⇦Chinese Physics Letters, 2019, Vol. 36, No. 8, Article code 080304 Detecting Quantumness in the $n$-cycle Exclusivity Graphs * Jie Zhou (周洁), Hui-Xian Meng (孟会贤), Jing-Ling Chen (陈景灵)** Affiliations Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071 Received 27 May 2019, online 22 July 2019 *Supported by the Nankai Zhide Foundation, the National Science Foundation for Post-doctoral Scientists of China under Grant No 2018M631726, the National Natural Science Foundation of China under Grant No 11875167, and the Fundamental Research Funds for the Central Universities under Grant No 63191507.
**Corresponding author. Email: chenjl@nankai.edu.cn Citation Text: Zhou J, Meng H X and Chen J L 2019 Chin. Phys. Lett. 36 080304    Abstract Quantum contextuality is one kind of quantumness that distinguishes quantum mechanics from classical theory. As the simplest exclusivity graph, quantum contextuality of the $n$-cycle graph has been reviewed, while only for odd $n$ the quantumness can be revealed. Motivated by this, we propose the degree of non-commutativity and the degree of uncertainty to measure the quantumness in the $n$-cycle graphs. As desired, these two measures can detect the quantumness of any $n$-cycle graph when $n\ge 4$. DOI:10.1088/0256-307X/36/8/080304 PACS:03.65.Ta, 03.65.Ud, 03.67.Mn © 2019 Chinese Physics Society Article Text Quantumness is an important feature that distinguishes quantum mechanics (QM) from classical theory. In different situations, quantumness may have different manifestations. For instance, quantumness can be quantum entanglement,[1] the Bell non-locality[2] and the Einstein–Podolsky–Rosen (EPR) steering[3] when one considers the non-classicality in the bipartite or multipartite systems. For a single-particle system, quantumness could be reflected by quantum contextuality (QC).[4] Similar to the Bell non-locality that can be revealed by the violation of the Bell inequality, QC can be revealed through the violation of the non-contextuality inequality, which is satisfied by any non-contextual-hidden-variable (NCHV) models.[5–10] Very recently, the Cabello–Severini–Winter graph-theoretical approach[11] has been introduced to study QC with the help of the exclusivity graph. For a single-particle system, suppose that we perform $n$ projective measurements ${\hat P}_i=|v_i\rangle\langle v_i|(i=1,2,\ldots, n)$ on it, and for some measurements they are mutually orthogonal (e.g., $|v_i\rangle\perp|v_j\rangle$), then we shall have an exclusivity graph $G$, which has $n$ vertices, and the $i$th vertex connects with the $j$th vertex if $|v_i\rangle\perp|v_j\rangle$ (namely, there is an edge between vertex $i$ and vertex $j$). The simplest exclusivity graph is the $n$-cycle exclusive graph as shown in Fig. 1, where $|v_i\rangle\perp|v_{i+1} \rangle$ for the $n$ projective measurements. The corresponding non-contextuality inequality for arbitrary $n$-cycle graph is given by $$\begin{align} I_{n}=\sum_{i=1}^{n}P_i\overset{\rm NCHV}\to{\leqslant} \alpha_n \overset{\rm QM}\to{\leqslant} Q_n,~~ \tag {1}, \end{align} $$ which is just the generalized Klyachko–Can–Binicioğlu–Shumovsky (KCBS) inequality.[7,12–15] Here $$\begin{align} \alpha_n=\frac{n-1}{2}~~ \tag {2} \end{align} $$ denotes the classical bound for the NCHV models, $P_i=|\langle \psi|v_i\rangle|^2$ is the probability of performing $\hat{P}_i$ on the state $|\psi\rangle$, and $Q_n=I^{\max}_n=\max_{\{|\psi\rangle, |v_i\rangle\}}\sum_{i=1}^{n}|\langle \psi|v_i\rangle|^2$ is the maximal violation of the inequality under the constraint of the orthogonal relations for the vectors $|v_i\rangle$. For the $n$-cycle graphs ($n\ge 4$), we have $$ Q_n=\left\{ \begin{array}{cc} \alpha_n, &n={\rm even},\\ \frac{n\cos(\frac{\pi}{n})}{1+\cos(\frac{\pi}{n})}, &n={\rm odd}. \end{array} \right.~~ \tag {3} $$
cpl-36-8-080304-fig1.png
Fig. 1. The $n$-cycle exclusivity graph. The $i$th projective measurement is orthogonal to the $(i+1)$th projective measurement, i.e., $|v_i\rangle\perp|v_{i+1}\rangle$, or ${\rm Tr}[\hat{P}_i\;\hat{P}_{i+1}]=0$.
