values greater than $0$ iff $\epsilon>0,c\neq 0,1$. Here,
$\rho^{\Gamma_{A}}$ represents the partial transpose with respect to subsystem
$A$, and $||X||_{1}$ denotes the trace norm of operator $X$. This condition
can be intuitively understood since the entangling perturbations are of
strength $\epsilon$, and we have also verified this numerically (data not
shown)444An analysis of exact eigenvalues is also possible since we need to
solve quartic equations, but tedious.. $\mathcal{C}_{2}$ with the choice of
operator $\mathcal{O}_{A,B}=\sigma^{z}$ gives

$\mathcal{C}_{2}=-2\epsilon c\sqrt{1-c^{2}}$ (20)

which agrees with $\mathcal{N}$ as an entanglement detection criterion. On the
other hand
$\mathcal{C}_{1}^{\epsilon=0}=\frac{1}{4}(\sqrt{1-r_{1}^{2}}\sqrt{1-s_{1}^{2}}-\sqrt{1-r_{1}^{2}}\sqrt{1-s_{2}^{2}}-\sqrt{1-r_{2}^{2}}\sqrt{1-s_{1}^{2}}+\sqrt{1-r_{2}^{2}}\sqrt{1-s_{2}^{2}})\neq
0$ which clearly does not correspond to the correct entanglement detection.
This method can be immediately generalized to perturbations in other
directions of the Hilbert space. Generalizations to more number of qubits will
involve higher dimensional representations of the Bloch sphere.

But one drawback of such a stringent condition lies in uncertainty of state
preparations, thus in practice the value of zero for separable states becomes
a strong lower bound instead of an exact result. However, in this framework we
can make also an estimate of the maximal lower bound for separable states due
a result of slight imperfections in state preparation. The result depends on
choice of the observable.

Let us assume the separated density matrix of the subsystem has both real and
imaginary parts $\rho_{q,r}=\rho^{R}_{q,r}+i\rho^{I}_{q,r}$. The we should
recompute the derivation from the second step of Eq. (14) as,

$\displaystyle\frac{1}{d}\sum_{p=1}^{d/2}\sum_{q,r}^{d}f(p)e^{-i\pi/2[(L_{A}-\phi_{p,q})-(L_{A}-\phi_{r,p})]}(\rho^{R}_{q,r}+i\rho^{I}_{q,r})$
$\displaystyle=\frac{1}{d}\sum_{p=1}^{d/2}\sum_{q,r}^{d}f(p)e^{-i\pi/2(\phi_{p,q}-\phi_{r,p})}\rho^{R}_{q,r}$
$\displaystyle+\frac{1}{d}\sum_{p=1}^{d/2}\sum_{\begin{subarray}{c}q,r\\\
q\neq
r\end{subarray}}^{d}f(p)ie^{i\pi/2(\phi_{p,q}-\phi_{r,p})}\rho^{I}_{q,r}$ (21)

We are going to try to find an upper bound of the value given by the extra
contribution from the imaginary part. Noting that a similar term will also
contribute from the first term of Eq. (13), we can bound its contribution as,

$\displaystyle|-\frac{2}{d}\sum_{p=1}^{d/2}\sum_{q,r}^{d}f(p)\sin(\pi/2(\phi_{p,q}-\phi_{r,p}))\rho^{I}_{q,r}|$
$\displaystyle=|-\frac{4}{d}\sum_{p=1}^{d/2}f(p)\sum_{\begin{subarray}{c}q,r\\\
q<r\end{subarray}}^{d}\sin(\pi/2(\phi_{p,q}-\phi_{r,p}))\rho^{I}_{q,r}|$

Clearly the bound is dependent strongly on the distribution of the imaginary
elements of the density matrix and only terms for which
$(\phi_{p,q}-\phi_{r,p})=2k+1,k\in\mathbb{Z}$ contribute to the sum.
Furthermore since $\sin((2k+1)\pi/2)=(-1)^{k}$, this sum contains terms of
opposite signs. If $|\rho^{I}_{q,r}|\sim O(\epsilon)$, as a rough estimate, we
can approximate the sum as the expected deviation from the starting point of a
one-dimensional classical random walk with step size $\epsilon$. Then we can
define a upper bound by counting terms contributing to the sum,

$\displaystyle|-\frac{4}{d}\sum_{p=1}^{d/2}f(p)\sum_{\begin{subarray}{c}q,r\\\
q<r\end{subarray}}^{d}\sin(\pi/2(\phi_{p,q}-\phi_{r,p}))\rho^{I}_{q,r}|$
$\displaystyle\leq\frac{4}{d}\sum_{p=1}^{d/2}|f(p)|\sum_{\begin{subarray}{c}q,r\\\
q<r\end{subarray}}^{d}|\sin(\pi/2(\phi_{p,q}-\phi_{r,p}))\rho^{I}_{q,r}|$
$\displaystyle\sim\frac{4\epsilon}{d}\sqrt{\frac{d^{2}}{4}}\sum_{p=1}^{d/2}|f(p)|=|2\epsilon\sum_{p=1}^{d/2}f(p)|$

ignoring some prefactors, since only $d^{2}/4$ terms from $\frac{d(d-1)}{2}$
terms will contribute. For large $d$, typically $\epsilon\sim O(1/d)$ for
random states, then the bound becomes,

$\frac{2}{d}\sum_{p=1}^{d/2}|f(p)|$ (23)

From this naive computation we expect that the error vanishes in the
thermodynamic limit. In fact, later in Example II where the purely real
(imaginary) conditions are not satisfied and we have to define a cutoff based
on numerical results, we actually show in Appendix D the cutoff actually
becomes smaller with system size.

