Violation of modified Wigner inequality by means binary bipartite quantum system allows the discrimination between the quantum world and the classical local-realistic one, and also ensures the security of Ekert-like quantum key distribution protocol. In this paper we study both theoretically and experimentally the bounds of quantum correlation associated to the modified Wigner's inequality finding the optimal experimental configuration for its maximal violation. We also extend this analysis to the implementation of Ekert's protocol.
In this report we discuss the insecurity with present implementations of the Ekert protocol for quantum-key distribution based on the Wigner Inequality. We propose a modified version of this inequality which guarantees safe quantum-key distribution.
We experimentally implemented an eavesdropping attack against the Ekert protocol for quantum key distribution based on the Wigner inequality. We demonstrate a serious lack of security of this protocol when the eavesdropper gains total control of the source. In addition we tested a modified Wigner inequality which should guarantee a secure quantum key distribution.
Wigner's argument inferring Bell-type inequality for the EPR-Bohm entangled state is generalized here for any N-partite state. This is based on assuming for the relevant dichotomic observables the existence of the overall joint probability distributions, satisfying the locality condition, that would yield the measurable marginal probabilities. For any N, such Generalized Wigner's Inequality (GWI) is violated by quantum mechanics for all pure entangled states. The efficacy of GWI is probed, comparing with the Seevinck-Svetlichny multipartite Bell-type inequality, by calculating threshold visibi…
We construct an explicit Wigner function for N-mode squeezed state. Based on a previous observation that the Wigner function describes correlations in the joint measurement of the phase-space displaced parity operator, we investigate the non-locality of multipartite entangled state by the violation of Zukowski-Brukner N-qubit Bell inequality. We find that quantum predictions for such squeezed state violate these inequalities by an amount that grows with the number N.
A Bell inequality violation (BIQV) allowed by the two-mode squeezed state (TMSS), whose Wigner function is nonnegative, is shown to hold only for correlations among dynamical variables (DV) that cannot be interpreted via a local hidden variable (LHV) theory. Explicit calculations and interpretation are given for Bell's suggestion that the EPR (Einstein, Podolsky and Rosen) state will not allow for BIQV in conjuction with its Wigner representative state being nonnegative. It is argued that Bell's theorem disallowing the violation of Bell's inequality within a local hidden-variable theory depend…
In the study of systems that cannot be described classically, the Wigner inequality, has received only a small amount of attention. In this paper we extend the Wigner inequality - originally derived in 1969 - and show how it may be used to contradict local realism with only coincidence detections in the absence of two-outcome measurements - that is, for any system where only one possible result of a pair of potential outcomes can be registered. It thus encapsulates a much broader class of measurement schemes than could previously violate a local-realistic inequality. This is possible due to an…
We propose a new Wigner inequality suitable for test of the hypothesis of realism. We show that this inequality is not identical neither to the well-known Wigner inequality nor to the Leggett-Garg inequality in Wigner form. The obtained inequality is suitable for test of realism not only in quantum mechanical systems, but also in quantum field systems. Also we propose a mathematically consistent derivation of the Leggett--Garg inequality in Wigner form, which was recently presented in the literature, for three and $n$ distinct moments of time. Contrary to these works, our rigor derivation uses…
A Bell-type inequality is proposed in terms of Wigner-Yanase skew information, which is quadratic and involves only one local spin observable at each site. This inequality presents a hierarchic classification of all states of multipartite quantum systems from separable to fully entangled states, which is more powerful than the one presented by quadratic Bell inequalities from two-entangled to fully entangled states. In particular, it is proved that the inequality provides an exact test to distinguish entangled from nonentangled pure states of two qubits. Our inequality sheds considerable light…
It is not generally known, that the inequality that Bell derived using three random variables must be identically satisfied by any three corresponding data sets of plus and minus 1s that are writable on paper.This surprising fact is not immediately obvious from Bell's inequality derivation based on causal random variables, but follows immediately if the same mathematical operations are applied to finite data sets.For laboratory data, the inequality is identically satisfied as a fact of pure algebra, and its satisfaction is independent of whether the processes generating the data are local, non…
Wigner's form of the local realist inequality is used to derive its temporal version for an oscillating two-level system involving two-time joint probabilities. Such an inequality may be regarded as a novel form of the Leggett-Garg inequality (LGI) constituting a necessary condition for macrorealism. The robustness of its quantum mechanical (QM) violation against unsharpness of measurement is investigated by using a suitable model of unsharp measurements. It is found that there exists a range of values of the sharpness parameter (characterizing precision of the relevant measurements) for which…
We perform the analysis of probabilistic assumptions of Bell's approach. We emphasize that J. Bell wrote about probability without to specify the concrete axiomatics of probability theory. The careful analysis demonstrated that (surprisingly) J. Bell did not apply the classical probability model (Kolmogorov) to describe ``classical physical framework.'' In fact, he created his own probabilistic model and compared it with the quantum one. The crucial point is that J. Bell did not pay attention to {\it conditional probabilities.} We show that conditional probability in his model cannot be define…
We introduce a generalized Wigner-Yanase skew information and then derive the trace inequality related to the uncertainty relation. This inequality is a non-trivial generalization of the uncertainty relation derived by S.Luo for the quantum uncertainty quantity excluding the classical mixure. In addition, several trace inequalities on our generalized Wigner-Yanase skew information are argued.
