Abstract
In recent years there has been a growing interest in treating manybody systems as Bell scenarios, where lattice sites play the role of distant parties and only nearneighbor statistics are accessible. We investigate contextuality arising from three Bell scenarios in infinite, translationinvariant 1D models: nearestneighbor with two dichotomic observables per site; nearest and nexttonearest neighbor with two dichotomic observables per site, and nearestneighbor with three dichotomic observables per site. For the first scenario, we give strong evidence that it cannot exhibit contextuality, not even in nonsignaling physical theories beyond quantum mechanics. For the second one, we identify several lowdimensional models that reach the ultimate quantum limits, paving the way for selftesting ground states of quantum manybody 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 manybody quantum system, correlations appear as one of the most common manifestations of the quantum nature of the system Fig. 1. 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^{1,2}. While experiments exploiting entanglement to teleport photons date back to the 1990s^{3}, nonlocality passed the most stringent experimental tests only in 2015^{4,5,6}. One of the key assumptions in any nonlocality experiment is keeping the different parties spacelike separated. In a manybody quantum system, however, this assumption may be too formidable to be overcome.
In recent years, the exploration of contextuality in quantum manybody systems has been a fruitful endeavor. Contextuality witnesses adapted from Bell inequalities have been tested in Bose–Einstein condensates^{7,8}. Translation and permutation symmetry allowed the full characterization of contextuality witnesses in manybody systems through bipartite correlators only^{9,10,11,12} (see ref. ^{13} 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 manybody systems^{9}, we investigate under which conditions the nnearest neighbor statistics of a translationinvariant 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 in the number of measurement settings available to each party and the size of the nearneighbor marginals considered.
Our results show that some a priori promising Bell scenarios are unlikely to show any form of contextuality, even if we allow greaterthanquantum 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 ref. ^{9}, 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 intermediatescale 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 klocal Bell functional opens the possibility to falsify quantum theory in the manybody 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 θ. By estimating the gradient of the Bell functional with respect to the variables θ, 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^{14}). If it so happened that the final value of the Bell functional was below the quantum limit, then we would have disproven the universal validity of the quantum theory, despite the lack of an alternative theoretical model.
Onedimensional quantum systems are the simplest manybody 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 nearneighbor correlators averaged over the whole chain (the socalled structure factors). As shown in ref. ^{9}, in the regime of large system size, structure factors correspond to the nearneighbor correlators of an infinite, translationinvariant chain. Comprehending Bell nonlocality in large 1D systems hence requires us to characterize nearneighbor correlations in classical, quantum, and supraquantum translationinvariant systems.
The correlations P(a_{1}, …, a_{n}∣x_{1}, …, x_{n}) generated by n spacelike separated classical systems with (classical) inputs x_{i}, i ∈ 1, …, n and outputs a_{i}, i ∈ 1, …, n admit a decomposition of the form:
where λ is a set of hidden variables with probability distribution P(λ). The distributions P(λ) and \({\{P({a}_{i} {x}_{i},\lambda )\}}_{i = 1}^{n}\) are hence a local hidden variable model for the observed correlations P(a_{1}, …, a_{n}∣x_{1}, …, x_{n}). In Bell tests, different parties are required to be spacelike separated, which can be seen as the physical realization of the independence of probabilities in Eq. (1). However, in a manybody quantum system, such a requirement is too formidable to overcome. As a result, when we assume that Eq. (1) holds in a quantum manybody system, what we are actually testing is contextuality^{15,16}. The connection between contextuality and Bell nonlocality had been known since the 1970s. Every Bell inequality can be regarded as a contextuality witness; the other direction is less systematic^{17}. The role of contextuality, especially of the Kochen–Specker type, in quantum computation has been actively investigated in recent years (for a review see ref. ^{16}). It can be shown to be the source of the quantum advantage in several scenarios in quantum computation^{18,19,20,21}. 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^{22}, the model itself is contextual^{16}.
