Multipartite Generalized Bell Inequality with an Arbitrary Number of Settings

We derive a generalized Bell inequality for N qubits which involves an arbitrary number of settings for each of the local measuring apparatuses. The inequality forms a necessary condition for the existence of a local realistic model for the values of a correlation function, given in a n-setting Bell experiment. We show that a local realistic model for the values of a correlation function, given in a two-setting Bell experiment, cannot construct a local realistic model for the values of a correlation function, given in an arbitrary number of n-setting Bell experiment, even though there exist two-setting models for the n measurement directions chosen in the given n-setting experiment. Therefore, the property of two-setting model is different from the property of n-setting model. We discuss classification of local realistic theories in further detail more than the result presented in (J. Phys. A: Math. Theor. 41:155308, 2008). The generalized Bell inequality covers the previous results correctly.     

Admissible states in quantum phase space

We address the question of which phase space functionals might represent a quantum state. We derive necessary and sufficient conditions for both pure and mixed phase space quantum states. From the pure state quantum condition we obtain a formula for the momentum correlations of arbitrary order and derive explicit expressions for the wave functions in terms of time-dependent and independent Wigner functions. We show that the pure state quantum condition is preserved by the Moyal (but not by the classical Liouville) time evolution and is consistent with a generic stargenvalue equation. As a by-product Baker's converse construction is generalized both to an arbitrary stargenvalue equation, associated to a generic phase space symbol, as well as to the time-dependent case. These results are properly extended to the mixed state quantum condition, which is proved to imply the Heisenberg uncertainty relations. Globally, this formalism yields the complete characterization of the kinematical structure of Wigner quantum mechanics. The previous results are then succinctly generalized for various quasi-distributions. Finally, the formalism is illustrated through the simple examples of the harmonic oscillator and the free Gaussian wave packet. As a by-product, we obtain in the former example an integral representation of the Hermite polynomials.     

Inequality relations for the hierarchy of quantum correlations in two-qubit systems

Entanglement, quantum steering and Bell nonlocality can be used to describe the distinct quantum correlations of quantum systems. Because of their different characteristics and application fields, how to divide them quantitatively and accurately becomes particularly important. Based on the sufficient and necessary criterion for quantum steering of an arbitrary two-qubit T-state, we derive the inequality relations between quantum steering and entanglement as well as between quantum steering and Bell nonlocality for the T-state. Additionally, we have verified those relations experimentally.     

Configuration of separability and tests for multipartite entanglement in bell-type experiments

We derive tight quadratic inequalities for all kinds of hybrid separable-inseparable n-particle density operators on an arbitrary dimensional space. This methodology enables us to derive a tight quadratic inequality as tests for full n-partite entanglement in various Bell-type correlation experiments on the systems that may not be identified as a collection of qubits, e.g., those involving photons measured by incomplete detectors. It is also proved that when the two measured observables are assumed to precisely anticommute, a stronger quadratic inequality can be used as a witness of full n-partite entanglement.     

Perturbative quantum error correction

We derive simple necessary and sufficient conditions under which a quantum channel obtained from an arbitrary perturbation from the identity can be reversed on a given code to the lowest order in fidelity. We find the usual Knill-Laflamme conditions applied to a certain operator subspace which, for a generic perturbation, is generated by the Lindblad operators. For a weak interaction with an environment, the error space to be corrected is a subspace of that spanned by the interaction operators, selected by the environment's initial state.     

Bell's inequalities and quantum communication complexity

We prove that for every Bell's inequality, including those which are not yet known, there always exists a communication complexity problem, for which a protocol assisted by states which violate the inequality is more efficient than any classical protocol. Violation of Bell's inequalities is the necessary and sufficient condition for quantum protocol to beat the classical ones.     

Data processing for the sandwiched Rényi divergence: a condition for equality

The \(\alpha \)-sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for \(\alpha \ge 1/2\). In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the \(\alpha \)-sandwiched Rényi divergence for all \(\alpha \ge 1/2\). For the range \(\alpha \in [1/2,1)\), our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb (J Math Phys 54(12):122201, 2013) using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a Rényi version of the Araki–Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb (Lett Math Phys 101(1):1–11, 2012) about equality in the original Araki–Lieb inequality. Furthermore, we prove a general lower bound on a Rényi version of the entanglement of formation and observe that it is attained by states saturating the Rényi version of the Araki–Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.     

Tight multipartite Bell's inequalities involving many measurement settings

We derive tight Bell's inequalities for N>2 observers involving more than two alternative measurement settings. We give a necessary and sufficient condition for a general quantum state to violate the new inequalities. The inequalities are violated by some classes of states, for which all standard Bell's inequalities with two measurement settings per observer are satisfied.     

Subadditivity Condition for Spin Tomograms and Density Matrices of Arbitrary Composite and Noncomposite Qudit Systems

We obtain a new quantum entropic inequality for the states of a system of n ≥ 1 qudits. The inequality has the form of the quantum subadditivity condition of a bipartite qudit system and coincides with the subadditivity condition for the system of two qudits. We formulate a general statement on the existence of the subadditivity condition for an arbitrary probability distribution and an arbitrary qudit-system tomogram. We discuss the nonlinear quantum channels creating the entangled states from separable states.     