It is also worth noting that the observable $\mathcal{O}^{\prime}_{A}$,
obtained via unitarily transforming $\mathcal{O}_{A}$ by $U_{A}$ that
satisfies our purpose, is not unique. However $U_{A}$, which also serves as
the eigenbasis to the new observable $\mathcal{O}^{\prime}_{A}$, cannot be any
arbitrary complete set of MUBs to the computational basis. Only certain
special rotations (unitary evolutions) serve to generate the correct
eigenbasis mutually unbiased to the original basis to satisfy Eq. 13. We have
not been able to find a general way to construct all such basis but in
Appendix B, we provide another explicit example of $U_{A}$, which although is
not as local as the current example, it also works for a collection of qudits.
Further note that if we were able to obtain $U_{A[B]}$ of the above form but
with the additional property,

$\phi_{p,q}-\phi_{p,r}=2m,\hskip 14.45377ptm\in\mathbb{Z}$ (24)

this technique would be applicable without the first condition. Unfortunately,
this renders the matrix non-unitary and thus is an invalid choice. However
with conserved local magnetization (the alternative condition), only specific
elements of $U_{A}$ are responsible for the rotation. In this set-up, Eq. (24)
is fulfilled, leading to applicability for generic density matrices. We
explain this aspect in details in the following.

## Proof of condition (ii):

First note that if indeed we obtain separable states with fixed total
magnetization, then the magnetization of $\rho^{A}_{i}$ and $\rho^{B}_{i}$
needs to be fixed. However, this does not imply the subsystems have fixed
magnetization, but just that the distribution of magnetization between the two
subsystems become explicit. For example if we have a separable state with
$\langle\sum\limits_{j=1}^{L}\sigma^{z}_{j}\rangle=1$, it could be written as,

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle
p_{1}\rho^{A}_{1,\Sigma^{A}=1}\otimes\rho^{B}_{1,\Sigma^{B}=0}+$
$\displaystyle
p_{2}\rho^{A}_{2,\Sigma^{A}=-1}\otimes\rho^{B}_{2,\Sigma^{B}=2}+\ldots+$
$\displaystyle
p_{N}\rho^{A}_{N,\Sigma^{A}=l}\otimes\rho^{B}_{N,\Sigma^{B}=1-l}$

where
$\sum_{i=1}^{N}p_{i}=1,\Sigma^{A[B]}=\langle\sum_{j=1}^{L_{A[B]}}\sigma^{z}_{j}\rangle$,
the subscripts denote the magnetization sector of each $\rho_{i}$ of the
corresponding subsystem, and the values it can take are limited by the total
number of sites in the subsystem ($l$ and $1-l$ are some allowed values within
these limits). It can easily be seen that if each such $\rho_{i}^{A[B]}$ did
not have a fixed magnetization, we would no longer be confined to the
$\langle\sum\limits_{j=1}^{L}\sigma^{z}_{j}\rangle=1$ magnetization sector of
$\rho$ of the full system.

Let us revisit Eq. (14) for such a situation. Because now non-zero elements of
$\rho_{i}^{A[B]}$ are confined to a particular $\langle\sum\sigma^{z}\rangle$,
only certain indices $q,r$ contributes to the sum— the indices corresponding
to the basis states with that particular $\langle\sum\sigma^{z}\rangle$. For
example if we have $3$ sites and
$\langle\sum\limits_{j=1}^{3}\sigma^{z}_{j}\rangle=1$ the relevant basis
states would be $\ket{011},\ket{101},\ket{110}$, since in all the three cases
$\langle\sum_{j=1}^{3}\sigma^{z}_{j}\rangle=\langle\sum_{j=1}^{3}(2n_{j}-1)\rangle=1$,
$n_{j}$ being the number of particles at the site $j$. We shall again drop the
subscript $i$ rom $\rho_{A}$ in what follows.

Since non-zero elements of $\rho^{A}$ are confined to certain indices, when we
multiply by $U_{A}$, the column indices which are relevant will also be
confined to those indices. Let us assume we have an element $U_{p,q_{M}}$
where $q_{M}$ denotes particular column indices corresponding to the conserved
magnetization $M$. A similar situation will arise for the indices of
$U_{A}^{*}$ of Eq. (14) where the row (or column, due to $U^{T}=U$) index $r$
can take limited values $r_{M}$.

Now we shall show that under these restrictions
$(\phi_{p,q_{M}}-\phi_{r_{M},p})=(\phi_{p,q_{M}}-\phi_{p,r_{M}})=2m$. If we
explicitly write out the computational basis, we realize that only two kinds
of operations are possible to go from any $q_{M}$ to any $r_{M}$. (i) The
trivial case of keeping the spin at a site constant. (ii) The non-trivial case
of switching positions of spins between two sites. From the properties of $U$
it is easy to show that the second case introduces a factor of $aa^{-1}=1$ or
$a^{\pm 2}$ (where $a=e^{-i\pi/2}$),(ignoring prefactors) since it involves
two spin flips, while the first introduces a trivial $1$. Due to the tensor
product structure of $U$, this means any the relevant indices are always
connected by factors of the form $a^{2m}$ as $a^{\pm 2}$ and $1$ are the only
operations allowed.

To take a concrete example, consider the case where $L=3,p=4,S_{z}=-1$. The
allowed values of $q_{M}$ and $r_{M}$ are $2,3,5$ which denotes the
computational states $\ket{001},\ket{010},\ket{100}$. Since
$p\in[1,d/2]\cap\mathbb{Z}$ we have chosen $p=4$ whose computational state is
$\ket{011}$, for generality. Ignoring the unimportant prefactors, element
$(4,2)$ is then given by
$\bra{0}U_{L=1}\ket{0}\bra{1}U_{L=1}\ket{0}\bra{1}U_{L=1}\ket{1}=1\times
a\times 1=a$, where we have used Eq. (11). Element $(4,5)$ is found as
$\bra{0}U_{L=1}\ket{1}\bra{1}U_{L=1}\ket{0}\bra{1}U_{L=1}\ket{0}=a\times
a\times a=a^{3}$. The factor of $a^{2}$ coming from the double spin flip is
apparent. And finally element $(4,3)$ is
$\bra{0}U_{L=1}\ket{0}\bra{1}U_{L=1}\ket{1}\bra{1}U_{L=1}\ket{0}=1\times
1\times a=a$. In this case the flips in the first two site from the previous
element gives a ratio of
$\frac{\bra{0}U_{L=1}\ket{0}\bra{1}U_{L=1}\ket{1}}{\bra{0}U_{L=1}\ket{1}\bra{1}U_{L=1}\ket{0}}=a^{-2}$.
Since $a=e^{i\pi/2}$, and the change $e^{i(\phi_{p,q_{M}}-\phi_{p,r_{M}})}$ is
by a factor of $a^{2m}=e^{im\pi}=e^{i(2m)\pi/2}$ for $m=1$ or $m=0$, Eq. (24)
is satisfied.

