# Contextuality in infinite one-dimensional translation-invariant local
Hamiltonians: strengths and limits

Kaiyan Yang  Institute of Fundamental and Frontier Sciences,
University of Electronic Science and Technology of China, Chengdu 610054,
China  School of Mathematical Sciences,
University of Electronic Science and Technology of China, Chengdu 611731,
China  Xiao Zeng  Institute of Fundamental and Frontier Sciences,
University of Electronic Science and Technology of China, Chengdu 610054,
China  School of Computer Science and Engineering,
University of Electronic Science and Technology of China, Chengdu 611731,
China  Yujing Luo  Institute of Fundamental and Frontier Sciences,
University of Electronic Science and Technology of China, Chengdu 610054,
China  Guowu Yang  School of Computer Science and Engineering,
University of Electronic Science and Technology of China, Chengdu 611731,
China  Lan Shu  School of Mathematical Sciences,
University of Electronic Science and Technology of China, Chengdu 611731,
China  Miguel Navascués  Institute for Quantum Optics and Quantum Information
- IQOQI Vienna,
Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria  Zizhu
Wang  Institute of Fundamental and Frontier Sciences,
University of Electronic Science and Technology of China, Chengdu 610054,
China

###### Abstract

In recent years there has been a growing interest in treating many-body
systems as Bell scenarios, where lattice sites play the role of distant
parties and only near-neighbor statistics are accessible. We investigate
contextuality arising from three Bell scenarios in infinite, translation-
invariant 1D models: nearest-neighbor with two dichotomic observables per
site; nearest- and next-to-nearest neighbor with two dichotomic observables
per site and nearest-neighbor with three dichotomic observables per site. For
the first scenario, we give strong evidence that it cannot exhibit
contextuality, not even in non-signaling physical theories beyond quantum
mechanics. For the second one, we identify several low-dimensional models that
reach the ultimate quantum limits, paving the way for self-testing ground
states of quantum many-body systems. For the last scenario, which generalizes
the Heisenberg model, we give strong evidence that, in order to exhibit
contextuality, the dimension of the local quantum system must be at least 3.

## Introduction

In a many-body quantum system, correlations appear as one of the most common
manifestations of the quantum nature of the system. Different types of
correlations, such as entanglement, EPR steering and nonlocality, were
identified over the years and found applications in various quantum
information processing tasks. Out of these types of correlations, nonlocality
is the strongest and most difficult to test [5, 33]. While experiments
exploiting entanglement to teleport photons date back to the 90s [2],
nonlocality passed the most stringent experimental tests only in 2015 [16, 12,
35]. One of the key assumptions in any nonlocality experiment is keeping the
different parties space-like separated. In a many-body quantum system,
however, this assumption may be too formidable to be overcome.

In recent years, the exploration of contextuality in quantum many-body systems
has been a fruitful endeavor. Contextuality witnesses adapted from Bell
inequalities have been tested in Bose-Einstein condensates [34, 40].
Translation and permutation symmetry allowed the full characterization of
contextuality witnesses in many-body systems through bipartite correlators
only [45, 39, 42, 41] (see [10] for a review). However, very little is known
about the strengths and limits of the quantum models which violate these
witnesses.

In this work, building on earlier characterizations of classical local
behavior in many-body systems [45], we investigate under which conditions the
$n$-nearest neighbor statistics of a translation-invariant 1D quantum system
evidence that the latter is contextual, by exploiting the connection between
contextuality and Bell nonlocality. We focus on three different Bell
scenarios, which differ on the number of measurement settings available to
each party and the size of the near-neighbor marginals considered.

Our results show that some _a priori_ promising Bell scenarios are unlikely to
show any form of contextuality, even if we allow greater-than-quantum
correlations. In other scenarios, we give evidence that some Bell inequalities
require quantum systems of high enough local dimension to be violated. More
interestingly, for several Bell inequalities, we find the maximum violation
compatible with the laws of quantum mechanics, and identify the Hamiltonians
achieving it. The ground states of contextual Hamiltonians are entangled, even
though locally they may appear separable. As shown in [45], it is possible to
estimate the size at which the reduced density matrix of a quantum state
becomes entangled, just by computing the difference between its average energy
and the classical bound. Since the former can be easily estimated in
essentially any Noisy Intermediate-Scale Quantum (NISQ) device, one can think
of our contextuality witnesses as robust entanglement benchmarks for future
quantum simulators. From a more fundamental perspective, identifying the
maximum quantum violation of a $k$-local Bell functional opens the possibility
to falsify quantum theory in the many-body regime. Indeed, given access to a
NISQ device, we can represent the local measurements and the state preparation
of the corresponding Bell test through a vector of lab controls $\theta$. By
estimating the gradient of the Bell functional with respect to the variables
$\theta$, we can sequentially update the latter so as to minimize the observed
value of the Bell functional in the device (effectively mimicking the working
principle of the variational quantum eigensolver [30]). If it so happened that
the final value of the Bell functional were below the quantum limit, then we
would have disproven the universal validity of quantum theory, despite the
lack of an alternative theoretical model.

One-dimensional quantum systems are the simplest many-body ensembles one can
control in the lab. They can be found in natural condensed matter systems, as
well as implemented via optical lattices or ion traps. For some such systems
the only experimentally available data are near-neighbor correlators averaged
over the whole chain (the so-called _structure factors_). As shown in [45], in
the regime of large system size, structure factors correspond to the near-
neighbor correlators of an infinite, translation-invariant chain.
Comprehending Bell nonlocality in large 1D systems hence requires us to
characterize near-neighbor correlations in classical, quantum and supra-
quantum translation-invariant systems.

The correlations $P(a_{1},\ldots,a_{n}|x_{1},\ldots,x_{n})$ generated by $n$
space-like separated classical systems with (classical) inputs
$x_{i},i\in{1,\ldots,n}$ and outputs $a_{i},i\in{1,\ldots,n}$ admit a
decomposition of the form:

$\displaystyle P(a_{1},\ldots,a_{n}$ $\displaystyle|x_{1},\ldots,x_{n})$
$\displaystyle=\int P(a_{1}|x_{1},\lambda)\ldots
P(a_{n}|x_{n},\lambda)P(\lambda)d\lambda,$ (1)

where $\lambda$ is a set of hidden variables with probability distribution
$P(\lambda)$. The distributions $P(\lambda)$ and
$\\{P(a_{i}|x_{i},\lambda)\\}_{i=1}^{n}$ are hence a _local hidden variable
model_ for the observed correlations
$P(a_{1},\ldots,a_{n}|x_{1},\ldots,x_{n})$. In Bell tests, different parties
are required to be space-like separated, which can be seen as the physical
realization of the independence of probabilities in (1). However, in a many-
body quantum system, such a requirement is too formidable to overcome. As a
result, when we assume that Eq. (1) holds in a quantum many-body system, what
we are actually testing is contextuality [22, 6]. The connection between
contextuality and Bell nonlocality had been known since the 70s. Every Bell
inequality can be regarded as a contextuality witness; the other direction is
less systematic [7]. The role of contextuality, especially of the Kochen-
Specker type, in quantum computation has been actively investigated in recent
years (for a review see [6]). It can be shown to be the source of the quantum
advantage in several scenarios in quantum computation [4, 3, 17, 1]. Most of
these scenarios are constructed from the stabilizer formalism with magic
states. While a hidden variable model for this formalism has been found
recently [47], the model itself is contextual [6].

Our starting point is thus a Bell scenario with infinitely many parties in a
chain, labeled by the integer numbers. At site $i\in\mathbb{Z}$, the
corresponding party can conduct a measurement $x_{i}\in\\{1,...,X\\}$,
obtaining the result $a_{i}\in A$. Since we assume translation invariance, the
measurement statistics observed by any $m$ consecutive parties equal those of
parties, $1,...,m$, that is, $P_{1,..,m}(a_{1},...,a_{m}|x_{1},...,x_{m})$.
Any Bell scenario in the translation-invariant chain can therefore be fully
specified by the three natural numbers $m,X,|A|$. Consequently, in this paper,
a Bell scenario where only nearest-neighbor correlations are available and
each party can conduct two dichotomic observables will be called the
_222-scenario_. Extending the interaction distance to next-to-nearest
neighbors gives us the _322-scenario_. Heisenberg-like Bell scenarios, with
nearest-neighbor interactions but three dichotomic observables per site
correspond to the _232-scenario_.

We say that an $m$-partite distribution
$P_{1,..,m}(a_{1},...,a_{m}|x_{1},...,x_{m})$ is no-signaling [31] if, for all
$i\in\\{1,...,m\\}$,

$\displaystyle\sum_{a_{i}}P_{1,..,m}(a_{1},...,a_{m}|x_{1},...,x_{i},...,x_{m})=$
$\displaystyle\sum_{a_{i}}P_{1,..,m}(a_{1},...,a_{m}|x_{1},...,\tilde{x}_{i},...,x_{m}),$
(2)

for all pairs of measurement settings $x_{i},\tilde{x}_{i}\in X$. Intuitively,
this condition signifies that the statistics of the remaining $m-1$ parties
are not affected by the choice of measurement setting of party $i$. Hence,
party $i$ cannot instantaneously transmit information to others.

Some no-signalling distributions $P_{1,..,m}(a_{1},...,a_{m}|x_{1},...,x_{m})$
can be shown not to arise out of an infinite no-signalling TI system. This
idea is formalized in the following definition: we say that
$P_{1,..,m}(a_{1},...,a_{m}|x_{1},...,x_{m})$ admits a TI no-signalling
extension if there exists a mapping $Q$ from finite sets $B\subset\mathbb{Z}$
to no-signalling $|B|$-partite measurement statistics $Q_{B}(a_{B}|x_{B})$
with the following properties:

  1. 1.

$\displaystyle\sum_{a_{B\setminus C}}Q_{B}(a_{B\setminus
C},a_{C}|x_{B\setminus C},x_{C})=$ $\displaystyle\sum_{a_{D\setminus
C}}Q_{D}(a_{D\setminus C},a_{C}|x_{D\setminus C},x_{C}),$

for all finite sets $B,C,D\subset\mathbb{Z}$, with $C\subset B,D$
(compatibility).

  2. 2.

$Q_{B}=Q_{B+z}$, for all $z\in\mathbb{Z}$ (translation invariance).

  3. 3.

$Q_{1,...,n}=P_{1,...,n}$ (consistency with observed statistics).

We call ${\operatorname{TI-NS}}$ the set of all distributions
$P_{1,..,m}(a_{1},...,a_{m}|x_{1},...,x_{m})$ admitting a no-signalling,
translation-invariant extension.

The existence of a no-signalling extension is just a pre-requisite for the
existence of an overall infinite translation-invariant state. Whether such an
entity exists at all depends also on the physics generating the observed
correlations. We say that $P_{1,..,m}(a_{1},...,a_{m}|x_{1},...,x_{m})$ admits
a TI classical extension if it admits a NS extension $Q$ and there exist
distributions $P(\lambda),\\{P_{i}(a|x,\lambda)\mathrel{\mathop{\mathchar
58\relax}}i\in\mathbb{Z}\\}$ such that, for all $N$,

$Q_{-N,..,N}(a_{-N},...,a_{N}|x_{-N},...,x_{N})=\sum_{\lambda}P(\lambda)\prod_{i=-N}^{N}P_{i}(a_{i}|x_{i},\lambda).$
(3)

We call ${\operatorname{TI-LHV}}$ the set of all distributions
$P_{1,..,m}(a_{1},...,a_{m}|x_{1},...,x_{m})$ admitting a TI classical
extension.