The experimental test of Bell's inequality is mainly focused on Clauser-Horne-Shimony-Holt (CHSH) form, which provides a quantitative bound, while little attention has been pained on the violation of Wigner inequality (WI). Based on the spin coherent state quantum probability statistics we in the present paper extend the WI and its violation to arbitrary two-spin entangled states with antiparallel and parallel spin-polarizations. The local part of density operator gives rise to the WI while the violation is a direct result of non-local interference between two components of the entangled state…
Distinct from Bell's approach, Wigner had derived a form of local realist (LR) inequality which is quantum mechanically violated for a bipartite maximally entangled state. Subsequently, this approach was generalized to obtain a multipartite LR inequality. However, the violation of such generalised Wigner's inequality does not guarantee nonlocality between all possible different bipartitions of the multipartite system. In the present work, this limitation has been overcome by formulating a further generalisation of Wigner's approach through the derivation of a set of LR inequalities with respec…
The presence of an additive conserved quantity imposes a limitation on the measurement process. According to the Wigner-Araki-Yanase theorem, the perfect repeatability and the distinguishability on the apparatus cannot be attained simultaneously. Instead of the repeatability, in this paper, the distinguishability on both systems is examined. We derive a trade-off inequality between the distinguishability of the final states on the system and the one on the apparatus. The inequality shows that the perfect distinguishability of both systems cannot be attained simultaneously.
Institute for Quantum Optics and Quantum Information - Vienna, Austrian Academy of Sciences（奥地利科学院维也纳量子光学与量子信息研究所）；Atominstitut, Technische Universität Wien（维也纳工业大学原子研究所）
Optical beams with topological singularities have a Schmidt decomposition. Hence, they display features typically associated with bipartite quantum systems; in particular, these classical beams can exhibit entanglement. This classical entanglement can be quantified by a Bell inequality formulated in terms of Wigner functions. We experimentally demonstrate the violation of this inequality for Laguerre-Gauss (LG) beams and confirm that the violation increases with increasing orbital angular momentum. Our measurements yield negativity of the Wigner function at the origin for $\LG_{10}$ beams, whe…
We give a trace inequality related to the uncertainty relation of Wigner-Yanase-Dyson skew information. This inequality corresponds to a generalization of the uncertainty relation derived by S. Luo for the quantum uncertainty quantity excluding the classical mixture.
In this work, we show that Bell's inequality violation of arise from the fact that the condition imposed upon the development of inequality is not respected when it is applied in the idealized experiment. Such a condition is that the quantities taken by probability must be non negative, and such a condition is represented by $|\cal{P(w)}|=\cal{P(w)}$. We will also show that, when trying to define the values of the joint probabilities of $ (Z_1, Z_2, Z_3) $, through the values obtained from the $ (Z_j, Z_k) $ pairs, we find that these values are negative, so not Kolmogorov's axiom is respected:…
We propose a refinement of the Robertson-Schrodinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product …
We present a relativistic generalization of the Wigner inequality for the scalar and pseudoscalar particles decaying to two particles with spin (fermions and photons.) We consider Wigner's inequality with the full spin anticorrelation (with the non-relativistic analog) as well as the case with the full spin correlation. The latter case may be obtained by a special choice of the plane of measurement of the spin projections on the direction of propagation of fermions. The possibility for relativistic testing of Bohr's complementarity principle is shown.
