State-independent quantum contextuality with projectors of nonunit rank

Experiments provide strong evidence that the measurements on quantum systems cannot be reproduced by any noncontextual hidden variable model (NCHV). In an NCHV model each outcome of any measurement has a preassigned value and this value in particular does not depend on which other properties are obtained alongside. This phenomenon is called quantum contextuality. Being closely connected to the incompatibility of observables [1], quantum contextuality is the underlying feature of quantum theory that enables, for example, the violation of Bell inequalities [2], enhanced quantum communication [3, 4], cryptographic protocols [5, 6], quantum enhanced computation [7, 8], and quantum key distribution [9].

The first example of quantum contextuality was found by Kochen and Specker [10] and required 117 rank-one projectors. Subsequently the number of projectors was reduced until it was proved that the minimal set has 18 rank-one projectors [11]. This analysis was based on the particular type of contradiction between value assignments and projectors that was already used in the original proof by Kochen and Specker. The situation changed with the introduction of state-independent noncontextuality inequalities, where any NCHV model obeys the inequality, while it is violated for any quantum state and a certain set of projectors. With this enhanced definition of state-independent contextuality (SIC), Yu and Oh [12] found an instance of SIC with only 13 rank-one projectors and subsequently it was proved that this set is minimal [13] provided that all projectors are of rank one. Note that the iconic example of the Peres–Mermin square [14, 15] uses 9 observables with two-fold degenerate eigenspaces, but they are combined to 6 measurements of 24 rank-one projectors.

In contrast, SIC involving nonunit rank projectors has been rarely considered. To the best of our knowledge, the only examples [15–18] which use nonunit rank are based on the Mermin star [19]. In these examples it was shown that nonunit projectors are sufficient for SIC, but it was not shown whether nonunit projectors are also necessary for SIC. Furthermore, in a graph theoretical analysis by Ramanathan and Horodecki [20] a necessary condition for SIC was provided which also allows one to study the case of nonunit rank.

In this article, we develop mathematical tools to analyze SIC for the case of nonunit rank. We first show that in certain situations nonunit rank is necessary for SIC. Then we approach the question whether projectors with nonunit rank enable SIC with less than 13 projectors. We find that in this case at least 9 projectors are required. For the special cases of SIC where all projectors are of rank 2 or rank 3 we find strong numerical evidence that 13 is indeed the minimal number of projectors.

This paper is structured as follows. In section 2 we give an introduction to quantum contextuality using the graph theoretic approach. We extend this discussion to SIC in section 3 and we give an example where rank-two projectors are necessary for SIC. In section 4 we provide a general analysis of the case of nonunit rank and show that scenarios with 8 or less projectors do not feature SIC, irrespective of the involved ranks. This analysis is used in section 5 to show in an exhaustive numerical search that all graphs smaller than the graph given by Yu and Oh do not have SIC, if the rank of all projectors is 2 or 3. We conclude in section 6 with a discussion of our results.

Our analysis is based on the graph theoretic approach to quantum contextuality [21]. In this approach an exclusivity graph G with vertices V(G) and edges E(G) specifies the exclusivity relations in a contextuality scenario. The vertices represent events and two events are exclusive if they are connected by an edge. The cliques of the graph form the contexts of the scenario. (In appendix A we give definitions of essential terms from graph theory.) Recall that an event is a class of outcomes in an experiment and two events are exclusive if they cannot be obtained simultaneously in any experiment. We consider now two types of models implementing the exclusivity graph, quantum models and noncontextual hidden variable models.

In a quantum model of the exclusivity graph G one assigns projectors Π_k to each event k such that ∑_k∈C Π_k is again a projector for every context C. This is equivalent to having Π_k Π_l = 0 for any two exclusive events k and l. With such an assignment and a quantum state ρ one obtains the probability for the event k as

$$\begin{equation}{P}_{\text{QT}}\left(k\right)=\text{tr}\left(\rho {{\Pi}}_{k}\right).\end{equation} \tag{ 1 }$$

The set of all probability assignments P_QT that can be reached with some projectors ${\left({{\Pi}}_{k}\right)}_{k}$ and some state ρ is a convex set which coincides [21] with the theta body TH(G) of the graph G.

In contrast, in an NCHV model for the exclusivity graph G the events are predetermined by a hidden variable λ ∈ Λ. That is, to each event k one associates a response function R_k : Λ → {0, 1}. For a context C the function λ ↦ ∑_k∈C R_k (λ) has to be again a response function, which is equivalent to R_k (λ)R_l (λ) = 0 for all λ and any pair of exclusive events k and l. The probability of an event k is now given by

$$\begin{equation}{P}_{\text{NCHV}}\left(k\right)=\sum\limits _{\lambda \in {\Lambda}}\mu \left(\lambda \right){R}_{k}\left(\lambda \right),\end{equation} \tag{ 2 }$$

where μ is some probability distribution over the hidden variable space Λ. The set of all probability assignments P_NCHV that can be reached with some response functions ${\left({R}_{k}\right)}_{k}$ and some distribution μ forms a polytope which can be shown [21] to be the stable set STAB(G) of the graph G.

Quantum models and NCHV models are both noncontextual in the sense that the computation of the probability P(k) of an event k does not depend on the context in which k is contained. Quantum contextuality occurs now for an exclusivity graph G if we can find a quantum model with probability assignment P_QT which cannot be achieved by any NCHV model and hence P_QT ∈ TH(G)\STAB(G). Since STAB(G) is convex, it is possible to find nonnegative numbers ${\left({w}_{k}\right)}_{k\in V\left(G\right)}\equiv \mathbf{w}$ such that

$$\begin{equation}{I}_{\mathbf{w}}:P{\mapsto}\sum\limits _{k}{w}_{k}P\left(k\right)\end{equation} \tag{ 3 }$$

separates all NCHV models from some of the quantum models. That is, there exists some α, such that I_w (P_NCHV) ⩽ α holds for any P_NCHV ∈ STAB(G), while I_w (P_QT) > α holds true for some P_QT ∈ TH(G). This can be further formalized by realizing that the weighted independence number [22] α(G, w) is exactly the maximal value that I_w attains within STAB(G) and similarly that the weighted Lovász number [23] ϑ(G, w) is exactly the maximum of I_w over TH(G). Consequently the inequality I_w (P_NCHV) ⩽ α(G, w) holds for all NCHV probability assignments and this inequality is violated by some quantum probability assignment if and only if [21] ϑ(G, w) > α(G, w) holds. In addition, one can show [21] that the value of ϑ(G, w) can always be attained for some quantum model employing only rank-one projectors.

The discussion so far concerns quantum models as being specified by the projectors assigned to each event together with a quantum state. In SIC one removes the quantum state from the specification of a quantum model and instead requires that probabilities from the quantum model cannot be reproduced by an NCHV model, independent of the quantum state. Therefore we consider the set of probability assignments formed by all quantum states and fixed projectors ${\left({{\Pi}}_{k}\right)}_{k}$,

$$\begin{equation}{\mathcal{P}}_{\text{SIC}}=\left\{P:k{\mapsto}\text{tr}\left(\rho {{\Pi}}_{k}\right)\vert \rho \enspace \text{is a quantum state}\right\}.\end{equation} \tag{ 4 }$$

This set is also convex, since P is linear and the set of quantum states is convex. Hence, in the case of SIC it is again possible to find nonnegative numbers ${\left({w}_{k}\right)}_{k}\equiv \mathbf{w}$ such that I_w separates STAB(G) from ${\mathcal{P}}_{\text{SIC}}$. Therefore, it holds that

$$\begin{equation}\sum\limits _{k}{w}_{k}\enspace \text{tr}\left(\rho {{\Pi}}_{k}\right){ >}\alpha \left(G,\mathbf{w}\right),\quad \text{for}\enspace \text{all}\enspace \rho ,\end{equation} \tag{ 5 }$$

or, equivalently, that the eigenvalues of

$$\begin{equation}\sum\limits _{k}{w}_{k}{{\Pi}}_{k}-\alpha \left(G,\mathbf{w}\right)\end{equation} \tag{ 6 }$$

are all strictly positive.

We say that the projectors ${\left({{\Pi}}_{k}\right)}_{k}$ of a quantum model of G form a rank-r projective representation (PR) ³ of G, when $\mathbf{r}={\left({r}_{k}\right)}_{k\in V\left(G\right)}$ with r_k the rank of Π_k . The smallest known contextuality scenario which allows SIC is given by the exclusivity graph G_YO with 13 vertices [12]. This graph is shown in figure 1(a). For this scenario it is sufficient to consider rank-one projective representations. It also has been shown that no exclusivity graph with 12 or less vertices allows SIC [13], provided that all projectors are of rank one, r = 1. But this does not yet show that SIC requires 13 projectors, since it is possible that a contextuality scenario features SIC only if some of the projectors are of nonunit rank.

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This rises the question whether projectors of nonunit rank can be of advantage regarding SIC. We now show that this is the case by analyzing the exclusivity graph G_Toh with 30 vertices [18]. This graph is shown in figure 1(b). One can find a rank-two PR of this graph [18], such that ${\sum }_{k}{{\Pi}}_{k}=7+\frac{1}{2}$. Since the independence number of G_Toh is 7, that is, α(G_Toh) ≡ α(G_Toh, 1) = 7, this shows that rank two is sufficient for SIC in this scenario.