Analogously, $P_{1,..,m}(a_{1},...,a_{m}|x_{1},...,x_{m})$ admits a TI quantum
extension if it admits a NS extension $Q$ and there exist a Hilbert space
${\cal H}$, measurement operators $E_{a|x}\mathrel{\mathop{\mathchar
58\relax}}{\cal H}\to{\cal H}$, with $\sum_{a}E_{a|x}=\mathbb{I}$, and a
translation-invariant quantum state $\rho$ on the infinite chain with local
Hilbert space ${\cal H}$ such that, for all $N$,

$Q_{-N,..,N}(a_{-N},...,a_{N}|x_{-N},...,x_{N})=\mbox{tr}\left\\{\bigotimes_{j=-N}^{N}E_{a_{j}|x_{j}}\rho_{-N,....,N}\right\\}.$
(4)

We call ${\operatorname{TI-Q}}$ the set of all distributions
$P_{1,..,m}(a_{1},...,a_{m}|x_{1},...,x_{m})$ admitting a TI quantum
extension.

In [45], two of us provided a full characterization of the set of $m$-nearest
neighbor correlations admitting a TI classical extension. This set happens to
be a polytope, i.e., a convex set defined by a finite number of linear
inequalities or _facets_. When all local measurements are dichotomic
($|A|=2$), one can regard any measurement $x$ by party $i$ as an observable
$\sigma^{i}_{x}$ with possible values $\pm 1$, and specify any no-signaling
$m$-nearest neighbor distribution $P(a_{1},...,a_{m}|x_{1},...,x_{m})$ through
the averages of the different products of the observables
$\sigma^{1}_{x_{1}},...,\sigma^{m}_{x_{m}}$. For $m=2$, in this ‘observable
representation’ a facet would take the form

$\sum_{x=1,...,X}J_{x}\langle\sigma^{1}_{x}\rangle+\sum_{x,y=1,...,X}J_{xy}\langle\sigma^{1}_{x}\sigma^{2}_{y}\rangle\geq
L_{J}.$ (5)

Should the observed one-particle averages
$\\{\langle\sigma^{1}_{x}\rangle\mathrel{\mathop{\mathchar 58\relax}}x\in
X\\}$ and nearest-neighbor two-point correlators
$\\{\langle\sigma^{1}_{x}\sigma^{2}_{y}\rangle\mathrel{\mathop{\mathchar
58\relax}}x,y\in X\\}$ of a TI system violate a facet of the classical (also
called ‘local’) polytope, the corresponding many-body system would be shown
not to admit a description compatible with classical physics.

The left hand side of Eq. (5) can be interpreted as a _Bell functional_ that
acts linearly on the distribution
$P_{1,...,m}(a_{1},...,a_{m}|x_{1},...,x_{m})$. Minimizing it over all
distributions admitting a TI quantum extension, we obtain the _quantum limit_
$Q_{J}$ of the Bell functional $J$.

In the following, we describe a method that, for any $d\in\mathbb{N}$, carries
such a minimization variationally over TI quantum systems of local dimension
$d$, thus obtaining an upper bound $\mathcal{Q}_{d}$ on $Q_{J}$. The method
also returns a concrete TI quantum system, with $\mbox{dim}(H)=d$, achieving
the Bell value $\mathcal{Q}_{d}$ with measurement operators
$\\{E_{a|x}\mathrel{\mathop{\mathchar 58\relax}}a,x\\}$. Most statistical
models studied in the literature use projective measurements, i.e.
$\\{E_{a|x}\mathrel{\mathop{\mathchar 58\relax}}a,x\\}$ are projectors. For
most of our results we only consider projective measurements, with one notable
exception. When we need to verify that no 232-type Hamiltonian can violate the
classical bound when $d=2$, only considering projective measurements is too
restrictive. Therefore for these Hamiltonians we allow fully general complex
positive operator-valued measurements (POVMs) as their local observables,
using a modified version of the algorithm presented in the following section
to perform the optimization.

## Results

### Upper bounding the ground state energy density

To minimize the left-hand side of expressions of the form (5), we start from
the following observation: let $\\{\sigma_{x}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{C}^{d}\to\mathbb{C}^{d}\mathrel{\mathop{\mathchar
58\relax}}x\in X\\}$ be a set of $d$-dimensional Hermitian operators with
spectrum contained in $\\{1,-1\\}$. Then, the minimum value of (5) over all TI
quantum states corresponds to the minimum energy-per site of the TI
Hamiltonian

$\mathcal{H}_{222}(\sigma_{1},...,\sigma_{m})=\sum_{i\in\mathbb{Z}}\sum_{x=1,...,X}J_{x}\sigma^{i}_{x}+\sum_{x,y=1,...,X}J_{xy}\sigma^{i}_{x}\sigma^{i+1}_{y}.$
(6)

Tools from condensed matter physics such as uniform matrix product states
(uMPS) [43] allow us to compute the desired energy density efficiently. In
order to minimize (5) for a given local dimension $d$, all we have to do is
suitably explore the manifold of the set of local observables, e.g.: via
gradient descent.

Our first step consists of finding a parametrization of all the local
observables. Consider observables $\\{\sigma_{a}|a\in X\\}$, each of which can
be diagonalized by an unitary matrix $U_{a}$ as

$\sigma_{a}=U_{a}\Lambda_{a}U_{a}^{\dagger},$ (7)

where $\Lambda_{a}$ is a diagonal matrix with entries $\pm 1$. To make this
more explicit, we use the vector $[n_{x},n_{y}]$ to describe number of $-1$ in
the eigenvalues $\sigma_{x}$ and $\sigma_{y}$.

We can then use the space of skew-Hermitian matrices to effectively
parameterize each $U_{a}$ as

$U_{a}=e^{S_{a}},$ (8)

where $S_{a}$ is skew-Hermitian. Let $\\{B_{1},B_{2},\dots,B_{n}\\}$ be a
basis of the vector space of skew-Hermitian matrices. Here $n=d^{2}-d$ denotes
the dimension of the space. Expanding $S_{a}$ in this basis gives

$S_{a}(W_{a})=\sum_{k=1}^{n}w_{ak}B_{k},$ (9)

where $W_{a}\equiv\\{w_{a1},w_{a2},\dots,w_{an}\\}$ are scalars. Our
optimization parameters are therefore $\\{w_{ak}|a\in X;k=1,\ldots,n\\}$.

Using the method above, observables $\sigma_{a}(a=x,y)$ in ${\cal{H}}_{222}$
can be parameterized as

$\sigma_{a}(W_{a})=(e^{\sum_{k=1}^{n}w_{ak}B_{k}})\Lambda_{a}(e^{\sum_{k=1}^{n}w_{ak}B_{k}})^{\dagger}.$
(10)

Consequently, ${\cal{H}}_{222}$ is parameterized as
${\cal{H}}_{222}(W_{x},W_{y})$.

Using Jordan’s lemma [33], the number of real parameters can be reduced when
$|X|=2$. For example, applying Jordan’s lemma to ${\cal{H}}_{222}$ when $d=4$
yields a basis in which both $\sigma_{x}$ and $\sigma_{y}$ are block-diagonal:

$\sigma_{x}=\begin{bmatrix}\sigma_{x,1}&\mbox{\large 0}\\\ \mbox{\large
0}&\sigma_{x,2}\\\
\end{bmatrix},\quad\sigma_{y}=\begin{bmatrix}\sigma_{y,1}&\mbox{\large 0}\\\
\mbox{\large 0}&\sigma_{y,2}\\\ \end{bmatrix}$ (11)

where $\sigma_{x,1},\sigma_{x,2},\sigma_{y,1},\sigma_{y,2}$ are ${2\times 2}$
Hermitian matrices.

We are now ready to present our MPS based gradient descent method. The method
is iterative. For $a=x,y$, let $W_{a}(k)$ denote the parametrization of
observable $\sigma_{a}$ at the $k$-th iteration. We will refer to the
parametrization $\\{W_{x}(k),W_{y}(k)\\}$ of both observables as $W(k)$. At
each iteration $k$, the parameters $W(k)$ are updated to $W(k+1)$ through the
following procedure.

First, we minimize the energy-per-site of the Hamiltonian
${\cal{H}}_{222}(k)\equiv{\cal{H}}_{222}(\sigma_{a}(k),\sigma_{b}(k))$ over
the manifold of uMPS. The result $e(k)$ can be computed using, e.g., the Time-
dependent Variational Principle (TDVP) algorithm [15, 27, 43] or the
Variational Uniform Matrix Product State (VUMPS) algorithm [46, 43]. We mainly
use the TDVP algorithm for its good numerical stability and reasonable speed
of convergence.

Following the TDVP algorithm, $e(k)$ can be expressed as

$\begin{split}e(k)=\sum_{s,t,u,v}h(k)_{st}^{uv}{\rm
Tr}[{A^{t}(k)}^{\dagger}{A^{s}(k)}^{\dagger}l(k){A^{u}(k)}{A^{v}(k)}r(k)],\end{split}$
(12)

Figure 1: Energy density

where $\sum_{s,t,u,v}h(k)^{uv}_{st}\ket{s}\bra{u}\otimes\ket{t}\bra{v}$ is the
local term of ${\cal{H}}_{222}(k)$;
$\\{A^{s}(k)\\}_{s}\subset\mathbb{C}^{D\times D}$ is the tensor defining the
optimal uMPS, and $l(k),r(k)$ are the left and right leading eigenvectors of
the transfer matrix $T(k)=\sum_{s=1}^{d}\bar{A}^{s}(k)\otimes A^{s}(k)$.

Next, we seek to find observables leading to a Hamiltonian with a smaller
energy-per-site, when evaluated over the uMPS with tensor $\\{A^{s}(k)\\}$
just identified. Hence, with $A(k)$ fixed, we replace the local term $h(k)$ by
$h(\sigma_{x}(W),\sigma_{y}(W))$ in Eq.(12). This leads to a function $e(W;k)$
of the parameters $W$ defining the observables. To update the parameters
$W(k)$, we move away from $W(k)$ in the direction of maximum function decrease
at point $W(k)$. That is, we move against the gradient of $e({W};k)$:

${W}(k+1)={W}(k)-{\gamma}(k)\cdot\nabla_{W}e({W};k).$ (13)

Here ${\gamma}(k)$ is a scaling parameter, which we take to be of the form
${\gamma}(k)={\rm max}(\gamma_{0}\alpha^{q(k)},\gamma_{\rm{min}})$, where
$\alpha\in(0,1)$ and $q(k)$ is linear with respect to the iteration number
$k$.

Starting from an initial seed $W(0)$, we iterate the two steps above, hence
generating a sequence of parameter values $(W(0),W(1),...)$. At every
iteration $k$, we check the condition ${\|\nabla e(W;k)\|}_{2}<\epsilon^{*}$,
for some desired convergence threshold $\epsilon^{*}$. If the condition holds,
we stop the algorithm and return the optimal parameters $W^{*}\equiv W(k)$.

In our experience, the quantity $e(W^{*})$ is typically a very good estimate
of the lowest quantum value of the considered contextuality functional over TI
quantum systems of local dimension $d$. If $e^{*}$ happens to be smaller than
the classical bound of the corresponding facet inequality, then we can state
that the found quantum system characterized by the TI Hamiltonian
${\cal{H}}_{222}({W}^{*})$ exhibits contextuality.

To test the algorithm, we apply it to compute the minimum ground state energy
densities ${\cal{Q}}_{d}$ $(d=2,3,4)$ of six 322-type TI quantum systems
introduced in Sec. Contextuality in 322-type Hamiltonians. All the results are
plotted in Fig. 2. We find that the initial ground state energy densities
determined by random parameters typically do not violate the classical bound
${\cal{L}}_{322}$ (red line). As the iteration number increases,
${\cal{Q}}_{2}$ and ${\cal{Q}}_{3}$ decrease approximately linearly and begin
to show contextuality. In Fig. 2(e) and Fig. 2(f), ${\cal{Q}}_{4}$ oscillates
during the first several iterations. As the optimization process continues,
${\cal{Q}}_{4}$ also begins to cross ${\cal{L}}_{322}$ after. The ground state
energy densities of all six models converge to values below their classical
bounds within 20 iterations.