The relativistic version of the Greenberger-Horne-Zeilinger experiment with massive particles is proposed. We point out that, in the moving frame, GHZ correlations of spins in original directions transfer to different directions due to the Wigner rotation. Its effect on the degree of violation of Bell-type inequality is also discussed.
It is a long-standing belief, as pointed out by Bell in 1986, that it is impossible to use a two-mode Gaussian state possessing a positive-definite Wigner function to demonstrate nonlocality as the Wigner function itself provides a local hidden-variable model. In particular, when one performs continuous-variable (CV) quadrature measurements upon a routinely generated CV entanglement, namely, the two-mode squeezed vacuum (TMSV) state, the resulting Wigner function is positive-definite and as such, the TMSV state cannot violate any Bell inequality using CV quadrature measurements. We show here, …
We give a trace inequality related to the uncertainty relation of generalized Wigner-Yanase-Dyson skew information which includes our result in JMAA, vol.365, pp.12-18, 2010.
We construct a wide class of bounded continuous variables observables that lead to violations of Bell inequalities for the EPR state and give an intuitive Wigner function explanation how to predetermine which operators won't ever exceed the bounds given by local theories.
We propose a new test of local realism based on correlation measurements of continuum valued functions of positions and momenta, known as modular variables. The Wigner representation of these observables are bounded in phase space, and therefore, the associated inequality holds for any state described by a non-negative Wigner function. This agrees with Bell's remark that positive Wigner functions, serving as a valid probability distribution over local (hidden) phase space coordinates, do not reveal non-locality. We construct a class of entangled states resulting in a violation of the inequalit…
Wigner negativity, as a well-known indicator of nonclassicality, plays an essential role in quantum computing and simulation using continuous-variable systems. Recently, it has been proven that Einstein-Podolsky-Rosen steering is a prerequisite to generate Wigner negativity between two remote modes. Motivated by the demand of real-world quantum network, here we investigate the shareability of generated Wigner negativity in the multipartite scenario from a quantitative perspective. By establishing a monogamy relation akin to the generalized Coffman-Kundu-Wootters inequality, we show that the am…
Bell's inequality violation (BIQV) for correlations of polarization is studied for a {\it product} state of two two-mode squeezed vacuum (TMSV) states. The violation allowed is shown to attain its maximal limit for all values of the squeezing parameter, $ζ$. We show via an explicit example that a state whose entanglement is not maximal allow maximal BIQV. The Wigner function of the state is non negative and the average value of either polarization is nil.
It remains an open question how realist view of macroscopic world emerges from quantum formalism. For testing the macrorealism in quantum domain, an interesting approach was put forward by Leggett and Garg in $1985$, by formulating a suitable inequality valid for any macrorealistic theory. Recently, by following the Wigner idea of local realist inequality, a probabilistic version of standard Leggett-Garg inequalities have also been proposed. While the Wigner form of local realist inequalities are equivalent to the two-party, two-measurements and two outcomes CHSH inequalities, in this paper we…
We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. This quantity is properly defined only for states that possess a positive Wigner function, which we name Wigner-positive states, but we argue that it is a proper measure of quantum uncertainty in phase space. It is invariant under symplectic transformations (displacements, rotations, and squeezing) and we conjecture that it is lower bounded by $\lnπ+1$ within the convex set of Wigner-positive states. It reaches this lower bound for Gaussian pure states, which are natural min…