Inseparability criteria for continuous bipartite quantum states

We provide necessary and sufficient conditions for the partial transposition of bipartite harmonic quantum states to be nonnegative. The conditions are formulated as an infinite series of inequalities for the moments of the state under study. The violation of any inequality of this series is a sufficient condition for entanglement. Previously known entanglement conditions are shown to be special cases of our approach.     

Entanglement in finitely correlated spin states

We consider entanglement properties of pure finitely correlated states (FCS). We derive bounds for the entanglement of a spin with an interval of spins in an arbitrary pure FCS. Finitely correlated states are also known as matrix product states or generalized valence-bond states. The bounds become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our bounds provide a proof of the recent conjecture by Benatti, Hiesmayr, and Narnhofer that their necessary condition for nonvanishing entanglement in terms of a single spin and the memory of the FCS is also sufficient. We also generalize the study of entanglement in the Affleck-Kennedy-Lieb-Tasaki model by Fan, Korepin, and Roychowdhury. Our result permits a more efficient calculation, numerically and in some cases analytically, of the entanglement of arbitrary finitely correlated quantum spin chains.     

Bell-type inequalities in horizontal sums of Boolean algebras

We give a necessary and sufficient condition for a Bell-type inequality to hold in a horizontal sum of finitely many finite Boolean algebras.     

Comparison between local Kolmogorovian models and Bell's inequality

The conditions on the relative frequencies of coincidence between the measurements on two physical systems are deduced, in the particular case of four different directions, from Kolmogorovian probability and the Gutkowski and Valdes-Franco computational method. These conditions are compared with those imposed by Bell's inequality. It is proved that Bell's inequality is a necessary but not a sufficient condition for local Kolmogorovian probability. The further assumptions to be added to Bell's inequality, in order to prove the equivalence with local Kolmogorovian probability, are studied. The connection with the results obtained by other authors on the subject is discussed.     

Greenberger-Horne-Zeilinger states and few-body Hamiltonians

The generation of Greenberger-Horne-Zeilinger (GHZ) states is a crucial problem in quantum information. We derive general conditions for obtaining GHZ states as eigenstates of a Hamiltonian. We find that a necessary condition for an n-qubit GHZ state to be a nondegenerate eigenstate of a Hamiltonian is the presence of m-qubit couplings with m≥[(n+1)/2]. Moreover, we introduce a Hamiltonian with a GHZ eigenstate and derive sufficient conditions for the removal of the degeneracy.     

Direct sampling of negative quasiprobabilities of a squeezed state

Although squeezed states are nonclassical states, so far, their nonclassicality could not be demonstrated by negative quasiprobabilities. In this work we derive pattern functions for the direct experimental determination of so-called nonclassicality quasiprobabilities. The negativities of these quantities turn out to be necessary and sufficient for the nonclassicality of an arbitrary quantum state and are therefore suitable for a direct and general test of nonclassicality. We apply the method to a squeezed vacuum state of light that was generated by parametric down-conversion in a second-order nonlinear crystal.     

Nonlocality of Two-Qubit and Three-Qubit Schmidt-Correlated States

We investigate the nonlocality of Schmidt-correlated (SC) states, and present analytical expressions of the maximum violation value of Bell inequalities. It is shown that the violation of Clauser-Horne-Shimony-Holt (CHSH) inequality is necessary and sufficient for the nonlocality of two-qubit SC states, whereas the violation of the Svetlichny inequality is only a sufficient condition for the genuine nonlocality of three-qubit SC states. Furthermore, the relations among the maximum violation values, concurrence, and relative entropy entanglement are discussed.     

Quantum correlation between two qubits induced by a common heat bath

In terms of geometric discord, we study quantum correlations between two qubits interacting with a common heat bath. A necessary and sufficient condition for zero discord for arbitrary two-qubit density matrix is derived. With this condition, we show that a common heat bath can always induce two-qubit quantum correlations if both qubits are initially prepared in arbitrary superposition of “pointer basis”.     

Some Characterizations of EPR Steering

Einstein-Podolsky-Rosen(EPR) steering is one of important quantum correlations of a composite quantum system, which was observed firstly by Schrödinger in the context of the famous EPR paradox and has been discussed recently. In this paper, we give some characterizations of EPR steerability of bipartite states by proving some necessary and sufficient conditions for a state to be unsteerable with a measurement assemblage of Alice. Based on one of the obtained characterizations, we derive an EPR steering inequality, which serves to check EPR steerability of the maximally entangled states.     

Quantum Fisher Information in Two-Qubit Pure States

In terms of quantum Fisher information (QFI), a quantity χ ² was introduced by Pezzé and Smerzi (Phys. Rev. Lett. 102 100401, 2009). They pointed out that the inequality χ ²<1 was a sufficient condition for multiparticle entanglement. For the two-qubit symmetric states, we found that the inequality χ ²<1 is a necessary and sufficient condition for entanglement and spin squeezing, and that χ ² is equal to the second kind of spin squeezing parameter x₂²\xi _{2}^{2}. For the two-qubit asymmetric states, it is only a sufficient condition. In order to make it a necessary and sufficient condition, we extend the concept of the QFI and χ ², and generalize the relations among the entanglement measurement, the spin squeezing parameters and χ ² in symmetric pure states to arbitrary pure states.