Figure 2: Convergences of ground state energy density ${\cal{Q}}_{d}$ (blue
line) and two-norm of gradient $\epsilon$ (black line) for 322-type TI quantum
systems with the local observable dimension $d=2,3,4$. The red dashed line
represents the classical bound, and the regions below show contextuality. (a)
${\cal{Q}}_{2}$ of No.1 in Table 6. (b) ${\cal{Q}}_{2}$ of No.38 in Table 6.
(c) ${\cal{Q}}_{3}$ of No.7 in Table 6. (d) ${\cal{Q}}_{3}$ of No.30 in Table
6. (e) ${\cal{Q}}_{4}$ of No.37 in Table 6. (f) ${\cal{Q}}_{4}$ of No.54 in
Table 6.

### Lower bounding the ground state energy density

Consider the scenario shown in Fig. 3: an infinite chain of elephants, each of
which represents a physical system, be it quantum, classical or else. Call
$\varepsilon$ the overall state of the chain. Depending on the context,
$\varepsilon$ will be a classical probability distribution, a quantum state or
a no-signaling box. Because $\varepsilon$ is TI, the marginal distribution or
the reduced state of each of the $5$ marked elephants, taken from an arbitrary
contiguous subset of the chain, should be equal:
$\varepsilon_{1}=\ldots=\varepsilon_{5}$. Moreover, the reduced state of any
contiguous subset of elephants should also be equal:
$\varepsilon_{1,\ldots,1+k}=\varepsilon_{2,\ldots,2+k},\forall 1\leq k\leq 3$.
When $k=3$, the marginals/reduced states are shown in Fig. 3 as green and red
rectangles. For any contiguous subset of $\varepsilon$ of length $l$, the
marginals/reduced states are said to be _locally translation-invariant_ (LTI)
if

$\displaystyle\varepsilon_{1,\ldots,l-1}=\varepsilon_{2,\ldots,l}.$ (14)

Figure 3: Local translation invariance. A necessary condition from the
requirement that the entire system is translation-invariant.

Clearly, LTI is a necessary condition for $\varepsilon$ to be TI. For
classical probability distributions in 1D, LTI is also sufficient: any LTI
marginal can be extended to an infinite TI distribution [13]. In fact, this
property is the key to the characterization the set ${\operatorname{TI-LHV}}$
presented in [45].

Unfortunately, LTI is not enough to characterize the near-neighbor density
matrices of TI quantum states or even the near-neighbor marginals of TI no-
signaling systems. In those scenarios LTI can be used use to relax the set of
such marginals, rather than to fully characterize it. Define thus
${\operatorname{LTI_{n}-NS}}$ as the set of boxes admitting an extension to an
$n$-partite no-signaling box with local translation invariance. As shown in
[45], the distance between any element of the set
${\operatorname{LTI_{n}-NS}}$ and its subset ${\operatorname{TI-NS}}$ is upper
bounded by $O\left(\frac{1}{n}\right)$.

A straightforward extension to bound ${\operatorname{TI-Q}}$ is impossible, as
the approximate characterization of general multipartite quantum correlations
is an undecidable problem [21]. One can, however, relax the existence of
quantum states and observables reproducing the observed correlations to that
of positive semidefinite _moment matrices_. Those are matrices $\Gamma$ whose
rows and columns are labeled by monomials of measurement operators with at
most $s$ (the order of the relaxation) measurement operators per party, and
where each entry $\Gamma_{\alpha\beta}$ is supposed to represent the quantity
$\langle\alpha^{\dagger}\beta\rangle$, see [29, 28] for details. In order to
bound ${\operatorname{TI-Q}}$, we demand the existence of a moment matrix for
an $n$-partite Bell scenario and then impose LTI over the said moment matrix.
Call ${\operatorname{LTI_{n}-NPA_{s}}}$ the corresponding relaxation.

For any Bell functional, we can thus find a lower bound on its minimal value
in ${\operatorname{TI-NS}}$ and ${\operatorname{TI-Q}}$ by respectively
optimizing over ${\operatorname{LTI_{n}-NS}}$ (with linear programming
techniques [25]) and ${\operatorname{LTI_{n}-NPA_{s}}}$ (with semidefinite
programming techniques [11]). Moreover, one can improve those lower bounds by
increasing the values of $n,s$.

### Contextuality in 222-type Hamiltonians

The LTI-LHV polytope for 2 dichotomic observables has 36 facets. Computing
their ${\operatorname{LTI_{4}-NS}}$ lower bounds reveals that most of them
coincide with the corresponding classical bounds. In fact, there is only one
inequality, up to local relabeling, which can potentially show contextuality.
In its 1D TI quantum Hamiltonian form, it reads

$\begin{split}{\cal
H}_{222}=&\sum_{i=1}^{\infty}(2\sigma_{x}^{i}+\sigma_{x}^{i}\sigma_{x}^{i+1}+\sigma_{x}^{i}\sigma_{y}^{i+1}-\sigma_{y}^{i}\sigma_{x}^{i+1}-\sigma_{y}^{i}\sigma_{y}^{i+1}),\end{split}$
(15)

with classical bound $-2$.

Lower bounding Eq. (15) with ${\operatorname{LTI_{n}-NS}}$ with increasing
$n$, we observed some curious phenomena. Because exact optimal solutions of
linear programs are rational numbers, we obtain the solutions in Table 1.

$n$ | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---
${\operatorname{LTI_{n}-NS}}$ lower bound | $-\frac{8}{3}$ | $-\frac{12}{5}$ | $-\frac{9}{4}$ | $-\frac{13}{6}$ | $-\frac{36}{17}$ | $-\frac{48}{23}$ | $-\frac{62}{30}$
Table 1: Exact solutions of ${\operatorname{LTI_{n}-NS}}$ approximations of
the lower bound of (15) as a function of $n$.

The numerators and denominators in the table form two integer sequences:
A027691 [37] and A152948 [38] in The On-Line Encyclopedia of Integer
Sequences. Moreover, the displaced inverse of a quadratic function

$\displaystyle-2-\frac{4}{n^{2}-3n+6},\,n\in\mathbb{N},\,n\geq 3$ (16)

perfectly fits the sequence of lower bounds in Table 1, see Fig. 4.

In the limit $n\to\infty$, this function converges to the classical bound
$-2$. In other words, if the solution of the optimization over
${\operatorname{LTI_{n}-NS}}$ satisfies (16) for all $n\geq 3$, then no
Hamiltonian of the form (15), quantum or otherwise, can possibly violate the
classical bound.

Figure 4: Exact solutions for the ${\operatorname{LTI_{n}-NS}}$ approximation
of the lower bound of Eq. 15 (orange dots), the fitted function (16) (blue
line) when $3\leq n\leq 40$, and the classical bound (red line).

Proving that a series of rational numbers, the solutions of linear programs of
exponentially increasing size, converges to a certain value is very hard.
However, we do have additional numerical evidence to support our claim that
the lowest possible ground state energy density of 1D TI quantum Hamiltonians
of the form (15) is $-2$. We used our algorithm, described in Section Results,
to search for the quantum Hamiltonian with the lowest ground state energy
density, for local observables of dimension $2\leq d\leq 6$. For each $d$,
$\sigma_{x}$ and $\sigma_{y}$ are parameterized by the method described below,
and we find the lowest quantum value ${\cal{Q}}_{d}$ among all possible
systems is $-2$. Moreover, the corresponding two-body reduced density matrix
of the quantum system for the ground state is a rank $1$ projector, which
shows that the ground state is in fact a product state. We present these
ground states and the parameters for observables in Table 2.

Table 2: Ground state and parameters of ${\cal{H}}_{222}$ under the different observable dimension settings when the ground state energy density equals to the classical bound $-2$. $d$ | $w_{1}$ | $w_{2}$ | $w_{3}$ | $[n_{x},n_{y}]$ | Ground state
---|---|---|---|---|---
2 | 0.00001 | – | – | $[1,1]$ | $\ket{1}^{\otimes\infty}$
3 | 0.65129 | – | – | $[2,2]$ | $\ket{2}^{\otimes\infty}$
4 | 0.00004 | 0.59946 | – | $[2,2]$ | $\ket{3}^{\otimes\infty}$
5 | 0.98662 | 0.00001 | – | $[2,2]$ | $\ket{4}^{\otimes\infty}$
6 | 0.90451 | 0.91913 | 0.00005 | $[3,3]$ | $\ket{5}^{\otimes\infty}$

To make sense of Table 2, we next explicitly write the parametrization of the
$d$-dimensional observables achieving the classical bound. Two $2\times 2$
matrices having eigenvalues one $1$ and one $-1$ will repeatedly appear below:
$\Lambda$ is the diagonal matrix with diagonal entries $\pm 1$, $B(w)$ is a
matrix governed by one parameter $\\{w\\}$:

$\Lambda=\begin{bmatrix}1&0\\\ 0&-1\\\ \end{bmatrix},\quad
B(w)=\begin{bmatrix}\cos(2w)&-\sin(2w)\\\ -\sin(2w)&-\cos(2w)\\\
\end{bmatrix}.$ (17)

When $d=2$ is assigned to local observables in ${\cal{H}}_{222}$, $\sigma_{x}$
is a diagonal matrix with diagonal entries $1$ and $-1$, i.e.,
$\sigma_{x}=\Lambda$, and $\sigma_{y}$ determined by one parameter
$\\{w_{1}\\}$ has the same parameterized form as $B(w)$, i.e.,
$\sigma_{y}(w_{1})=B(w_{1})$. In this case, $[n_{x},n_{y}]=[1,1]$ and
$\sigma_{x},\sigma_{y}$ are of the form

$\sigma_{x}=\Lambda,\quad\sigma_{y}(w_{1})=B(w_{1}).$ (18)

When $d=3$ is assigned to local observables in ${\cal{H}}_{222}$, $\sigma_{x}$
is a diagonal matrix with diagonal entries $(1,-1,-1)$. $\sigma_{y}$ is a
block diagonal matrix, where the main-diagonal blocks are one matrix
$B(w_{1})$ and one numerical value $-1$. Then, $[n_{x},n_{y}]=[2,2]$ and
$\sigma_{x},\sigma_{y}$ are given by

$\sigma_{x}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\ \mbox{\large 0}&-1\\\
\end{bmatrix},\quad\sigma_{y}(w_{1})=\begin{bmatrix}B(w_{1})&\mbox{\large
0}\\\ \mbox{\large 0}&-1\\\ \end{bmatrix}.$ (19)

When $d=4$ is assigned to local observables in ${\cal{H}}_{222}$, $\sigma_{x}$
is a diagonal matrix with diagonal entries two $1$ and two $-1$, and
$\sigma_{y}$ is a block diagonal matrix with main-diagonal blocks being two
$2\times 2$ matrices $B(w_{1})$ and $B(w_{2})$. In this case,
$[n_{x},n_{y}]=[2,2]$ and $\sigma_{x},\sigma_{y}$ are of the forms

$\sigma_{x}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\ \mbox{\large
0}&\Lambda\\\
\end{bmatrix},\quad\sigma_{y}(w_{1},w_{2})=\begin{bmatrix}B(w_{1})&\mbox{\large
0}\\\ \mbox{\large 0}&B(w_{2})\\\ \end{bmatrix}.$ (20)

When $d=5$ is assigned to local observables in ${\cal{H}}_{222}$, $\sigma_{x}$
is a diagonal matrix with diagonal entries two $-1$ and three $1$.
$\sigma_{y}$ is a block diagonal matrix, where the main-diagonal blocks are
two $2\times 2$ matrices $B(w_{1})$ and $B(w_{2})$ and one numerical number
$1$. Then, $[n_{x},n_{y}]=[2,2]$ and $\sigma_{x},\sigma_{y}$ are given by

$\sigma_{x}=\begin{bmatrix}\Lambda&&\lx@intercol\hfil\hbox{\multirowsetup{\Large
0}}\hfil\lx@intercol\\\ &\Lambda&\\\
\lx@intercol\hfil\hbox{\multirowsetup{\Large 0}}\hfil\lx@intercol&&1\\\
\end{bmatrix},\quad\sigma_{y}(w_{1},w_{2})=\begin{bmatrix}B(w_{1})&&\lx@intercol\hfil\hbox{\multirowsetup{\Large
0}}\hfil\lx@intercol\\\ &B(w_{2})&\\\
\lx@intercol\hfil\hbox{\multirowsetup{\Large 0}}\hfil\lx@intercol&&1\\\
\end{bmatrix}.$ (21)

