The question of whether quantum phenomena can be explained by classical models with hidden variables is the subject of a long lasting debate [1]. In 1964, Bell showed that certain types of classical models cannot explain the quantum mechanical predictions for specific states of distant particles [2]. Along this line, some types of hidden variable models have been experimentally ruled out [3,4,5,6,7,8,9]. An intuitive feature for classical models is non-contextuality: the property that any measurement has a value which is independent of other compatible measurements being carried out at the same time. However, the results of Kochen, Specker, and Bell[10,11,12] show that non-contextuality is in conflict with quantum mechanics. The conflict resides in the structure of the theory and is independent of the properties of special states. It has been debated whether the Kochen-Specker theorem could be experimentally tested at all [13,14]. Only recently, first tests of quantum contextuality have been proposed and undertaken with photons [15] and neutrons [16,17]. Yet these tests required the generation of special quantum states and left various loopholes open. Here, using trapped ions, we experimentally demonstrate a state-independent conflict with non-contextuality. The experiment is not subject to the detection loophole and we show that, despite imperfections and possible measurement disturbances, our results cannot be explained in non-contextual terms. PACS numbers:Hidden variable models assert that the result v(A) of measuring the observable A on an individual quantum system is predetermined by a hidden variable λ. Two observables A and B are mutually compatible, if the result of A does not depend on whether B is measured before, after, or simultaneously with A and vice versa. Non-contextuality is the property of a hidden variable model that the value v(A) is determined, regardless of which other compatible observable is measured jointly with A. As a consequence, for compatible observables the relation v(AB) = v(A)v(B) holds. Kochen and Specker showed that the assumption of noncontextuality cannot be reconciled with quantum mechanics. A considerable simplification of the original Kochen-Specker argument by Mermin and Peres [18,19] uses a 3 × 3 square of observables A ij with possible outcomes v(A ij ) = ±1, where the observables in each row or column are mutually compatible. Considering the products of rows, the total product would be k=1,2,3 R k C k = 1, since any v(A ij ) appears twice in the total product.In quantum mechanics, however, one can take a fourlevel quantum system, for instance two spin-1 2 -particles, * Electronic address: christian.roos@uibk.ac.at and the following array of observables,(1) Here, σ (k) i denotes the Pauli matrix acting on the k-th particle, and all the observables have the outcomes ±1. Moreover, in each of the rows or columns of (1), the observables are mutually commuting and can be measured simultaneously or in any order. In any row or column, their measurement product R k or C k equa...

Is the closest product state to a symmetric entangled multiparticle state also symmetric? This question has appeared in the recent literature concerning the geometric measure of entanglement. First, we show that a positive answer can be derived from results concerning symmetric multilinear forms and homogeneous polynomials, implying that the closest product state can be chosen to be symmetric. We then prove the stronger result that the closest product state to any symmetric multiparticle quantum state is necessarily symmetric. Moreover, we discuss generalizations of our result and the case of translationally invariant states, which can occur in spin models.

Sequential measurements on a single particle play an important role in fundamental tests of quantum mechanics. We provide a general method to analyze temporal quantum correlations, which allows us to compute the maximal correlations for sequential measurements in quantum mechanics. As an application, we present the full characterization of temporal correlations in the simplest Leggett-Garg scenario and in the sequential measurement scenario associated with the most fundamental proof of the Kochen-Specker theorem.