Stateindependent proof of KochenSpecker theorem with 13 rays
Abstract
Quantum contextuality, as proved by Kochen and Specker, and also by Bell, should manifest itself in any state in any system with more than two distinguishable states and recently has been experimentally verified on various physical systems. However for the simplest system capable of exhibiting contextuality, a qutrit, the quantum contextuality is verified only statedependently in experiment because too many (at least 31) observables are involved in all the known stateindependent tests. Here we report an experimentally testable inequality involving only 13 observables that is satisfied by all noncontextual realistic models while being violated by all qutrit states. Thus our inequality will practically facilitate a stateindependent test of the quantum contextuality for an indivisible quantum system. We provide also a recordbreaking stateindependent proof of the KochenSpecker theorem with 13 directions determined by 26 points on the surface of a three by three magic cube.
It is believed, almost religiously, that every effect has its own cause and the same cause shall lead to the same effect. The predictions of quantum mechanics (QM) are however probabilistic and the effect that different outcomes appear in different runs of a measurement seem to have no definite cause, at least unexplainable using QM alone. Einstein, Podolsky, and Rosen (1) initiated a longlasting quest for a quantum reality by questioning the completeness of quantum mechanics. Hidden variable (HV) models are introduced in order to explain why a certain outcome appears in each run of a measurement, attempting to make QM complete. Years later Kochen, Specker (2), and Bell (3) discovered that quantum mechanics can be completed only by a hidden variable model that is contextual: the outcome of a measurement depends on which compatible observable might be measured alongside. Simply put, KochenSpecker (KS) theorem states that noncontextual HV models cannot reproduce all the predictions of QM or quantum mechanics is contextual.
In any noncontextual HV model all observables have definite values determined only by some HVs that are distributed according to a given probability distribution with normalization . Two observables are compatible if they can be measured in a single experimental setup and a maximal set of mutually compatible observables defines a context. Noncontextuality is a typical classical property: the value of an observable revealed by a measurement is predetermined by HVs only regardless of which compatible observable might be measured alongside. Local realism is a form of noncontextuality enforced by the locality and thus Bell’s inequalities (4) are a special form of KS inequalities (5); (6); (7); (8); (9), experimentally testable inequalities that are satisfied by all noncontextual HV models, some of which have been tested in recent measurements (10); (11); (12); (13); (14); (15); (17); (16); (18). In general KS inequalities reveal the nonclassical nature of single systems demanding neither spacelike separation nor entanglement, i.e., independent of state.
However for the simplest system capable of exhibiting the quantum contextuality, a qutrit, only a statedependent verification has been made in a recent experiment (18). This is because all the known stateindependent KS inequalities for a qutrit (9) are based on the proofs of KS theorem and involve too many observables to be tested practically. For examples the original KS proof involves 117 directions (2) and Peres’ (19) and Schuttle’s (20) proofs involve 33 directions while the best KS proof known so far is due to Conway and Kochen (21) in which 31 directions are still involoved. Also there are many statedependent KS proofs among which a 5direction proof (7) has been used in the recent statedependent experimental verification of quantum contextuality for qutrit. In this Letter we report a stateindependent proof of KS theorem for qutrit with only 13 directions. Based on this proof we propose an experimental testable inequality that involves only 13 observables and twoobservable correlations and is satisfied by all noncontextual HV models while being violate by all qutrit state. Thus our inequality will make a stateindependent test of quantum contextuality for an indivisible system practical.
How can we exclude noncontextual HV models for QM or prove the quantum contextuality? Obviously the answer depends on what kinds of quantum mechanical predictions we want the HV model to reproduce. For example if we only want the predictions on nonsequential measurements, i.e., correlations not included, to be reproduced, then a noncontextual HV model does exist according to Kochen and Specker (2). Because of this toy model Kochen and Specker imposed a rather strong constraint on the HV models as a way out (2): the algebraic structure of compatible observables must be preserved. Especially the value assigned to the product or the sum of two compatible observables must be equal to the product or the sum of the values assigned to these two compatible observables, which will be referred to as the product rule and the sum rule respectively. As we shall see later this constraint can be lifted if we consider sequential measurements.
As a result of the product rule the value assigned to the product of two orthogonal rays, normalized rank1 projections, which are compatible, must be zero. As a result of the sum rule there is one and only one ray that is assigned to value 1 among all the rays in a complete orthonormal basis since the identity is always assigned to value 1. Thus in every noncontextual HV model preserving the partial algebraic structure of compatible observables there exists a KS value assignment to all rays in the corresponding Hilbert space satisfying:

The value assigned to a ray is independent of which bases it finds itself in;

One and only one ray is assigned to value 1 among all the rays in a complete orthonormal basis.