For necessity, we show that no rank-one PR featuring SIC of G_Toh exists. We first note that such a representation would be necessarily constructed in a four-dimensional Hilbert space. This is the case because the largest clique of G_Toh has size four and hence any PR must contain at least four mutually orthogonal projectors of rank one. For an upper bound on the dimension d of any PR featuring SIC we use the result [20, 24]

$$\begin{equation}d{< }{\chi }_{\mathrm{f}}\left(G\right),\end{equation} \tag{ 7 }$$

where χ_f(G) denotes the fractional chromatic number of G. One finds ${\chi }_{\mathrm{f}}\left({G}_{\text{Toh}}\right)=4+\frac{2}{7}$ implying d ⩽ 4. We do not find any rank-one PR of G_Toh in dimension d = 4 using the numerical methods discussed in section 5.2 and in appendix B we prove also analytically that no such representation exists.

The example of the previous section showed that considering projective representations of nonunit rank can be necessary for the existence of a quantum model with SIC. Since the case of rank-one has already been analyzed in detail, it is helpful to reduce the case of nonunit rank to the case of rank one. To this end we adapt the notation [25] G^r for the graph where each vertex k is replaced by a clique C_k of size r_k and all vertices between two cliques C_k and C_ℓ are connected when [k, ℓ] is an edge. See figure 2 for an illustration. That is,

$$\begin{equation}V\left({G}^{\mathbf{r}}\right)=\left\{\left(k,i\right)\vert k\in V\left(G\right),\enspace i=1,2,\dots ,{r}_{k}\right\},\end{equation} \tag{ 8 }$$

$$\begin{equation}E\left({G}^{\mathbf{r}}\right)=\left\{\left[\left(k,i\right),\left(\ell ,j\right)\right]\vert \left[k,\ell \right]\in E\left(G\right)\enspace \text{or}\enspace \left(k=\ell \;\text{and}\;i\ne j\right)\right\}.\end{equation} \tag{ 9 }$$

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The construction of G^r is such that if ${\left({{\Pi}}_{k,i}\right)}_{k,i}$ is a rank-one PR of G^r , then evidently Π_k = ∑_i Π_k,i defines a rank-r PR of G. Vice versa, if ${\left({{\Pi}}_{k}\right)}_{k}$ is a rank-r PR of G, then one can immediately construct a rank-one PR of G^r by decomposing each projector Π_k into rank-one projectors ${\left({{\Pi}}_{k,i}\right)}_{i}$ such that Π_k = ∑_i Π_k,i .

For a given graph G we denote by d_π (G, r) the minimal dimension which admits a rank-r PR and by χ_f(G, r) the fractional chromatic number for the graph G with vertex weights $\mathbf{r}\in {\mathbb{N}}^{\vert V\left(G\right)\vert }$. In addition we abbreviate the Lovász function of the complement graph by $\bar{\vartheta }\left(G,\mathbf{r}\right)=\vartheta \left(\bar{G},\mathbf{r}\right)$. For these three functions we omit the second argument if r_k = 1 for all k, that is, χ_f(G) ≡ χ_f(G, 1), etc.

Theorem 1. For any graph G and vertex weights $\mathbf{r}\in {\mathbb{N}}^{\vert V\left(G\right)\vert }$ we have d_π (G^r ) = d_π (G, r), χ_f(G^r ) = χ_f(G, r), and $\bar{\vartheta }\left({G}^{\mathbf{r}}\right)=\bar{\vartheta }\left(G,\mathbf{r}\right)$. In addition, χ_f(G, m r) = mχ_f(G, r) and $\bar{\vartheta }\left(G,m\mathbf{r}\right)=m\bar{\vartheta }\left(G,\mathbf{r}\right)$ hold for any $m\in \mathbb{N}$.

The proof is provided in appendix C. As a consequence we extend the relation [23] $\bar{\vartheta }\left(G\right){\leqslant}{d}_{\pi }\left(G\right)$ (see also appendix A) to the case of nonunit rank,

$$\begin{equation}\bar{\vartheta }\left(G,\mathbf{r}\right){\leqslant}{d}_{\pi }\left(G,\mathbf{r}\right).\end{equation} \tag{ 10 }$$

Similarly we have generalization of the condition in equation (7): whenever a graph G has a rank-r PR featuring SIC, then it holds that

$$\begin{equation}{d}_{\pi }\left(G,\mathbf{r}\right){< }{\chi }_{\mathrm{f}}\left(G,\mathbf{r}\right).\end{equation} \tag{ 11 }$$

Following the ideas from references [20] and [26], we consider quantum models that use the maximally mixed state ${\rho }_{\mathrm{m}\mathrm{m}}=\mathbb{1}/d$, where d is the dimension of the Hilbert space. For a rank-r PR, the corresponding probability assignment is then simply given by

$$\begin{equation}{P}_{\mathrm{m}\mathrm{m}}\left(k\right)={r}_{k}/d.\end{equation} \tag{ 12 }$$

If the representation features SIC, then P_mm ∉ STAB(G), since, by definition, ${P}_{\text{mm}}\in {\mathcal{P}}_{\text{SIC}}$ while ${\mathcal{P}}_{\text{SIC}}$ and STAB(G) are disjoint sets. This is the motivation to define the set RANK(G) of all probability assignments P_mm which arise from any PR of G. That is,

$$\begin{equation}\text{RANK}\left(G\right)=\left\{\mathbf{r}/\ell \vert \mathbf{r}\in {\mathbb{N}}^{\vert V\left(G\right)\vert }\text{,}\;\ell \in \mathbb{N},\enspace \text{such that}\enspace \ell {\geqslant}{d}_{\pi }\left(G,\mathbf{r}\right)\right\}.\end{equation} \tag{ 13 }$$

Denoting by $\bar{\text{RANK}}\left(G\right)$ the topological closure of RANK(G) we show in appendix D the following inclusions.

Theorem 2. For any graph G, the set $\bar{\text{RANK}}\left(G\right)$ is convex and $\text{STAB}\left(G\right)\subseteq \bar{\text{RANK}}\left(G\right)\subseteq \text{TH}\left(G\right)$.

This implies that any NCHV probability assignment can be arbitrarily well approximated by a quantum probability assignment using the maximally mixed state. Conversely, if RANK(G) ⊂ STAB(G) for an exclusivity graph G, then any quantum probability assignment using the maximally mixed state can be reproduced by an NCHV model and hence no PR of G can feature SIC. This is the case for all graphs with at most 8 vertices, as we show in appendix E by using a linear relaxation of RANK(G).

Theorem 3.  $\text{STAB}\left(G\right)=\bar{\text{RANK}}\left(G\right)$ for any graph G with 8 vertices or less.

Since any exclusivity graph allowing SIC must have $\overline{\mathrm{R}\mathrm{A}\mathrm{N}\mathrm{K}}\left(G\right){\supsetneq}\mathrm{S}\mathrm{T}\mathrm{A}\mathrm{B}\left(\mathrm{G}\right)$, this implies the following statement.

Corollary 4. Any scenario allowing SIC requires more than 8 events.

We now aim to find the smallest scenario allowing SIC, that is, the smallest exclusivity graph which has a PR featuring SIC. Here, we say that a graph G' is smaller than a graph G if either G' has less vertices than G or if both have the same number of vertices and G' has less edges than G. With this notion, the smallest known graph allowing SIC is ${G}_{{\text{YO}}^{\prime }}$ with 13 vertices and 23 edges ⁴ , where ${G}_{{\text{YO}}^{\prime }}$ is G_YO but with one edge removed as shown in figure 1(a). Due to corollary 4 it remains to consider the graphs with 9 and up to 12 vertices as well as all graphs with 13 vertices and 23 edges or less.

Instead of testing for a PR featuring SIC, we use the weaker condition in equation (11) and we limit our considerations to rank-r representations where all projectors have the same rank and r = 1, r = 2, or r = 3. We now aim to establish the following.

Assertion 5. For r = 1, 2, 3, the smallest graph G with d_π (G, r 1) < χ_f(G, r 1) is ${G}_{{\text{YO}}^{\prime }}$.

This assertion implies that ${G}_{{\text{YO}}^{\prime }}$ is the smallest graph admitting SIC when considering rank-r projective representations for r = 1, 2, 3.

Our approach to assertion 5 consists of two steps. First we identify four conditions that are easy to compute and necessary for d_π (G, r) < χ_f(G, r) to hold. For graphs which satisfy all these conditions and for r = r 1 with r = 1, 2, 3, we then implement a numerical optimization algorithm in order to compute d_π (G, r 1). We then confirm assertion 5, aside from the uncertainty that is due to the numerical optimization.

5.1. Conditions

5.2. Numerical estimate of the dimension

If an exclusivity graph G has a rank-r PR with SIC, then, according to theorem 1 and the subsequent discussion, there must be a rank-one PR of G^r in dimension d = ⌈χ_f(G, r)⌉ − 1. At this point, we do not further exploit the structure of the problem. We rather consider methods which allow us to verify or falsify the existence of a rank-one PR in dimension d of an arbitrary graph G with n vertices.