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} ∈ {1, …, X}, obtaining the result a_{i} ∈ 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 translationinvariant chain can therefore be fully specified by the three natural numbers m, X, ∣A∣. Consequently, in this paper, a Bell scenario where only nearestneighbor correlations are available and each party can conduct two dichotomic observables will be called the 222scenario. Extending the interaction distance to nexttonearest neighbors gives us the 322scenario. Heisenberglike Bell scenarios, with nearestneighbor interactions but three dichotomic observables per site correspond to the 232scenario.
We say that an mpartite distribution P_{1,..,m}(a_{1}, …, a_{m}∣x_{1}, …, x_{m}) is nosignaling^{23} if, for all i ∈ {1, …, m},
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 nosignalling distributions P_{1,..,m}(a_{1}, …, a_{m}∣x_{1}, …, x_{m}) can be shown not to arise out of an infinite nosignalling 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 nosignalling extension if there exists a mapping Q from finite sets \(B\subset {\mathbb{Z}}\) to nosignalling ∣B∣partite measurement statistics Q_{B}(a_{B}∣x_{B}) with the following properties:

1.
\(\begin{array}{l}\mathop{\sum}\limits_{{a}_{B\setminus C}}{Q}_{B}({a}_{B\setminus C},{a}_{C} {x}_{B\setminus C},{x}_{C})=\\ \mathop{\sum}\limits_{{a}_{D\setminus C}}{Q}_{D}({a}_{D\setminus C},{a}_{C} {x}_{D\setminus C},{x}_{C}),\end{array}\)
for all finite sets \(B,C,D\subset {\mathbb{Z}}\), with C ⊂ B, D (compatibility).

2.
Q_{B} = Q_{B+z}, for all \(z\in {\mathbb{Z}}\) (translation invariance).

3.
Q_{1,…,n} = P_{1,…,n} (consistency with observed statistics).
We call TINS the set of all distributions P_{1,..,m}(a_{1}, …, a_{m}∣x_{1}, …, x_{m}) admitting a nosignalling, translationinvariant extension.
The existence of a nosignalling extension is just a prerequisite for the existence of an overall infinite translationinvariant 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 an NS extension Q and there exist distributions \(P(\lambda ),\{{P}_{i}(a x,\lambda ):i\in {\mathbb{Z}}\}\) such that, for all N,
We call TILHV 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 \({{{\mathcal{H}}}}\), measurement operators \({E}_{a x}:{{{\mathcal{H}}}}\to {{{\mathcal{H}}}}\), with \({\sum }_{a}{E}_{a x}={\mathbb{I}}\), and a translationinvariant quantum state ρ on the infinite chain with local Hilbert space \({{{\mathcal{H}}}}\) such that, for all N,
We call TIQ the set of all distributions P_{1,..,m}(a_{1}, …, a_{m}∣x_{1}, …, x_{m}) admitting a TI quantum extension.
In ref. ^{9}, two of us provided a full characterization of the set of mnearest 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 \({\sigma }_{x}^{i}\) with possible values ±1, and specify any nosignaling mnearest neighbor distribution P(a_{1}, …, a_{m}∣x_{1}, …, x_{m}) through the averages of the different products of the observables \({\sigma }_{{x}_{1}}^{1},\ldots ,{\sigma }_{{x}_{m}}^{m}\). For m = 2, in this ‘observable representation’ a facet would take the form
Should the observed oneparticle averages \(\{\langle {\sigma }_{x}^{1}\rangle :x\in X\}\) and nearestneighbor twopoint correlators \(\{\langle {\sigma }_{x}^{1}{\sigma }_{y}^{2}\rangle :x,y\in X\}\) of a TI system violate a facet of the classical (also called ‘local’) polytope, the corresponding manybody system would be shown not to admit a description compatible with classical physics.