To complete the proof, notice that in this case we can write
$e^{-i\pi/2(\phi_{pq}-\phi_{rp})}=e^{-im\pi}=e^{im\pi}=e^{-i\pi/2(\phi_{rp}-\phi_{pq})}=e^{-i\pi/2[(L-\phi_{pq})-(L-\phi_{rp})]}$.
Then Eq. (14) follows without any assumption on $\rho$. Similar arguments can
be provided for other locally conserved charges.

It is worthy to note that, an additional intuitive argument exists to prove
the requirement of only one set of measurements to detect entanglement in
generic density matrices with locally conserved charges. If a locally
conserved charge is removed from a subsystem of a bipartite system, it has to
necessarily appear in the other subsystem, i.e. the two halves are always
correlated if one measures the conserved charge. Then we borrow the result
from Eq. (5) and choose $\mathcal{O}_{A[B]}$ to be the conserved local charge,
say for example the magnetization, i.e.
$\sum_{i=1}^{L_{A}}\langle\sigma^{z}_{i}\rangle+\sum_{i=L_{A}+1}^{L}\langle\sigma^{z}_{i}\rangle=M$,
then by choosing $\mathcal{O_{A}}=\sum_{i=1}^{L_{A}}\sigma^{z}_{i}$ and
$\mathcal{O_{B}}=\sum_{i=L_{A}+1}^{L}\sigma^{z}_{i}$,
$|\mathcal{P}_{\mathcal{O}}|=1$ as the two observables are always completely
linearly correlated (an analytic proof of this statement is provided in
Appendix C). This reduces Eq. (5) to,

$|\mathcal{P}_{\mathcal{O^{\prime}}}|>0$ (25)

Since $\mathcal{P}_{\mathcal{O^{\prime}}}$ is just the rescaled connected
correlation, a non-zero connected correlation in any MUB to the computational
basis will indicate entanglement . Since the unitary evolution in Eq. (10)
generates such a basis, we shall be able to detect entanglement between $A$
and $B$ by measuring correlation between the total magnetization of the
subsystems after the evolution. Thus we have,

Figure 1:  Comparison of behaviour of $\mathcal{C}_{2}$ [Eq. (8)] with
$\mathcal{N}$ [Eq. (19)] for top: different values of $\beta$ for states
defined in Eq. (27) of the model defined in Eq. (26) for two different $W$ and
bottom: for dissipative evolution via Eq. (28) starting from Néel state.
$\mathcal{O}_{A}=\sum_{i=1}^{L_{A}}\sigma^{z}_{i}$ and
$\mathcal{O}_{B}=\sum_{i=L_{A}+1}^{L_{B}}\sigma^{z}_{i}$, $U_{A[B]}$ is given
by Eq. (10)

Figure 2: Comparison of behaviour of $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$
defined in Eqs. (6),(8), to $\mathcal{N}$[Eq. (19)] and the criterion in Eq.
(5) for the model given in Eq. (29), for(a) $\kappa=0$ at $\beta=1/T=100$, (b)
$\kappa=0$ at $\beta=1$, (c) $\kappa=0.3$ at $\beta=100$ and (d) $\kappa=0.3$
at $\beta=1$. The system size is $L=12$, the blue shaded region in (b) and (d)
shows the region where $\mathcal{C}_{2}$ is above the cut-off and entanglement
detection is possible. The brown dashed line denotes the cut-off $1$ above
with entanglement detection is possible via Eq. (5). A small longitudinal
field of strength $10^{-3}$ is added to break degeneracy.

In presence of a locally conserved charge, a single measurement of connected
correlation in mutually unbiased bases (MUBs) to the eigenbasis of the of the
bipartite system is sufficient to detect entanglement of arbitrary states.

## Numerical tests:

To test the efficacy of this technique in experimentally detecting
entanglement of mixed states generated by generic Hamiltonians, we numerically
compare $\mathcal{C}_{2}$ with a quantity originating from the PPT
criterion[47, 48], entanglement negativity[49, 50, 51, 52, 53]
$\mathcal{N}$.We provide two examples below, (i) Heisenberg models, which
conserves magnetization, (ii) Ising models, which do not conserve
magnetization. A third example of PXP model, a constrained model which does
not conserve magnetization, is provided in Appendix E. In these examples, the
quantum state whose entanglement we intend to measure is typically in thermal
equilibrium with a corresponding initial Hamiltonian. Then to measure
$\mathcal{C}_{2}$ which requires a specific time evolution of the state, we
need to perform a ‘quantum quench’, where we unitarily evolve the equilibrium
state with a new effective Hamiltonian having a totally different eigenbasis
which in our case is simply $\sum_{j}\sigma^{x}_{j}$.