If such a PR exists, then one can assign normalized vectors ${\mathbf{y}}_{k}\in {\mathbb{C}}^{d}$ to each vertex k ∈ V(G) such that ${\mathbf{y}}_{\ell }^{{\dagger}}{\mathbf{y}}_{k}=0$ for all edges [ℓ, k] ∈ E(G). Collecting these vectors in the columns of a matrix Y, we obtain the feasibility problem

$$\begin{equation}\begin{aligned}\hfill & \text{find} X={Y}^{{\dagger}}Y\quad \text{with}\quad Y\in {\mathbb{C}}^{d{\times}n},\hfill \\ \hfill & \text{subject}\;\text{to}\quad {X}_{k,k}=1\quad \text{for}\enspace \text{all}\quad j\in V\left(G\right),\hfill \\ \hfill & \enspace {X}_{\ell ,k}=0\quad \text{for}\enspace \text{all}\quad \left[\ell ,k\right]\in E\left(G\right).\hfill \end{aligned}\end{equation} \tag{ 14 }$$

This problem is equivalent to the optimization problem

$$\begin{equation}\begin{aligned}\hfill & \text{minimize}\quad {\sum }_{k\in V\left(G\right)}{\left({X}_{k,k}-1\right)}^{2}+{\sum }_{\left[\ell ,k\right]\in E\left(G\right)}{X}_{\ell ,k}^{2},\hfill \\ \hfill & \text{with}\quad X={Y}^{{\dagger}}Y\quad \text{and}\quad Y\in {\mathbb{C}}^{d{\times}n},\hfill \end{aligned}\end{equation} \tag{ 15 }$$

where the problem in equation (14) is feasible if and only if the problem in equation (15) yields zero. The optimization can be executed using a standard algorithm like the conjugate-gradient method [28]. However, the obtained value can be from a local minimum and depend on the initial value used in the optimization. Hence obtaining a value greater than zero does not conclusively exclude the existence of a PR, but this problem can be mitigated by performing the minimization for many different initial values.

Instead of employing one of the standard optimization algorithms, we use a faster method that allows us to repeat the minimization with many different initial values. For this we denote by $\mathcal{L}$ the set of all (n × n)-matrices X which satisfy the constraints of the problem in equation (14) and we write $\mathcal{R}$ for the set of all matrices X for which X = Y^† Y for some (d × n)-matrix Y. In an alternating optimization, we generate a sequence ${\left({X}^{\left(j\right)}\right)}_{j}$ from an initial value X⁽⁰⁾ such that

$$\begin{equation}\begin{aligned}\hfill {X}^{\left(2i+1\right)}& ={\mathrm{arg}\mathrm{min}}_{R}\left\{{\Vert}R-{X}^{\left(2i\right)}{\Vert}\vert R\in \mathcal{R}\right\},\hfill \\ \hfill {X}^{\left(2i\right)}& ={\mathrm{arg}\mathrm{min}}_{L}\left\{{\Vert}L-{X}^{\left(2i-1\right)}{\Vert}\vert L\in \mathcal{L}\right\}.\hfill \end{aligned}\end{equation} \tag{ 16 }$$

By construction, δ_j = ∥X^(j) − X^(j−1)∥ is a nonincreasing sequence and hence δ_∞ = lim_j→∞  δ_j exists. Consequently, for the existence of a PR it is sufficient if δ_∞ = 0 because then X^(∞) = lim_j→∞  X^(j) exists with ${X}^{\left(\infty \right)}\in \mathcal{R}\cap \mathcal{L}$. In appendix F we show that this alternating optimization can be implemented efficiently for the Frobenius norm ∥M∥_F = ∑_i,j |M_i,j |².

We run the optimization with 100 randomly chosen initial values X⁽⁰⁾ for each of the remaining graphs with corresponding rank r. We stop the optimization if δ_k−2/δ_k < 1 + 10⁻⁵. For all graphs and all repetitions the optimization converges with a final value of δ_k in the order of 1. In comparison, we test the algorithm for many graphs with known d_π where the graphs have up to 40 vertices. In all these cases, the algorithm converges to δ_k in the order of 10⁻⁹, which gives us confidence that the alternating optimization is reliable. In summary this constitutes strong numerical evidence that none of the remaining graphs with corresponding rank has a PR with SIC.

The search for a primitive entity of contextuality has not yet reached a conclusion despite of decades of research on this topic. Of course, one can argue that the pentagon scenario by Klyachko et al [29] does provide a provably minimal scenario. But the drawback of the pentagon scenario is that it is state-dependent. That is, contextuality is here a feature of both, the state and the measurements. In contrast, in the state-independent approach, contextuality is a feature exclusively of the measurements and we argue that a primitive entity of contextuality should embrace state-independence. Among the known SIC scenarios, the one by Yu and Oh [12] is minimal and this has also been proved rigorously for the case where all measurement outcomes are represented by rank-one projectors.

As we pointed out here, there is no guarantee that the actual minimal scenario will also be of rank one: we showed that a scenario by Toh [18]—albeit far from minimal—requires projectors of rank two. This motivated our search for the minimal SIC scenario for the case of nonunit rank. Due to theorem 3, we can exclude the case where the exclusivity graph has 8 or less vertices. For the remaining cases of 9 to 12 vertices, we also obtain a negative result, however, under the restriction that the PR is uniformly of rank two or uniformly of rank three. A key to this result is a fast and empirically reliable numerical method to find or exclude projective representations of a graph, which might be also a useful method for related problems in graph theory.

Curiously, there is no simple argument that shows that the scenario by Yu and Oh is minimal, even when assuming unit rank. This in contrast to the case of state-dependent contextuality, where the reason that the pentagon scenario is the simplest scenario beautifully has the origin in graph theory [21]. For the future it will be interesting to develop additional methods for SIC, in particular for the case of heterogeneous rank. It will be particularly interesting whether this problem can be solved using more methods from graph theory, whether it can be solved using new numerical methods, or whether the problem turns out to be genuinely hard to decide.

The data that support the findings of this study are available upon reasonable request from the authors.

We thank A Cabello, N Tsimakuridze, YY Wang for discussions, A Ganesan for pointing out reference [25] to us, and the University of Siegen for enabling our computations through the HoRUS cluster. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation - 447948357), the ERC (ConsolidatorGrant 683107/TempoQ), and the Alexander von Humboldt Foundation.

A graph G is a collection of vertices V(G) connected by edges E(G). Each edge [i, j] ∈ is an unordered pair of the vertices i ≠ j ∈ V(G). Conversely, for a given vertex set V and edge set E the pair (V, E) forms the graph denoted by G(V, E). For a given subset W of V and subset F of E, the graph G(W, F) is a subgraph of G(V, E). In the case where

$$\begin{equation}F=E\cap {\left\{\left[i,j\right]\right\}}_{i,j\in W},\end{equation} \tag{ A1 }$$

G(W, F) is a subgraph of G(V, E) induced by the subset W. In the case where

$$\begin{equation}F={\left\{\left[{i}_{t},{i}_{t+1}\right]\right\}}_{t},\end{equation} \tag{ A2 }$$

G(W, F) is a path in G(V, E). A graph G(V, E) is connected if any two vertices can be connected by a path. A subset of vertices C is a clique, if in the induced subgraph all vertices are mutually connected by an edge. A clique C is maximal, if any strict superset of C is not clique. The complement graph $\bar{G}$ of G has an edge [i, j] if and only if i ≠ j and [i, j] is not an edge in G. A clique in $\bar{G}$ is an independent set of G. Independent sets are also called stable sets. If any strict superset of W is not an independent set, then W is a maximally independent set.

Now, the index vector of a given subset of vertices W is defined as

$$\begin{equation}{{\Delta}}_{W}={\left[{\delta }_{W}\left(k\right)\right]}_{k\in V},\end{equation} \tag{ A3 }$$

where δ_W (k) = 1 if k ∈ W and δ_W (k) = 0 otherwise. Let $\mathcal{I}$ denote the set of all independent sets of graph G, then the stable set polytope STAB(G) is the convex hull of the set $\left\{{{\Delta}}_{W}\vert W\in \mathcal{I}\right\}$.

A collection of real vectors ${\left({\mathbf{v}}_{i}\right)}_{i\in V}$ is an orthogonal representation of G, provided that [i, j] ∉ E implies v_i ⋅v_j = 0. The Lovász theta body of a given graph G can be defined as [30]

$$\begin{equation}\text{TH}\left(G\right)=\left\{{\left[{\left(\mathbf{s}\cdot {\mathbf{v}}_{i}\right)}^{2}\right]}_{i\in V}\vert {\left({\mathbf{v}}_{i}\right)}_{i\in V}\enspace \text{is}\;\text{an}\;\text{OR}\enspace \text{of}\enspace \bar{G}\right\},\end{equation} \tag{ A4 }$$

where s = (1, 0, ..., 0). We also use the following, equivalent definition of TH(G). A collection of projectors ${\left({{\Pi}}_{k}\right)}_{k\in V}$ (over a complex Hilbert space) is a PR of G if Π_i Π_j = 0 whenever [i, j] ∈ E(G). Then, one can also write [21]

$$\begin{equation}\text{TH}\left(G\right)=\left\{{\left[\text{tr}\left(\rho {{\Pi}}_{i}\right)\right]}_{i\in V}\vert {\left({{\Pi}}_{i}\right)}_{i\in V}\enspace \text{is a PR of}\enspace G,\text{tr}\left(\rho \right)=1,\rho {\geqslant}0\right\}.\end{equation} \tag{ A5 }$$

Note that in the definition, the projectors might be of any rank.

