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We use a variant of the fountain Theorem to prove the existence of infinitely many weak solutions for the following fractional p-Laplace equation (-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u=f(x,u) in R^N, where $s \in (0,1)$,$ p \geq 2$,$ N \geq 2$, $(-\Delta)^{s}_{p}$ is the fractional $p$-Laplace operator, the nonlinearity f is $p$-superlinear at infinity and the potential V(x) is allowed to be sign-changing.

In this work we study the following fractional scalar field equation \begin{equation*}\label{P} \left\{ \begin{array}{ll} (-\Delta)^{s} u = g'(u) \mbox{ in } \mathbb{R}^{N} \\ u> 0 \end{array} \right. \end{equation*} where $N\geq 2$, $s\in (0,1)$, $(-\Delta)^{s}$ is the fractional Laplacian and the nonlinearity $g\in C^{2}(\mathbb{R})$ is such that $g''(0)=0$. By using variational methods, we prove the existence of a positive solution which is spherically symmetric and decreasing in $r=|x|$.

In this paper we prove the existence, regularity and symmetry of a ground state for a nonlinear equation in the whole space, involving a pseudo-relativistic Schr\"odinger operator.

In this paper we study ground states of the following fractional Schr\"odinger equation (- \Delta)^{s} u + V(x) u = f(x, u) \, \mbox{ in } \, \R^{N}, u\in \H^{s}(\R^{N}) where $s\in (0,1)$, $N>2s$ and $f$ is a continuous function satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition. We consider the cases when the potential $V(x)$ is $1$-periodic or has a bounded potential well.

The purpose of this paper is to study $T$-periodic solutions to [(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{in} (0,T)^{N} (P) u(x+Te_{i})=u(x) &\mbox{for all} x \in \R^{N}, i=1, \dots, N where $s\in (0,1)$, $N>2s$, $T>0$, $m> 0$ and $f(x,u)$ is a continuous function, $T$-periodic in $x$ and satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition. The nonlocal operator $(-\Delta_{x}+m^{2})^{s}$ can be realized as the Dirichlet to Neumann map for a degenerate elliptic problem posed on the half-cylinder $\mathcal{S}_{T}=(0,T)^{N}\times (0,\infty)$. By using a variant of the Linking Theorem, we show that the extended problem in $\mathcal{S}_{T}$ admits a nontrivial solution $v(x,\xi)$ which is $T$-periodic in $x$. Moreover, by a procedure of limit as $m\rightarrow 0$, we also prove the existence of a nontrivial solution to (P) with $m=0$.

Greenberger-Horne-Zeilinger (GHZ) states and their mixtures exhibit fascinating properties. A complete basis of GHZ-states can be constructed by properly choosing local basis rotations. We demonstrate this experimentally for the Hilbert space C2x4 by entangling two photons in polarisation and orbital angular momentum. Mixing GHZ-states unmasks different entanglement features based on their particular local geometrical connectedness. In particular, a specific GHZ-state in a complete orthonormal basis has a twin GHZ-state for which equally mixing leads to full separability in opposition to any other basis-state. Exploiting these local geometrical relations provides a toolbox for generating specific types of multipartite entanglement, each providing different benefits in outperforming classical devices. Our experiment, based on hybrid entangled entanglement, investigates these GHZs properties showing a good stability and fidelity and allowing a scaling in degrees of freedom and advanced operational manipulations., Comment: 8 pages, 4 figures

The importance of integrated quantum photonics in the telecom band resides on the possibility of interfacing with the optical network infrastructure developed for classical communications. In this framework, femtosecond laser written integrated photonic circuits, already assessed for quantum information experiments in the 800 nm wavelength range, have great potentials. In fact these circuits, written in glass, can be perfectly mode-matched at telecom wavelength to the in/out coupling fibers, which is a key requirement for a low-loss processing node in future quantum optical networks. In addition, for several applications quantum photonic devices will also need to be dynamically reconfigurable. Here we experimentally demonstrate the high performance of femtosecond laser written photonic circuits for quantum experiments in the telecom band and we show the use of thermal shifters, also fabricated by the same femtosecond laser, to accurately tune them. State-of-the-art manipulation of single and two-photon states is demonstrated, with fringe visibilities greater than 95%. This opens the way to the realization of reconfigurable quantum photonic circuits on this technological platform.

