Zeqian Chen

Non-Markovian effect on the quantum discord

We study the non-Markovian effect on the dynamics of the quantum discord by exactly solving a model consisting of two independent qubits subject to two zero-temperature non-Markovian reservoirs, respectively. Considering the two qubits initially prepared in Bell-like or extended Werner-like states, we show that there is no occurrence of the sudden death, but only instantaneous disappearance of the quantum discord at some time points, in comparison to the entanglement sudden death in the same range of the parameters of interest. This implies that the quantum discord is more useful than the entanglement to describe the quantum correlation involved in quantum systems.

Harmonic Analysis on Quantum Tori

This paper is devoted to the study of harmonic analysis on quantum tori. We consider several summation methods on these tori, including the square Fejér means, square and circular Poisson means, and Bochner-Riesz means. We first establish the maximal inequalities for these means, then obtain the corresponding pointwise convergence theorems. In particular, we prove the noncommutative analogue of the classical Stein theorem on Bochner-Riesz means. The second part of the paper deals with Fourier multipliers on quantum tori. We prove that the completely bounded Lp Fourier multipliers on a quantum torus are exactly those on the classical torus of the same dimension. Finally, we present the Littlewood-Paley theory associated with the circular Poisson semigroup on quantum tori. We show that the Hardy spaces in this setting possess the usual properties of Hardy spaces, as one can expect. These include the quantum torus analogue of Fefferman’s H1-BMO duality theorem and interpolation theorems. Our analysis is based on the recent developments of noncommutative martingale/ergodic inequalities and Littlewood-Paley-Stein theory.

On the Cauchy problem for Gross-Pitaevskii hierarchies

The purpose of this paper is to investigate the Cauchy problem for the Gross–Pitaevskii infinite linear hierarchy of equations on Rn, n ⩾ 1. We prove local existence and uniqueness of solutions in certain Sobolev-type spaces Hξα of sequences of marginal density operators with α > n/2. In particular, we give a clear discussion of all cases α > n/2, which covers the local well-posedness problem for the Gross–Pitaevskii hierarchy in this situation.

Wigner-Yanase skew information as tests for quantum entanglement

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 on relationships between quantum entanglement and information theory.

Characterizations of Arveson's Hardy Space

A Hardy-type space H 2 d in the unit ball Bd of Cd , which was recently introduced by Arveson [W. Arveson (1998). Subalgebras of C*-algebras III: multivariable operator theory. Acta Math., 181, 159-228.], is appropriate for the operator theory of d-contractions. In this article, it is proved that H 2 d actually coincides with a Hardy-Sobolev space. This yields almost immediately some of the related results obtained in [W. Arveson (1998). Subalgebras of C*-algebras III: multivariable operator theory. Acta Math., 181, 159-228.], including the facts that H 2 d is not associated with any measure on C d ; and that the corresponding algebra of multipliers M ⊂ H ∞(Bd ) and the inclusion is proper. Finally, a function-theoretic version of von Neumanns inequality for the d-contractions is presented.

Measurement contextuality is implied by macroscopic realism

Ontological theories of quantum mechanics provide a realistic description of single systems by means of well-defined quantities conditioning the measurement outcomes. In order to be complete, they should also fulfill the minimal condition of macroscopic realism. Under the assumption of outcome determinism and for Hilbert space dimension greater than 2, they were all proved to be contextual for projective measurements. In recent years a generalized concept of noncontextuality was introduced that applies also to the case of outcome indeterminism and unsharp measurements. It was pointed out that the Beltrametti-Bugajski model is an example of measurement noncontextual indeterminist theory. Here we provide a simple proof that this model is the only one with such a feature for projective measurements and Hilbert space dimension greater than 2. In other words, there is no extension of quantum theory providing more accurate predictions of outcomes and simultaneously preserving the minimal labeling of events through projective operators. As a corollary, noncontextuality for projective measurements implies noncontextuality for unsharp measurements. By noting that the condition of macroscopic realism requires an extension of quantum theory, unless a breaking of unitarity is invoked, we arrive at the conclusion that the only way to solve the measurement problem in the framework of an ontological theory is by relaxing the hypothesis of measurement noncontextuality in its generalized sense.

PARTIAL BOCHNER INTEGRABILITY (I)

This paper is concerned with partial Bochner integrals with emphasis on a geometrical study of partial Bochner integrability. Convex sets involving the range of L-1-bounded functions are constructed. These constructions provide a complete characterization of partial Bochner integrable functions.

AREA INTEGRAL FUNCTIONS FOR SECTORIAL OPERATORS ON L p SPACES

Area integral functions are introduced for sectorial operators on LP-spaces. We establish the equivalence between the square and area integral functions for sectorial operators on LP spaces. This follows that the results of Cowling, Doust, McIntosh, Yagi, and Le Merdy on H-infinity functional calculus of sectorial operators on L-P-spaces hold true when the square functions are replaced by the area integral functions.

Characterization of multiqubit pure-state entanglement

A necessary and sufficient entanglement criterion based on variances of Mermin-Klyshkos Bell operators is proved for multiqubit pure states. Contrary to Bells inequalities, entangled pure states strictly satisfy a quadratic inequality but product ones can attain the equality under some local unitary transformations, which can be obtained by solving a quadratic maximum problem. This presents a characterization of multiqubit pure-state entanglement.

Characterization of maximally entangled two-qubit states via the Bell-Clauser-Horne-Shimony-Holt inequality

Maximally entangled states should maximally violate the Bell inequality. It is proved that all two-qubit states that maximally violate the Bell-Clauser-Horne-Shimony-Holt inequality are exactly Bell states and the states obtained from them by local transformations. The proof is obtained by using the certain algebraic properties that Paulis matrices satisfy. The argument is extended to the three-qubit system. Since all states obtained by local transformations of a maximally entangled state are equally valid entangled states, we thus give the characterizations of maximally entangled states in both the two-qubit and three-qubit systems in terms of the Bell inequality.