For a vector r of nonnegative real numbers,

$$\begin{equation}\alpha \left(G,\mathbf{r}\right)={\mathrm{max}}_{\mathbf{x}}\left\{\mathbf{r}\cdot \mathbf{x}\vert \mathbf{x}\in \text{STAB}\left(G\right)\right\}\end{equation} \tag{ A6 }$$

is the weighted independence number [30] and the weighted Lovász number is given [31] by

$$\begin{equation}\vartheta \left(G,\mathbf{r}\right)={\mathrm{max}}_{\mathbf{x}}\left\{\mathbf{r}\cdot \mathbf{x}\vert \mathbf{x}\in \text{TH}\left(G\right)\right\}.\end{equation} \tag{ A7 }$$

For convenience, we write $\bar{\vartheta }\left(G,\mathbf{r}\right)=\vartheta \left(\bar{G},\mathbf{r}\right)$.

The weighted chromatic number χ(G, r) can be defined as [25]

$$\begin{equation}\begin{aligned}\hfill {\mathrm{min}}_{{\left({c}_{I}\right)}_{I\in \mathcal{I}}}& \quad \hfill & \hfill & \sum\limits _{I\in \mathcal{I}}{c}_{I},\hfill \\ \hfill \text{such}\;\text{that}& \hfill & \hfill & \sum\limits _{I\ni i}{c}_{I}{\geqslant}{r}_{i},\quad \text{for}\enspace \text{all}\enspace i\in V,\hfill \end{aligned}\end{equation} \tag{ A8 }$$

where c_I are nonnegative integers. Equivalently, if C = χ(G, r), then there exists an r-coloring of G with C colors, that is, C is the minimal number of colors such that r_k colors are assigned to each vertex k and two vertices i and j do not share a common color if they are connected.

The weighted fractional chromatic number χ_f(G, r) is a relaxation of the integer program in equation (A8) to a linear program [25]

$$\begin{equation}\begin{aligned}\hfill {\mathrm{min}}_{{\left({x}_{I}\right)}_{I\in \mathcal{I}}}& \quad \hfill & \hfill & \sum\limits _{I\in \mathcal{I}}{x}_{I},\hfill \\ \hfill \text{such}\;\text{that}& \hfill & \hfill & \sum\limits _{I\ni i}{x}_{I}{\geqslant}{r}_{i},\quad \text{for}\enspace \text{all}\enspace i\in V,\hfill \end{aligned}\end{equation} \tag{ A9 }$$

where x_I are now nonnegative real numbers. Being a linear program with rational coefficients, all x_I can be chosen to be rational numbers and hence one can find a $b\in \mathbb{N}$ such that all bx_I are integer. This yields the relation

$$\begin{equation}{\chi }_{f}\left(G,\mathbf{r}\right)={\mathrm{min}}_{b\in \mathbb{N}}\frac{\chi \left(G,b\mathbf{r}\right)}{b}.\end{equation} \tag{ A10 }$$

Finally, we use d_π (G, r) as defined in the main text, that is, d_π (G, r) is the minimal dimension admitting a rank-r PR. We also omit the weights r for the functions d_π , χ_f, and $\bar{\vartheta }$, if r = 1. We now show the known relation [23] $\bar{\vartheta }\left(G\right){\leqslant}{d}_{\pi }\left(G\right)$, which is extended to the case of r = 1 in equation (10) in the main text.

Lemma 6.  $\bar{\vartheta }\left(G\right){\leqslant}{d}_{\pi }\left(G\right)$

Proof. For a given d-dimensional rank-1 PR ${\left({{\Pi}}_{k}\right)}_{k}$ of G, a d²-dimensional rank-1 PR ${\left({P}_{k}\right)}_{k}$ of G can be constructed as

$$\begin{equation}{P}_{k}={{\Pi}}_{k}^{{\ast}}\otimes {{\Pi}}_{k},\end{equation} \tag{ A11 }$$

where complex conjugation is with respect to some arbitrary, but fixed orthonormal basis |1⟩, |2⟩, ..., |d⟩. Using Ψ = ∑_j,l |jj⟩⟨ll|, we have tr(ΨP_k ) = 1 and tr(Ψ) = d.

We consider now an arbitrary rank-1 PR ${\left({Q}_{k}\right)}_{k}$ of $\bar{G}$ together with an arbitrary density operator ρ acting on the same Hilbert space as the PR. Then ${\left({P}_{i}\otimes {Q}_{j}\right)}_{i,j}$ is a PR of $G\otimes \bar{G}$ and (i, i) is connected with (j, j) either within G or within $\bar{G}$, for any two vertices i ≠ j. Here G ⊗ K denotes the graph with vertices V(G) × V(K) and where [(v, w), (v', w')] is an edge, if [v, v'] or [w, w'] is an edge.

Therefore, ${\sum }_{k}{P}_{k}\otimes {Q}_{k}{\leqslant}\mathbb{1}$ and consequently,

$$\begin{equation}d=\text{tr}\left({\Psi}\otimes \rho \right){\geqslant}\sum\limits _{k}\text{tr}\left(\left[{\Psi}\otimes \rho \right]\left[{P}_{k}\otimes {Q}_{k}\right]\right)=\sum\limits _{k}\text{tr}\left(\rho {Q}_{k}\right).\end{equation} \tag{ A12 }$$

By virtue of equation (A5) we obtain ∑x_i ⩽ d for all $\mathbf{x}\in \text{TH}\left(\bar{G}\right)$, which then yields the desired inequality due to equation (A7). □

The disjoint union G = G₁ ∪ G₂ of two graphs consists of the disjoint union of the vertices, V(G) = V(G₁) ⊎ V(G₂), and [i, j] is an edge in G if it is an edge in either G₁ or G₂. For condition 1 in section 5.1 we use the following observation.

Lemma 7. If G = ⋃_i G_i , then d_π (G, r) = max_i d_π (G_i , r_i ) and χ_f (G, r) = max_i χ_f (G_i , r_i ), where r_i is the part of r for G_i .

Proof. By definition, d_π (G, r) ⩾  max_i d_π (G_i , r_i ). Conversely, if d = max_i d_π (G_i , r_i ) then we can find a d-dimensional rank-r_i PR for each G_i . Since the subgraphs are mutually disjoined, these PRs jointly form already a d-dimensional rank-r PR of G. Thus d ⩾ d_π (G, r).

For the fractional chromatic number, one first observes that ${G}^{\mathbf{r}}={\bigcup }_{i}{G}_{i}^{{\mathbf{r}}_{i}}$. Hence the assertion reduces to ${\chi }_{\mathrm{f}}\left({\bigcup }_{i}{G}_{i}^{{\mathbf{r}}_{i}}\right)={\mathrm{max}}_{i}{\chi }_{\mathrm{f}}\left({G}_{i}^{{\mathbf{r}}_{\mathbf{i}}}\right)$, which is a well-known relation for disjoint unions of graphs [32]. □

The join G = G₁ + G₂ of two graphs is similar to the disjoint union, however with an additional edge between any two vertices [i, j] if i ∈ V(G₁) and j ∈ V(G₂). For condition 2 in section 5.1 we use then the following observation.

Lemma 8. If G = ∑_i G_i , then d_π (G, r) = ∑_i d_π (G_i , r_i ) and χ_f (G, r) = ∑_i χ_f (G_i , r_i ).

Proof. For given d_i -dimensional rank-r_i PRs ${\left({{\Pi}}_{j}\right)}_{j\in V\left({G}_{i}\right)}$ of G_i , we define

$$\begin{equation}{P}_{j,i}=\left(\underset{k{< }i}{\bigoplus}{\mathbb{O}}_{k}\right)\oplus {{\Pi}}_{j}\oplus \left(\underset{k{ >}i}{\bigoplus}{\mathbb{O}}_{{d}_{k}}\right),\end{equation} \tag{ A13 }$$

where j ∈ G_i and ${\mathbb{O}}_{k}$ is the zero-operator acting on the space of the PR of G_k . This construction achieves that ${\left({\left({P}_{j,i}\right)}_{j\in V\left({G}_{i}\right)}\right)}_{i}$ is a (∑_i d_i )-dimensional rank-r PR of G and therefore d_π (G, r) ⩽ ∑_i d_π (G_i , r_i ) holds. Conversely, from a given d-dimensional rank-r PR of G, we can deduce a d_i -dimensional rank-r_i PR of each G_i , where d_i is the dimension of the subspace S_i where ${\left({{\Pi}}_{j}\right)}_{j\in {G}_{i}}$ acts nontrivially. Since each of subspace S_i is orthogonal to the other subspaces S_j , we obtain d ⩾ ∑_i d_i ⩾ ∑_i d_π (G_i , r_i ).

For the fractional chromatic number, we note that ${G}^{\mathbf{r}}={\sum }_{i}{G}_{i}^{{\mathbf{r}}_{i}}$ and since χ_f is additive under the join of graphs [32], the assertion follows. □

It can be verified numerically that there is no four-dimensional rank-1 PR of G_Toh with our numerical methods in appendix F. Here, we give an analytical proof with the help of the computer algebra system Mathematica.