Let us take a brief detour to elaborate a bit more about quantum quench. When
we evolve a quantum state by a Hamiltonian, we can take two extreme
approaches. In one case we slowly change the Hamiltonian parameter such that
during the evolution the state always remains the ground state of the changing
Hamiltonian. This is adiabatic evolution and is typically utilized in quantum
annealing. [54] However the other scenario is starting from the ground (eigen)
state of a Hamiltonian we suddenly change the parameters of the Hamiltonian
and then allow the system to relax with the new parameters. Clearly this
induces non-adiabaticity in the system, and results in a completely different
non-trivial evolution of the state. For example, if $\ket{\psi(0)}$ is the
initial pure state we start from, since it is not an eigenstate of the new
Hamiltonian, the time evolution can be expressed as,

$\displaystyle\ket{\psi(t)}=e^{-iHt}\ket{\psi(0)}=\sum_{j=1}^{d}e^{-iE_{j}t}\ket{E_{j}}\braket{E_{j}}{\psi(0)}$

where $E_{j}$ and $\ket{E_{j}}$ are the eigenvalues and corresponding
eigenvectors of the new Hamiltonian $H$. This non-trivial dynamics allows us
to reach our desired MUB to measure $\mathcal{C}_{2}$. We shall see that while
for the Heisenberg model it immediately detects entanglement, for the Ising
models, additional information about the state is needed to detect
entanglement.

###  Example I— Heisenberg model with quasi periodic disorder:

The Heisenberg model with on-site quasiperiodic disorder is described by the
following Hamiltonian[55],

$H=-J\sum_{k}\bm{\sigma}_{k}\cdot\bm{\sigma}_{k+1}+\sum_{k}h_{k}\sigma_{k}^{z}.$
(26)

In this equation, $\bm{\sigma}=(\sigma^{x},\sigma^{y},\sigma^{z})$ represents
a vector of Pauli matrices. The term $h_{k}$ is the quasiperiodic component
given by $h_{k}=W\cos(2\pi\eta k)$, where $\eta=\frac{\sqrt{5}-1}{2}$ and $W$
represents the strength of the disorder. We take $J=1$. The inclusion of the
quasiperiodic term allows for the study of a more general model, while
eliminating the need for averaging over realizations that would be required
with random disorder.

We consider computing the entanglement in the following two cases, (i)for the
equilibrium density matrix at any temperature $T=1/\beta$, given by the Gibbs
state,

$\rho=\sum_{i=1}^{d^{2}}\frac{e^{-\beta\epsilon_{i}}}{\sum_{i=1}^{d^{2}}e^{-\beta\epsilon_{i}}}\ket{E_{i}}\bra{E_{i}}$
(27)

where $\epsilon_{i}$ denotes the eigenenergies and $\ket{E_{i}}$ denotes the
corresponding eigenvectors; (ii) a time evolution in an open quantum system
where the unitary evolution is governed via the Hamiltonian in Eq. (26) and
the influence of the environment is effectively included via the Lindblad
equation of motion [56]:

$\frac{d\rho}{dt}=i[H,\rho]+\sum_{k}([L_{k}\rho,L_{k}^{\dagger}]+[L_{k},\rho
L_{k}^{\dagger}]),$ (28)

where $\rho$ represents the density matrix describing the system, and $H$ is
the Hamiltonian from Eq. (26). The Lindblad operators $L_{k}$ capture the
interactions between the system and the environment. We consider only on-site
dephasing terms of strength $\gamma$,
$L_{k}=\sqrt{\frac{\gamma}{2}}\sigma^{z}_{k}$, which causes the system to
eventually evolve to the maximally mixed state, and ensures conservation of
magnetization. We choose the initial state to be the Néel state[57, 58]. We
have selected these specific cases in order to examine and analyse mixed
states, thereby avoiding the trivial scenario of pure state entanglement
detection. We restrict our system to the $\sum_{k=1}^{L}\sigma^{z}_{k}=0$
sector.

In Fig.1(a), we present the results for case (i). Notably, $\mathcal{C}_{2}$
(Eq.(8)) with observables
$\mathcal{O}_{A[B]}=\sum_{i=1}^{L_{A[B]}}\sigma^{z}_{i}$, and unitary
transformations $U_{A[B]}$ [Eq.(10)], successfully detects entanglement in the
same parameter regimes where $\mathcal{N}$ is greater than $0$. This
correspondence demonstrates that our criterion of non-zero $\mathcal{C}_{2}$
aligns with the violation of the positive partial transpose (PPT) criterion
for both instances of disorder considered. Additionally, it is worth noting
that $\mathcal{C}_{2}$ exhibits a value of $0$ at $\beta=1$, consistent with
separability according to the PPT criterion. Moreover, the qualitative
agreement between $\mathcal{C}_{2}$ and $\mathcal{N}$ becomes more pronounced
at higher $\beta$ values, corresponding to purer states.

In Fig. 1(b), where we examine case (ii), there is no separability during the
time evolution. However, $\mathcal{C}_{2}$ exhibits the ‘entanglement barrier’
feature previously observed in other measures of entanglement and also in
$\mathcal{N}$ [59]. For larger disorder, we observe that the barrier shifts to
lower time-scales due to the earlier decay of long-range correlations in such
systems. Furthermore, the entanglement decay slows down on addition of
disorder, i.e. disorder helps protect coherence during the transient dynamics
[60]. Thus, the measurement of $\mathcal{C}_{2}$ provides a reliable technique
to detect entanglement. It is worth noting that in the presence of locally
conserved charge, one can adjust the quench procedure to suit the experiment
as long as an observable is generated whose eigenbasis is mutually unbiased
with respect to the computational basis.

###  Example II— Anisotropic next-nearest neighbour Transverse Ising model:

Next, consider the model Hamiltonian,

$H=-J\sum_{k}\sigma^{z}_{k}\sigma^{z}_{k+1}+\kappa\sum_{k}\sigma^{z}_{k}\sigma^{z}_{k+2}-h\sum_{k}\sigma^{x}_{k}$
(29)

This is the non-integrable anisotropic next-nearest neighbour Transverse Ising
(ANNNI) model, which reduces to the integrable Transverse Ising (TI) model
when $\kappa=0$. Note that this model does not have a simple locally conserved
charge such as magnetization, so quantities such as number entropy or particle
fluctuations cannot detect entanglement.