The lefthand 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 \(\,{{\mbox{dim}}}\,({{{\mathcal{H}}}})=d\), achieving the Bell value \({{{{\mathcal{Q}}}}}_{d}\) with measurement operators {E_{a∣x}: a, x}. Most statistical models studied in the literature use projective measurements, i.e. {E_{a∣x}: 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 232type 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 operatorvalued 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 lefthand side of expressions of the form (5), we start from the following observation: let \(\{{\sigma }_{x}:{{\mathbb{C}}}^{d}\to {{\mathbb{C}}}^{d}:x\in X\}\) be a set of ddimensional Hermitian operators with spectrum contained in {1,−1}. Then, the minimum value of Eq. (5) over all TI quantum states corresponds to the minimum energyper site of the TI Hamiltonian
Tools from condensed matter physics such as uniform matrix product states (uMPS)^{24} 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 {σ_{a}∣a ∈ X}, each of which can be diagonalized by a unitary matrix U_{a} as
where Λ_{a} is a diagonal matrix with entries ±1. To make this more explicit, we use the vector [n_{x}, n_{y}] to describe a number of −1 in the eigenvalues of σ_{x} and σ_{y}.
We can then use the space of skewHermitian matrices to effectively parameterize each U_{a} as
where S_{a} is skewHermitian. Let {B_{1}, B_{2}, …, B_{n}} be a basis of the vector space of skewHermitian matrices. Here n = d^{2}−d denotes the dimension of the space. Expanding S_{a} in this basis gives
where W_{a} ≡ {w_{a1}, w_{a2}, …, w_{an}} are scalars. Our optimization parameters are therefore {w_{ak}∣a ∈ X; k = 1, …, n}.
Using the method above, observables σ_{a}(a = x, y) in \({{{{\mathcal{H}}}}}_{222}\) can be parameterized as
Consequently, \({{{{\mathcal{H}}}}}_{222}\) is parameterized as \({{{{\mathcal{H}}}}}_{222}({W}_{x},{W}_{y})\).
Using Jordan’s lemma^{2}, the number of real parameters can be reduced when ∣X∣ = 2. For example, applying Jordan’s lemma to \({{{{\mathcal{H}}}}}_{222}\) when d = 4 yields a basis in which both σ_{x} and σ_{y} are blockdiagonal:
where σ_{x,1}, σ_{x,2}, σ_{y,1}, σ_{y,2} are 2 × 2 Hermitian matrices.
We are now ready to present our MPSbased gradient descent method. The method is iterative. For a = x, y, let W_{a}(k) denote the parametrization of observable σ_{a} at the kth 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 energypersite of the Hamiltonian \({{{{\mathcal{H}}}}}_{222}(k)\equiv {{{{\mathcal{H}}}}}_{222}({\sigma }_{a}(k),{\sigma }_{b}(k))\) over the manifold of uMPS. The result e(k) can be computed using, e.g., the timedependent variational principle (TDVP) algorithm^{24,25,26} or the variational uniform matrix product state (VUMPS) algorithm^{24,27}. 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
where \({\sum }_{s,t,u,v}h{(k)}_{st}^{uv}\lefts\right\rangle \left\langle u\right\otimes \leftt\right\rangle \left\langle v\right\) is the local term of \({{{{\mathcal{H}}}}}_{222}(k)\); \({\{{A}^{s}(k)\}}_{s}\subset {{\mathbb{C}}}^{D\times D}\) is the tensor defining the optimal uMPS, and l(k), r(k) are the left and right leading eigenvectors of the transfer matrix \(T(k)=\mathop{\sum }\nolimits_{s = 1}^{d}{\bar{A}}^{s}(k)\otimes {A}^{s}(k)\).
Next, we seek to find observables leading to a Hamiltonian with a smaller energypersite, 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(σ_{x}(W), σ_{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):
Here γ(k) is a scaling parameter, which we take to be of the form \(\gamma (k)=\max ({\gamma }_{0}{\alpha }^{q(k)},{\gamma }_{\min })\), where α ∈ (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 ∥∇e(W; k)∥_{2} < ϵ^{*}, for some desired convergence threshold ϵ^{*}. If the condition holds, we stop the algorithm and return the optimal parameters W^{*} ≡ 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 \({{{{\mathcal{H}}}}}_{222}({W}^{* })\) exhibits contextuality.