At first sight, the use of an everywhere positive Wigner function as a probability density to perform stochastic simulations in quantum optics seems equivalent to the introduction of local hidden variables, thus preventing any violation of Bell inequalities. However, because of the difference between symmetrically and normally ordered operators, some trajectories in stochastic simulations can imply negative intensities, despite a positive mean. Hence, Bell inequalities do not apply. Here, we retrieve for a weakly squeezed Gaussian state the maximum violation on polarization states allowed by q…
Banaszek, Wódkiewicz and others (\cite{Banaszek},\cite{Chen},\cite{Chen-Zhang}) made the surprising discovery that Einstein-Bell locality inequalities can be violated by the two mode squeezed vacuum by a factor $\sqrt{2}$, in spite of the fact that the state has a positive Wigner function. I use here the more general Gleason-Kochen-Specker assumption of non-contextuality \cite{Gleason} to express classicality. I then derive non-contextuality Bell inequalities for correlations of $N$ pseudo spins embedded in an infinite dimensional continuous variable Hilbert space, and show that their maximum …
By extending the usual two-mode squeezing operator $S_{2}=\exp [ iλ(Q_{1}P_{2}+Q_{2}P_{1}) ] $ to the three-mode squeezing operator $S_{3}=\exp {iλ[ Q_{1}(P_{2}+P_{3}) +Q_{2}(P_{1}+P_{3}) +Q_{3}(P_{1}+P_{2}) ]} $, we obtain the corresponding three-mode squeezed coherent state. The state's higher-order properties, such as higher-order squeezing and higher-order sub-Possonian photon statistics, are investigated. It is found that the new squeezed state not only can be squeezed to all even orders but also exhibits squeezing enhancement comparing with the usual cases. In addition, we examine the vi…
We demonstrate that intensity correlations of second order in the fluorescence light of N > 2 single- photon emitters may violate locality while the visibility of the signal remains below 71%. For this, we derive a homogeneous Bell-Wigner-type inequality, which can be applied to a broad class of experimental setups. We trace the violation of this inequality back to path entanglement created by the process of detection.
We formulate the Einstein-Podolsky-Rosen (EPR) gedankenexperiment within the framework of relativistic quantum theory to analyze a situation in which measurements are performed by moving observers. We point out that under certain conditions the perfect anti-correlation of an EPR pair of spins in the same direction is deteriorated in the moving observers' frame due to the Wigner rotation, and show that the degree of the violation of Bell's inequality prima facie decreases with increasing the velocity of the observers if the directions of the measurement are fixed. However, this does not imply a…
Distinct from the type of local realist inequality (known as the Collins-Gisin-Linden-Massar-Popescu or CGLMP inequality) usually used for bipartite qutrit systems, we formulate a new set of local realist inequalities for bipartite qutrits by generalizing Wigner's argument that was originally formulated for the bipartite qubit singlet state. This treatment assumes existence of the overall joint probability distributions in the underlying stochastic hidden variable space for the measurement outcomes pertaining to the relevant trichotomic observables, satisfying the locality condition and yieldi…
We study various formulations of Leggett-Garg inequality (LGI), specifically, the Wigner and Clauser-Horne forms of LGI, in the context of subatomic systems, in particular, three flavor neutrino as well as meson systems. The optimal forms of various LGIs for either neutrinos or mesons are seen to depend on measurement settings. For the neutrinos, some of these inequalities can be written completely in terms of experimentally measurable probabilities. Hence, the Wigner and Clauser-Horne forms of LGI are found to be more suitable as compared to the standard LGI from the experimental point of vie…
We introduce an efficient method to reconstruct the Wigner function of many-mode continuous variable systems. It is based on convex optimization with semidefinite programs, and also includes a version of the maximum entropy principle, in order to yield unbiased states. A key ingredient of the proposed approach is the representation of the state in a truncated Fock basis. As a bonus, the discrete finite representation allows to easily quantify the entanglement.