In the limit of $\beta\rightarrow\infty$, the Transverse Ising (TI) model
undergoes a second-order phase transition at $h=h_{c}=1$ for
$L\rightarrow\infty$. This phase transition is also observed in the
anisotropic next-nearest neighbour Transverse Ising (ANNNI) model, but within
a limited parameter range. As we gradually increase the parameter $\kappa$ in
the ANNNI model, $h_{c}$ decreases, and at $\kappa=0.5$, the critical value
becomes $h_{c}=0$. [61, 62, 63]. Once again, we investigate the behaviour of
$\mathcal{C}_{2}$ in comparison to $\mathcal{N}$ in the thermal states of the
models described by the ANNNI Hamiltonian. The thermal states are given by Eq.
(27), but with the Hamiltonian $H$ now specified by the ANNNI model.

From Fig. 2, it is evident that in all scenarios, the behaviour of
$\mathcal{C}_{2}$ for the same observables as before is qualitatively similar
to that of the entanglement measure $\mathcal{N}$. However, the agreement of
$\mathcal{C}_{1}$ is dependent on the chosen parameters.555Since we work with
just the connected correlation functions we sometimes may obtain large values
of $\mathcal{C}_{1}$ and $\mathcal{C_{2}}$. Because we are mostly concerned
with detection and not measurement we deem rescaling unnecessary. Bounds can
however be found depending on the system size an operators involved. For the
transverse Ising model in our work where we use total magnetization operator,
the peak occurs at $h=0$ and takes a value $L^{2}/4$. This can be seen by
computing the Gibbs state in the presence of infinitesimal longitudinal field
at small but finite temperature—
$\frac{1}{2}\ket{\uparrow\uparrow\ldots\uparrow}\bra{\uparrow\uparrow\ldots\uparrow}+\frac{1}{2}\ket{\downarrow\downarrow\ldots\downarrow}\bra{\downarrow\downarrow\ldots\downarrow}$.
This state gives the maximum possible unnormalized connected correlation for
this observable, and in fact $\mathcal{C}_{2}$ is loosely upper bounded by
this value as well since we always rotate away from this special state.
Furthermore, the criteria in Eq. (5),
$|\mathcal{P}_{\mathcal{O}}|+|\mathcal{P}_{\mathcal{O}}^{\prime}|>1$, denoted
by the brown dashed line, is not fulfilled almost everywhere, thus being a
poor entanglement witness in this set-up. An improvement could potentially be
achieved by increasing the number of mutually unbiased basis (MUB)
measurements, although the analysis of this is beyond the scope of the present
study. However, a complete agreement between $\mathcal{C}_{2}$ and
$\mathcal{N}$ for all values of $h$ with $\kappa=0$ and $\kappa=0.3$ only
occurs at large values of $\beta$, where the state becomes minimally mixed.

For smaller $\beta$, there is a region where $\mathcal{N}=0$ while
$\mathcal{C}_{2}$ has a small value. Here the system is separable via PPT
criterion, but not to purely real states (or there exists bound
entanglement).666It is not separable to states with purely imaginary off
diagonal elements either as taking
$\mathcal{O}=\sum\langle|\sigma_{z}|\rangle$ does not yield
$\mathcal{C}_{2}=0$ in this region However there still exists a qualitative
similarity of $\mathcal{C}_{2}$ with $\mathcal{N}$, which is due to the
stability of the correlator to perturbation with imaginary separable states.
We can utilize this qualitative similarity to define a lower non-zero cut-off
of $\mathcal{C}_{2}$ at each $\beta$ which can be used to detect entanglement.
The cut-off can be found using numerical simulations for small size systems
and extrapolated to larger sizes, due to weak dependence on system size (See
Appendix D) and thus provides a viable route to detect entanglement of states
at thermal equilibrium. Note that an arbitrary MUB to computational basis does
not yield qualitatively similar results to $\mathcal{N}$ in this case unlike
the previous one.

The experimental realization of the Transverse Ising model is possible, for
example, in quantum wires by tuning the chemical potential appropriately.
[66]. After letting the system reach thermal equilibrium, to perform the
rotation one would need perform a quench by ‘pinching’ between neighbouring
electrons to ensure $J=\kappa=0$ and let the system evolve for time
$t=\frac{\pi}{4h}+2k\pi$,for $k\in\mathbb{Z}$ and measure the total
$\sigma_{z}$ correlations between the corresponding parts of the system. After
the ‘pinch’ the evolution occurs via an on-site Hamiltonian, hence the
evolution does not generate entanglement. Then by using the thresholds for the
specific temperature, we can detect entanglement.

## Conclusion:

In this work we have demonstrated through an analytic proof how measuring the
one-connected correlation in the appropriate basis can provide valuable
insights into the separability of the state, and effectively detect
entanglement by leveraging specific properties of the state. This approach
offers a substantial reduction in the experimental burden for a wide range of
systems, as compared to previous correlation measurement methods that often
require multiple basis measurements to achieve efficient detection. Moreover,
our method presents an alternative means of detecting entanglement in charge-
conserved systems.

We have showcased the immediate applicability of our technique in detecting
entanglement in Heisenberg chains, where our technique provides an efficient
entanglement detection pathway for both pure and mixed states and in quantum
wires, where conventional correlation measurements face challenges in
detecting entanglement. Although the detection in the latter case may not be
immediate, the qualitative similarity between our correlator and the
Negativity measure allows for the establishment of thresholds based on
experimental conditions to effectively detect entanglement. Additionally, we
can also detect and quantify certain errors in full state preparation by
computing $\mathcal{C}_{2}$ between two uncoupled thermal states which should
be identically $0$ for perfectly prepared states. For example, lack of
equilibriation will give us $\mathcal{C}_{2}\neq 0$ as then we will have
imaginary elements in the off-diagonal density matrix due to unitary time
evolution.