When $d=6$ is assigned to local observables in ${\cal{H}}_{222}$, $\sigma_{x}$
is a diagonal matrix, where three $-1$ and three $1$ are alternately arranged
in the diagonal. $\sigma_{y}$ is a block diagonal matrix with main-diagonal
blocks being three $2\times 2$ matrices $B(w_{1})$, $B(w_{2})$ and $B(w_{3})$
. Then, $[n_{x},n_{y}]=[3,3]$ and $\sigma_{x},\sigma_{y}$ have the forms

$\begin{split}\sigma_{x}&=\begin{bmatrix}\Lambda&&\lx@intercol\hfil\hbox{\multirowsetup{\Large
0}}\hfil\lx@intercol\\\ &\Lambda&\\\
\lx@intercol\hfil\hbox{\multirowsetup{\Large 0}}\hfil\lx@intercol&&\Lambda\\\
\end{bmatrix},\\\
\sigma_{y}(w_{1},w_{2},w_{3})&=\begin{bmatrix}B(w_{1})&&\lx@intercol\hfil\hbox{\multirowsetup{\Large
0}}\hfil\lx@intercol\\\ &B(w_{2})\\\
\lx@intercol\hfil\hbox{\multirowsetup{\Large 0}}\hfil\lx@intercol&&B(w_{3})\\\
\end{bmatrix}.\end{split}$ (22)

### Contextuality in 322-type Hamiltonians

The ${\operatorname{TI-LHV}}$ polytope for the 322-type Hamiltonians has been
characterized in [45]: it has 32372 facets which can be sorted into 2102
equivalence classes. The general form of the 322-type Hamiltonian is given by

$\begin{split}{\cal
H}_{322}&=\sum_{i=1}^{\infty}J_{x}{\sigma_{x}^{i}}+J_{y}{\sigma_{y}^{i}}+J_{xx}^{AB}{\sigma_{x}^{i}\sigma_{x}^{i+1}}+J_{xy}^{AB}{\sigma_{x}^{i}\sigma_{y}^{i+1}}\\\
&+J_{yx}^{AB}{\sigma_{y}^{i}\sigma_{x}^{i+1}}+J_{yy}^{AB}{\sigma_{y}^{i}\sigma_{y}^{i+1}}+J_{xx}^{AC}{\sigma_{x}^{i}\sigma_{x}^{i+2}}\\\
&+J_{xy}^{AC}{\sigma_{x}^{i}\sigma_{y}^{i+2}}+J_{yx}^{AC}{\sigma_{y}^{i}\sigma_{x}^{i+2}}+J_{yy}^{AC}{\sigma_{y}^{i}\sigma_{y}^{i+2}},\end{split}$
(23)

where
$\\{J_{x},J_{y},J_{xx}^{AB},J_{xy}^{AB},J_{yx}^{AB},J_{yy}^{AB},J_{xx}^{AC},J_{xy}^{AC},J_{yx}^{AC},J_{yy}^{AC}\\}$
are the couplings given by the facet inequalities and $\sigma_{x},\sigma_{y}$
are local observables.

Using our uMPS based gradient descent algorithm, a total of 63 Hamiltonians
exhibit contextuality. The explicit parameterization of observables
$\sigma_{x}$ and $\sigma_{y}$ is explained at the end of this section. All the
contextual Hamiltonians and ground state energy densities are listed in Table
5 and Table 6 respectively. Among them, we identify some quantum models whose
ground state energy density matches the ${\operatorname{LTI_{5}-NPA_{1}}}$
lower bounds. For all these contextuality witnesses, we have thus identified
translation-invariant quantum models exhibiting the strongest quantum
violation. All the matched models are summarized in Table 3. As the reader can
appreciate, the first five inequalities seem to require local dimension $d=3$
to be saturated; inequality $6$, dimension $4$; and the last four
inequalities, dimension $5$.

Table 3:  The ground state energy density ${\cal{Q}}_{d}$ of ten 322-type TI quantum systems matches the ${\operatorname{LTI_{5}-NPA_{1}}}$ lower bound. The second column gives the number of each model in this table in Table 6.  No. |  Table 6 | ${\cal{L}}$ | ${\cal{Q}}_{2}$ | ${\cal{Q}}_{3}$ | ${\cal{Q}}_{4}$ | ${\cal{Q}}_{5}$ | ${\operatorname{LTI_{5}-NPA_{1}}}$
---|---|---|---|---|---|---|---
1 | No.1 | -6 | -6.08108 | -6.32747 |  |  | -6.32747
2 | No.2 | -6 | -6.10943 | -6.33712 |  |  | -6.33712
3 | No.3 | -3 | -3.04150 | -3.20711 |  |  | -3.20711
4 | No.4 | -4 | – | -4.14623 |  |  | -4.14623
5 | No.5 | -8 | – | -8.12123 |  |  | -8.12123
6 | No.6 | -4 | – | -4.02415 | -4.10310 |  | -4.10310
7 | No.7 | -5 | – | -5.09951 | -5.09951 | -5.29852 | -5.29852
8 | No.8 | -4 | – | -4.18655 | -4.18655 | -4.33137 | -4.33137
9 | No.9 | -4 | – | -4.11581 | -4.11581 | -4.41421 | -4.41421
10 | No.10 | -5 | – | -5.07058 | -5.07058 | -5.26969 | -5.26969

In Fig. 5, the reader can see the trajectories in parameter space followed by
two quantum systems, of dimensions $d=3$ and $d=4$, undergoing our gradient
descent method. This is possible because the number of free continuous
parameters in one and another case are $1$ and $2$.

(a) No.3 in Table 3, ${\cal{L}}=-3$

(b) No.6 in Table 3, ${\cal{L}}=-4$

Figure 5: The trajectory of the ground state energy density (black dotted
line) of two models in Table 3. (a) has two subplots, and the right one is an
enlarged view for the trajectory in the left one when ground state energy
density converges to the minimum. (b) has three subplots, the the middle one
is the top view of the leftmost one, and the rightmost one only depicts the
trajectory.

In Fig. 5(a), ${\cal{Q}}_{3}$ surfaces (blue lines) show that the Hamiltonian
exhibits contextuality no matter which values the parameter takes. The
trajectory of ground state energy density (black dotted line) in the left
subplot decreases along the ${\cal{Q}}_{3}$ surface to the bottom. Besides,
the right enlarged subplot of the trajectory demonstrates that ground state
energy density eventually converges to the respective
${\operatorname{LTI_{n}-NPA_{s}}}$ lower bound. In Fig. 5(b), the leftmost
subplot shows the trajectory of the ground state energy density on the 3D
${\cal{Q}}_{4}$ surface, the middle 2D-subplot is the top view of the leftmost
one, and the rightmost one only depicts the trajectory of the ground state
energy density. Iteratively, our methods guide the initial random ground state
energy density converging to the lowest possible one.

We plot the ground state energy density as functions of the parameters
defining the local observables for some of the Hamiltonians in Table 3 and
Table 6 to gauge the robustness of the contextuality violations. Five models
for $d=4$ and another five models for $d=5$ are shown in Fig. 6 and Fig. 7
respectively. Note that the first two models in Fig. 6(a)-(b) and the first
four models in Fig. 7(a)-(d) exhibit the strongest contextuality.

(a) No.1 in Table 3, ${\cal{L}}=-6$

(b) No.5 in Table 3, ${\cal{L}}=-8$

(c) No.8 in Table 6, ${\cal{L}}=-4$

(d) No.19 in Table 6, ${\cal{L}}=-11$

(e) No.23 in Table 6, ${\cal{L}}=-8$

Figure 6: ${\cal{Q}}_{4}$ surface of five models in Table 3 and Table 6, where
$w_{1},w_{2}$ both take discrete values on $[-2,2]$ at the interval $0.1$.
Each model has three subplots. The leftmost is a 3D-surface, the middle figure
is from an another perspective of the left side one to view the internal
structure, and the third one is the top view of the leftmost image.

(a) No.7 in Table 3, ${\cal{L}}=-5$

(b) No.8 in Table 3, ${\cal{L}}=-4$

(c) No.9 in Table 3, ${\cal{L}}=-4$

(d) No.10 in Table 3, ${\cal{L}}=-5$

(e) No.11 in Table 6, ${\cal{L}}=-3$

Figure 7: ${\cal{Q}}_{5}$ surface of five models in Table 3 and Table 6, where
$w_{1},w_{2}$ both take discrete values on $[-2,2]$ at the interval $0.1$.
Each model has three subplots. The leftmost is a 3D-surface, the middle figure
is from an another perspective of the left side one to view the internal
structure, and the third one is the top view of the leftmost image.

It can be seen that some Hamiltonians are much more susceptible to small
changes in parameters that define the local observables than others. For the
Hamiltonians in Fig. 6(b), Fig. 7(b) and Fig. 7(c), keeping the ground state
energy density above the classical bound is unstable, small perturbations in
the parameters will make them violate it. In contrast, the remaining
Hamiltonians need carefully engineered parameters to violate the classical
bound. Especially for the Hamiltonian in Fig. 7(e), square-like parameter
regions exist in which the corresponding ground state energy density could not
violate the classical bound no matter how many times the perturbations are
given. These plots help us find suitable Hamiltonians for simulation in
trapped-ion or optical lattice systems, where witnessing contextuality (or the
strongest contextuality) simply involves cooling the corresponding Hamiltonian
to the ground state.

We next describe the parameterization of the 322-type Hamiltonians achieving
the minimum quantum values in Table 3. For $d=2$ and $d=4$, $\sigma_{x}$,
$\sigma_{y}$ have the exact same parameterized forms as the observables in the
222-type Hamiltonians in Eq. (18) and Eq. (20) respectively.

For $d=3$ and $d=5$, depending on the number of $1$’s and $-1$’s of each
matrix $\Lambda_{a}(a=x,y)$ in (7), two different classes of pairs of local
observables $\sigma_{x}$ and $\sigma_{y}$ are considered. Here, we continue
using the notations $\Lambda$ and $B(w)$ introduced in (17).

