Quantum discord characterizes "nonclassicality" of correlations in quantum mechanics. It has been proposed as the key resource present in certain quantum communication tasks and quantum computational models without containing much entanglement. We obtain a necessary and sufficient condition for the existence of nonzero quantum discord for any dimensional bipartite states. This condition is easily experimentally implementable. Based on this, we propose a geometrical way of quantifying quantum discord. For two qubits this results in a closed form of expression for discord. We apply our results to the model of deterministic quantum computation with one qubit, showing that quantum discord is unlikely to be the reason behind its speedup.

The idea that events obey a definite causal order is deeply rooted in our understanding of the world and at the basis of the very notion of time. But where does causal order come from, and is it a necessary property of nature? Here, we address these questions from the standpoint of quantum mechanics in a new framework for multipartite correlations that does not assume a pre-defined global causal structure but only the validity of quantum mechanics locally. All known situations that respect causal order, including space-like and time-like separated experiments, are captured by this framework in a unified way. Surprisingly, we find correlations that cannot be understood in terms of definite causal order. These correlations violate a 'causal inequality' that is satisfied by all space-like and time-like correlations. We further show that in a classical limit causal order always arises, which suggests that space-time may emerge from a more fundamental structure in a quantum-to-classical transition.

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Our common understanding of the physical world deeply relies on the notion that events are ordered with respect to some time parameter, with past events serving as causes for future ones. Nonetheless, it was recently found that it is possible to formulate quantum mechanics without any reference to a global time or causal structure. The resulting framework includes new kinds of quantum resources that allow performing tasks-in particular, the violation of causal inequalities-which are impossible for events ordered according to a global causal order. However, no physical implementation of such resources is known. Here we show that a recently demonstrated resource for quantum computationthe quantum switch-is a genuine example of 'indefinite causal order'. We do this by introducing a new tool-the causal witness-which can detect the causal nonseparability of any quantum resource that is incompatible with a definite causal order. We show however that the quantum switch does not violate any causal inequality.

The physics of low-energy quantum systems is usually studied without explicit consideration of the background spacetime. Phenomena inherent to quantum theory in curved spacetime, such as Hawking radiation, are typically assumed to be relevant only for extreme physical conditions: at high energies and in strong gravitational fields. Here we consider low-energy quantum mechanics in the presence of gravitational time dilation and show that the latter leads to the decoherence of quantum superpositions. Time dilation induces a universal coupling between the internal degrees of freedom and the centre of mass of a composite particle. The resulting correlations lead to decoherence in the particle position, even without any external environment. We also show that the weak time dilation on Earth is already su cient to a ect micrometre-scale objects. Gravity can therefore account for the emergence of classicality and this e ect could in principle be tested in future matterwave experiments.O ne of the most striking features of quantum theory is the quantum superposition principle. It has been demonstrated in numerous experiments with diverse systems, such as neutrons 1 , atoms 2 and even large molecules 3 . However, quantum superpositions are not observed on everyday, macroscopic scales. The origin of the quantum-to-classical transition is still an active field of research. A prominent role in this transition is commonly attributed to decoherence 4,5 : owing to interaction with an external environment, a particle gets entangled with its environment and loses its quantum coherence. Many specific models have been studied in which a particle interacts with its surroundings, such as a bath of phonons 6 , photons 7,8 , spins 9,10 and gravitational waves [11][12][13] . An alternative route to explain classicality is taken in socalled wavefunction collapse models, which postulate an inherent breakdown of the superposition principle at some scale without any external environment [14][15][16] . Such models are often inspired by general relativity, but they rely on a fundamental modification of quantum theory. In contrast, here we derive the existence of decoherence due to time dilation without any modification of quantum mechanics and which takes place even for isolated composite systems. We show that even the weak time dilation on Earth is already sufficient to decohere micro-scale quantum systems.We consider standard quantum mechanics in the presence of time dilation, with the focus on gravitational time dilation which causes clocks to run slower near a massive object. In the Methods, we derive the Hamiltonian governing the quantum dynamics of a composite system on an arbitrary, static background spacetime (and show that the same result is obtained as a limit of a quantum field model). As we consider slowly moving particles and weak gravitational fields (that is, to lowest order in c −2 , where c is the speed of light), the results can also be obtained directly from the mass-energy equivalence 17 : any internal energy contributes t...