Since each (row) vector v corresponds to a rank-1 projector P(v) = v^† v/|v|², we can use vectors instead of projectors in the case of rank-1 PR. Also, two non-zero vectors v₁ and v₂ are called equal if P(v₁) = P(v₂). For three independent vectors v₁, v₂, v₃ in the four-dimensional Hilbert space, from Cramer's rule we know that their common orthogonal vector is proportional to ${\Lambda}\left({\mathbf{v}}_{1},{\mathbf{v}}_{2},{\mathbf{v}}_{3}\right)={\left({\lambda }_{1},{\lambda }_{2},{\lambda }_{3},{\lambda }_{4}\right)}^{{\ast}}$, with v_i = {v_i,1, v_i,2, v_i,3, v_i,4},

$$\begin{equation}{\lambda }_{i}={\left(-1\right)}^{i}\left\vert \begin{matrix}\hfill {v}_{1,i+1}\hfill & \hfill {v}_{1,i+2}\hfill & \hfill {v}_{1,i+3}\hfill \\ \hfill {v}_{2,i+1}\hfill & \hfill {v}_{2,i+2}\hfill & \hfill {v}_{2,i+3}\hfill \\ \hfill {v}_{3,i+1}\hfill & \hfill {v}_{3,i+2}\hfill & \hfill {v}_{3,i+3}\hfill \end{matrix}\right\vert ,\enspace \end{equation} \tag{ B1 }$$

where the sum i + j is modulo 4. The proof that there is no 4-dimensional rank-1 SIC set for G₃₀ can be divided in two cases.

Case 1: Let ${\left\{{\mathbf{v}}_{i}\right\}}_{i\in V\left({G}_{30}\right)}$ be a four-dimensional rank-1 PR. We first consider the case where

$$\begin{equation}{\mathbf{v}}_{i}\ne {\mathbf{v}}_{j}\enspace \text{for}\enspace \left(i,j\right)\in \left\{\left(5,21\right),\left(4,24\right),\left(14,23\right),\left(3,10\right),\left(3,22\right),\left(4,22\right)\right\}.\end{equation} \tag{ B2 }$$

We can have the following process of parametrization in the basis of {v₂₈, v₁₄, v₁, v₂₂}:

$$\begin{equation}{\mathbf{v}}_{28}=\left(1,0,0,0\right);\qquad {\mathbf{v}}_{14}=\left(0,1,0,0\right);\qquad {\mathbf{v}}_{1}=\left(0,0,1,0\right);\qquad {\mathbf{v}}_{22}=\left(0,0,0,1\right);\end{equation} \tag{ B3 }$$

$$\begin{equation}{\mathbf{v}}_{2}=\left(\mathrm{cos}\enspace {x}_{1},\mathrm{sin}\enspace {x}_{1},0,0\right);\qquad {\mathbf{v}}_{13}=\left(0,0,\mathrm{cos}\enspace {x}_{2},{\text{e}}^{\text{i}{\theta }_{1}}\enspace \mathrm{sin}\enspace {x}_{2}\right);\end{equation} \tag{ B4 }$$

$$\begin{equation}{\mathbf{v}}_{17}=\left(-\mathrm{sin}\enspace {x}_{1}\enspace \mathrm{cos}\enspace {x}_{3},\mathrm{cos}\enspace {x}_{1}\enspace \mathrm{cos}\enspace {x}_{3},{\text{e}}^{\text{i}{\theta }_{2}}\enspace \mathrm{sin}\enspace {x}_{3},0\right);\end{equation} \tag{ B5 }$$

$$\begin{equation}{\mathbf{v}}_{29}=\left(-\mathrm{sin}\enspace {x}_{1}\enspace \mathrm{sin}\enspace {x}_{3},\mathrm{cos}\enspace {x}_{1}\enspace \mathrm{sin}\enspace {x}_{3},-{\text{e}}^{\text{i}{\theta }_{2}}\enspace \mathrm{cos}\enspace {x}_{3},0\right);\end{equation} \tag{ B6 }$$

$$\begin{equation}{\mathbf{v}}_{20}=\left(\mathrm{cos}\enspace {x}_{4}\enspace \mathrm{cos}\enspace {x}_{1},\mathrm{cos}\enspace {x}_{4}\enspace \mathrm{sin}\enspace {x}_{1},0,\mathrm{sin}\enspace {x}_{4}\right);\end{equation} \tag{ B7 }$$

$$\begin{equation}{\mathbf{v}}_{3}=\left(-\mathrm{sin}\enspace {x}_{1}\enspace \mathrm{cos}\enspace {x}_{5},\mathrm{cos}\enspace {x}_{1}\enspace \mathrm{cos}\enspace {x}_{5},0,{\text{e}}^{\text{i}{\theta }_{3}}\enspace \mathrm{sin}\enspace {x}_{5}\right);\end{equation} \tag{ B8 }$$

$$\begin{equation}{\mathbf{v}}_{4}=\left(-\mathrm{sin}\enspace {x}_{1}\enspace \mathrm{sin}\enspace {x}_{5},\mathrm{cos}\enspace {x}_{1}\enspace \mathrm{sin}\enspace {x}_{5},0,-{\text{e}}^{\text{i}{\theta }_{3}}\enspace \mathrm{cos}\enspace {x}_{5}\right).\end{equation} \tag{ B9 }$$

We claim that v₃ is not on the plane spanned by v₂₀, v₂₉, otherwise v₅ ⊥ v₃. Thus, v₅ = v₂₁ since they are orthogonal to v₃, v₆, v₂₄ in the four-dimensional space. This is conflicted with the assumption in equation (B2). Hence, we get v₂₄ = Λ(v₃, v₂₀, v₂₉), which further leads to v₅ = Λ(v₂₀, v₂₄, v₂₉). Note that v₄ ⊥ v₃, v₄ ⊥ v₂₁, hence v₄ is on the plane spanned by v₆, v₂₄. Since v₄ ≠ v₂₄, we get ${\mathbf{v}}_{6}=\left({\mathbf{v}}_{24}{\mathbf{v}}_{24}^{{\dagger}}\right){\mathbf{v}}_{4}-\left({\mathbf{v}}_{4}{\mathbf{v}}_{24}^{{\dagger}}\right){\mathbf{v}}_{24}$ and hence v₂₁ = Λ(v₃, v₆, v₂₄). Since v₁₄ ≠ v₂₃, we have that v₇ = Λ(v₅, v₆, v₁₄), v₈ = Λ(v₅, v₆, v₇), and v₂₃ = Λ(v₈, v₁₃, v₂₈). Since v₃ ≠ v₁₀, we have that v₉ = Λ(v₃, v₇, v₂₃), v₂₇ = Λ(v₇, v₉, v₂₃), and v₁₀ = Λ(v₄, v₂₁, v₂₇).

For the following proof, we make use of the computer algebra system Mathematica. Since v₄ ⊥v₅, direct computation shows that sin(2x₅)sin(x₃ − x₄)sin(x₃ + x₄) = 0. As sin(2x₅) = 0 will result in either v₃ = v₂₂ or v₄ = v₂₂, which conflicts with the assumption in equation (B2), we have that x₃ = ±x₄ mod π. Because of the freedom of choosing θ₂, we can, without loss of generality, assume that x₃ = x₄. Then |v₈|² > 0 implies that sin x₄ cos x₄ ≠ 0. Further, v₈ ⊥ v₂₈ implies that ${\mathrm{cos}}^{2}\enspace {x}_{1}={e}^{2i{\theta }_{3}}\enspace {\mathrm{sin}}^{2}\enspace {x}_{1}$, i.e., θ₃ = 0 mod π and x₁ = ±π/4 mod π. Without loss of generality, we can assume that x₁ = π/4. Then |v₈|² > 0 also implies that sin(x₄ + x₅) ≠ 0. Since v₇ ⊥v₂₃, v₈ ⊥v₁₃, we can find that $\mathrm{cos}\enspace {x}_{2}+{\text{e}}^{\text{i}\left({\theta }_{1}+{\theta }_{2}\right)}\enspace \mathrm{sin}\enspace {x}_{2}=0$. Without loss of generality, we can assume x₂ = −π/4, θ₁ = −θ₂. All the above arguments result in that

$$\begin{equation}{\mathbf{v}}_{8}{\mathbf{v}}_{10}^{{\dagger}}=-{\text{e}}^{\text{i}{\theta }_{2}}\enspace {\mathrm{sin}}_{\enspace 4}^{10}\enspace {\mathrm{cos}}_{\enspace 4}^{29}\enspace {\mathrm{sin}}^{5}\left({x}_{4}+{x}_{5}\right)/\sqrt{2}\ne 0,\end{equation} \tag{ B10 }$$

which conflicts with the exclusivity relations. Thus, v_i = v_j should hold for at least one pair of (i, j) ∈{(5, 21), (4, 24), (14, 23), (3, 10), (3, 22), (4, 22)}.

Case 2: Let {a, b, c, d} be an orthogonal basis and x, y are another two vectors in the 4-dimensional space, then

• (a)  
  x ⊥a, x ⊥b, y ⊥c, y ⊥d implies that x ⊥ y;
• (b)  
  x ⊥ a, x ⊥ b, x ⊥ c implies that x = d;
• (c)  
  x ⊥ a, x ⊥ b, y ⊥ x, y ⊥ d implies that either x ⊥ c or y ⊥c .