For local dimension $d=3$, the first class of pairs of local observables is of
the form $[n_{x},n_{y}]=[1,2]$. More specifically:

$\sigma_{x}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\ \mbox{\large 0}&1\\\
\end{bmatrix},\quad\sigma_{y}(w_{1})=\begin{bmatrix}B(w_{1})&\mbox{\large
0}\\\ \mbox{\large 0}&-1\\\ \end{bmatrix}.$ (24)

The second class is of the form $[n_{x},n_{y}]=[1,1]$, with

$\sigma_{x}=\begin{bmatrix}1&\mbox{\large 0}\\\ \mbox{\large 0}&\Lambda\\\
\end{bmatrix},\quad\sigma_{y}(w_{1})=\begin{bmatrix}1&\mbox{\large 0}\\\
\mbox{\large 0}&B(w_{1})\\\ \end{bmatrix}.$ (25)

For local dimension $d=5$, the first class of observable pairs is of the form
$[n_{x},n_{y}]=[2,2]$, with

$\sigma_{x}=\begin{bmatrix}\Lambda&&\lx@intercol\hfil\hbox{\multirowsetup{\Large
0}}\hfil\lx@intercol\\\ &\Lambda&\\\
\lx@intercol\hfil\hbox{\multirowsetup{\Large 0}}\hfil\lx@intercol&&1\\\
\end{bmatrix},\quad\sigma_{y}(w_{1},w_{2})=\begin{bmatrix}B(w_{1})&&\lx@intercol\hfil\hbox{\multirowsetup{\Large
0}}\hfil\lx@intercol\\\ &B(w_{2})&\\\
\lx@intercol\hfil\hbox{\multirowsetup{\Large 0}}\hfil\lx@intercol&&1\\\
\end{bmatrix}.$ (26)

The second class of observable pairs satisfies $[n_{x},n_{y}]=[3,3]$, and
$\sigma_{x}$ and $\sigma_{y}$ are given by

$\sigma_{x}=\begin{bmatrix}\Lambda&&\lx@intercol\hfil\hbox{\multirowsetup{\Large
0}}\hfil\lx@intercol\\\ &\Lambda&\\\
\lx@intercol\hfil\hbox{\multirowsetup{\Large 0}}\hfil\lx@intercol&&-1\\\
\end{bmatrix},\quad\sigma_{y}(w_{1},w_{2})=\begin{bmatrix}B(w_{1})&&\lx@intercol\hfil\hbox{\multirowsetup{\Large
0}}\hfil\lx@intercol\\\ &B(w_{2})&\\\
\lx@intercol\hfil\hbox{\multirowsetup{\Large 0}}\hfil\lx@intercol&&-1\\\
\end{bmatrix}.$ (27)

### Contextuality in 232-type Hamiltonians

The LTI-LHV polytope for 3 dichotomic observables has 92694 facets, which can
be classified into 652 equivalent classes [32]. The general form of this type
of Hamiltonian is given by

$\displaystyle{\cal H}_{232}$
$\displaystyle=\sum_{i=1}^{\infty}J_{x}{\sigma_{x}^{i}}+J_{y}{\sigma_{y}^{i}}+J_{z}{\sigma_{z}^{i}}+J_{xx}{\sigma_{x}^{i}\sigma_{x}^{i+1}}+J_{xy}{\sigma_{x}^{i}\sigma_{y}^{i+1}}$
$\displaystyle+J_{xz}{\sigma_{x}^{i}\sigma_{z}^{i+1}}+J_{yx}{\sigma_{y}^{i}\sigma_{x}^{i+1}}+J_{yy}{\sigma_{y}^{i}\sigma_{y}^{i+1}}+J_{yz}{\sigma_{y}^{i}\sigma_{z}^{i+1}}$
$\displaystyle+J_{zx}{\sigma_{z}^{i}\sigma_{x}^{i+1}}+J_{zy}{\sigma_{z}^{i}\sigma_{y}^{i+1}}+J_{zz}{\sigma_{z}^{i}\sigma_{z}^{i+1}}.$
(28)

We consider ${\cal{H}}_{232}$ when $d=2,3,4$ and perform the optimizations on
one representative facet from each of the 652 classes. For $d=3,4$, we only
consider real parameters. For $d=2$, we allow the most general measurements in
quantum theory: complex POVMs. In the Method section, we present a projected
gradient descent algorithm to optimize over the set of complex POVMs. All 652
Hamiltonians can only reach the classical bound up to numerical precision of
$10^{-5}$ when $d=2$, while for $d\geq 3$ there are many Hamiltonians which
can violate the classical bound. The ground state energy densities of some
contextual Hamiltonians are shown in Table 4, see Table 7 for the couplings
defining the contextuality witnesses. In addition, the parameters specifying
the optimal local observables are listed in Table 8 for $d=3$ and in Table 9
for $d=4$.

Table 4: Ground state energy density for 5 232-type TI Hamiltonians under the dimension of local observables $d=2,3,4$. No. | ${\cal{L}}$ | ${\cal{Q}}_{2}$ | ${\cal{Q}}_{3}$ | ${\cal{Q}}_{4}$ | ${\operatorname{LTI_{4}-NPA_{1}}}$
---|---|---|---|---|---
1 | -9 | -9.00000 | -9.01875 | -9.01875 | -9.27833
2 | -4 | -4.00000 | -4.03928 | -4.03928 | -4.20626
3 | -5 | -5.00000 | -5.04162 | -5.04162 | -5.14754
4 | -4 | -4.00000 | -4.01336 | -4.11562 | -4.23786
5 | -2 | -2.00000 | -2.08094 | -2.08749 | -2.28767

The local observables $\sigma_{x},\sigma_{y}$ and $\sigma_{z}$ are
parametrized using the method presented in Section Upper bounding the ground
state energy density: for given $\Lambda_{a},S_{a}$ the local observable
$\sigma_{a}$ can be written as

$\sigma_{a}=e^{S_{a}}\Lambda_{a}(e^{S_{a}})^{\dagger}.$ (29)

While this step is straightforward, different combinations of $\pm 1$ in
$\Lambda_{a}$ may lead to different ground state energy density.

Since the number of $1$ and $-1$ on the diagonal of $\Lambda_{a}$ in three
local observables is not necessarily the same, there is more than one
combination of three parameterized local observables. We consider every
possible combination of $1$ and $-1$ in $\Lambda_{a}$ for each
$a\in\\{x,y,z\\}$. We only show combinations of parameterized local
observables used in Table 4 below. Here, we denote the $2\times 2$ identity
matrix by $I$, and continue using the notation $\Lambda$ introduced in (17).

When $d=2$, the classical bound can be achieved via three local observables
determined by two parameters, where $[n_{x},n_{y},n_{z}]=[2,1,1]$. The first
local observable $\sigma_{x}$ is minus the identity matrix:

$\sigma_{x}=\begin{bmatrix}-1&0\\\ 0&-1\\\ \end{bmatrix}.$ (30)

For the second and third local observables $\sigma_{a}(a=y,z)$, $\Lambda_{a}$
has entries one $1$ and one $-1$ on the main diagonal, and $S_{a}$ is
determined by one parameter $\\{w\\}$. Hence, $\sigma_{y}$ and $\sigma_{z}$
are specified by

$\begin{split}&\Lambda_{y}=\begin{bmatrix}1&0\\\ 0&-1\\\ \end{bmatrix},\quad
S_{y}(w_{1})=\begin{bmatrix}0&w_{1}\\\ -w_{1}&0\\\ \end{bmatrix};\\\
&\Lambda_{z}=\begin{bmatrix}1&0\\\ 0&-1\\\ \end{bmatrix},\quad
S_{z}(w_{2})=\begin{bmatrix}0&w_{2}\\\ -w_{2}&0\\\ \end{bmatrix}.\end{split}$
(31)

For $d=3$, two different combinations of three parameterized local observables
are used, where the difference arises from $\Lambda_{a}$, but $S_{a}$ of each
local observable share the same parameterized form as

$\begin{split}&S_{x}(w_{1},w_{2},w_{3})=\begin{bmatrix}0&w_{1}&w_{2}\\\
-w_{1}&0&w_{3}\\\ -w_{2}&-w_{3}&0\\\ \end{bmatrix},\\\
&S_{y}(w_{4},w_{5},w_{6})=\begin{bmatrix}0&w_{4}&w_{5}\\\ -w_{4}&0&w_{6}\\\
-w_{5}&-w_{6}&0\\\ \end{bmatrix},\\\
&S_{z}(w_{7},w_{8},w_{9})=\begin{bmatrix}0&w_{7}&w_{8}\\\ -w_{7}&0&w_{9}\\\
-w_{8}&-w_{9}&0\\\ \end{bmatrix}.\end{split}$ (32)

In the first combination, $[n_{x},n_{y},n_{z}]=[2,1,1]$, where $\Lambda_{x}$
has one $1$ and two $-1$ on the main diagonal, and $\Lambda_{y}$ and
$\Lambda_{z}$ both have two $1$ and one $-1$ being main diagonal entries.
Then, $\Lambda_{x}$, $\Lambda_{y}$, and $\Lambda_{z}$ are given by

$\Lambda_{x}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\ \mbox{\large 0}&-1\\\
\end{bmatrix},\quad\Lambda_{y}=\begin{bmatrix}1&\mbox{\large 0}\\\
\mbox{\large 0}&\Lambda\\\
\end{bmatrix},\quad\Lambda_{z}=\begin{bmatrix}1&\mbox{\large 0}\\\
\mbox{\large 0}&\Lambda\\\ \end{bmatrix}.$ (33)

In the second combination, $[n_{x},n_{y},n_{z}]=[2,1,2]$, where $\Lambda_{x}$
and $\Lambda_{z}$ both have one $1$ and two $-1$ on the main diagonal, and
$\Lambda_{y}$ has two $1$ and one $-1$ being main diagonal entries. Then,
$\Lambda_{x}$, $\Lambda_{y}$, and $\Lambda_{z}$ are given by

$\Lambda_{x}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\ \mbox{\large 0}&-1\\\
\end{bmatrix},\quad\Lambda_{y}=\begin{bmatrix}1&\mbox{\large 0}\\\
\mbox{\large 0}&\Lambda\\\
\end{bmatrix}\quad\Lambda_{z}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\
\mbox{\large 0}&-1\\\ \end{bmatrix}.$ (34)

For $d=4$, four different classes of triples of local observables are used.
These classes differ from each other on the structure of the matrices
$\Lambda_{a}$ in (29). The matrices $S_{a}$ have, nonetheless, the same form
in the three classes, namely:

$\begin{split}S_{x}(w_{1},w_{2},w_{3},w_{4},w_{5},w_{6})&=\begin{bmatrix}0&w_{1}&w_{2}&w_{3}\\\
-w_{1}&0&w_{4}&w_{5}\\\ -w_{2}&-w_{4}&0&w_{6}\\\ -w_{3}&-w_{5}&-w_{6}&0\\\
\end{bmatrix},\\\
S_{y}(w_{7},w_{8},w_{9},w_{10},w_{11},w_{12})&=\begin{bmatrix}0&w_{7}&w_{8}&w_{9}\\\
-w_{7}&0&w_{10}&w_{11}\\\ -w_{8}&-w_{10}&0&w_{12}\\\
-w_{9}&-w_{11}&-w_{12}&0\\\ \end{bmatrix},\\\
S_{z}(w_{13},w_{14},w_{15},w_{16},w_{17},w_{18})&=\begin{bmatrix}0&w_{13}&w_{14}&w_{15}\\\
-w_{13}&0&w_{16}&w_{17}\\\ -w_{14}&-w_{16}&0&w_{18}\\\
-w_{15}&-w_{17}&-w_{18}&0\\\ \end{bmatrix}.\end{split}$ (35)

The first class is of the form $[n_{x},n_{y},n_{z}]=[3,1,1]$, where
$\Lambda_{x}$ has one $1$ and three $-1$ on the main diagonal, and
$\Lambda_{y}$ and $\Lambda_{z}$ both have three $1$ and one $-1$ being main
diagonal entries. Then, $\Lambda_{x}$, $\Lambda_{y}$, and $\Lambda_{z}$ are
given by

$\Lambda_{x}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\ \mbox{\large 0}&-I\\\
\end{bmatrix},\quad\Lambda_{y}=\begin{bmatrix}I&\mbox{\large 0}\\\
\mbox{\large 0}&\Lambda\\\
\end{bmatrix},\quad\Lambda_{z}=\begin{bmatrix}I&\mbox{\large 0}\\\
\mbox{\large 0}&\Lambda\\\ \end{bmatrix}.$ (36)

The second class is of the form $[n_{x},n_{y},n_{z}]=[3,2,3]$, where
$\Lambda_{x}$ and $\Lambda_{z}$ both have one $1$ and three $-1$ on the main
diagonal, and $\Lambda_{y}$ has two $1$ and two $-1$ being main diagonal
entries. Then, $\Lambda_{x}$, $\Lambda_{y}$, and $\Lambda_{z}$ are given by

$\Lambda_{x}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\ \mbox{\large 0}&-I\\\
\end{bmatrix},\quad\Lambda_{y}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\
\mbox{\large 0}&\Lambda\\\
\end{bmatrix}\quad\Lambda_{z}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\
\mbox{\large 0}&-I\\\ \end{bmatrix}.$ (37)

The third class is of the form $[n_{x},n_{y},n_{z}]=[3,2,1]$, where
$\Lambda_{x}$ has one $1$ and three $-1$ on the main diagonal, $\Lambda_{y}$
has two $1$ and two $-1$ being main diagonal entries, and $\Lambda_{z}$ takes
three $1$ and one $-1$ on the main diagonal. Then, $\Lambda_{x}$,
$\Lambda_{y}$, and $\Lambda_{z}$ are given by