To test the algorithm, we apply it to compute the minimum ground state energy densities \({{{{\mathcal{Q}}}}}_{{d}}\) (d = 2, 3, 4) of six 322type TI quantum systems. 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 \({{{{\mathcal{L}}}}}_{322}\) (red line). As the iteration number increases, \({{{{\mathcal{Q}}}}}_{2}\) and \({{{{\mathcal{Q}}}}}_{3}\) decrease approximately linearly and begin to show contextuality. In Fig. 2e and f, \({{{{\mathcal{Q}}}}}_{4}\) oscillates during the first several iterations. As the optimization process continues, \({{{{\mathcal{Q}}}}}_{4}\) also begins to cross \({{{{\mathcal{L}}}}}_{322}\) after. The ground state energy densities of all six models converge to values below their classical bounds within 20 iterations.
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 ε the overall state of the chain. Depending on the context, ε will be a classical probability distribution, a quantum state, or a nosignaling box. Because ε 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: ε_{1} = … = ε_{5}. Moreover, the reduced state of any contiguous subset of elephants should also be equal: ε_{1,…,1+k} = ε_{2,…,2+k}, ∀1 ≤ k ≤ 3. When k = 3, the marginals/reduced states are shown in Fig. 3 as green and red rectangles. For any contiguous subset of ε of length l, the marginals/reduced states are said to be locally translationinvariant (LTI) if
Clearly, LTI is a necessary condition for ε to be TI. For classical probability distributions in 1D, LTI is also sufficient: any LTI marginal can be extended to an infinite TI distribution^{28}. In fact, this property is the key to the characterization of the set TILHV presented in ref. ^{9}.
Unfortunately, LTI is not enough to characterize the nearneighbor density matrices of TI quantum states or even the nearneighbor marginals of TI nosignaling systems. In those scenarios, LTI can be used to relax the set of such marginals, rather than to fully characterize it. Define thus LTI_{n}−NS as the set of boxes admitting an extension to an npartite nosignaling box with local translation invariance. As shown in ref. ^{9}, the distance between any element of the set LTI_{n}−NS and its subset TINS is upper bounded by \(O\left(\frac{1}{n}\right)\).
A straightforward extension to bound TIQ is impossible, as the approximate characterization of general multipartite quantum correlations is an undecidable problem^{29}. 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 Γ 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 Γ_{αβ} is supposed to represent the quantity 〈α^{†}β〉 (see refs. ^{30,31} for details). In order to bound TIQ, we demand the existence of a moment matrix for an npartite Bell scenario and then impose LTI over the said moment matrix. Call LTI_{n}−NPA_{s} the corresponding relaxation.
For any Bell functional, we can thus find a lower bound on its minimal value in TINS and TIQ by, respectively, optimizing over LTI_{n}−NS (with linear programming techniques^{32}) and LTI_{n}−NPA_{s} (with semidefinite programming techniques^{33}). Moreover, one can improve those lower bounds by increasing the values of n, s.
Contextuality in 222type Hamiltonians
The LTILHV polytope for 2 dichotomic observables has 36 facets. Computing their 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
with classical bound −2.
Lower bounding Eq. (15) with 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.
The numerators and denominators in the table form two integer sequences: A027691^{34} and A152948^{35} in The OnLine Encyclopedia of Integer Sequences. Moreover, the displaced inverse of a quadratic function
perfectly fits the sequence of lower bounds in Table 1 (see Fig. 4).
In the limit n → ∞, this function converges to the classical bound −2. In other words, if the solution of the optimization over LTI_{n}−NS satisfies (16) for all n ≥ 3, then no Hamiltonian of the form (15), quantum or otherwise, can possibly violate the classical bound.
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 to search for the quantum Hamiltonian with the lowest ground state energy density, for local observables of dimension 2 ≤ d ≤ 6. For each d, σ_{x} and σ_{y} are parameterized by the method described below, and we find the lowest quantum value \({{{{\mathcal{Q}}}}}_{d}\) among all possible systems is −2. Moreover, the corresponding twobody 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.