In Ref.  [16], the degree of quantum contextuality has been introduced to describe the quantumness of the exclusivity graph. The definition is given by $$\begin{align} r_n=\frac{Q_n}{\alpha_n},~~ \tag {4} \end{align} $$ which is the ratio between the maximal violation $Q_n$ and the classical bound $\alpha_n$. In general, one obtains $r_n\ge 1$. The larger the $r_n$, the stronger the quantum contextuality. Here $r_n=1$ means no quantum contextuality. Based on Eqs. (3) and (4), it is easy to have $$ r_n=\left\{\begin{array}{cc} 1, & n={\rm even},\\ \frac{2n \cos(\pi/n)}{(n-1)(1+\cos(\pi/n))}>1,& n={\rm odd}. \end{array}\right.~~ \tag {5} $$ The physical meaning of Eq. (5) implies that if one uses quantum contextuality to describe the quantumness of the $n$-cycle graph (or, the quantumness of a single-particle system performed by $n$ projective measurements), then the quantumness will exist for the $n$=odd case, but will not exist for the $n$=even case. This gives rise to a natural question: can one find an alternative definition such that for any $n$-cycle graph the quantumness can always be detected? The answer is positive, and this is also the motivation of our present work. In the next step, we shall detect the quantumness of an arbitrary $n$-cycle ($n\ge 4$) graph through the degree of non-commutativity. The degree of uncertainty is also discussed to reveal the quantumness. For a pair of operators $\hat{A}$ and $\hat{B}$, the commutator can be written as $$\begin{align} [\hat{A}, \hat{B}]=\hat{A} \hat{B}-\hat{B} \hat{A}.~~ \tag {6} \end{align} $$ In the classical world, all mechanical quantities are commutative, i.e., $[\hat{A}, \hat{B}]=0$, but in quantum mechanics, some mechanical quantity operators are non-commutative, i.e., $[\hat{A}, \hat{B}]\neq 0$. Therefore, the commutator becomes the key property to distinguish between quantum mechanics and classical theory. For the $n$-cycle exclusive graph, as shown in Fig. 1, there are totally $n$ projective measurements $\hat{P}_j\; (j=1, 2,\ldots, n)$. For any two adjacent vertices $j$ and $j+1$, because $|v_j\rangle\perp|v_{j+1}\rangle$ or $|\langle v_j|v_{j+1}\rangle|=0$, $\hat{P}_j$ commutes with $\hat{P}_{j+1}$, i.e., $$\begin{align} [\hat{P}_j, \hat{P}_{j+1}]=0,~~(j=1,2,\ldots,n),~~ \tag {7} \end{align} $$ and $\hat{P}_{n+1}\equiv\hat{P}_{1}$. For any two non-adjacent vertices $j$ and $k\neq j+1$, in general, one has the non-zero commutator as $[\hat{P_j}, \hat{P_k}]\neq 0$. When the single-particle system is in the state $|\psi\rangle$, we can have the average value for commutator relation between operators $\hat{P_j}$ and $\hat{P_k}$ as $$\begin{align} \mathcal{C}_{j,k}=\Big|\frac{1}{2 i}\langle[\hat{P}_j, \hat{P}_k]\rangle\Big|=\frac{1}{2 i}{\rm Tr}([\hat{P}_j, \hat{P}_k]\cdot\rho),~~ \tag {8} \end{align} $$ where $i=\sqrt{-1}$ is the imaginary unit, and $\rho=|\psi\rangle\langle\psi|$. Based on this, we then define the degree of non-commutativity for the $n$-cycle graph as follows: $$\begin{align} \mathcal{C}_n=\max_{\{|\psi\rangle, |v_j\rangle, |v_k\rangle\}}\sum_{j=1}^{n-1}\sum_{k>j}^{n}\mathcal{C}_{j,k}.~~ \tag {9} \end{align} $$ When we numerically compute the maximal value of $\mathcal{C}_n$, it is sufficient to consider the four-dimensional Hilbert space. In this case, the state of the quantum system is always taken as $$\begin{align} |\psi\rangle=(1,0,0,0)^{\rm T},~~ \tag {10} \end{align} $$ and the form of the vectors is parameterized as $$\begin{align} |v_j\rangle =\,&(e^{i x_j}\sin \theta_j \sin \phi_j \sin \gamma_j, e^{i y_j}\sin \theta_j \sin \phi_j \cos \gamma_j,\\ &e^{i z_j}\sin \theta_j \cos \phi_j, \cos \theta_j)^{\rm T} \\ =&(a, b, c, d)^{\rm T},~~ \tag {11} \end{align} $$ where superscript T represents the matrix transpose. By running all possible $|v_j\rangle$ under the constraint $|\langle v_j|v_{j+1}\rangle|=0$, one may obtain the optimal measurement directions for $|v_j\rangle$ and $\mathcal{C}_n^{\max}$ in Eq. (9). For the first few $n$, we obtain the degree of non-commutativity $\mathcal{C}_n^{\max}$ as listed in Table 1. The optimal measurement vectors are listed in Tables 3–8. It can be observed that (i) the degree of non-commutativity $\mathcal{C}_n^{\max}$ increases with $n$, and (ii) even for $n$ = even, the quantumness of the $n$-cycle graph can be detected with the non-zero $\mathcal{C}_n^{\max}$.