$\Lambda_{x}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\ \mbox{\large 0}&-I\\\
\end{bmatrix},\quad\Lambda_{y}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\
\mbox{\large 0}&\Lambda\\\
\end{bmatrix}\quad\Lambda_{z}=\begin{bmatrix}I&\mbox{\large 0}\\\ \mbox{\large
0}&\Lambda\\\ \end{bmatrix}.$ (38)

The fourth class is of the form $[n_{x},n_{y},n_{z}]=[2,1,2]$, where
$\Lambda_{x}$ and $\Lambda_{z}$ has two $1$ and two $-1$ on the main diagonal,
$\Lambda_{y}$ has three $1$ and one $-1$ being main diagonal entries. Then,
$\Lambda_{x}$, $\Lambda_{y}$, and $\Lambda_{z}$ are given by

$\Lambda_{x}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\ \mbox{\large
0}&\Lambda\\\ \end{bmatrix},\quad\Lambda_{y}=\begin{bmatrix}I&\mbox{\large
0}\\\ \mbox{\large 0}&\Lambda\\\
\end{bmatrix}\quad\Lambda_{z}=\begin{bmatrix}\Lambda&\mbox{\large 0}\\\
\mbox{\large 0}&\Lambda\\\ \end{bmatrix}.$ (39)

## Discussion

In this paper, we investigate the contextuality of several types of infinite
one-dimensional translation-invariant local quantum Hamiltonians. We found
that it is very likely that all quantum Hamiltonians with nearest-neighbor
only interactions and two dichotomic observables per site admit local hidden
variable models. Violation of contextuality witnesses are only possible when
we either increase the interaction distance to include next-to-nearest
neighbor terms or have three dichotomic observables per site. In the former
case, we identified several Hamiltonians with the lowest possible ground state
energy density in quantum theory. In the latter case, we give strong evidence
that contextuality is only present if the dimension of local observables is
greater than 2, which excludes the usual Heisenberg-type models where local
observables are Pauli matrices.

States and measurements which exhibit the strongest violations of Bell
inequalities are essential ingredients in device-independent certifications
and self-testing [26, 36]. So far the possibility of self-testing in quantum
many-body systems has not been thoroughly established, due to a lack of tools
to certify the strongest violation of Bell inequalities or contextuality
witnesses, without having to solve the corresponding quantum model
analytically. Our results pave the way for self-testing quantum many-body
systems in the thermodynamic limit.

The ground states of our models are computed using uMPS, and they are global
approximations of the true ground state of the corresponding quantum models.
However, in applications such as quantum simulation, we will only have access
to local approximations of the ground state. Moreover, the qualities we are
interested in, such as the ground state energy density and the expectation
values of local observables all depended on the accuracy of the local
description. Finding locally accurate approximations of properties of one-
dimensional local quantum Hamiltonians has yielded many interesting results
[24, 23, 19, 18, 9]. However, most of these results assume the models to have
nearest-neighbor interactions. As we can see from our results, the models with
next-to-nearest neighbor interactions are surprisingly the most interesting in
terms of contextuality.

In two dimensions, very little is known about the contextuality of
translation-invariant local Hamiltonians. We know that when the number of
inputs and outputs is unrestricted, the set of local hidden variable models
becomes non-semi-algebraic and eventually characterization of the set is
impossible [44]. Properties of 2D classical and quantum models differ so
markedly from their 1D counterparts that most intuitions and tools we gained
in 1D break down. However, in 2D a powerful mathematical tool, tiling, has
been repeatedly employed to solve question about computability and complexity
of classical and quantum models [8, 14, 44, 20]. The number of tiles in an
aperiodic tiling would correspond to the number of states in a local hidden
variable model, so it would be interesting to explore the connection between
tiling and contextuality.

## Methods

We describe an extension of the algorithm used to minimize the ground state
energy density to include the most general quantum measurements: positive
operator-valued measure (POVM) measurements. The extended algorithm is used to
minimize 232-type Hamiltonians when the dimension of local observables is 2.
These local observables $\\{\sigma_{a}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{C}^{2}\to\mathbb{C}^{2},a\in X\\}$ are constructed from POVM
elements $M_{a0},\;M_{a1}$. In a gradient descent algorithm, at iteration $k$
the current gradient is subtracted from the parameters, which may take the
local observables out of the space of POVMs. To correct this issue we project
the local observables after the gradient has been subtracted onto the closest
POVM found via semidefnite programming:

$\begin{split}min\quad&\|\tilde{\sigma}_{a}-\sigma_{a}\|_{2}\\\
s.t.\quad&\sigma_{a}=M_{a0}-M_{a1}\\\ &M_{a0}+M_{a1}=I\\\ &M_{a0}\succeq
0,\;M_{a1}\succeq 0.\end{split}$ (40)

Here, $\tilde{\sigma}_{a}=\tilde{\sigma}_{a}(k+1)$ and
$\sigma_{a}=\sigma_{a}(k+1)$. The parameters $W(k)$ are complex decision
variables for the SDP, whose value at each iteration will be given by the
solver.

Even though the extended algorithm based on projected gradient descent works
in principle, we have encountered a number of numerical issues which require
additional tweaks. The main issue affecting convergence is that it takes many
iterations to traverse a nearly flat region in the parameter space. It is one
of the most common problems affecting the performance of gradient descent
algorithms, and it is very common to encounter such regions in our
Hamiltonians. We use a well-known remedy, using momentum to speed up the
traversal of nearly flat regions. At each iteration $k$, the parameters $W(k)$
defining the local observables $\\{\sigma_{a}(k)|a\in X\\}$ are updated by

$\displaystyle V(k+1)$ $\displaystyle=\eta\cdot
V(k)-{\gamma}(k)\cdot\nabla_{W}e(W;k),$ (41) $\displaystyle\tilde{W}(k+1)$
$\displaystyle=W(k)+V(k+1),$ (42)

where $V(k)$ is the momentum and $\eta$ is the decay factor.

Beginning with random initial $W(0)$ and $V(0)=0$, iterating the steps above,
we obtain the a sequence of parameter values $(W(0),W(1),\dots)$ defining a
sequence of local observables $(\sigma_{a}(0),\sigma_{a}(1),\dots)$, each of
which is constructed from POVM elements. If the convergence criterion
$|e(k+1)-e(k)|\leq\epsilon^{*}$ is met at a iteration $k$, then the algorithm
stops and returns the optimal parameters $W^{*}\equiv W(k+1)$. For 100 out of
the 652 Hamiltonianswe tested, even though the random initial parameters are
allowed to be complex, the converged values are all real. Having nonzero
imaginary parts in the local observables meant the ground state energy of the
Hamiltonian will stall at a value higher than the classical bound, often
resulting in 10000 iterations only decreasing the ground state energy
marginally. When this happens, a new set of random initial complex parameters
are generated and the algorithm restarts. When the algorithm converges, the
imaginary parts of all the parameters are smaller than $10^{-10}$.

Acknowledgments—This work is supported by the National Key R&D Program of
China (No.2018YFA0306703, No.2021YFE0113100). Z.W. is supported by the Sichuan
Innovative Research Team Support Fund (No. 2021JDTD0028). G.Y. is supported by
the National Natural Science Foundation of China (No.62172075).

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##  Appendix A Couplings and quantum values of the 322-type Hamiltonians

Out of the 2102 equivalent classes of 322-type Hamiltonians, our algorithm
found 63 which can violate the classical bound. Table 5 lists their couplings
and the corresponding classical bounds (${\cal{L}}$). Table 6 gives lowest the
ground state energy density (${\cal{Q}}$) with $d$-dimensional local
observables. The optimal parameters to reconstruct these local observables
($w$), the numbers of $-1$ in the eigenvalues of each local observable
($[n_{x},n_{y}]$), the bond dimension of the uMPS describing the ground state
($D$), and the ${\operatorname{LTI_{5}-NPA_{1}}}$ lower bound are also given.