Overall, our work provides a valuable approach for entanglement detection that
offers both efficiency and applicability across various systems. It
significantly simplifies possibilities of entanglement detection in
experiments.

###### Acknowledgements.

We thank Felix Fritzsch, Daniele Toniolo, Madhumita Sarkar, Marko Žnidarič,
Alexander Nico-Katz, Crispin Barnes and Debarshi Das for discussions and
suggestions. We also thank three anonymous referees for their valuable
comments which have helped improve our work considerably. The authors
acknowledge the EPSRC grant Nonergodic quantum manipulation EP/R029075/1 for
support.

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##  Appendix A Properties of $U$ in Eq. (10)

We start with the following definition of $U$,

$U=(e^{-i\sigma^{x}\pi/4})^{\otimes L},$ (30)

where $L$ denotes the length of the subsystem and,

$\sigma_{x}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$

For $L=1$, we have,

$U=\left(\begin{array}[]{cc}\frac{1}{\sqrt{2}}&-\frac{i}{\sqrt{2}}\\\
-\frac{i}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\ \end{array}\right)$ (31)

This can be written for future convenience as,

$U=\frac{1}{\sqrt{2}}\begin{pmatrix}1&a\\\ a&1\end{pmatrix}$ (32)

with $a=e^{-i\pi/2}$. Then for $L=2,3$ respectively we have,

$\displaystyle U=\frac{1}{2}\left(\begin{array}[]{cccc}1&a&a&a^{2}\\\
a&1&a^{2}&a\\\ a&a^{2}&1&a\\\ a^{2}&a&a&1\\\ \end{array}\right),$ (37)
$\displaystyle
U=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{cccccccc}1&a&a&a^{2}&a&a^{2}&a^{2}&a^{3}\\\
a&1&a^{2}&a&a^{2}&a&a^{3}&a^{2}\\\ a&a^{2}&1&a&a^{2}&a^{3}&a&a^{2}\\\
a^{2}&a&a&1&a^{3}&a^{2}&a^{2}&a\\\ a&a^{2}&a^{2}&a^{3}&1&a&a&a^{2}\\\
a^{2}&a&a^{3}&a^{2}&a&1&a^{2}&a\\\ a^{2}&a^{3}&a&a^{2}&a&a^{2}&1&a\\\
a^{3}&a^{2}&a^{2}&a&a^{2}&a&a&1\\\ \end{array}\right)$ (46)

Clearly $U=U^{T}$, which comes from $\sigma_{x}^{T}=\sigma_{x}$. Additionally
we observe for these examples, $U_{p,q}U_{d-p+1,q}=\frac{1}{L}a^{L}$, where
$d=2^{L}$. This can be verified for a general case as follows.

Since we use the computational basis, the basis states in Eq. 32 will be
$(\ket{0},\ket{1})$, hence
$U_{L=1}=\frac{1}{\sqrt{2}}(\ket{0}\bra{0}+\ket{1}\bra{1}+a\ket{0}\bra{1}+a\ket{1}\bra{0})$.
Any element of $U$ for a generic $L$ can be constructed out of these elements
due to the tensor product structure. For example in $L=3$, the
$\ket{110}\bra{010}$ element can be constructed as
$\bra{1}U_{L=1}\ket{0}\bra{1}U_{L=1}\ket{1}\bra{0}U_{L=1}\ket{0}=a\times
1\times 1=a$, which matches with the corresponding element in $U_{L=3}$
(row-$7$, column-$3$) in Eq. (46). Now for an element of an arbitrary row
index $p$, the basis state for the column indices $q$ and $d-q+1$ will be just
bit flipped states of each other. For example, at $L=3$, if we consider $p=2$,
then the $q=3$ and $q=6$ basis states would be $\ket{001}\bra{010}$ and
$\ket{001}\bra{101}$. Then it can be seen that
$U_{23}U_{26}=\frac{a^{3}}{2\sqrt{2}}$ using Eq. (32).

It follows that, $U_{p,q}U_{d-p+1,q}=\frac{1}{L}e^{-i\pi L/2}$, and any
element of $U$ can be written as an integer power of $a$, i.e.
$U_{p,q}=\frac{1}{\sqrt{L}}e^{-i\pi(\phi_{p,q})/2}$, where
$\phi_{p,q}\in[0,L]\cap\mathbb{Z}$.

##  Appendix B An alternative unitary rotation

In this section we shall provide another example of the rotation which
accomplishes the same task as the one described in the main text. While this
rotation provides the advantage of applicability to qudit systems, it is
significantly more non-local than the example in the main text, and thus
harder to implement in practice.

The rotation matrix to be considered is closely related to the many-body
Fourier transform local to the subsystems,

$(U_{A})_{pq}=\frac{1}{\sqrt{d}}(e^{\frac{2i\pi(p-1/2)q}{d}})$ (47)

and similarly for $U_{B}$. The purpose of adding the factor of $-2i\pi q/2$
with the usual Fourier transform will be clear in the following computation.
Using the specific nature of $\mathcal{O}$ we have,

$\displaystyle{\rm Tr}[\rho^{\prime}\mathcal{O}_{A}]={\rm Tr}[\rho^{\prime
A}\mathcal{O}_{A}]$ (48) $\displaystyle=\frac{1}{d}\sum_{p,q,r,s}^{d}e^{2\pi
iq\frac{p-1/2}{d}}\rho_{qr}e^{-2\pi ir\frac{s-1/2}{d}}(f(s)+c_{1})\delta_{sp}$
$\displaystyle=$
$\displaystyle\frac{1}{d}\sum_{p=1}^{d/2}\sum_{q,r}^{d}(f(p)+c_{1})e^{2\pi
i(\frac{p(q-r)}{d}-\frac{q-r}{2d})}\rho_{qr}$ $\displaystyle+$
$\displaystyle\frac{1}{d}\sum_{p=1}^{d/2}\sum_{q,r}^{d}(-[+]f(p)+c_{1})e^{2\pi
i(-\frac{p(q-r)}{d}+\frac{q-r}{2d})}\rho_{qr}$