To make sense of Table 2, we next explicitly write the parametrization of the ddimensional observables achieving the classical bound. Two 2 × 2 matrices having eigenvalues one 1 and one −1 will repeatedly appear below: Λ is the diagonal matrix with diagonal entries ±1, B(w) is a matrix governed by one parameter {w}:
When d = 2 is assigned to local observables in \({{{{\mathcal{H}}}}}_{222}\), σ_{x} is a diagonal matrix with diagonal entries 1 and −1, i.e., σ_{x} = Λ, and σ_{y} determined by one parameter {w_{1}} has the same parameterized form as B(w), i.e., σ_{y}(w_{1}) = B(w_{1}). In this case, [n_{x}, n_{y}] = [1, 1] and σ_{x}, σ_{y} are of the form
When d = 3 is assigned to local observables in \({{{{\mathcal{H}}}}}_{222}\), σ_{x} is a diagonal matrix with diagonal entries (1,−1,−1). σ_{y} is a block diagonal matrix, where the maindiagonal blocks are one matrix B(w_{1}) and one numerical value −1. Then, [n_{x}, n_{y}] = [2, 2] and σ_{x}, σ_{y} are given by
When d = 4 is assigned to local observables in \({{{{\mathcal{H}}}}}_{222}\), σ_{x} is a diagonal matrix with diagonal entries two 1 and two −1, and σ_{y} is a block diagonal matrix with maindiagonal blocks being two 2 × 2 matrices B(w_{1}) and B(w_{2}). In this case, [n_{x}, n_{y}] = [2, 2] and σ_{x}, σ_{y} are of the forms
When d = 5 is assigned to local observables in \({{{{\mathcal{H}}}}}_{222}\), σ_{x} is a diagonal matrix with diagonal entries two −1 and three 1. σ_{y} is a block diagonal matrix, where the maindiagonal blocks are two 2 × 2 matrices B(w_{1}) and B(w_{2}) and one numerical number 1. Then, [n_{x}, n_{y}] = [2, 2] and σ_{x}, σ_{y} are given by
When d = 6 is assigned to local observables in \({{{{\mathcal{H}}}}}_{222}\), σ_{x} is a diagonal matrix, where three −1 and three 1 are alternately arranged in the diagonal. σ_{y} is a block diagonal matrix with maindiagonal blocks being three 2 × 2 matrices B(w_{1}), B(w_{2}) and B(w_{3}). Then, [n_{x}, n_{y}] = [3, 3] and σ_{x}, σ_{y} have the forms
Contextuality in 322type Hamiltonians
The TILHV polytope for the 322type Hamiltonians has been characterized in ref. ^{9}: it has 32,372 facets which can be sorted into 2102 equivalence classes. The general form of the 322type Hamiltonian is given by
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 σ_{x}, σ_{y} are local observables.
Using our uMPSbased gradient descent algorithm, a total of 63 Hamiltonians exhibit contextuality. The explicit parameterization of observables σ_{x} and σ_{y} is explained at the end of this section. All the contextual Hamiltonians and ground state energy densities are listed in Supplementary Tables 1 and 2, respectively. Among them, we identify some quantum models whose ground state energy density reaches the LTI_{5}−NPA_{1} lower bounds. For all these contextuality witnesses, we have thus identified translationinvariant 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.
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.
In Fig. 5a, \({{{{\mathcal{Q}}}}}_{3}\) 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 \({{{{\mathcal{Q}}}}}_{3}\) surface to the bottom. Besides, the right enlarged subplot of the trajectory demonstrates that ground state energy density eventually converges to the respective LTI_{n}−NPA_{s} lower bound. In Fig. 5b, the leftmost subplot shows the trajectory of the ground state energy density on the 3D \({{{{\mathcal{Q}}}}}_{4}\) surface, the middle 2Dsubplot 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 a function of the parameters defining the local observables for some of the Hamiltonians in Table 3 and Supplementary Table 2 to gauge the robustness of the contextuality violations. The couplings and local observables achieving these values can be found in Supplementary Tables 1 and 2. The first column in Table 3 indicates the position of each model in these Supplementary Tables. Five models for d = 4 and another five models for d = 5 are shown in Figs. 6 and 7, respectively. Note that the first two models in Fig. 6a, b and the first four models in Fig. 7a–d exhibit the strongest contextuality.