Table 5:  Couplings and classical bounds for 63 322-type Hamiltonians. No. | ${\cal{L}}$ | $J_{x}$ | $J_{y}$ | $J_{xx}^{AB}$ | $J_{xy}^{AB}$ | $J_{yx}^{AB}$ | $J_{yy}^{AB}$ | $J_{xx}^{AC}$ | $J_{xy}^{AC}$ | $J_{yx}^{AC}$ | $J_{yy}^{AC}$
---|---|---|---|---|---|---|---|---|---|---|---
1 | -6 | -6 | 0 | 2 | 3 | 3 | -2 | 3 | -1 | -1 | 1
2 | -6 | -4 | 2 | 2 | 2 | 2 | -4 | 1 | -1 | -1 | 3
3 | -3 | -3 | 1 | 1 | 1 | 1 | -1 | 1 | 0 | -1 | 1
4 | -4 | -2 | -2 | -2 | 1 | -1 | -2 | 1 | 0 | 2 | 1
5 | -8 | -11 | 1 | 5 | 2 | 2 | -1 | 4 | -1 | -2 | 1
6 | -4 | -3 | -3 | 2 | 2 | -1 | 2 | 1 | 1 | -1 | 0
7 | -5 | -3 | -3 | 2 | 2 | 2 | -3 | 1 | 0 | -1 | 2
8 | -4 | -2 | -4 | -2 | 2 | 2 | 2 | 1 | 0 | 0 | 1
9 | -4 | 0 | -4 | 2 | 2 | -2 | 2 | 0 | 1 | -1 | 0
10 | -5 | -2 | 2 | 2 | -2 | -2 | -4 | 1 | 1 | 1 | 2
11 | -3 | -1 | 1 | 2 | -1 | -1 | -2 | 1 | 1 | 0 | 1
12 | -6 | -2 | -6 | -2 | 4 | 3 | 3 | 1 | -1 | 0 | 2
13 | -11 | 6 | -8 | 4 | 2 | -8 | 5 | -2 | 1 | -7 | 0
14 | -16 | 8 | -12 | 5 | 2 | -12 | 7 | -4 | 1 | -11 | 0
15 | -6 | 3 | -5 | 2 | 1 | -4 | 3 | -1 | 1 | -3 | 1
16 | -7 | -6 | 4 | 4 | 2 | -4 | 3 | 0 | 1 | -3 | 0
17 | -14 | -12 | 6 | 7 | 2 | -10 | 4 | 0 | 1 | -9 | -3
18 | -9 | -8 | 4 | 5 | 2 | -6 | 3 | 0 | 1 | -5 | -1
19 | -11 | -4 | -12 | -4 | 6 | 6 | 6 | 1 | -1 | -1 | 4
20 | -4 | -1 | -5 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | 2
21 | -4 | -3 | 1 | 3 | -2 | -2 | -1 | 2 | 1 | 0 | 1
22 | -5 | -6 | 0 | 2 | 2 | 1 | -2 | 2 | 0 | -1 | 1
23 | -8 | -8 | 2 | 4 | -4 | -4 | -4 | 3 | 1 | 1 | 1
24 | -7 | -5 | -5 | 2 | 3 | 2 | -4 | 1 | 1 | -1 | 3
25 | -8 | -3 | -7 | -2 | 4 | 5 | 4 | 2 | 0 | -2 | 3
26 | -4 | 2 | -4 | 1 | 1 | -3 | 2 | -1 | 0 | -2 | 0
27 | -3 | -2 | -2 | 2 | 2 | -1 | 1 | 0 | 1 | 0 | 0
28 | -3 | -2 | 0 | 1 | 2 | 1 | -2 | 1 | -1 | 0 | 1
29 | -4 | -4 | 2 | 2 | -1 | -2 | -2 | 1 | 0 | -1 | 1
30 | -4 | -1 | -5 | 1 | 2 | -1 | 3 | 0 | 1 | -1 | 1
31 | -8 | -6 | -6 | 3 | 6 | -2 | 3 | -1 | 3 | 1 | -1
32 | -7 | -3 | 3 | -5 | 3 | 2 | 2 | 4 | -1 | -1 | 1
33 | -3 | -1 | -3 | -1 | 2 | 1 | 2 | 0 | 0 | 0 | 1
34 | -6 | -4 | -4 | 2 | 5 | -1 | 2 | -1 | 4 | 0 | -1
35 | -6 | -4 | -6 | -3 | 3 | 2 | 2 | 2 | 1 | 0 | 1
36 | -6 | -4 | -4 | 2 | 5 | -1 | 2 | -1 | 3 | 1 | -1
37 | -6 | -4 | 4 | 3 | 2 | -3 | 4 | 0 | 1 | -2 | 1
38 | -4 | -4 | 0 | 1 | 2 | 2 | -1 | 2 | -1 | -1 | 0
39 | -5 | -3 | 1 | -4 | 2 | 1 | 1 | 3 | -1 | -1 | 0
40 | -4 | -1 | 1 | -3 | 2 | 2 | 1 | 2 | 0 | -1 | 1
41 | -15 | 5 | -3 | 4 | 9 | 7 | -9 | 5 | -5 | 2 | 6
42 | -5 | -1 | -5 | -1 | 3 | 1 | 4 | 0 | 0 | -1 | 3
43 | -6 | 2 | 0 | -5 | 3 | 3 | 1 | 3 | -1 | -1 | 1
44 | -11 | -3 | -11 | -3 | 7 | 5 | 6 | 2 | -2 | -1 | 5
45 | -6 | -2 | -6 | -2 | 3 | 3 | 4 | 1 | -1 | -1 | 3
46 | -6 | -4 | 0 | -5 | 2 | 2 | 1 | 4 | -1 | -1 | 0
47 | -4 | -2 | 2 | 1 | 1 | 1 | -3 | 0 | -1 | -1 | 2
48 | -10 | -4 | 4 | -5 | 3 | -2 | -6 | 2 | -6 | 1 | 3
49 | -8 | -4 | 2 | -6 | 3 | 3 | 2 | 5 | -1 | -1 | 1
50 | -7 | -6 | 2 | 5 | 1 | -5 | 1 | 1 | 0 | -4 | -2
51 | -10 | 4 | -8 | 3 | 2 | -7 | 6 | -2 | 1 | -6 | 1
52 | -4 | -2 | 2 | 1 | 0 | -1 | 2 | 2 | 1 | -2 | 1
53 | -13 | 12 | 4 | 7 | 8 | 6 | -3 | 6 | -3 | -1 | 3
54 | -6 | -6 | 2 | 5 | 2 | -3 | 2 | 1 | 1 | -2 | 0
55 | -8 | 5 | -3 | 1 | -3 | -5 | -1 | 4 | 1 | -4 | 1
56 | -7 | -7 | 1 | 6 | 1 | -4 | 1 | 2 | 1 | -3 | -1
57 | -5 | -1 | 5 | 0 | 0 | 0 | 4 | -1 | -1 | -2 | 3
58 | -7 | -6 | 2 | 5 | 0 | -2 | -1 | 4 | 1 | -3 | 1
59 | -8 | 6 | -2 | 5 | -1 | -1 | 1 | 4 | -2 | -4 | -2
60 | -8 | -6 | -8 | -4 | 3 | 3 | 2 | 3 | 1 | 1 | 1
61 | -5 | -3 | 3 | -3 | -1 | -2 | 2 | 2 | 1 | -1 | 1
62 | -7 | 6 | -2 | 6 | 0 | -1 | -1 | 3 | -1 | -4 | -1
63 | -5 | -3 | -3 | 0 | 3 | 1 | -1 | 1 | -2 | 1 | 2
Table 6:  Ground state energy density, optimal parameters, type of parameterized observables and bond dimension of 63 322-type Hamiltonians under the different dimensions of local observables. No. | ${\cal{L}}$ | $d=2$ | $d=3$ | $d=4$ | $d=5$ | ${\operatorname{LTI_{5}-NPA_{1}}}$
---|---|---|---|---|---|---
${\cal{Q}}_{2}$ | $w_{1}$ | $[n_{x},n_{y}]$ | $D$ | ${\cal{Q}}_{3}$ | $w_{1}$ | $[n_{x},n_{y}]$ | $D$ | ${\cal{Q}}_{4}$ | $w_{1}$ | $w_{2}$ | $[n_{x},n_{y}]$ | $D$ | ${\cal{Q}}_{5}$ | $w_{1}$ | $w_{2}$ | $[n_{x},n_{y}]$ | $D$
1 | -6 | -6.08108 | 1.2472 | $[1,1]$ | 7 | -6.32747 | 0.7811 | $[1,2]$ | 5 |  |  |  |  |  |  |  |  |  |  | -6.32747
2 | -6 | -6.10943 | 1.2224 | $[1,1]$ | 7 | -6.33712 | 0.9117 | $[1,2]$ | 5 |  |  |  |  |  |  |  |  |  |  | -6.33712
3 | -3 | -3.04150 | 1.2508 | $[1,1]$ | 8 | -3.20711 | 0.7852 | $[1,2]$ | 5 |  |  |  |  |  |  |  |  |  |  | -3.20711
4 | -4 | – | – | – | – | -4.14623 | 0.7925 | $[1,1]$ | 5 |  |  |  |  |  |  |  |  |  |  | -4.14623
5 | -8 | – | – | – | – | -8.12123 | 0.6735 | $[1,2]$ | 5 |  |  |  |  |  |  |  |  |  |  | -8.12123
6 | -4 | – | – | – | – | -4.02415 | 0.3641 | $[1,1]$ | 16 | -4.10310 | -0.0000 | 0.7031 | $[2,2]$ | 6 |  |  |  |  |  | -4.10310
7 | -5 | – | – | – | – | -5.09951 | 1.1158 | $[1,1]$ | 12 | -5.09951 | 0.0000 | -1.1156 | $[2,2]$ | 12 | -5.29852 | 1.5704 | 0.8509 | $[2,2]$ | 6 | -5.29852
8 | -4 | – | – | – |  | -4.18655 | 0.9483 | $[1,1]$ | 14 | -4.18655 | -0.9483 | 0.0000 | $[2,2]$ | 14 | -4.33137 | 1.5699 | 0.7854 | $[2,2]$ | 6 | -4.33137
9 | -4 | – | – | – | – | -4.11581 | 1.1034 | $[1,1]$ | 14 | -4.11581 | 0.4671 | 1.5708 | $[2,2]$ | 15 | -4.41421 | 0.7854 | 1.5702 | $[2,2]$ | 5 | -4.41421
10 | -5 | – | – | – | – | -5.07058 | 0.3947 | $[1,2]$ | 12 | -5.07058 | 0.3948 | 1.5708 | $[2,2]$ | 12 | -5.26969 | 1.5708 | 0.6590 | $[3,3]$ | 6 | -5.26969
11 | -3 | – | – | – | – | -3.00628 | 0.3540 | $[1,2]$ | 8 | -3.00628 | 0.3533 | 1.5708 | $[2,2]$ | 8 | -3.11696 | 0.7857 | 1.5704 | $[3,3]$ | 6 | -3.11698
12 | -6 | – | – | – | – | -6.06459 | 0.4885 | $[1,1]$ | 14 | -6.20259 | 0.0000 | 0.7000 | $[2,2]$ | 6 |  |  |  |  |  | -6.20261
13 | -11 | – | – | – | – | – | – | – | – | -11.19018 | -1.5708 | -2.2493 | $[2,2]$ | 6 |  |  |  |  |  | -11.19036
14 | -16 | – | – | – | – | – | – | – | – | -16.17635 | 0.8977 | 1.5708 | $[2,2]$ | 6 |  |  |  |  |  | -16.17654
15 | -6 | – | – | – | – | – | – | – | – | -6.09323 | 1.5708 | 0.8916 | $[2,2]$ | 6 |  |  |  |  |  | -6.09347
16 | -7 | – | – | – | – | – | – | – | – | -7.21677 | 1.5708 | 0.8683 | $[2,2]$ | 6 |  |  |  |  |  | -7.21747
17 | -14 | – | – | – | – | – | – | – | – | -14.18216 | -1.5708 | 2.2462 | $[2,2]$ | 6 |  |  |  |  |  | -14.18291
18 | -9 | – | – | – | – | – | – | – | – | -9.20253 | 1.5708 | 0.8852 | $[2,2]$ | 6 |  |  |  |  |  | -9.20406
19 | -11 | – | – | – | – | -11.40483 | 0.6551 | $[1,1]$ | 14 | -11.47642 | 0.0000 | -0.7229 | $[2,2]$ | 6 |  |  |  |  |  | -11.47839
20 | -4 | -4.00682 | 1.2170 | $[1,1]$ | 4 | -4.02137 | 0.4173 | $[1,1]$ | 16 | -4.12887 | 1.5708 | 0.7030 | $[2,2]$ | 5 |  |  |  |  |  | -4.13251
21 | -4 | – | – | – | – | – | – | – | – | -4.06491 | 1.5708 | 0.9817 | $[2,2]$ | 6 |  |  |  |  |  | -4.06890
22 | -5 | -5.00062 | 1.3959 | $[1,1]$ | 4 | -5.12457 | 0.7855 | $[1,2]$ | 5 | -5.12457 | 1.5708 | 0.7854 | $[2,2]$ | 5 |  |  |  |  |  | -5.12887
23 | -8 | -8.01053 | 0.2887 | $[1,1]$ | 5 | -8.13694 | 0.6619 | $[1,1]$ | 5 | -8.13694 | 0.0000 | 0.6618 | $[2,2]$ | 5 |  |  |  |  |  | -8.14126
24 | -7 | – | – | – |  | -7.11178 | 1.1000 | $[1,1]$ | 12 | -7.11178 | 0.0000 | 1.1002 | $[2,2]$ | 12 | -7.25367 | 0.8707 | 1.5704 | $[2,2]$ | 6 | -7.25827
25 | -8 | – | – | – | – | -8.00096 | 0.1630 | $[1,1]$ | 16 | -8.14914 | 0.0000 | 0.6379 | $[2,2]$ | 6 |  |  |  |  |  | -8.15428
26 | -4 | – | – | – | – | – | – | – | – | -4.07800 | 1.5708 | 1.0105 | $[2,2]$ | 6 | -4.14354 | 0.8877 | 1.5707 | $[2,2]$ | 5 | -4.14877
27 | -3 | – | – | – | – | -3.11927 | 0.8353 | $[1,1]$ | 8 | -3.11932 | 0.0000 | 0.7027 | $[2,2]$ | 6 | -3.14915 | 0.7027 | 1.5694 | $[2,2]$ | 5 | -3.15470
28 | -3 | -3.04030 | 1.2555 | $[1,1]$ | 7 | -3.14915 | 0.8685 | $[1,2]$ | 5 | -3.14915 | 0.8682 | 1.5708 | $[2,2]$ | 5 |  |  |  |  |  | -3.15470
29 | -4 | – | – | – | – | – | – | – | – | -4.00935 | 1.3742 | 0.3202 | $[2,2]$ | 12 | -4.09966 | 1.5708 | 0.7860 | $[3,3]$ | 7 | -4.10571
30 | -4 | – | – | – | – | -4.03262 | 0.3966 | $[1,1]$ | 19 | -4.11218 | 0.0000 | 0.7054 | $[2,2]$ | 6 | -4.14023 | 0.7054 | 1.5699 | $[2,2]$ | 5 | -4.14877
31 | -8 | – | – | – | – | – | – | – | – | -8.22037 | 0.0000 | 0.6827 | $[2,2]$ | 6 |  |  |  |  |  | -8.23071
32 | -7 | -7.08653 | 1.2383 |  | 6 | -7.25928 | 0.9558 | $[1,2]$ | 5 | -7.25928 | 0.9558 | 1.5708 | $[2,2]$ | 5 |  |  |  |  |  | -7.27038
33 | -3 | – | – | – | – | -3.11932 | 0.7018 | $[1,1]$ | 6 | -3.11932 | 0.0000 | 0.7026 | $[2,2]$ | 6 |  |  |  |  |  | -3.13058
34 | -6 | – | – | – | – | – | – | – | – | -6.03951 | 0.0000 | 0.4637 | $[2,2]$ | 6 |  |  |  |  |  | -6.05216
35 | -6 | – | – | – | – | -6.19072 | 0.9694 | $[1,1]$ | 14 | -6.19072 | 0.0000 | 0.9700 | $[2,2]$ | 14 | -6.26479 | 0.8478 | 1.5702 | $[2,2]$ | 6 | -6.27801
36 | -6 | – | – | – | – | – | – | – | – | -6.03951 | 0.0000 | 0.4637 | $[2,2]$ | 6 |  |  |  |  |  | -6.05327
37 | -6 | – | – | – | – | – | – | – | – | -6.17482 | 1.5708 | 0.8827 | $[2,2]$ | 6 |  |  |  |  |  | -6.18885
38 | -4 | -4.03281 | 1.2745 | $[1,1]$ | 8 | -4.14623 | 0.7767 | $[1,2]$ | 5 | -4.14623 | 1.5708 | 0.7775 | $[2,2]$ | 5 |  |  |  |  |  | -4.16445
39 | -5 | -5.06166 | 1.9574 | $[1,1]$ | 7 | -5.12466 | 0.9504 | $[1,2]$ | 7 | -5.12466 | 1.5708 | 0.9503 | $[2,2]$ | 7 |  |  |  |  |  | -5.14311
40 | -4 | -4.02193 | 1.3015 | $[1,1]$ | 5 | -4.06591 | 1.0457 | $[1,2]$ | 5 | -4.06591 | 1.0459 | 1.5708 | $[2,2]$ | 5 |  |  |  |  |  | -4.08498
41 | -15 | -15.01150 | 1.3600 | $[1,1]$ | 5 | -15.01150 | 1.7814 | $[1,1]$ | 5 | -15.13717 | 4.1651 | 1.5708 | $[2,2]$ | 5 |  |  |  |  |  | -15.15803
42 | -5 | – | – | – | – | – | – | – | – | -5.01755 | 0.7989 | 0.0884 | $[2,2]$ | 16 | -5.04311 | 0.2159 | 0.7668 | $[2,2]$ | 12 | -5.06400
43 | -6 | -6.01649 | 1.8170 | $[1,1]$ | 5 | -6.01649 | 1.8170 | $[1,1]$ | 6 | -6.04536 | 1.1166 | 1.5708 | $[2,2]$ | 8 |  |  |  |  |  | -6.06803
44 | -11 | – | – | – | – | -11.00119 | 0.1638 | $[1,1]$ | 12 | -11.15383 | 0.0000 | 0.5575 | $[2,2]$ | 6 |  |  |  |  |  | -11.17826
45 | -6 | – | – | – | – | -6.18037 | 0.6128 | $[1,1]$ | 16 | -6.22875 | 0.0000 | 0.7013 | $[2,2]$ | 6 |  |  |  |  |  | -6.25396
46 | -6 | -6.06990 | 1.9751 | $[1,1]$ | 6 | -6.12318 | 0.9210 | $[1,2]$ | 7 | -6.12318 | 1.5708 | 0.9210 | $[2,2]$ | 7 |  |  |  |  |  | -6.14907
47 | -4 | -4.07352 | 1.1798 | $[1,1]$ | 7 | -4.15468 | 0.9590 | $[1,2]$ | 5 | -4.15469 | 1.5708 | 0.9587 | $[2,2]$ | 5 |  |  |  |  |  | -4.18142
48 | -10 | – | – | – | – | – | – | – | – | -10.14203 | 0.9241 | -1.5708 | $[2,2]$ | 5 |  |  |  |  |  | -10.17428
49 | -8 | -8.07815 | 1.2431 | $[1,1]$ | 7 | -8.18189 | 1.0153 | $[1,2]$ | 5 | -8.18189 | 1.5708 | 1.0152 | $[2,2]$ | 5 |  |  |  |  |  | -8.21614
50 | -7 | – | – | – | – | – | – | – | – | -7.00917 | 0.7748 | -1.6163 | $[2,2]$ | 12 |  |  |  |  |  | -7.04393
51 | -10 | – | – | – | – | – | – | – | – | -10.10510 | -0.9808 | 1.5708 | $[2,2]$ | 6 |  |  |  |  |  | -10.14126
52 | -4 | – | – | – | – | – | – | – | – | -4.08523 | 1.5708 | 0.8681 | $[2,2]$ | 9 |  |  |  |  |  | -4.12479
53 | -13 | – | – | – | – | – | – | – | – | -13.10714 | 0.0000 | -0.5039 | $[2,2]$ | 7 |  |  |  |  |  | -13.14828
54 | -6 | – | – | – | – | – | – | – | – | -6.16786 | 0.8899 | 1.5708 | $[2,2]$ | 6 |  |  |  |  |  | -6.21564
55 | -8 | – | – | – | – | – | – | – | – | -8.00371 | 2.3012 | -1.5949 | $[2,2]$ | 16 |  |  |  |  |  | -8.05192
56 | -7 | – | – | – | – | – | – | – | – | -7.01064 | 0.8015 | 1.6238 | $[2,2]$ | 12 |  |  |  |  |  | -7.05935
57 | -5 | – | – | – | – | – | – | – | – | -5.05572 | 1.5708 | 1.0099 | $[2,2]$ | 9 |  |  |  |  |  | -5.10699
58 | -7 | – | – | – | – | – | – | – | – | -7.00144 | 0.3372 | 1.5603 | $[2,2]$ | 11 |  |  |  |  |  | -7.05714
59 | -8 | – | – | – | – | – | – | – | – | -8.17045 | -1.5708 | 0.8676 | $[2,2]$ | 9 | -8.17045 | 0.8686 | 1.5708 | $[2,2]$ | 9 | -8.24220
60 | -8 | – | – | – | – | -8.22013 | 0.9539 | $[1,1]$ | 5 | -8.22013 | 0.0000 | 0.9539 | $[2,2]$ | 5 |  |  |  |  |  | -8.29902
61 | -5 | – | – | – | – | -5.00822 | 0.3766 | $[1,2]$ | 11 | -5.00822 | 0.3774 | 1.5708 | $[2,2]$ | 11 |  |  |  |  |  | -5.09824
62 | -7 | – | – | – | – | – | – | – | – | -7.00638 | 2.1879 | 1.5295 | $[2,2]$ | 24 |  |  |  |  |  | -7.10456
63 | -5 | – | – | – | – | -5.00639 | 1.3547 | $[1,1]$ | 8 | -5.00639 | 1.3556 | 0.0000 | $[2,2]$ | 8 |  |  |  |  |  | -5.13037