Table 1. The values of $\mathcal{C}_n^{\max}$ for $n$ = 4–9.
$n$ 4 5 6 7 8 9
$\mathcal{C}_n^{\max}$ 0.25 0.8274 1.36164 2.13731 2.95301 3.95931
Quantum contextuality and the degree of non-commutativity are not the only ways to detect the quantumness of the $n$-cycle graph. Next, we propose a new measure, the degree of uncertainty, to characterize the quantumness of the $n$-cycle graph. In 1927, Heisenberg stated that the more precisely the position $\hat{X}$ of particle is determined, the less precisely its momentum $\hat{P}$ can be known, and vice versa.[17] Later, Kennard gave the mathematical expression of the uncertainty relationship between position and momentum $\Delta \hat{X} \Delta \hat{P} \ge \hbar/2$, where $\hbar$ is the reduced Planck constant.[18] In 1929, Robertson gave the most common general form of the uncertainty principle.[19] The uncertainty principle[19] is expressed as $$\begin{align} \Delta A \Delta B \ge \Big|\frac{1}{2i}\langle[\hat{A}, \hat{B}]\rangle\Big|,~~ \tag {12} \end{align} $$ where $\Delta A\equiv \sqrt{\langle (\Delta \hat{A})^2\rangle}$, and $\Delta B\equiv \sqrt{\langle (\Delta \hat{B})^2\rangle}$. For any state $|\psi\rangle$, the average value of mechanical quantity $\hat{A}$ is $\langle \hat{A}\rangle=\langle \psi |\hat{A}| \psi \rangle$, and the degree of dispersion of the observed values can be written as the standard deviation of $\hat{A}$, i.e., $\langle (\Delta \hat{A})^2\rangle=\langle \psi |(\hat{A}-\langle \hat{A}\rangle)^2|\psi \rangle$. Similar to Eq. (9), we can define the total degree of uncertainty for the $n$-cycle graph as follows: $$\begin{align} \mathcal{U}_n=\max_{\{|\psi\rangle, |v_j\rangle, |v_k\rangle\}}\sum_{j=1}^{n-1}\sum_{k>j}^{n}\mathcal{U}_{j,k},~~ \tag {13} \end{align} $$ with the degree of uncertainty between two projectors $\hat{P_j}$ and $\hat{P_k}$ as $$\begin{align} \mathcal{U}_{j,k}=\Delta P_j \Delta P_k.~~ \tag {14} \end{align} $$ The uncertainty relation (12) implies that $\mathcal{U}_{j,k} \ge \mathcal{C}_{j,k}$, thus we have $$\begin{align} \mathcal{U}_n^{\max} \ge \mathcal{C}_n^{\max},~~ \tag {15} \end{align} $$ which indicates that the degree of non-commutativity is the lower bound of the degree of uncertainty. However, even numerically it is very hard to obtain the maximal value $\mathcal{U}_n^{\max}$. In Table 2, we list the non-optimal values of $\mathcal{U}_n$, which share the same measurement vectors that yield $\mathcal{C}_n^{\max}$. Based on Table 2, we also plot Fig. 2. The result shows that (i) both the degree of uncertainty (red line) and the degree of non-commutativity (yellow line) increase with $n$ and non-zero (due to commutativity, the classic value are 0), and (ii) for a fixed $n$, the degree of uncertainty is larger than the degree of non-commutativity.
cpl-36-8-080304-fig2.png
Fig. 2. The degree of uncertainty (red line) and the degree of non-commutativity (yellow line) versus the number of vertices $n$.
