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 state-dependently in experiment because too many (at least 31) observables are involved in all the known state-independent tests. Here we report an experimentally testable inequality involving only 13 observables that is satisfied by all non-contextual realistic models while being violated by all qutrit states. Thus our inequality will practically facilitate a state-independent test of the quantum contextuality for an indivisible quantum system. We provide also a record-breaking state-independent proof of the Kochen-Specker 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, Kochen-Specker (KS) theorem states that non-contextual HV models cannot reproduce all the predictions of QM or quantum mechanics is contextual.In any non-contextual HV model all observables have definite values determined only by some HVs λ that are distributed according to a given probability distribution ̺ λ with normalization dλ̺ λ = 1. 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. Non-contextuality is a typical classical property: the value of an observable revealed by a measurement is predetermined by HVs λ only regardless of which compatible observable might be measured alongside. Local realism is a form of non-contextuality 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][16][17][18]. In general KS inequalities reveal the nonclassical nature of single systems demanding neither space-like separation nor entanglement, i.e....

Quantum discord provides a measure for quantifying quantum correlations beyond entanglement and is very hard to compute even for two-qubit states because of the minimization over all possible measurements. Recently a simple algorithm to evaluate the quantum discord for two-qubit X-states is proposed by Ali, Rau and Alber [Phys. Rev. A 81, 042105 (2010)] with minimization taken over only a few cases. Here we shall at first identify a class of X-states, whose quantum discord can be evaluated analytically without any minimization, for which their algorithm is valid, and also identify a family of X-states for which their algorithm fails. And then we demonstrate that this special family of X-states provides furthermore an explicit example for the inequivalence between the minimization over positive operator-valued measures and that over von Neumann measurements. For an important family of two-qubit states, the so called X-states [25], an algorithm has been proposed to calculate their quantum discord with minimization taken over only a few simple cases [26], which is unfortunately impeded by a counter example [27]. In this paper we shall at first identify a vast class of X-states, whose quantum discord can be evaluated analytically without any minimization at all, for which their algorithm is valid, and also identify a family of X-states X m , the so-called maximally discordant mixed states [24], for which the above mentioned algorithm fails. And then for this family of Xstates X m we construct a POVM showing that the quantum discord obtained by minimization over all POVMs is strictly smaller than that over all possible von Neumann measurements.For a given quantum state ̺ of a composite system AB the total amount of correlations, including classical and quantum correlations, is quantified by the quantum mutual information I(ρ) = S(̺ A ) + S(̺ B ) − S(̺) where S(̺) = −Tr(̺ log 2 ̺) denotes the von Neumann entropy and ̺ A , ̺ B are reduced density matrices for subsystem A, B respectively. An alternative version of the mutual information can be defined aswhere the minimum is taken over all possible POVMs {E defines the quantum discord that quantifies the quantum correlation. Also the minimum in Eq.(1) can be taken over all von Neumann measurements [3] and we