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 Figs. 6b, 7b and 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. 7e, squarelike 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 trappedion 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 322type Hamiltonians achieving the minimum quantum values in Table 3. For d = 2 and d = 4, σ_{x}, σ_{y} have the exact same parameterized forms as the observables in the 222type Hamiltonians in Eqs. (18) and (20), respectively.
For d = 3 and d = 5, depending on the number of 1’s and −1’s of each matrix Λ_{a}(a = x, y) in Eq. (7), two different classes of pairs of local observables σ_{x} and σ_{y} are considered. Here, we continue using the notations Λ and B(w) introduced in Eq. (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
The second class is of the form [n_{x}, n_{y}] = [1, 1], with
For local dimension d = 5, the first class of observable pairs is of the form [n_{x}, n_{y}] = [2, 2], with
The second class of observable pairs satisfies [n_{x}, n_{y}] = [3, 3], and σ_{x} and σ_{y} are given by
Contextuality in 232type Hamiltonians
The LTILHV polytope for 3 dichotomic observables has 92,694 facets, which can be classified into 652 equivalent classes^{36}. The general form of this type of Hamiltonian is given by
We consider \({{{{\mathcal{H}}}}}_{232}\) 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 ≥ 3 there are many Hamiltonians that can violate the classical bound. The ground state energy densities of some contextual Hamiltonians are shown in Table 4, see Supplementary Table 3 for the couplings defining the contextuality witnesses. In addition, the parameters specifying the optimal local observables are listed in Supplementary Table 4 for d = 3 and in Supplementary Table 5 for d = 4.
The local observables σ_{x}, σ_{y} and σ_{z} are parameterized using the method presented above: for given Λ_{a}, S_{a} the local observable σ_{a} can be written as
While this step is straightforward, different combinations of ±1 in Λ_{a} may lead to different ground state energy density.
Since the number of 1 and −1 on the diagonal of Λ_{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 Λ_{a} for each a ∈ {x, y, z}. We only show combinations of parameterized local observables used in Table 4. Here, we denote the 2 × 2 identity matrix by I, and continue using the notation Λ introduced in Eq. (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 σ_{x} is minus the identity matrix:
For the second and third local observables σ_{a}(a = y, z), Λ_{a} has entries one 1 and one −1 on the main diagonal, and S_{a} is determined by one parameter {w}. Hence, σ_{y} and σ_{z} are specified by
For d = 3, two different combinations of three parameterized local observables are used, where the difference arises from Λ_{a}, but S_{a} of each local observable shares the same parameterized form as
In the first combination, [n_{x}, n_{y}, n_{z}] = [2, 1, 1], where Λ_{x} has one 1 and two −1 on the main diagonal, and Λ_{y} and Λ_{z} both have two 1 and one −1 being the main diagonal entries. Then, Λ_{x}, Λ_{y}, and Λ_{z} are given by
In the second combination, [n_{x}, n_{y}, n_{z}] = [2, 1, 2], where Λ_{x} and Λ_{z} both have one 1 and two −1 on the main diagonal, and Λ_{y} has two 1 and one −1 being the main diagonal entries. Then, Λ_{x}, Λ_{y}, and Λ_{z} are given by
For d = 4, four different classes of triples of local observables are used. These classes differ from each other in the structure of the matrices Λ_{a} in Eq. (29). The matrices S_{a} have, nonetheless, the same form in the three classes, namely:
The first class is of the form [n_{x}, n_{y}, n_{z}] = [3, 1, 1], where Λ_{x} has one 1 and three −1 on the main diagonal, and Λ_{y} and Λ_{z} both have three 1 and one −1 being main diagonal entries. Then, Λ_{x}, Λ_{y}, and Λ_{z} are given by
The second class is of the form [n_{x}, n_{y}, n_{z}] = [3, 2, 3], where Λ_{x} and Λ_{z} both have one 1 and three −1 on the main diagonal, and Λ_{y} has two 1 and two −1 being main diagonal entries. Then, Λ_{x}, Λ_{y}, and Λ_{z} are given by
The third class is of the form [n_{x}, n_{y}, n_{z}] = [3, 2, 1], where Λ_{x} has one 1 and three −1 on the main diagonal, Λ_{y} has two 1 and two − 1 being main diagonal entries, and Λ_{z} takes three 1 and one −1 on the main diagonal. Then, Λ_{x}, Λ_{y}, and Λ_{z} are given by
The fourth class is of the form [n_{x}, n_{y}, n_{z}] = [2, 1, 2], where Λ_{x} and Λ_{z} have two 1 and two −1 on the main diagonal, Λ_{y} has three 1 and one −1 being the main diagonal entries. Then, Λ_{x}, Λ_{y}, and Λ_{z} are given by
Discussion
In this paper, we investigate the contextuality of several types of infinite onedimensional translationinvariant local quantum Hamiltonians. We found that it is very likely that all quantum Hamiltonians with nearestneighboronly interactions and two dichotomic observables per site admit local hidden variable models. Violation of contextuality witnesses is only possible when we either increase the interaction distance to include nexttonearest 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 >2, which excludes the usual Heisenbergtype models where local observables are Pauli matrices.
