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

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

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

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

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

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

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:

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

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 $e({W};k)$:

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.

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

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 $n,s$.

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

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.

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.

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.

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 $\{w\}$:

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

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

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

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

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

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

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 $5$.

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. 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.

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:

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 $\sigma_{x}$ and $\sigma_{y}$ are given by

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

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 $d=4$.

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

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:

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