Quantum correlations of observables for two particle states have demonstrated the nonlocal character of the quantum mechanics. However nonlocality can be exhibited even for noncommuting observables of a single particle system. In this paper we show nonlocality of position-momentum correlations of a single particle in the double-slit experiment modeled by an initially correlated Gaussian wavepacket. The positivity or negativity of the Wigner function for the state of the particle at the detection screen is related with the $σ_{xp}$ covariances. A Bell's inequality is constructed from the Wigner…
In this paper we propose a time-independent \textit{equality} and time-dependent \textit{inequality}, suitable for an experimental test of the hypothesis of realism. The derivation of these relations is based on the concept of conditional probability and on Bayes' theorem in the framework of Kolmogorov's axiomatics of probability theory. The equality obtained is intrinsically different from the well known GHZ-equality and its variants, because violation of the new equality might be tested in experiments with only two microsystems in a maximally entangled Bell state $\big\vert\, Ψ^-\,\big\rangl…
In this thesis, entanglement under fully relativistic settings are discussed. The thesis starts with a brief review of the relativistic quantum mechanics. In order to describe the effects of Lorentz transformations on the entangled states, quantum mechanics and special relativity are merged by construction of the unitary irreducible representations of Poincar{é} group on the infinite dimensional Hilbert space of state vectors. In this framework, the issue of finding the unitary irreducible representations of Poincar{é} group is reduced to that of the little group. Wigner rotation for the massi…
We propose a method for characterizing a photodetector by directly reconstructing the Wigner functions of the detector's Positive-Operator-Value-Measure (POVM) elements. This method extends the works of S. Wallentowitz and Vogel [Phys. Rev. A 53, 4528 (1996)] and Banaszek and Wódkiewicz [Phys. Rev. Lett. 76, 4344 (1996)] for quantum state tomography via weak-field homodyne technique to characterize quantum detectors. The scheme uses displaced thermal mixtures as probes to the detector and reconstructs the Wigner function of the photodetector POVM elements from its outcome statistics. In order …
Optimizing Wigner Negativity in Scattering Processes Using Energetic Cost Functions
MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Laboratory（MajuLab，CNRS-UCA-SU-NUS-NTU国际联合研究实验室）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学物理与数学科学学院）；Université de Lorraine, CNRS, LPCT（洛林大学，法国国家科学研究中心，LPCT）
We consider optical beams with topological singularities which possess Schmidt decomposition and show that such classical beams share many features of two mode entanglement in quantum optics. We demonstrate the coherence properties of such beams through the violations of Bell inequality for continuous variables using the Wigner function. This violation is a consequence of correlations between the $(x, p_x)$ and $(y, p_y)$ spaces which mathematically play the same role as nonlocality in quantum mechanics. The Bell violation for the LG beams is shown to increase with higher orbital angular momen…
We derive a Bell inequality based on a generalized quasiprobability function which is parameterized by one non-positive real value. Two types of known Bell inequalities formulated in terms of the Wigner and Q functions are included as limiting cases. We investigate violations of our Bell inequalities for single photon entangled states and two-mode squeezed vacuum states when varying the detector efficiency. We show that the Bell inequality for the Q function allows the lowest detection efficiency for violations of local realism.
We have considered optical beams with phase singularity and experimentally verified that these beams, although being classical, have properties of two mode entanglement in quantum states. We have observed the violation of Bell's inequality for continuous variables using the Wigner distribution function (WDF) proposed by Chowdhury et al. [Phys. Rev. A \textbf{88}, 013830 (2013)]. Our experiment establishes a new form of Bell's inequality in terms of the WDF which can be used for classical as well as quantum systems.
Extended Wigner's Friend Scenarios with Agent-like Observers on Quantum Computers
Institute for Theoretical Physics, ETH Zürich（苏黎世联邦理工学院理论物理研究所）；Quantum and Advanced Technologies Research Institute, Griffith University（格里菲斯大学量子与先进技术研究院）
The scientific method relies on facts, established through repeated measurements and agreed upon universally, independently of who observed them. In quantum mechanics, the objectivity of observations is not so clear, most dramatically exposed in Eugene Wigner's eponymous thought experiment where two observers can experience seemingly different realities. The question whether these realities can be reconciled in an observer-independent way has long remained inaccessible to empirical investigation, until recent no-go-theorems constructed an extended Wigner's friend scenario with four observers t…
The Wigner rotation angle for a particle in a circular motion in the Schwarzschild spacetime is obtained via the Fermi-Walker transport of spinors. Then, by applying the WKB approximation, a possible application of the Fermi-Walker transport of spinors in relativistic EPR correlations is discussed, where it is shown that the spins of the correlated particle undergo a precession in an analogous way to that obtained by Terashima and Ueda [H. Terashima and M. Ueda, Phys. Rev. A {\bf69}, 032113 (2004)] via the application of successive infinitesimal Lorentz transformations. Moreover, from the WKB …