States and measurements which exhibit the strongest violations of Bell inequalities are essential ingredients in deviceindependent certifications and selftesting^{37,38}. So far the possibility of selftesting in quantum manybody 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 selftesting quantum manybody 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 onedimensional local quantum Hamiltonians has yielded many interesting results^{39,40,41,42,43}. However, most of these results assume the models to have nearestneighbor interactions. As we can see from our results, the models with nexttonearest neighbor interactions are surprisingly the most interesting in terms of contextuality.
In two dimensions, very little is known about the contextuality of translationinvariant local Hamiltonians. We know that when the number of inputs and outputs is unrestricted, the set of local hidden variable models becomes nonsemialgebraic 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 questions about the computability and complexity of classical and quantum models^{44,45,46,47}. 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
Optimization of 1D TI Hamiltonians with POVMs as local observables
We describe an extension of the algorithm used to minimize the ground state energy density to include the most general quantum measurements: POVMs. The extended algorithm is used to minimize 232type Hamiltonians when the dimension of local observables is 2. These local observables \(\{{\sigma }_{a}:{{\mathbb{C}}}^{2}\to {{\mathbb{C}}}^{2},a\in X\}\) 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 semidefinite programming:
Here, \({\tilde{\sigma }}_{a}={\tilde{\sigma }}_{a}(k+1)\) and σ_{a} = σ_{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 wellknown remedy, using momentum to speed up the traversal of nearly flat regions. At each iteration k, the parameters W(k) defining the local observables {σ_{a}(k)∣a ∈ X} are updated by
where V(k) is the momentum and η is the decay factor.
Beginning with random initial W(0) and V(0) = 0, iterating the steps above, we obtain the sequence of parameter values (W(0), W(1), … ) defining a sequence of local observables (σ_{a}(0), σ_{a}(1),…), each of which is constructed from POVM elements. If the convergence criterion ∣e(k + 1)−e(k)∣ ≤ ϵ^{*} is met at iteration k, then the algorithm stops and returns the optimal parameters W^{*} ≡ W(k + 1). For 100 out of the 652 Hamiltonians we 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 <10^{−10}.
Data availability
Data will be made available upon reasonable request to the corresponding author.
Code availability
Code will be made available upon reasonable request to the corresponding author.
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Acknowledgements
This work is supported by the National Key R&D Program of China (Nos. 2018YFA0306703, 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). We thank Wei He for helping with some of the illustrations.
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Z.W. and M.N. conceived the project. K.Y., X.Z. and Y.L. performed the numerical computation. Z.W., G.Y. and L.S. supervised the project. All authors contributed to the development of the manuscript.
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Yang, K., Zeng, X., Luo, Y. et al. Contextuality in infinite onedimensional translationinvariant local Hamiltonians. npj Quantum Inf 8, 89 (2022). https://doi.org/10.1038/s41534022005980
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DOI: https://doi.org/10.1038/s41534022005980