In the language of graph theory, if a given graph G has a rank-1 PR in dimension 4, then the graph obtained from the following rules should also have rank-1 PR in dimension 4: let {a, b, c, d} be a clique and x, y are two other vertices in G,

• (a)  
  if (x, a), (x, b), (y, c), (y, d) ∈ E(G), then add (x, y) to E(G);
• (b)  
  if (x, a), (x, b), (x, c) ∈ E(G), then combine x, d into one vertex whose neighbors is the union of the ones of x and the ones of d;
• (c)  
  if (x, a), (x, b), (x, y), (y, d) ∈ E(G), then add either (y, c) or (x, c) to E(G).

When we apply these rules repeatedly to G₃₀ after combining any pair in {(5, 21), (4, 24), (14, 23), (3, 10), (3, 22), (4, 22)}, we either end up with a graph which contains a clique with size larger than 4 or a self-loop. This can be done automatically again with Mathematica. It is obvious that a clique of size larger than 4 has no PR in 4-dimensional space and a self-loop has no rank-1 PR.

The rank-2 PR of G₃₀ is made up of Kernaghan and Peres' 40 rays as shown in table 2. Denote ${P}_{\left\{a,b\right\}}{:=}{\mathbf{r}}_{a}^{{\dagger}}{\mathbf{r}}_{a}/\vert {\mathbf{r}}_{a}{\vert }^{2}+{\mathbf{r}}_{b}^{{\dagger}}{\mathbf{r}}_{b}/\vert {\mathbf{r}}_{b}{\vert }^{2}$ where r_a r_b = 0, then the vertices from v₁ to v₃₀ are represented by the following rank-2 projectors:

$$\begin{equation}\begin{gathered}{P}_{\left\{1,7\right\}},{P}_{\left\{2,8\right\}},{P}_{\left\{3,4\right\}},{P}_{\left\{5,6\right\}},{P}_{\left\{9,12\right\}},{P}_{\left\{13,16\right\}},{P}_{\left\{14,10\right\}},{P}_{\left\{15,11\right\}},\\ {P}_{\left\{19,20\right\}},{P}_{\left\{21,22\right\}},{P}_{\left\{23,17\right\}},{P}_{\left\{24,18\right\}},{P}_{\left\{28,27\right\}},{P}_{\left\{30,29\right\}},{P}_{\left\{31,25\right\}},\\ {P}_{\left\{32,26\right\}},{P}_{\left\{33,35\right\}},{P}_{\left\{34,40\right\}},{P}_{\left\{36,37\right\}},{P}_{\left\{38,39\right\}},{P}_{\left\{1,2\right\}},{P}_{\left\{3,5\right\}},{P}_{\left\{9,13\right\}},\\ {P}_{\left\{14,15\right\}},{P}_{\left\{19,21\right\}},{P}_{\left\{28,30\right\}},{P}_{\left\{23,24\right\}},{P}_{\left\{31,32\right\}},{P}_{\left\{34,36\right\}},{P}_{\left\{33,38\right\}}.\end{gathered}\end{equation} \tag{ B11 }$$

Table 2. Kernaghan and Peres' 40 rays, where the ray r_i is represented by the vector in the ith column and $\bar{1}$ stands for −1.

1	0	0	0	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1	0	0	0	0
0	1	0	0	0	0	0	0	1	1	$\bar{1}$	$\bar{1}$	0	0	0	0	1	1	$\bar{1}$	$\bar{1}$	0	0	0	0	0	0	0	0	1	1	1	1	0	0	0	0	1	1	$\bar{1}$	$\bar{1}$
0	0	1	0	0	0	0	0	1	$\bar{1}$	1	$\bar{1}$	0	0	0	0	0	0	0	0	1	1	1	1	1	1	$\bar{1}$	$\bar{1}$	0	0	0	0	0	0	0	0	1	$\bar{1}$	1	$\bar{1}$
0	0	0	1	0	0	0	0	1	$\bar{1}$	$\bar{1}$	1	0	0	0	0	0	0	0	0	1	1	$\bar{1}$	$\bar{1}$	0	0	0	0	1	1	$\bar{1}$	$\bar{1}$	1	$\bar{1}$	1	$\bar{1}$	0	0	0	0
0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	$\bar{1}$	1	$\bar{1}$	0	0	0	0	1	$\bar{1}$	1	$\bar{1}$	0	0	0	0	0	0	0	0	$\bar{1}$	1	1	$\bar{1}$
0	0	0	0	0	1	0	0	0	0	0	0	1	1	$\bar{1}$	$\bar{1}$	1	$\bar{1}$	$\bar{1}$	1	0	0	0	0	0	0	0	0	1	$\bar{1}$	1	$\bar{1}$	1	1	$\bar{1}$	$\bar{1}$	0	0	0	0
0	0	0	0	0	0	1	0	0	0	0	0	1	$\bar{1}$	1	$\bar{1}$	0	0	0	0	1	$\bar{1}$	1	$\bar{1}$	1	$\bar{1}$	$\bar{1}$	1	0	0	0	0	$\bar{1}$	1	1	$\bar{1}$	0	0	0	0
0	0	0	0	0	0	0	1	0	0	0	0	1	$\bar{1}$	$\bar{1}$	1	0	0	0	0	1	$\bar{1}$	$\bar{1}$	1	0	0	0	0	1	$\bar{1}$	$\bar{1}$	1	0	0	0	0	1	1	1	1

The theorem consists of the following statements for any graph G, vertex weights $\mathbf{r}\in {\mathbb{N}}^{\vert V\left(G\right)\vert }$, and $m\in \mathbb{N}$. (i) d_π (G^r , 1) = d_π (G, r), (ii) χ_f(G^r , 1) = χ_f(G, r), (iii) $\bar{\vartheta }\left({G}^{\mathbf{r}},1\right)=\bar{\vartheta }\left(G,\mathbf{r}\right)$, (iv) χ_f(G, m r) = mχ_f(G, r), and (v) $\bar{\vartheta }\left(G,m\mathbf{r}\right)=m\bar{\vartheta }\left(G,\mathbf{r}\right)$.

• (a)  
  In the main text, above theorem 1, it was already shown, that any rank-one PR of G^r induces a rank-r PR of G and vice versa. Hence the assertion follows.
• (b)  
  For the chromatic number we also have χ(G^r , 1) = χ(G, r), as it follows by an argument completely analogous to the proof of d_π (G^r , 1) = d_π (G, r) (using colorings instead of projectors). This implies,

  $$\begin{equation}{\chi }_{f}\left(G,\mathbf{r}\right)={\mathrm{min}}_{b\in \mathbb{N}}\enspace \frac{\chi \left(G,b\mathbf{r}\right)}{b}={\mathrm{min}}_{b\in \mathbb{N}}\enspace \frac{\chi \left({G}^{\mathbf{r}},b\right)}{b}={\chi }_{f}\left({G}^{\mathbf{r}}\right).\end{equation} \tag{ C1 }$$
• (c)  
  By definition, the weighted Lovász number of $\bar{G}$ is calculated as

  $$\begin{equation}\vartheta \left(\bar{G},\mathbf{r}\right)={\mathrm{max}}_{\rho ,{\left({{\Pi}}_{k}\right)}_{k}}\sum\limits _{k\in V\left(\bar{G}\right)}{r}_{k}\text{tr}\left(\rho {{\Pi}}_{k}\right),\end{equation} \tag{ C2 }$$

  where the maximum is taken over all states ρ and all PRs ${\left({{\Pi}}_{k}\right)}_{k}$ of $\bar{G}$. However, if ${\left({{\Pi}}_{k}\right)}_{k}$ is a PR of $\bar{G}$ then ${\left({{\Pi}}_{k}\right)}_{k,i}$ is a (r-fold degenerate) PR of $\bar{{G}^{\mathbf{r}}}$, due to

  $$\begin{equation}E\left(\bar{{G}^{\mathbf{r}}}\right)=\left\{\left[\left(v,i\right),\left(w,j\right)\right]\vert \left[v,w\right]\in E\left(\bar{G}\right)\right\}.\end{equation} \tag{ C3 }$$

  Thus, $\vartheta \left(\bar{{G}^{\mathbf{r}}}\right){\geqslant}\vartheta \left(\bar{G},\mathbf{r}\right)$. Conversely, let ${\left({P}_{k,i}\right)}_{k,i}$ be any PR of $\bar{{G}^{\mathbf{r}}}$. For any state ρ we let ${P}_{k}^{\prime }={P}_{k,\hat{{\imath}}}$ for $\hat{{\imath}}$ the index that maximizes tr(ρP_k,i ). Then ${\left({P}_{k}^{\prime }\right)}_{k}$ is a PR of $\bar{G}$ and hence $\vartheta \left(\bar{G},\mathbf{r}\right){\geqslant}\vartheta \left(\bar{{G}^{\mathbf{r}}}\right)$.
• (d)  
  This follows directly from the definition in equation (A9) by substituting x_I by mx_I and r by m r.
• (e)  
  This follows at once from the definition in equation (A7).

The theorem consists of three statements: (i) $\bar{\text{RANK}}\left(G\right)$ is convex, (ii) $\text{STAB}\left(G\right)\subseteq \bar{\text{RANK}}\left(G\right)$, and (iii) $\bar{\text{RANK}}\left(G\right)\subseteq \text{TH}\left(G\right)$.