The first condition reflects the noncontextuality and the second condition arises from the requirement that the algebraic structure of compatible observables be preserved. For a Hilbert space of a dimension greater than 2 there always exists a finite set of rays to which the KS value assignment is impossible. For qutrits, a stateindependent proof originally involves 117 rays (2) and the number is reduced to 33 by Peres (19) and Schütte as reported by K. Svozil in 1994 and pointed out by Bub (20). The best KS proof known so far is given by Conway and Kochen (21) with 31 rays. For 4state systems the best stateindependent proof is due to Cabello, Estebaranz, and GarcíaAlcaine (22) with 18 rays, the smallest stateindependent KS proof known so far.
To warm up let us present a stateindependent proof of KS theorem for qutrit using only 13 rays. In a given basis we shall represent a qutrit ray by a triple such that . Consider the following 13 rays
(1) 
that are determined by 26 points on the surface of a magic cube as illustrated in Fig.1. If we regard those 13 rays as 13 vertices and link two vertices if and only if the corresponding rays are orthogonal, then we obtain the orthogonality graph as shown in Fig.2. Obviously a given set of rays determines uniquely the orthogonality graph and usually not vice versa. However those 13 rays are determined uniquely by the orthogonality relationships specified by the graph up to a global unitary transformation.
In fact without loss of generality we can choose as in Eq.(1) since they form a basis. Because are mutually orthogonal for each there exist nonzero such that and , and , and . As a result we have , , and . Since is orthogonal to for we have , , and from which it follows that and , i.e., for some real . Finally we obtain which is orthogonal to . The diagonal unitary transformation taking to leaves unchanged so that the standard form of 13 rays in Eq.(1) is obtained.
The KS value assignments to the 13ray set are possible, i.e., no logical contradiction can be extracted by considering conditions 1 and 2 only. However in any possible KS value assignment there is at most one ray among that can be assigned to value 1. Suppose that this is not true, i.e., there are two or more that are assigned to value 1. Due to the symmetry of the graph as shown in Fig.2, we need only to consider the following two cases:

if and are assigned to value then and must be assigned to 0 so that both and must be assigned to value 1 which is impossible;

If and are assigned to 1 then and must be zero so that both and must be assigned to value 1, also a contradiction.
In the reasonings above we have taken into account of condition 2 which demands that linked rays not be assigned simultaneously to value 1 and in a triangle one and only one ray be assigned to value 1. A set of eight rays in each case considered above, e.g., , , and , constitutes in fact a 3box paradox (23) which appears also in Cliffton’s statedependent proof of KS theorem (24) and a recent proposal to close the compatibility loopholes (25).
Denoting by the value assigned to for given , we then have from the above arguments. As a result the following inequality
(2) 
must be satisfied by all noncontextual HV models that admit a KS value assignment. However the quantum mechanics predicts due to the identity with being the identity operator of qutrit (see Appendix). This proves the original KS theorem: the noncontextual HV model satisfying conditions 1 and 2 cannot reproduce all the predictions of QM on nonsequential measurements.
Usually the KS theorem is proved by finding a set of rays to which the KS value assignment does not exist so that we need not to check other predictions of QM. Our proof here is a set of 13 rays, to which all possible KS value assignments, which do exist, fail to reproduce a certain prediction of QM. Notably the inequality Eq.(2), involving only 4 observables explicitly, provides a stateindependent test for the noncontextual HV models that preserve the algebraic structure of compatible observables, in the spirit of Kochen and Specker. In an experimental test of the inequality Eq.(2) each average must be measured on different subensembles prepared in the same state, similar to a standard test of a Bell inequality as pointed out by Cabello (8).
It turns out that the requirement of preserving the algebraic structure, i.e., the product and sum rules, of compatible observables is too strong and unnecessary. Peres already noticed that the sum rule can be abandoned and kept the product rule (27) as in KS proofs via the MerminPeres square (19); (26) for 2 qubits and Mermin’s pentagram (26) for 3 qubits. However the product rule is not necessary either. Instead we need only to require that the quantum correlations of compatible observables, the quantum mechanical predictions on sequential measurements, be reproduced. This imposes no additional constraints on the noncontextual HV models since they must reproduce all the predictions of QM in the first place. As noticed earlier, the toy model by Kochen and Specker did not take into account of the quantum correlations of compatible observables.