By using variational methods we investigate the existence of T-periodic solutions to [(-Delta_x + m^2)^s -m^(2s)]u= f(x,u) in (0,T)^N u(x+Te_i)=u(x) for all x in R^N, i=1,...,N where s in (0,1), N>2s, T>0, m>=0 and f(x,u) is a continuous function, T-periodic in x, verifying the Ambrosetti-Rabinowitz condition and a polynomial growth at rate p in (1, (N+2s)/(N-2s)).

Measurement scenarios containing events with relations of exclusivity represented by pentagons, heptagons, nonagons, etc., or their complements are the only ones in which quantum probabilities cannot be described classically. Interestingly, quantum theory predicts that the maximum values for any of these graphs cannot be achieved in Bell inequality scenarios. With the exception of the pentagon, this prediction remained experimentally unexplored. Here we test the quantum maxima for the heptagon and the complement of the heptagon using three- and five-dimensional quantum states, respectively. In both cases, we adopt two different encodings: linear transverse momentum and orbital angular momentum of single photons. Our results exclude maximally noncontextual hidden-variable theories and are in good agreement with the maxima predicted by quantum theory., Comment: REVTeX4, 7 pages, 2 figures, 4 experiments in 2 continents

Light beams having a vectorial field structure - or polarization - that varies over the transverse profile and a central optical singularity are called vector-vortex (VV) beams and may exhibit specific properties, such as focusing into "light needles" or rotation invariance, with applications ranging from microscopy and light trapping to communication and metrology. Individual photons in such beams exhibit a form of single-particle quantum entanglement between different degrees of freedom. On the other hand, the quantum states of two photons can be also entangled with each other. Here we combine these two concepts and demonstrate the generation of quantum entanglement between two photons that are both in VV states - a new form of quantum "entangled entanglement". This result may lead to quantum-enhanced applications of VV beams as well as to quantum-information protocols fully exploiting the vectorial features of light.

Recent schemes to encode quantum information into the total angular momentum of light, defining rotation-invariant hybrid qubits composed of the polarization and orbital angular momentum degrees of freedom, present interesting applications for quantum information technology. However, there remains the question as to how detrimental effects such as random spatial perturbations affect these encodings. Here, we demon- strate that alignment-free quantum communication through a turbulent channel based on hybrid qubits can be achieved with unit transmission fidelity. In our experiment, alignment-free qubits are produced with q-plates and sent through a homemade tur- bulence chamber. The decoding procedure, also realized with q-plates, relies on both degrees of freedom and renders an intrinsic error-filtering mechanism that maps errors into losses., Comment: 9 pages, 7 figures. published in Scientific Reports

We demonstrate a scheme to generate noncoherent and coherent correlations, i.e., a tunable degree of entanglement, between degrees of freedom of a single photon. Its nature is analogous to the tuning of the purity (first-order coherence) of a single photon forming part of a two-photon state by tailoring the correlations between the paired photons. Therefore, well-known tools such as the Clauser-Horne-Shimony-Holt (CHSH) Bell-like inequality can also be used to characterize entanglement between degrees of freedom. More specifically, CHSH inequality tests are performed, making use of the polarization and the spatial shape of a single photon. The four modes required are two polarization modes and two spatial modes with different orbital angular momentum., Comment: 7 pages, 5 figures. Updated to be consistent with published version

An important problem in quantum information processing is the certification of the dimension of quantum systems without making assumptions about the devices used to prepare and measure them, that is, in a device-independent manner. A crucial question is whether such certification is experimentally feasible for high-dimensional quantum systems. Here we experimentally witness in a device-independent manner the generation of six-dimensional quantum systems encoded in the orbital angular momentum of single photons and show that the same method can be scaled, at least, up to dimension 13., Comment: REVTeX4, 5 pages, 2 figures

Twisted photons are photons carrying a well-defined nonzero value of orbital angular momentum (OAM). The associated optical wave exhibits a helical shape of the wavefront (hence the name) and an optical vortex at the beam axis. The OAM of light is attracting a growing interest for its potential in photonic applications ranging from particle manipulation, microscopy and nanotechnologies, to fundamental tests of quantum mechanics, classical data multiplexing and quantum communication. Hitherto, however, all results obtained with optical OAM were limited to laboratory scale. Here we report the experimental demonstration of a link for free-space quantum communication with OAM operating over a distance of 210 meters. Our method exploits OAM in combination with optical polarization to encode the information in rotation-invariant photonic states, so as to guarantee full independence of the communication from the local reference frames of the transmitting and receiving units. In particular, we implement quantum key distribution (QKD), a protocol exploiting the features of quantum mechanics to guarantee unconditional security in cryptographic communication, demonstrating error-rate performances that are fully compatible with real-world application requirements. Our results extend previous achievements of OAM-based quantum communication by over two orders of magnitudes in the link scale, providing an important step forward in achieving the vision of a worldwide quantum network.

