Correlations in Bell and noncontextuality inequalities can be expressed as a positive linear combination of probabilities of events. Exclusive events can be represented as adjacent vertices of a graph, so correlations can be associated to a subgraph. We show that the maximum value of the correlations for classical, quantum, and more general theories is the independence number, the Lovász number, and the fractional packing number of this subgraph, respectively. We also show that, for any graph, there is always a correlation experiment such that the set of quantum probabilities is exactly the Grötschel-Lovász-Schrijver theta body. This identifies these combinatorial notions as fundamental physical objects and provides a method for singling out experiments with quantum correlations on demand.
We present a ``state-independent'' proof of the Bell-Kochen-Specker theorem using only 18 four-dimensional vectors, which is a record for this kind of proof. This set of vectors contains subsets which allow us to develop a ``state-specific'' proof with 10 vectors (also a record) and a ``probabilistic'' proof with 7 vectors which reflects the algebraic structure of Hardy's nonlocality theorem.Comment: LaTeX, 10 page
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...
We show that there are Bell-type inequalities for noncontextual theories that are violated by any quantum state. One of these inequalities between the correlations of compatible measurements is particularly suitable for testing this state-independent violation in an experiment.
Cabello Replies: As Fahmi and Golshani correctly point out in the preceding Comment [1], a protocol introduced in [2] to show that a quantum key distribution protocol can have efficiency one (i.e., can achieve the Holevo limit), where efficiency is defined as the number of secret bits per transmitted bit plus qubit, does indeed not have efficiency one. This error was already corrected in [3,4]. For completeness' sake, the corrected protocol introduced in [3,4], with efficiency one, is reproduced here.Suppose that the quantum channel is composed of two qubits (1 and 2) prepared with equal probabilities in one of four orthogonal pure states: PRL 100, 018902 (2008)
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