##  Appendix B Couplings and parameters of the 232-type Hamiltonians

Table 7 gives the couplings, classical bounds (${\cal{L}}$) of five 232-type
TI Hamiltonians presented in the main text. The parameters defining local
observables and the numbers of $-1$ in the eigenvalues of each local
observables ($[n_{x},n_{y},n_{z}]$) are presented in Table 8 when $d=3$ and in
Table 9 when $d=4$ .

Table 7:  Couplings for 5 232-type TI Hamiltonians presented in Table 4. No. | ${\cal{L}}$ | $J_{x}$ | $J_{y}$ | $J_{z}$ | $J_{xx}$ | $J_{xy}$ | $J_{xz}$ | $J_{yx}$ | $J_{yy}$ | $J_{yz}$ | $J_{zx}$ | $J_{zy}$ | $J_{zz}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | -9 | 3 | -2 | 1 | 2 | -2 | 5 | -1 | 1 | 3 | 1 | -2 | -2
2 | -4 | 2 | 1 | 1 | 2 | 2 | 1 | 0 | -1 | 0 | -1 | 1 | 0
3 | -5 | 2 | -1 | 1 | 1 | -1 | 2 | -2 | 0 | 2 | -1 | 0 | 0
4 | -4 | 1 | -1 | 0 | 1 | 1 | 2 | 0 | 1 | -1 | 0 | 1 | -1
5 | -2 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | 0
Table 8:  Parameters of 5 232-type TI Hamiltonians under the bond dimension $D=10$ and the dimension of local observables $d=3$. No. | ${\cal{L}}$ | $d=3$
---|---|---
$w_{1}$ | $w_{2}$ | $w_{3}$ | $w_{4}$ | $w_{5}$ | $w_{6}$ | $w_{7}$ | $w_{8}$ | $w_{9}$ | $[n_{x},n_{y},n_{z}]$
1 | -9 | 0.3392 | 0.1055 | 0.4115 | 0.4133 | 0.3652 | 0.9916 | 0.4810 | -0.0351 | -0.3163 | $[2,1,1]$
2 | -4 | 0.5943 | 0.0469 | 0.2011 | 0.2756 | 0.4268 | 0.1822 | 0.0277 | 0.7213 | 0.1413 | $[2,1,2]$
3 | -5 | 1.0061 | -0.0081 | 0.2303 | 0.3148 | 0.0616 | 0.5666 | -0.2498 | 0.2271 | 0.0076 | $[2,1,2]$
4 | -4 | 0.1303 | -0.0032 | 0.5139 | 0.2574 | 0.4723 | 0.6582 | 0.3285 | -0.1727 | 0.2820 | $[2,1,1]$
5 | -2 | 0.4899 | -0.1481 | 0.1905 | 0.4931 | 0.4919 | 0.3638 | -0.1067 | 0.7160 | 0.2088 | $[2,1,2]$
Table 9:  Parameters of 5 232-type TI Hamiltonians under the bond dimension $D=10$ and the dimension of local observables $d=4$. No. | ${\cal{L}}$ | $d=4$
---|---|---
$w_{1}$ | $w_{2}$ | $w_{3}$ | $w_{4}$ | $w_{5}$ | $w_{6}$ | $w_{7}$ | $w_{8}$ | $w_{9}$ | $w_{10}$ | $w_{11}$ | $w_{12}$ | $w_{13}$ | $w_{14}$ | $w_{15}$ | $w_{16}$ | $w_{17}$ | $w_{18}$ | $[n_{x},n_{y},n_{z}]$
1 | -9 | -0.0270 | 0.2185 | 0.0934 | 0.2068 | 0.3829 | 0.3010 | 0.3126 | 0.3287 | 0.2716 | 0.1611 | -0.2841 | -0.4393 | 0.1235 | 0.1030 | 0.1933 | 0.1480 | 0.6689 | 1.0601 | $[3,1,1]$
2 | -4 | 0.4490 | 0.5450 | 0.4828 | 0.0387 | 0.1335 | 0.1225 | 0.2003 | 0.2535 | 0.2878 | 0.0661 | 0.2800 | 0.1061 | 0.0102 | -0.0804 | 0.0821 | 0.0320 | 0.3449 | 0.0535 | $[3,2,3]$
3 | -5 | 0.5565 | 1.2916 | 0.8706 | 0.1371 | 0.1162 | 0.9160 | 0.4551 | 0.5068 | 0.1782 | 0.4305 | 0.8979 | 0.1097 | 0.2928 | -0.4064 | 0.1088 | 0.7717 | 0.6469 | 0.9466 | $[3,2,3]$
4 | -4 | -0.2912 | 0.2321 | 0.0211 | 0.0911 | 0.2954 | 0.0986 | 0.6905 | 0.3049 | 0.3104 | 0.2755 | 0.1093 | 0.1728 | 0.0995 | 0.4044 | -0.0148 | 0.1701 | 0.2056 | 0.4232 | $[3,2,1]$
5 | -2 | 0.4998 | 0.0447 | -0.3366 | -0.0197 | 0.3869 | -0.0971 | 0.2528 | 0.2196 | 0.5018 | 0.0579 | 0.1594 | 0.2374 | 0.0932 | 0.1598 | 0.5637 | 0.5072 | 0.2408 | 0.2743 | $[2,1,2]$