In contrast to classical physics, quantum theory demands that not all properties can be simultaneously well defined; the Heisenberg uncertainty principle is a manifestation of this fact. Alternatives have been explored--notably theories relying on joint probability distributions or non-contextual hidden-variable models, in which the properties of a system are defined independently of their own measurement and any other measurements that are made. Various deep theoretical results imply that such theories are in conflict with quantum mechanics. Simpler cases demonstrating this conflict have been found and tested experimentally with pairs of quantum bits (qubits). Recently, an inequality satisfied by non-contextual hidden-variable models and violated by quantum mechanics for all states of two qubits was introduced and tested experimentally. A single three-state system (a qutrit) is the simplest system in which such a contradiction is possible; moreover, the contradiction cannot result from entanglement between subsystems, because such a three-state system is indivisible. Here we report an experiment with single photonic qutrits which provides evidence that no joint probability distribution describing the outcomes of all possible measurements--and, therefore, no non-contextual theory--can exist. Specifically, we observe a violation of the Bell-type inequality found by Klyachko, Can, Binicioğlu and Shumovsky. Our results illustrate a deep incompatibility between quantum mechanics and classical physics that cannot in any way result from entanglement.

We show that magnetic susceptibility can reveal spin entanglement between individual constituents of a solid, while magnetization describes their local properties. We then show that magnetization and its variance (equivalent to magnetic susceptibility for a wide class of systems) satisfy complementary relation in the quantum-mechanical sense. It describes sharing of (quantum) information in the solid between spin entanglement and local properties of its individual constituents. Magnetic susceptibility is shown to be a macroscopic (thermodynamical) spin entanglement witness that can be applied without complete knowledge of the specific model (Hamiltonian) of the solid. PACS numbers: 03.67.Hk, 03.65.Ta, 03.65.UdThermodynamical properties, such as heat capacity, magnetization or magnetic susceptibility, are normally ascribed to macroscopic objects with the number of individual constituent of the order of 10 23 . In contrast, genuine quantum features like quantum superposition or entanglement are generally not seen beyond molecular scales. As mass, size, complexity and/or temperature of systems increase the observability of their quantum effects is gradually limited by decoherence -an interaction of the system with its environment -that turns them into classical phenomena. This raises several questions: under which conditions can quantum features of individual constituents of a solid have an effect on its global properties? Can one detect existence of quantum entanglement in a solid by observing its thermodynamical properties only? Can one consider macroscopic properties as quantum-mechanical observables in the sense that they obey complementary relations like position and momentum?The complementarity principle is the assertion that there exist observables which are mutually exclusive in the sense that they cannot be precisely defined simultaneously. One of them, for example, the z component of the spin-1 2 (σ z ), might be well defined at the expense of maximum uncertainty about the other orthogonal directions (σ x and σ y , σ's are respective Pauli matrices). One can speak about sharing of (quantum) information between mutually complementary observables [1]. In the case of a qubit this can quantitatively be described by the relation σ x 2 + σ y 2 + σ z 2 ≤ 1, where the average is taken over an arbitrary state. When extended to composite systems the principle of complementarity asserts the mutual exclusiveness between entanglement and local properties of individual constituents of the composite system. In the case of two qubits this can be described by the relation i=x,y,z σ 1 i 2 + σ 2 i 2 + σ 1 i σ 2 i 2 ≤ 3, where the upper indices indicate qubits. The maximal value of 3 can be achieved, e.g., with product states (e.g. σ 1 z = σ 2 z = σ 1 z σ 2 z = 1; others are zero) for which local properties of the qubits are well-defined, but there is no entanglement. Alternatively, their joint properties could be well-defined at the expense of a complete indefiniteness of the local properties (e.g. for a singlet state σ 1 x ...

Any pure entangled state of two particles violates a Bell inequality for two-particle correlation functions (Gisin's theorem). We show that there exist pure entangled N > 2 qubit states that do not violate any Bell inequality for N particle correlation functions for experiments involving two dichotomic observables per local measuring station. We also find that Mermin-Ardehali-BelinskiiKlyshko inequalities may not always be optimal for refutation of local realistic description.PACS numbers: 3.65. Ud, Quantum mechanics violates Bell type inequalities [1,2,3,4], which hold for any local realistic theory. In a realistic model the measurement results are determined by "hidden" properties the particles carry prior to and independent of observation. In a local model the results obtained at one location are independent of any measurements or actions performed at space-like separation. The theorem of Gisin [5] states that any pure nonproduct state of two particles violates a Clauser-HorneShimony-Holt (CHSH) [2] inequality, which involves only two-particle correlation functions, for two alternative dichotomic measurements for each of the local observers.Can Gisin's theorem be generalized to all N -particle pure entangled states? We show here that this is not the case for Bell inequalities involving only correlation functions in experiments in which local observers can choose between two dichotomic observables. We find a family of pure entangled states of N qubits which do not violate any such Bell inequality. This family is a subset of a larger one of the generalized GHZ states given by |ψ = cos α|0, ..., 0 + sin α|1, ..., 1 .( 1) with 0 ≤ α ≤ π/4. The GHZ states [3] are for α = π/4 Scarani and Gisin [9] noticed a surprising feature of such states. They show that for sin 2α ≤ 1/ √ 2 N −1 the states (1) do not violate the Mermin-Ardehali-BelinskiiKlyshko (MABK) inequalities [4]. This has been obtained numerically for N = 3, 4, 5 and conjectured for N > 5. Their result contrasts the case of N = 2 of two qubits and is surprising as the states (1) are a generalization of the GHZ states [3] which violate maximally the MABK inequalities.Scarani and Gisin write that "this analysis suggest that MK [here, MABK] inequalities, and more generally the family of Bell's inequalities with two observables per qubit, may not be the 'natural' generalizations of the CHSH inequality to more than two qubits" [9]. Concerning this question we find here an interesting discrepancy between the case of even and odd number of qubits. We prove that for all N odd and for sin 2α ≤ 1/ √ 2 N −1 the generalized GHZ states satisfy all possible Bell inequalities for N -particle correlation functions, which involve two alternative dichotomic observables at each local measurement station. Note also, that since the reduced density matrices of all proper subsystems of the N -qubit system described by (1) are separable, no Bell inequality for K < N particle correlation functions can be violated. Thus, Gisin's theorem cannot be straightforwardly generalized in thi...

Abstract. One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon manifests itself most strongly when observables are related to mutually unbiased bases. In this paper, we shed some light on the connection between mutually unbiased bases and another essential feature of quantum mechanics, quantum entanglement. It is shown that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independent of the choice of the set. This has implications for entanglement distribution among the states of a complete set. In prime-squared dimensions we present an explicit experiment-friendly construction of a complete set with a particularly simple entanglement distribution. Finally, we describe the basic properties of mutually unbiased bases composed of product states only. The constructions are illustrated with explicit examples in low dimensions. We believe that the properties of entanglement in mutually unbiased bases may be one of the ingredients to be taken into account to settle the question of the existence of complete sets. We also expect that they will be relevant to applications of bases in the experimental realization of quantum protocols in higher-dimensional Hilbert spaces.