• (a)  
  For any vector p ∈ RANK(G) we can find a d-dimensional PR ${\left({{\Pi}}_{k}\right)}_{k}$ such that p_k = tr(Π_k )/d. With Π'_k and d' accordingly for p' ∈ RANK(G), we let

  $$\begin{equation}{{\Gamma}}_{k}=\left(1 {\mathbb{1}}_{{d}^{\prime }}\otimes {{\Pi}}_{k}\right)\oplus \left(\mathbb{1} {1}_{d}\otimes {{\Pi}}_{k}^{\prime }\right),\end{equation} \tag{ D1 }$$

  where A ⊕ B denotes the block-diagonal matrix with blocks A and B. By construction, ${\left({{\Gamma}}_{k}\right)}_{k}$ is a (2dd')-dimensional PR of G. Due to tr(Γ_k )/(2dd') = (d'tr(Π_k ) + dtr(Π'_k ))/(2dd') = (p_k + p'_k )/2 we have (p + p')/2 ∈ RANK(G). Iterating this argument, any point q p + (1 − q)p' with 0 ⩽ q ⩽ 1 is arbitrarily close to some element of RANK(G) since any such q can be arbitrarily well approximated by a fraction x/2ⁿ with $x,n\in \mathbb{N}$. Hence $\bar{\text{RANK}}\left(G\right)$ is convex.
• (b)  
  Any extremal point a of STAB(G) is given by some independent set I of G via a_v = 1 if v ∈ I and a_v = 0 else. Then ${\left({a}_{v}1 {1}_{d}\right)}_{v}$ is a d-dimensional PR with r = a d, that is, a ∈ RANK(G). Since STAB(G) is the convex hull of its extremal points and RANK(G) is convex, the assertion follows.
• (c)  
  By definition, RANK(G) consists of all probability assignments involving the completely depolarized state and TH(G) consists of all probability assignments for any quantum state. Since TH(G) is closed [30], the assertion follows.

The proof of theorem 3 is based on an exhaustive test of all graphs with no more than 8 vertices. Since the exact description of RANK(G) is difficult, we propose a linear relaxation of RANK(G) by using the dimension relations of union and intersection of subspaces. Note that each rank-r projector corresponds to an r-dimensional subspace. More explicitly, for a given d-dimensional projector Π, denote Π^s as the subspace spanned by all the vectors {Πv|∀v}. Then we know that

$$\begin{equation}\begin{gathered}\mathrm{dim}\left({{\Pi}}^{s}\right)=\text{rank}\left({\Pi}\right)=\text{tr}\left({\Pi}\right){\leqslant}d,\\ \mathrm{dim}\enspace {{\Pi}}_{1}^{s}+\mathrm{dim}\enspace {{\Pi}}_{2}^{s}=\mathrm{dim}\left({{\Pi}}_{1}^{s}+{{\Pi}}_{2}^{s}\right)+\mathrm{dim}\left({{\Pi}}_{1}^{s}\cap {{\Pi}}_{2}^{s}\right),\\ \mathrm{dim}\left({{\Pi}}_{1}^{s}\cap {{\Pi}}_{2}^{s}\right){\leqslant}\enspace \mathrm{min}\enspace \left\{\mathrm{dim}\enspace {{\Pi}}_{1}^{s},\mathrm{dim}\enspace {{\Pi}}_{2}^{s}\right\},\end{gathered}\end{equation} \tag{ E1 }$$

where ${{\Pi}}_{1}^{s}+{{\Pi}}_{2}^{s}=\left\{{\mathbf{v}}_{1}+{\mathbf{v}}_{2}\vert \forall {\mathbf{v}}_{1}\in {{\Pi}}_{1}^{s},{\mathbf{v}}_{2}\in {{\Pi}}_{2}^{s}\right\}$ and ${{\Pi}}_{1}^{s}\cap {{\Pi}}_{2}^{s}=\left\{\mathbf{v}\vert \mathbf{v}\in {{\Pi}}_{1}^{s}\quad \text{and}\quad \mathbf{v}\in {{\Pi}}_{2}^{s}\right\}$. To take more advantage of these relations, we consider the intersections of subspaces which are related to the projectors in the PR. Denote Π_I = ∩_i∈I Π_i for a given set I of vertices in G and let Π_∅ = 1 1. By definition, Π_I = 0 if I is not an independent set. This implies that ${{\Pi}}_{{I}_{1}}$ and ${{\Pi}}_{{I}_{2}}$ are orthogonal if I₁ ∪ I₂ is no longer an independent set for two given independent sets I₁, I₂.

For a given graph G, denote the set of all independent sets as $\mathcal{I}$. Then define the corresponding independent set graph $\mathcal{G}$ as the graph such that

$$\begin{equation}V\left(\mathcal{G}\right)={\left\{{v}_{I}\right\}}_{I\in \mathcal{I}},\quad E\left(\mathcal{G}\right)=\left\{\left[{v}_{{I}_{1}},{v}_{{I}_{2}}\right]\vert \quad \text{if}\quad {I}_{1}\cup {I}_{2}\notin \mathcal{I},{I}_{1},{I}_{2}\in \mathcal{I}\right\}.\end{equation} \tag{ E2 }$$

For example, if G = C₅ is the five-cycle graph, then the independent set graph $\mathcal{G}$ is as shown in figure 3.

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Denote $\mathcal{C}$ as the set of all cliques in $\mathcal{G}$. For a given clique $C\in \mathcal{C}$, denote H_C as the set of vertices in $V\left(\mathcal{G}\right)$ which are connected to all vertices in C. That is,

$$\begin{equation}{H}_{C}{:=}\enspace \left\{{v}_{I}\vert {v}_{I}\in V\left(\mathcal{G}\right),C\cup \left\{{v}_{I}\right\}\in \mathcal{C}\right\}.\end{equation} \tag{ E3 }$$

Then we have the following constraints on the PRs of G:

$$\begin{equation}\begin{gathered}{{\Pi}}_{{I}_{1}}^{s}\perp {{\Pi}}_{{I}_{2}}^{s}\quad \text{if}\enspace {v}_{{I}_{1}},{v}_{{I}_{2}}\in C{\Rightarrow}\sum\limits _{{v}_{I}\in C}\bar{\mathrm{dim}}\left({{\Pi}}_{I}^{s}\right){\leqslant}1,\forall C\in \mathcal{C},\\ {{\Pi}}_{{I}_{1}}^{s}+{{\Pi}}_{{I}_{2}}^{s}\subseteq {{\Pi}}_{{I}_{1}\cap {I}_{2}}^{s}{\Rightarrow}\sum\limits _{i=1,2}\bar{\mathrm{dim}}\left({{\Pi}}_{{I}_{i}}^{s}\right){\leqslant}\bar{\mathrm{dim}}\left({{\Pi}}_{{I}_{1}\cap {I}_{2}}^{s}\right)+\bar{\mathrm{dim}}\left({{\Pi}}_{{I}_{1}\cup {I}_{2}}^{s}\right),\forall {I}_{1},{I}_{2},\\ \forall {v}_{{I}_{1}},{v}_{{I}_{2}}\in {H}_{C}{\Rightarrow}{{\Pi}}_{{I}_{1}}^{s}+{{\Pi}}_{{I}_{2}}^{s}\perp \sum\limits _{{v}_{I}\in C}{{\Pi}}_{I}^{s}{\Rightarrow}\sum\limits _{{v}_{I}\in C}\bar{\mathrm{dim}}\left({{\Pi}}_{I}^{s}\right)+\sum\limits _{i=1,2}\bar{\mathrm{dim}}\left({{\Pi}}_{{I}_{i}}^{s}\right){\leqslant}1+\bar{\mathrm{dim}}\left({{\Pi}}_{{I}_{1}\cup {I}_{2}}^{s}\right),\end{gathered}\end{equation} \tag{ E4 }$$

where $\bar{\mathrm{dim}}\left({\Pi}\right)=\mathrm{dim}\left({\Pi}\right)/d$.

By combining all the constraints in equation (E4) with the non-negativity constraints, we have a polytope whose elements are possible values for ${\left\{\bar{\mathrm{dim}}\left({{\Pi}}_{I}^{s}\right)\right\}}_{I\in \mathcal{I}}$. If we only consider the possible values of ${\left\{\bar{\mathrm{dim}}\left({{\Pi}}_{\left\{{v}_{i}\right\}}\right)\right\}}_{{v}_{i}\in V\left(G\right)}$, then we have a linear relaxation of RANK(G). We denote such a linear relaxation as LRANK(G). Note that we can add extra constraints that $\mathrm{dim}\left({{\Pi}}_{I}\right)\in \mathbb{N},\forall I\in \mathcal{I}$ if we only focus on a specific dimension d.

For a given graph, we can calculate LRANK(G) as described above with computer programs. If LRANK(G) = STAB(G), then we know that RANK(G) = STAB(G). As it turns out, LRANK(G) = STAB(G) if G is a graph with no more than 8 vertices. Thus, we have proved theorem 3.