We derive a single general Bell inequality which is a necessary and sufficient condition for the correlation function for N particles to be describable in a local and realistic picture, for the case in which measurements on each particle can be chosen between two arbitrary dichotomic observables. We also derive a necessary and sufficient condition for an arbitrary N-qubit mixed state to violate this inequality. This condition is a generalization and reformulation of the Horodeccy family condition for two qubits.PACS numbers: 03.65. Ud, 42.50.Ar Local realism imposes constraints on statistical correlations of measurements on multiparticle systems. They are in the form of Bell-type inequalities [1,2,3,4,5,6,7,8]. In a realistic theory the measurement results are determined by "hidden" properties the particles carry prior to and independent of observation. In a local realistic theory the results obtained at one location are independent of any measurements, or actions, performed at space-like separation. Quantum mechanics predicts violation of these constraints. This is known as Bell's theorem [1] .However the problems a) what are the most general constraints on correlations imposed by local realism, and b) which quantum states violate these constraints, are still open. The latter has been solved in general only in the case of two particles in pure states [9,10] and for two-qubit mixed states [11]. Only recently bounds for local realistic description of a higher-dimensional system have been found in some simple cases [12,13,14].Here the answer to the two long-standing questions (a) and (b) is presented for the case of a standard Bell type experiment on N qubits. By a standard Bell experiment we mean, one in which each local observer is given a choice between two dichotomic observables. We first derive a single general Bell inequality that summarizes all possible local realistic constraints on the correlation functions for a N-particle system. From this inequality one obtains as corollaries the Clauser-Horne-ShimonyHolt (CHSH) inequality [2] for two-particle systems and the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequalities for N particles [4,5,6]. We show that the correlation functions in a standard Bell experiment can be described by a local realistic model if and only if the general Bell inequality is satisfied. Therefore the general Bell inequality is a sufficient and necessary condition for correlation functions, in such an experiment, to be describable within a local realistic model. We also find a necessary and sufficient condition for correlation functions for N qubits in an arbitrary (mixed) quantum state to violate the general Bell inequality in direct measurements. This condition is generalization and reformulation of the one given by the Horodeccy family [11] for two qubits.These results are not only of importance from the fundamental point of view, but also as a research towards identifying ultimate resources for quantum information processing. Recently it was shown [15], that there is a direct lin...

Most working scientists hold fast to the concept of 'realism'--a viewpoint according to which an external reality exists independent of observation. But quantum physics has shattered some of our cornerstone beliefs. According to Bell's theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space-like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenable. Maintaining realism as a fundamental concept would therefore necessitate the introduction of 'spooky' actions that defy locality. Here we show by both theory and experiment that a broad and rather reasonable class of such non-local realistic theories is incompatible with experimentally observable quantum correlations. In the experiment, we measure previously untested correlations between two entangled photons, and show that these correlations violate an inequality proposed by Leggett for non-local realistic theories. Our result suggests that giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned.

Quantum entanglement is widely recognized as one of the key resources for the advantages of quantum information processing, including universal quantum computation 1 , reduction of communication complexity 2,3 or secret key distribution 4 . However, computational models have been discovered, which consume very little or no entanglement and still can efficiently solve certain problems thought to be classically intractable 5,6 . The existence of these models suggests that separable or weakly entangled states could be extremely useful tools for quantum information processing as they are much easier to prepare and control even in dissipative environments. It has been proposed that a requirement for useful quantum states is the generation of so-called quantum discord 7,8 , a measure of non-classical correlations that includes entanglement as a subset. Although a link between quantum discord and few quantum information tasks has been studied, its role in computation speed-up is still open and its operational interpretation remains restricted to only few somewhat contrived situations 9-12 . Here we show that quantum discord is the optimal resource for the remote quantum state preparation 13 , a variant of the quantum teleportation protocol 14 . Using photonic quantum systems, we explicitly show that the geometric measure of quantum discord 15 is related to the fidelity of this task, which provides an operational meaning. Moreover, we demonstrate that separable states with non-zero quantum discord can outperform entangled states. Therefore, the role of quantum discord might provide fundamental insights for resource-efficient quantum information processing.Introduction.-Quantum computation and quantum communication is believed to allow for information processing with an efficiency that cannot be achieved by any classical device. It is usually assumed that a key resource for this enhanced performance is quantum entanglement 16 . The creation and manipulation of entanglement, however, is a very demanding task, as it requires extremely precise quantum control and isolation from the environment. Thus, current experimental achievements are limited to rather small scale entangled systems [17][18][19] . On the other hand there is no proof that quantum entanglement is necessary for quantum information processing (QIP) that can outperform its classical counterpart. The investigation of QIP protocols that allow for significant enhancements in the efficiency of data processing by only using separable states is of high interest. Obviously, such states have the benefit of being easier to prepare and more robust against losses and experimental imperfections. In fact, there are quantum computational models based on mixed, separable states, most notably the so-called deterministic quantum computation with one qubit (DQC1) 5 , which has recently been demonstrated experimentally [20][21][22] . In this context, quantum discord has been proposed as the resource that can provide the enhancement for the computation 23,24 , but its relation to...