Why do correlations between the results of measurements performed on physical systems violate Bell and non-contextuality inequalities up to some specific limits? The answer may follow from the observation that in quantum theory, unlike in other theories, whenever there is an experiment to measure $A$ simultaneously with $B$, another to measure $B$ with $C$, and another to measure $A$ with $C$, there is always an experiment to measure all of them simultaneously. This property implies that quantum theory satisfies a seemingly irrelevant restriction called the exclusivity (E) principle which, surprisingly, explains the set of quantum correlations in some fundamental scenarios. An open problem is whether the E principle explains the maximum quantum violation of the Bell-CHSH inequality. Here we show experimentally that the E principle imposes an upper bound to the violation of the Bell-CHSH inequality that matches the maximum predicted by quantum theory. For that, we use the result of an independent experiment testing a specific non-contextuality inequality. We perform both experiments: the Bell-CHSH inequality experiment on polarization entangled states of pairs of photons in Stockholm and, to demonstrate independence, the non-contextuality inequality experiment on single photons' orbital angular momentum states in Rome. The observed results provide the first experimental evidence that the E principle determines the limits of quantum correlations and prove that hypothetical super-quantum violations for either experiment would violate the E principle. This supports the conclusion that the E principle captures a fundamental limitation of nature. If this is true, much of quantum theory trivially follow from merely taking the E principle to be a fundamental truth, and various information-theoretic postulates are also simplified and/or strengthened., Comment: 7+7 pages, 3+1 figures

In quantum mechanics, observing is not a passive act. Consider a system of two quantum particles A and B: if a measurement apparatus M is used to make an observation on B, the overall state of the system AB will typically be altered. When this happens no matter which local measurement is performed, the two objects A and B are revealed to possess peculiar correlations known as quantum discord. Here we demonstrate experimentally that the very act of local observation gives rise to an activation protocol which converts discord into distillable entanglement, a stronger and more useful form of quantum correlations, between the apparatus M and the composite system AB. We adopt a flexible two-photon setup to realize a three-qubit system (A,B,M) with programmable degrees of initial correlations, measurement interaction, and characterization processes. Our experiment demonstrates the fundamental mechanism underpinning the ubiquitous act of observing the quantum world, and establishes the potential of discord in entanglement generation., Comment: 5 + 6 pages, 3 + 2 figures. Close to published version

In quantum information, complementarity of quantum mechanical observables plays a key role. If a system resides in an eigenstate of an observable, the probability distribution for the values of a complementary observable is flat. The eigenstates of these two observables form a pair of mutually unbiased bases (MUBs). More generally, a set of MUBs consists of bases that are all pairwise unbiased. Except for specific dimensions of the Hilbert space, the maximal sets of MUBs are unknown in general. Even for a dimension as low as six, the identification of a maximal set of MUBs remains an open problem, although there is strong numerical evidence that no more than three simultaneous MUBs do exist. Here, by exploiting a newly developed holographic technique, we implement and test different sets of three MUBs for a single photon six-dimensional quantum state (a qusix), encoded either in a hybrid polarization-orbital angular momentum or a pure orbital angular momentum Hilbert space. A close agreement is observed between theory and experiments. Our results can find applications in state tomography, quantitative wave-particle duality, quantum key distribution and tests on complementarity and logical indeterminacy., Comment: 8 pages, 4 figures