Indeed every KS proof mentioned previously can be turned into a stateindependent KS inequality (9) that is obeyed by all noncontextual HV models, no matter whether the algebraic structure of compatible observables is preserved or not. All known stateindependent KS inequality involve the correlations of at least three compatible observables, i.e., predictions on three or more sequential measurements must be reproduced by the HV models. In the case of qutrit the resulting KS inequalities involve too many observables, e.g., at least 31, to be tested experimentally. Recently Klyachko, Can, Binicioglu, and Shumovskya (7) propose a simple KS inequality, called pentagram inequality since it is based on the graph of a pentagon, to test noncontextual HVs with only 5 dichotomic observables and correlations of two compatible observables. However this pentagram inequality, which is based on the assumption of noncontextuality only and whose violation has been verified in a recent experiment (18), has the drawback of being statedependent.
Our stateindependent inequality is based on the orthogonality graph as shown in Fig.2. We denote by its vertex set and by Its adjacency matrix which is a symmetric matrix with vanishing diagonal elements and if two vertices are neighbors and otherwise. For arbitrary 13 variables with it holds
(3) 
which can be easily verified with the help of a laptop by exhausting all possibilities and an analytic proof is provided in Appendix. Let be a set of 13 dichotomic observables taking values for given . Then from inequality Eq.(3) we obtain our magiccube inequality
(4) 
where we have denoted and . Though the correlation of two observables is always well defined in a noncontextual HV model regardless of whether they are compatible or not, in order to compare with the predictions of QM those observables labeled with linked vertices in the orthogonality graph should be compatible, i.e, they can be measured simultaneously or be commuting, so that their correlation is also well defined in QM.
Fortunately if we define 13 observables from 13 rays , then and are commuting. i.e., compatible, whenever two vertices are linked, i.e., . Therefore all the expectation values appear on the lefthandside of Eq.(4) are also well defined in QM and therefore should be reproduced. Due to the linearity of QM the lefthandside of Eq.(4) must reproduce the expectation value of
(5) 
Simple calculations yield and therefore for any qutrit state, meaning that the magiccube inequality Eq.(4) is violated by all qutrit states. Thus we have proved in a stateindependent fashion that every noncontextual HV model, no matter whether the algebraic structure of compatible observables is preserved or not, cannot reproduce all the predictions of QM, especially those quantum correlations of two compatible observables, by using only 13 observables.
To summarize, our magiccube inequality Eq.(4) provides the simplest stateindependent proof of quantum contextuality: Firstly it is a test for the contextuality for an indivisible system capable of exhibiting the quantum contextuality, i.e., smallest system; Secondly it involves a smallest number of observables so far and we conjecture that KS theorem does not have a stateindependent proof with less than 13 rays; Thirdly only correlations of at most two, the smallest number, of compatible observables are involved so that only two observables have to be measured sequentially. In comparison correlations of at least three compatible observables are involved in all the stateindependent tests of contextuality known so far. If we impose a minimal requirement on the noncontextual HV models, i.e, only the correlations of at most two compatible observables are required to be reproduced, then our magiccube inequality will be the first stateindependent proof of KS theorem in this case. We believe that there are also proofs of KS theorem for higher dimensional systems that involve only twoobservable correlations. Finally we have proved the KS theorem by finding a 13ray set, which is the smallest stateindependent proof so far, for which all possible KS value assignments, which do exist, fail to reproduce a certain prediction of QM. Since only a relatively small number of observables and correlations of only two compatible observables are involved, an experimental test of our inequality is well within the reach of the current technologies. We believe that the nitrogenvacancy center in diamond (28) should be a promising choice to test our inequality.
This work is supported by National Research Foundation and Ministry of Education, Singapore (Grant No. WBS: R710000008271) and NSF of China (Grant No. 11075227).
Appendix
Explicit rays for Eq.(1)— By denoting we have
Thus we have identities
(6) 
Let with . Since is the number of the edges in , is the degree of the vertex , and if , it holds
(7)  
(8)  
(9)  
(10) 
Proof of Eq.(4)— There are 9 vertices of degree 4 and 4 vertices of degree 3 in , where the degree of a vertex denotes the number its neighbors. Let be the total number of that take value , be the number of that take value with being of degree 4, and be the number of pairs of vertices such that and . We then have . Since the quadratic term in is unchanged under replacements , we need only to consider . If , since and , then we have . If , then from since it is impossible to have and , i.e., among any 4 vertices of degree 4 there is at least a connected pair, it follows , which is attained when and , e.g., and . In the case of , if then so that . If then so that , which is attained by, e.g., and . In the case of , if then so that ; If then we have again because ; If then since .
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