HuaiXin Cao

Distinguishing classical correlations from quantum correlations

The aim of this paper is to distinguish classical correlations from quantum ones. First, a new characterization of a classical correlated (CC) state is established. Corresponding results for the left and right classical correlations are also obtained. Second, a sufficient and necessary condition for a convex combination of two CC states to be CC is given. It is proved that the set CC(HA ⊗ HB) of all CC states on HA ⊗ HB is a perfect, nowhere dense and compact subset of the metric space D(HA ⊗ HB) of all states (density operators) on HA ⊗ HB. Based on our new characterization of CC states, a quantity Q(ρ) is associated with a state ρ. It is shown that a state ρ is CC if and only if Q(ρ) = 0. In particular, we prove that the Werner state Wλ in a two-qubit system with a single real parameter λ is CC if and only if λ = 0.25.

An upper bound for the adiabatic approximation error

In this paper, we derive an upper bound for the adiabatic approximation error, which is the distance between the exact solution to a Schrödinger equation and the adiabatic approximation solution. As an application, we obtain an upper bound for 1 minus the fidelity of the exact solution and the adiabatic approximation solution to a Schrödinger equation.

A generalization of Schrödinger's uncertainty relation described by the Wigner---Yanase skew information

The uncertainty principle in quantum mechanics is a fundamental relation with different forms, including Heisenberg’s uncertainty relation and Schrodinger’s uncertainty relation. We discuss the generalized Wigner–Yanase correlation and the generalized covariance of operators and establish a generalization of Schrodinger’s uncertainty relation expressed in terms of Wigner–Yanase information.

Generalized duality quantum computers acting on mixed states

A generalized duality quantum computer acting on mixed states (GDQC-MS) is established, which is a device consisting of a generalized quantum wave divider, a finite number of generalized quantum operations, and a generalized quantum wave combiner. Some of the interesting properties of a GDQC are explored. For example, it is proved that the divider and the combiner of a GDQC-MS are mutually dual contractions, and when the generalized quantum operations used in a GDQC-MS are contractions, the GDQC-MS is also a contraction. In that case, the loss of an input state passing through a GDQC-MS is measured and the corresponding operator of a GDQC-MS is a generalized quantum operation.

Generalized Robustness of Contextuality

Motivated by the importance of contextuality and a work on the robustness of the entanglement of mixed quantum states, the robustness of contextuality (RoC) R C ( e ) of an empirical model e against non-contextual noises was introduced and discussed in Science China Physics, Mechanics and Astronomy (59(4) and 59(9), 2016). Because noises are not always non-contextual, this paper introduces and discusses the generalized robustness of contextuality (GRoC) R g ( e ) of an empirical model e against general noises. It is proven that R g ( e ) = 0 if and only if e is non-contextual. This means that the quantity R g can be used to distinguish contextual empirical models from non-contextual ones. It is also shown that the function R g is convex on the set of all empirical models and continuous on the set of all no-signaling empirical models. For any two empirical models e and f such that the generalized relative robustness of e with respect to f is finite, a fascinating relationship between the GRoCs of e and f is proven, which reads R g ( e ) R g ( f ) ≤ 1 . Lastly, for any n-cycle contextual box e, a relationship between the GRoC R g ( e ) and the extent Δ e of violating the non-contextual inequalities is established.

Robustness of quantum correlations against linear noise

Relative robustness of quantum correlations (RRoQC) of a bipartite state is firstly introduced relative to a classically correlated state. Robustness of quantum correlations (RoQC) of a bipartite state is then defined as the minimum of RRoQC of the state relative to all classically correlated ones. It is proved that as a function on quantum states, RoQC is nonnegative, lower semi-continuous and neither convex nor concave; especially, it is zero if and only if the state is classically correlated. Thus, RoQC not only quantifies the endurance of quantum correlations of a state against linear noise, but also can be used to distinguish between quantum and classically correlated states. Furthermore, the effects of local quantum channels on the robustness are explored and characterized.

Estimations of the errors between the evolving states generated by two Hamiltonians with the same initial state

Time evolution of a quantum system is described by Schrödinger equation with initial pure state, or von Neumann equation with initial mixed state. In this paper, we estimate the error between the evolving states generated by two Hamiltonians with the same initial pure state. Secondly, according to the method of operator–vector correspondence, we give a relation of the Schrödinger equation and von Neumann equation and then estimate the error between the evolving states generated by two Hamiltonians with the same initial mixed state.

Uncertainty relations with the generalized Wigner–Yanase–Dyson skew information

The uncertainty principle in quantum mechanics is a fundamental relation with different forms, including Heisenberg’s uncertainty relation and Schrödinger’s uncertainty relation. We introduce the generalized Wigner–Yanase–Dyson correlation and the related quantities. Various properties of them are discussed. Finally, we establish several generalizations of uncertainty relation expressed in terms of the generalized Wigner–Yanase–Dyson skew information.

Adiabatic approximation for the evolution generated by an A -uniformly pseudo-Hermitian Hamiltonian

We discuss an adiabatic approximation for the evolution generated by an A-uniformly pseudo-Hermitian Hamiltonian H(t). Such a Hamiltonian is a time-dependent operator H(t) similar to a time-dependent Hermitian Hamiltonian G(t) under a time-independent invertible operator A. Using the relation between the solutions of the evolution equations H(t) and G(t), we prove that H(t) and H† (t) have the same real eigenvalues and the corresponding eigenvectors form two biorthogonal Riesz bases for the state space. For the adiabatic approximate solution in case of the minimum eigenvalue and the ground state of the operator H(t), we prove that this solution coincides with the system state at every instant if and only if the ground eigenvector is time-independent. We also find two upper bounds for the adiabatic approximation error in terms of the norm distance and in terms of the generalized fidelity. We illustrate the obtained results with several examples.