Then using the symmetry [antisymmetry] of $\rho$ (the first condition), i.e.,
$\rho_{qr}=[-]\rho_{rq}$ we can simplify Eq. 48 to obtain,

$\displaystyle{\rm Tr}[\rho^{\prime A}\mathcal{O}_{A}]$ $\displaystyle=$
$\displaystyle\frac{1}{d}\sum_{p=1}^{d}\sum_{q,r}^{d}c_{1}e^{2\pi
i(\frac{p(q-r)}{d}-\frac{q-r}{2d})}\rho_{qr}$ (49) $\displaystyle=c_{1}{\rm
Tr}[\rho^{\prime A}]=c_{1}$

It is worth noting here, that for odd $d$ Eq. (48) will have an additional
term and will be,

$\displaystyle{\rm Tr}[\rho^{\prime}\mathcal{O}_{A}]={\rm Tr}[\rho^{\prime
A}\mathcal{O}_{A}]$ $\displaystyle=$
$\displaystyle\frac{1}{d}\sum_{p=1}^{(d-1)/2}\sum_{q,r}^{d}(f(p)+c_{1})e^{2\pi
i(\frac{p(q-r)}{d}-\frac{q-r}{2d})}\rho_{qr}$ $\displaystyle+$
$\displaystyle\frac{1}{d}\sum_{p=1}^{(d-1)/2}\sum_{q,r}^{d}(-[+]f(p)+c_{1})e^{2\pi
i(-\frac{p(q-r)}{d}+\frac{q-r}{2d})}\rho_{qr}$ $\displaystyle+$
$\displaystyle\frac{1}{d}\sum_{q,r}^{d}(f([d+1]/2)+c_{1})e^{2\pi
i(\frac{(d+1)(q-r)}{2d}-\frac{q-r}{2d})}\rho_{qr}$

Choosing $f([d+1]/2)=0$ gives us Eq. (49) back. This naturally arises for
separable real matrices from $f(p)=-f(d-p+1)$, but becomes an additional
condition for the purely imaginary off-diagonal elements case. The rest of the
proof follows as in the main text. Thus this rotation serves to obtain the
appropriate observable in both even and odd dimensional Hilbert spaces.

##  Appendix C Proof of $\mathcal{P}_{\mathcal{O}}=1$ for states with
conservation of total magnetization

Let us consider a mixed state of the form
$\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$. $|\psi_{i}\rangle$ can
be any pure state with the constraint
$\langle\psi_{i}|\sum_{j=1}^{L}\sigma^{j}_{z}|\psi_{i}\rangle=M$. We choose
$\mathcal{O}_{A[B]}=\sum_{i=1}^{L_{A[B]}}\sigma^{z}_{i}$, hence the various
terms of the Pearson correlation for this state $|\psi_{i}\rangle$ are,

$\displaystyle\langle\mathcal{O}_{A}\otimes\mathcal{O}_{B}\rangle$
$\displaystyle=$ $\displaystyle\sum_{k=0}^{M}|c_{k}|^{2}(M-k)k$ (50)
$\displaystyle\langle\mathcal{O}_{A}\otimes\mathbb{I}\rangle$ $\displaystyle=$
$\displaystyle(\sum_{k=0}^{M}(M-k)|c_{k}|^{2})$ (51)
$\displaystyle\langle\mathbb{I}\otimes\mathcal{O}_{B}\rangle$ $\displaystyle=$
$\displaystyle(\sum_{k=0}^{M}k|c_{k}|^{2})$ (52)
$\displaystyle\langle\mathcal{O}_{A}^{2}\otimes\mathbb{I}\rangle$
$\displaystyle=$ $\displaystyle\sum_{k=0}^{M}(M-k)^{2}|c_{k}|^{2}$ (53)
$\displaystyle\langle\mathbb{I}\otimes\mathcal{O}_{B}^{2}\rangle$
$\displaystyle=$ $\displaystyle\sum_{k=0}^{M}k^{2}|c_{k}|^{2}$ (54)

where $|c_{k}|^{2}$ is the probability of measuring $k$ spins up in subsystem
$B$. Now $\mathcal{P}_{\mathcal{O}}$ can be written as,

$\displaystyle\mathcal{P}_{\mathcal{O}}$ $\displaystyle=$
$\displaystyle\frac{\langle\mathcal{O}_{A}\otimes\mathcal{O}_{B}\rangle-\langle\mathcal{O}_{A}\otimes\mathbb{I}\rangle\langle\mathbb{I}\otimes\mathcal{O}_{B}\rangle}{\sqrt{(\langle\mathcal{O}_{A}^{2}\otimes\mathbb{I}\rangle-\langle\mathcal{O}_{A}\otimes\mathbb{I}\rangle^{2})(\langle\mathbb{I}\otimes\mathcal{O}_{B}^{2}\rangle-\langle\mathbb{I}\otimes\mathcal{O}_{B}\rangle^{2})}}$
(55) $\displaystyle=$
$\displaystyle\frac{\sqrt{-[\sum_{k=0}^{M}(M-k)|c_{k}|^{2}]^{2}+M\sum_{k=0}^{M}(M-k)|c_{k}|^{2}-\sum_{k=0}^{M}(M-k)|c_{k}|^{2}\sum_{l=0}^{M}l|c_{l}|^{2}}}{\sqrt{\sum_{k=0}^{M}(M-k)^{2}|c_{k}|^{2}-(\sum_{k=0}^{M}(M-k)|c_{k}|^{2})^{2}}}\times$
$\displaystyle\frac{\sqrt{-[\sum_{k=0}^{M}(M-k)|c_{k}|^{2}]^{2}+M\sum_{k=0}^{M}k|c_{k}|^{2}-\sum_{l=0}^{M}(M-l)|c_{l}|^{2}\sum_{k=0}^{M}k|c_{k}|^{2}}}{\sqrt{\sum_{k=0}^{M}(M-k)^{2}|c_{k}|^{2}-(\sum_{k=0}^{M}k|c_{k}|^{2})^{2}}}=-1$

where we have used $\sum_{k=1}^{M}|c_{k}|^{2}=1$ and separately computed the
ratios between the two terms in the denominator by introducing a square root
in the numerator. This result is valid unless
$(\langle\mathcal{O}_{A[B]}^{2}\otimes\mathbb{I}\rangle-\langle\mathbb{I}\otimes\mathcal{O}_{A[B]}\rangle^{2}$
yields $0$, i.e. the subsystem has a fixed number of particles as well.