To have a closer look at this linear relaxation method, we illustrate it with odd cycles. It is known that STAB(G) = TH(G) if G is perfect [23], which means that those graphs cannot be used to reveal quantum contextuality. Recall that a graph is called perfect if all the induced subgraph of G are not odd cycles or odd anti-cycles [33]. Hence, odd cycles and odd anti-cycles are basic in the study of quantum contextuality [34]. Note that STAB(G) is a polytope which can be determined by the set of its facets I(G, w) = α(G, w), where w ⩾ 0. Each point outside of STAB(G) violates at least one of the tight inequalities, i.e., the inequalities defining the facets. For a given facet I(G, w) = α(G, w), if the subgraph of {i|w_i > 0} is a clique, then we say that this facet is trivial. This is because max  I(G, w) = 1 in both the NCHV case and the quantum case. Thus, we only need to consider the non-trivial tight inequalities one by one. For the odd cycle C_2n+1 in figure 4, the only non-trivial facet is [33]

$$\begin{equation}\sum\limits _{i=1}^{2n+1}P\left({\varepsilon }_{i}\right){\leqslant}n=\alpha \left({C}_{2n+1}\right).\end{equation} \tag{ E5 }$$

If ${\left\{{{\Pi}}_{i}\right\}}_{i=1}^{2n+1}$ is a PR of the odd cycle C_2n+1, then equation (E4) implies that

$$\begin{equation}\begin{gathered}\bar{\mathrm{dim}}\enspace \left({{\Pi}}_{1}^{s}\right)+\bar{\mathrm{dim}}\left({{\Pi}}_{2}^{s}\right)+\bar{\mathrm{dim}}\left({{\Pi}}_{2n+1}^{s}\right){\leqslant}1+\bar{\mathrm{dim}}\left({{\Pi}}_{\left\{2,2n+1\right\}}^{s}\right),\\ \bar{\mathrm{dim}}\left({{\Pi}}_{{I}_{k}}^{s}\right)+\bar{\mathrm{dim}}\left({{\Pi}}_{k+1}^{s}\right)+\bar{\mathrm{dim}}\left({{\Pi}}_{2n-k}^{s}\right){\leqslant}1+\bar{\mathrm{dim}}\left({{\Pi}}_{{I}_{k+1}}^{s}\right),\enspace \forall \enspace k=1,\dots ,n-2,\\ \bar{\mathrm{dim}}\left({{\Pi}}_{{I}_{n-1}}^{s}\right)+\bar{\mathrm{dim}}\left({{\Pi}}_{n+1}^{s}\right)+\bar{\mathrm{dim}}\left({{\Pi}}_{n+2}^{s}\right){\leqslant}1,\end{gathered}\end{equation} \tag{ E6 }$$

where ${I}_{k}={\cup }_{j=1}^{k}\left\{2j,2\left(n-j\right)+3\right\}$. Equation (E6) implies that, for any PR ${\left\{{{\Pi}}_{i}\right\}}_{i=1}^{2n+1}$,

$$\begin{equation}\sum\limits _{i=1}^{2n+1}\bar{\mathrm{dim}}\enspace \left({{\Pi}}_{i}^{s}\right){\leqslant}n.\end{equation} \tag{ E7 }$$

Thus, STAB(G) = RANK(G) if G is an odd cycle.

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Note that there exists a (d × n)-matrix Y such that R = Y^† Y if and only if R ⩾ 0 and rank(R) ⩽ d. Then, the fast implementation of the alternating optimization is based on the fact that the following two optimizations can be evaluated analytically:

$$\begin{equation}\begin{aligned}\hfill {\mathrm{min}}_{R}& \quad \hfill & \hfill & {\Vert}R-X{{\Vert}}_{F}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \hfill & \hfill & R{\geqslant}0,\enspace \text{rank}\left(R\right){\leqslant}d,\hfill \end{aligned}\end{equation} \tag{ F1 }$$

$$\begin{equation}\begin{aligned}\hfill {\qquad \mathrm{min}}_{L}& \quad \hfill & \hfill & {\Vert}L-X{{\Vert}}_{F}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \hfill & \hfill & {L}_{kk}=1,\enspace {L}_{k\ell }=0\enspace \forall \enspace \left[k,\ell \right]\in E\left(G\right),\hfill \end{aligned}\end{equation} \tag{ F2 }$$

where the Frobenius norm is defined as ∥M∥_F = tr(M^† M) = ∑_kℓ |M_kℓ |².

The first optimization can be solved using a semidefinite variant of the Eckart–Young–Mirsky theorem [35], which states that for any n × n matrix M, the best rank-d (more precisely, rank no larger than d) approximation with respect the Frobenius norm (that is, ${\mathrm{min}}_{\text{rank}\left({M}_{d}\right){\leqslant}d}{\Vert}{M}_{d}-M{{\Vert}}_{F}$) is achieved by

$$\begin{equation}{M}_{d}=U\enspace \mathrm{diag}\left({s}_{1},{s}_{2},\dots ,{s}_{d},0,\dots ,0\right){V}^{{\dagger}},\end{equation} \tag{ F3 }$$

where M = U diag(s₁, s₂, ..., s_n )V^† is the singular value decomposition of M, and the singular values satisfy that s₁ ⩾ s₂ ⩾⋯⩾ s_n ⩾ 0. We mention that M_d is not unique if s_d is a degenerate singular value. Now, let us consider the optimization in equation (F1). As X is Hermitian, it admits the decomposition X = X⁺ − X⁻, where X⁺ = P⁺ XP⁺ ⩾ 0, X⁻ = −P⁻ XP⁻⩾ 0, and

$$\begin{equation}\begin{aligned}\hfill {P}^{+}& =\sum\limits _{{\lambda }_{k}{\geqslant}0}\vert {\varphi }_{k}\rangle \langle {\varphi }_{k}\vert ,\hfill \\ \hfill {P}^{-}& =\sum\limits _{{\lambda }_{k}{< }0}\vert {\varphi }_{k}\rangle \langle {\varphi }_{k}\vert .\hfill \end{aligned}\end{equation} \tag{ F4 }$$

Here λ₁ ⩾ λ₂ ⩾⋯⩾ λ_n are the eigenvalues of X, and |φ_i ⟩ are the corresponding eigenvectors. Furthermore, let R⁺ = P⁺ RP⁺, R⁻ = P⁻ RP⁻, and

$$\begin{equation}{X}_{d}^{+}=\sum\limits _{k{\leqslant}d,{\lambda }_{k}{\geqslant}0}{\lambda }_{k}\vert {\varphi }_{k}\rangle \langle {\varphi }_{k}\vert ,\end{equation} \tag{ F5 }$$

then the optimization in equation F1 satisfies that

$$\begin{align}\hfill {\Vert}R-X{{\Vert}}_{F}& {\geqslant}{\Vert}{R}^{+}+{R}^{-}-{X}^{+}+{X}^{-}{{\Vert}}_{F}\hfill \\ \hfill & ={\Vert}{R}^{+}-{X}^{+}{{\Vert}}_{F}+{\Vert}{R}^{-}+{X}^{-}{{\Vert}}_{F}\hfill \\ \hfill & {\geqslant}{\Vert}{X}_{d}^{+}-{X}^{+}{{\Vert}}_{F}+{\Vert}{X}^{-}{{\Vert}}_{F},\hfill \end{align} \tag{ F6 }$$

where the first two lines follow from that ∥M∥_F ⩾ ∥P⁺ MP⁺ + P⁻ MP⁻∥_F = ∥P⁺ MP⁺∥_F + ∥P⁻ MP⁻∥_F , and the last line follows from the Eckart–Young–Mirsky theorem as well as the facts that rank(R⁺) = rank(P⁺ RP⁺) ⩽ rank(R)⩽d and ∥M₁ + M₂∥_F ⩾ ∥M₁∥_F when M₁, M₂ ⩾ 0. Moreover, one can easily verify that all inequalities in equation (F6) are saturated when $R={X}_{d}^{+}$, because ${P}^{+}{X}_{d}^{+}{P}^{+}={X}_{d}^{+}$ and ${P}^{-}{X}_{d}^{+}{P}^{-}=0$. By noting that ${X}_{d}^{+}$ satisfies that ${X}_{d}^{+}{\geqslant}0$ and $\text{rank}\left({X}_{d}^{+}\right){\leqslant}d$, we get that the optimization in equation (F1) is achieved when $R={X}_{d}^{+}$, which gives the solution

$$\begin{equation}\sum\limits _{k{\geqslant}d+1,{\lambda }_{k}{\geqslant}0}{\lambda }_{k}^{2}+\sum\limits _{{\lambda }_{k}{< }0}{\lambda }_{k}^{2}.\end{equation} \tag{ F7 }$$

The solution of the second optimization in equation (F2) follows directly from the definition of the Frobenius norm ∥M∥_F = ∑_kℓ |M_kℓ |². One can easily verify that the minimization is achieved when

$$\begin{equation}\begin{aligned}\hfill & {L}_{kk}=1,\quad \hfill & \hfill & k=1,2,\dots ,n\hfill \\ \hfill & {L}_{k\ell }=0,\quad \hfill & \hfill & \left[k,\ell \right]\in E\left(G\right),\hfill \\ \hfill & {L}_{k\ell }={X}_{k\ell },\quad \hfill & \hfill & k\ne \ell \quad \text{and}\quad \left[k,\ell \right]\notin E\left(G\right),\hfill \end{aligned}\end{equation} \tag{ F8 }$$

and the solution is

$$\begin{equation}\sum\limits _{k=1}^{d}{\left(1-{X}_{kk}\right)}^{2}+\sum\limits _{\left[k,\ell \right]\in E\left(G\right)}\vert {X}_{k\ell }{\vert }^{2}.\end{equation} \tag{ F9 }$$