Table 2. The values of $\mathcal{U}_n$ and $\mathcal{C}_n^{\max}$ for $n$=4–9.
$n$ 4 5 6 7 8 9
$\mathcal{U}_n$ 0.81269 1.870073 3.1505 4.99842 6.89215 9.40963
$\mathcal{C}_n^{\max}$ 0.25 0.8274 1.36164 2.13731 2.95301 $3.95931$
Additionally, we present the optimal measurement vectors of the $n$-cycle graphs for $n$=4–9.
Table 3. The optimal measurement vectors for $n=4$.
$n=4$ $a$ $b$ $c$ $d$
$|v_1\rangle$ $0.70665+0.0254066 i$ $-0.273116-0.652233 i$ 0 0
$|v_2\rangle$ 0 0 $0.868503+0.297489 i$ $0.396488$
$|v_3\rangle$ $-0.466525-0.53137 i$ $-0.646286-0.286906 i$ 0 0
$|v_4\rangle$ 0 0 $0.586517+0.515392 i$ $-0.624795$
Table 4. The optimal measurement vectors for $n=5$.
$n=5$ $a$ $b$ $c$ $d$
$|v_1\rangle$ $0.18768+0.490924 i$ $0.12785+0.403276 i$ $-0.206286+0.581194 i$ $0.405527$
$|v_2\rangle$ $0.397579+0.367669 i$ $0.0879838+0.198232 i$ $-0.457812+ 0.285688 i$ $0.607046$
$|v_3\rangle$ $0.398287-0.432446 i$ $0.371809+0.312584 i$ $0.456885+ 0.45049 i$ $0.0819882$
$|v_4\rangle$ $0.366946-0.398246 i$ $0.423918+0.185457 i$ $-0.0139384+ 0.267278 i$ $0.648859$
$|v_5\rangle$ $0.0607413-0.522054 i$ $0.299907+0.382791 i$ $0.0207791+ 0.551671 i$ $0.427229$
Table 5. The optimal measurement vectors for $n=6$.
$n=6$ $a$ $b$ $c$ $d$
$|v_1\rangle$ $-0.436403+0.252662 i$ $0.095235+0.276497 i$ $-0.057475+ 0.444261 i$ $0.677881$
$|v_2\rangle$ $-0.147543+0.490076 i$ $0.273338+0.411367 i$ $0.354141+ 0.36426 i$ $0.485818$
$|v_3\rangle$ $-0.210603+0.594956 i$ $0.152935+0.267333 i$ $0.0598766+ 0.0937419 i$ $0.703167$
$|v_4\rangle$ $0.221295+0.591063 i$ $-0.115864+0.153754 i$ $-0.151426+ 0.241731 i$ $0.695158$
$|v_5\rangle$ $0.174355+0.48119 i$ $-0.398965+0.470242 i$ $-0.182013+ 0.328311 i$ $0.46566$
$|v_6\rangle$ $0.444776+0.237613 i$ $0.0264764+0.494313 i$ $-0.11967+ 0.172508 i$ $0.675713$
Table 6. The optimal measurement vectors for $n=7$.
$n=7$ $a$ $b$ $c$ $d$
$|v_1\rangle$ $-0.394285+0.296713 i$ $0.546027+0.30655 i$ $-0.468963+ 0.380074 i$ 0
$|v_2\rangle$ $0.307976-0.448579 i$ $-0.4599-0.608007 i$ $0.145245-0.318829 i$ 0
$|v_3\rangle$ $-0.045416+0.589955 i$ $0.61183+0.0287591 i$ $-0.460073+ 0.25112 i$ 0
$|v_4\rangle$ $0.0322225-0.582536 i$ $-0.301126-0.475756 i$ $0.519879- 0.268921 i$ 0
$|v_5\rangle$ $0.111495+0.581101 i$ $0.196411+0.220302 i$ $-0.750142+ 0.008229 i$ 0
$|v_6\rangle$ $-0.410023-0.357705 i$ $-0.0810111-0.535883 i$ $0.358147- 0.530966 i$ 0
$|v_7\rangle$ $-0.467464-0.15804 i$ $-0.34308-0.383825 i$ $0.615884- 0.334906 i$ 0
Table 7. The optimal measurement vectors for $n=8$.