By substituting this result for each $\ket{\psi_{i}}$ and using $Tr[\rho]=1$
we can show $P_{\mathcal{O}}=1$ for the full density matrix. Using the result
of Ref. [31] it immediately follows that any correlation found in the MUB
basis denotes the state is entangled. Thus, measuring correlation in one basis
is enough to detect entanglement.

##  Appendix D Full phase diagram of TI model

Figure 3: Top:Behaviour of $\mathcal{C}_{2}$ for model in Eq. (29) with
$\kappa=0$ for different values if $\beta$ and $h$. The red line denotes the
contour of $\mathcal{N}=10^{-4}$ to be used as an indicator to define lower
bound of $\mathcal{C}_{2}$ to detect entanglement. Middle: Threshold
$\mathcal{C}_{2}$ values which can be used to detect entanglement.
Bottom:Threshold $\beta$ values above with $\mathcal{C}_{2}^{th}=0$ for
different system sizes $L$. The black solid line denotes the moving average
over $\delta h=0.1$

In the top panel of Fig. 3 we show the behaviour of $\mathcal{C}_{2}$ for a
range of $\beta$ and $h$, while denoting the contour where $\mathcal{N}$ shows
a non-zero value by red. Clearly for all values of $h$, $\mathcal{C}_{2}$ is
an excellent witness of entanglement for $\beta>1$ i.e. at low temperature.
For smaller $\beta$ we need to define a threshold value of $\mathcal{C}_{2}$,
$\mathcal{C}_{2}^{th}$ to accurately detect entanglement. We show the
numerically computed threshold values in the middle panel of Fig. 3. Two
things are immediately apparent. The first is there is no significant finite
size effect, in fact the threshold values reduce with system size which allows
one to numerically compute the threshold for small system sizes and use them
for experiments with a larger number of spins. Further accuracy can be
achieved by a proper finite size analysis of the values which are beyond the
scope of this work. The second prominent feature is the discrete ‘groupings’
of the values, which occurs due to discrete grid of $h$ and the finite
tolerance available to detect $\mathcal{N}$. Finally in the bottom panel of
Fig. 3, we plot the threshold value of $\beta$, $\beta^{th}$ for different
$h$, which is defined as the smallest value of $\beta$ which allows for
accurate detection of entanglement with $\mathcal{C}_{2}^{th}=0$. This also
shows reduction with system size $L$.

Figure 4: Plots showing comparison of behaviour of $\mathcal{C}_{2}$ and
$\mathcal{C}_{1}$ defined in Eqs. (6) and (8), to Negativity $\mathcal{N}$ and
the criterion in Eq. (5) for the model given in Eq. (56), (a) at
$\beta=1/T=100$, (b) at $\beta=1$, The system size is $L=12$. The green grid
line in panel (a) denotes the critical point.

##  Appendix E Example III — PXP model

As a final example, we consider the kinetically constrained Rydberg atom model
which has recently gained a lot of attention due to presence of ‘quantum
scars’. In particular we study entanglement detection for states in thermal
equilibrium [Eq. (27)] obtained from the Hamiltonian,

$H=\sum_{i=1}^{L}(\Omega\tilde{\sigma}_{i}^{x}+\Delta\sigma_{i}^{z})$ (56)

and show that our method has the potential to be applied for entanglement
witnessing. Mapping an empty Rydberg site to a down spin and a filled site to
an up spin, we denote here $\tilde{\sigma}_{i}^{x}=P\sigma_{i}^{x}P$ operation
where $P$ denotes projection to only those states in Hilbert space with no
neighbouring up spins. It is known that this model has a ferromagnetic ground
state with all spins pointing down, i.e., presence of zero excited Rydberg
atoms when $\Delta\gg 0$ and $Z_{2}$ symmetry broken ground state for
$\Delta\ll 0$. As shown in Fig. 4, at large $\beta=100$, different correlators
behave similarly. In Fig. 4(a) we observe that while $\mathcal{N}$ shows a
distinct peak at the known critical point of $\delta/\Omega=0.65$ and the
maxima of $\mathcal{C}_{2}$ occurs at a slightly larger value. This is not
unexpected as $\mathcal{C}_{2}$ is not an entanglement measure but a witness,
hence we can only expect qualitative agreement. However, using Eq. (5) we
cannot detect entanglement anywhere in the system as we never reach the
threshold value of $1$. Expectedly, $\mathcal{C}_{1}$ shows completely
different behaviour to $\mathcal{N}$ at low $\beta$ as seen Fig. 4(b) for a
highly mixed state, where $\mathcal{C}_{2}$ qualitatively follows
$\mathcal{N}$, but still there exists a region where $\mathcal{N}=0$ but
$\mathcal{C}_{2}\neq 0$. Note that the agreement between $\mathcal{C}_{2}$ and
$\mathcal{N}$ is more pronounced in this case than the Ising model, thus
allowing for smaller cut-offs and providing better detection at larger
temperatures. In Rydberg atoms, the change of basis can be simply executed
experimentally via switching off the detuning field and the kinetic
constraints by reducing the Rydberg blockade radius, and then taking a
snapshot of the atomic density at different sites at
$t=\frac{\pi}{4h}+2k\pi$,for $k\in\mathbb{Z}$, thus computing the correlation
between the relevant subsystems.