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We show that the state-independent violation of inequalities for noncontextual hidden variable theories introduced in [Phys. Rev. Lett. 101, 210401 (2008)] is universal, i.e., occurs for any quantum mechanical system in which noncontextuality is meaningful. We describe a method to obtain state-independent violations for any system of dimension d> or =3. This universality proves that, according to quantum mechanics, there are no "classical" states.

A simple geometrical criterion gives experimentally friendly sufficient conditions for entanglement. Its generalization gives a necessary and sufficient condition. It is linked with a family of entanglement identifiers, which is strictly richer than the family of entanglement witnesses.PACS numbers: 03.65.Ud Entanglement is one of the basic features of quantum physics and it is a resource for quantum information science [1]. Thus, detection of entanglement belongs to the mainstream of this field [2]. Today, the most widely used and experimentally feasible detectors of this resource are entanglement witnesses [3]. They are linked with positive but not completely positive maps [4], which are the most universal entanglement identifiers.We present an alternative approach to entanglement detection. It is rooted in an elementary geometrical fact: if a scalar product of two real vectors s and e satisfies s · e < e · e, then s = e. This fact was used in, e.g., [5] to derive a powerful series of Bell inequalities, and in [6] it led to sufficient condition for entanglement. Here, it inspires a new family of entanglement identifiers, which are naturally expressed in terms of the correlation functions [7], easily determined by local measurements. This makes them friendly to experiments. The family of our identifiers is richer than the family of the entanglement witnesses and leads to a necessary and sufficient criterion for entanglement.The bulk of our presentation uses systems of many spin-1 2 particles (qubits), but the method is applicable to composite systems of arbitrary dimensions. For that in our formulae, one needs to substitute Pauli operators by their Gell-Mann-type generalizations. This allows a complete separability analysis of a multi-partite state, and will be illustrated by an example. Even if the underlying system consists of many qubits, analysis of the so-called k-separability (k < N ) requires identification of entanglement in the system partitioned into k parts only [8]. Clearly, at least one part will contain two or more qubits, and can be considered as a multi-level system.