$n=8$ $a$ $b$ $c$ $d$
$|v_1\rangle$ $-0.499006+0.0216048 i$ $-0.211461+0.463812 i$ $0.439241+ 0.303319 i$ $-0.453602$
$|v_2\rangle$ $0.52228-0.00710324 i$ $0.144103-0.612403 i$ $0.00415181- 0.483616 i$ $0.312199$
$|v_3\rangle$ $0.520612-0.356394 i$ $-0.0904396-0.265975 i$ $-0.361235- 0.462783 i$ $0.422333$
$|v_4\rangle$ $-0.21583+0.524012 i$ $-0.114434+0.273583 i$ $0.434874+ 0.426784 i$ $-0.468643$
$|v_5\rangle$ $0.227886-0.518882 i$ $0.221532-0.204758 i$ $-0.349157- 0.382708 i$ $0.5652$
$|v_6\rangle$ $-0.149217-0.613015 i$ $0.200654-0.261108 i$ $-0.236598- 0.202355 i$ $0.629746$
$|v_7\rangle$ $-0.0416309-0.520667 i$ $0.308124-0.238458 i$ $0.211678- 0.28708 i$ $0.669438$
$|v_8\rangle$ $-0.262461-0.424956 i$ $0.181185-0.0945268 i$ $-0.155976- 0.598985 i$ $0.570659$
Table 8. The optimal measurement vectors for $n=9$.
$n=9$ $a$ $b$ $c$ $d$
$|v_1\rangle$ $-0.00251992+0.486816 i$ $0.238577-0.0176134 i$ $-0.0970935+ 0.663679 i$ $0.505844$
$|v_2\rangle$ $0.0692973+0.543341 i$ $0.388129+0.19344 i$ $-0.0698194+ 0.261011 i$ $0.662505$
$|v_3\rangle$ $-0.0230431+0.604249 i$ $0.274295+0.166844 i$ $-0.482856+ 0.302757 i$ $0.454386$
$|v_4\rangle$ $0.418232+0.391532 i$ $0.349656+0.092302 i$ $-0.244857+ 0.524417 i$ $0.453913$
$|v_5\rangle$ $0.410871+0.39902 i$ $0.143813+0.138889 i$ $-0.477104+ 0.569262 i$ $0.283387$
$|v_6\rangle$ $-0.572339+0.0253528 i$ $-0.355996-0.133033 i$ $-0.0301123- 0.636497 i$ $-0.348309$
$|v_7\rangle$ $0.521094-0.306771 i$ $0.203391+0.231058 i$ $0.0928997+ 0.675602 i$ $0.272999$
$|v_8\rangle$ $0.379493-0.394974 i$ $0.517077+0.312333 i$ $0.157038+ 0.356729 i$ $0.427953$
$|v_9\rangle$ $0.303225-0.380859 i$ $0.459073+0.148583 i$ $-0.168143+ 0.662693 i$ $0.250483$
In conclusion, we have studied quantumness in the $n$-cycle graphs. The degree of non-commutativity is presented, which can be used to detect all quantumness of an arbitrary $n$-cycle graph ($n\ge 4$). The degree of uncertainty is also discussed, which is an upper bound of the degree of non-commutativity, and can also be used to naturally detect all quantumness of an arbitrary $n$-cycle graph. These two new measures have an advantage over quantum contextuality, which can merely detect the odd $n$-cycle graph. Our result sheds new light in understanding the non-classicality in quantum mechanics. References Quantum entanglementBell nonlocalitySteering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen ParadoxA Kochen-Specker inequalityOptimal Inequalities for State-Independent ContextualityAll noncontextuality inequalities for the n -cycle scenarioMinimum detection efficiency for the loophole-free confirmation of quantum contextualityConnecting Quantum Contextuality and Genuine Multipartite Nonlocality with the Quantumness WitnessExperimental simulation of violation of the Wright inequality by coherent lightGraph-Theoretic Approach to Quantum CorrelationsSimple Test for Hidden Variables in Spin-1 SystemsPentagrams and ParadoxesSpecker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementaritySimple Hardy-Like Proof of Quantum ContextualityQuantum theory allows for absolute maximal contextualityThe Uncertainty Principle
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