The conflict between classical and quantum physics can be identified through a series of yes-no tests on quantum systems, without it being necessary that these systems be in special quantum states. Kochen-Specker (KS) sets of yes-no tests have this property and provide a quantum-versus-classical advantage that is free of the initialization problem that affects some quantum computers. Here, we report the first experimental implementation of a complete KS set that consists of 18 yes-no tests on four-dimensional quantum systems and show how to use the KS set to obtain a state-independent quantum advantage. We first demonstrate the unique power of this KS set for solving a task while avoiding the problem of state initialization. Such a demonstration is done by showing that, for 28 different quantum states encoded in the orbital-angular-momentum and polarization degrees of freedom of single photons, the KS set provides an impossible-to-beat solution. In a second experiment, we generate maximally contextual quantum correlations by performing compatible sequential measurements of the polarization and path of single photons. In this case, state independence is demonstrated for 15 different initial states. Maximum contextuality and state independence follow from the fact that the sequences of measurements project any initial quantum state onto one of the KS set's eigenstates. Our results show that KS sets can be used for quantum-information processing and quantum computation and pave the way for future developments., Comment: REVTeX, 15 pages, 4 figures

Quantum communication employs the counter-intuitive features of quantum physics to perform tasks that are im- possible in the classical world. It is crucial for testing the foundations of quantum theory and promises to rev- olutionize our information and communication technolo- gies. However, for two or more parties to execute even the simplest quantum transmission, they must establish, and maintain, a shared reference frame. This introduces a considerable overhead in communication resources, par- ticularly if the parties are in motion or rotating relative to each other. We experimentally demonstrate how to circumvent this problem with the efficient transmission of quantum information encoded in rotationally invariant states of single photons. By developing a complete toolbox for the efficient encoding and decoding of quantum infor- mation in such photonic qubits, we demonstrate the fea- sibility of alignment-free quantum key-distribution, and perform a proof-of-principle alignment-free entanglement distribution and violation of a Bell inequality. Our scheme should find applications in fundamental tests of quantum mechanics and satellite-based quantum communication., Comment: Main manuscript: 7 pages, 3 figures; Supplementary Information: 7 pages, 3 figures

Quantum resources outperform classical ones for certain communication and computational tasks. Remarkably, in some cases, the quantum advantage cannot be improved using hypothetical postquantum resources. A class of tasks with this property can be singled out using graph theory. Here we report the experimental observation of an impossible-to-beat quantum advantage on a four-dimensional quantum system defined by the polarization and orbital angular momentum of a single photon. The results show pristine evidence of the quantum advantage and are compatible with the maximum advantage allowed using postquantum resources., Comment: REVTeX4, 5 pages, 2 figures

The extraction of information from a quantum system unavoidably implies a modification of the measured system itself. It has been demonstrated recently that partial measurements can be carried out in order to extract only a portion of the information encoded in a quantum system, at the cost of inducing a limited amount of disturbance. Here we analyze experimentally the dynamics of sequential partial measurements carried out on a quantum system, focusing on the trade-off between the maximal information extractable and the disturbance. In particular we consider two different regimes of measurement, demonstrating that, by exploiting an adaptive strategy, an optimal trade-off between the two quantities can be found, as observed in a single measurement process. Such experimental result, achieved for two sequential measurements, can be extended to N measurement processes., Comment: 5 pages, 3 figures

In this work we experimentally implement a deterministic transfer of a generic qubit initially encoded in the orbital angular momentum of a single photon to its polarization. Such transfer of quantum information, completely reversible, has been implemented adopting a electrically tunable q-plate device and a Sagnac interferomenter with a Dove's prism. The adopted scheme exhibits a high fidelity and low losses., Comment: 3 pages, 2 figures

Quantum cryptographic protocols based on complementarity are nonsecure against attacks in which complementarity is imitated with classical resources. The Kochen-Specker (KS) theorem provides protection against these attacks, without requiring entanglement or spatially separated composite systems. We analyze the maximum tolerated noise to guarantee the security of a KS-protected cryptographic scheme against these attacks, and describe a photonic realization of this scheme using hybrid ququarts defined by the polarization and orbital angular momentum of single photons., Comment: REVTeX4-1, 5 pages, 2 figures

Exploiting electro-optic effects in liquid crystals, we achieved real-time control of the retardation of liquid- crystal-based $q$-plates through an externally applied voltage. The newly conceived electro-optic $q$-plates can be operated as electrically driven converters of photon spin into orbital angular momentum, enabling a variation of the orbital angular momentum probabilities of the output photons over a time scale of milliseconds., Comment: 4 pages, 5 figures, submitted