We study both the wave-like behavior and particle-like behavior in a general Mach-Zehnder interferometer with its asymmetric beam splitter. A error-free measurement in the detector is used to extract the which-path information. The fringe visibility V and the which-path information I path are derived: their complementary relation V + I path ≤ 1 are found, and the condition for the equality is also presented.

Ground state properties and level statistics of the Dicke model for a finite number of atoms are investigated based on a progressive diagonalization scheme (PDS). Particle number statistics, the entanglement measure and the Shannon information entropy at the resonance point in cases with a finite number of atoms as functions of the coupling parameter are calculated. It is shown that the entanglement measure defined in terms of the normalized von Neumann entropy of the reduced density matrix of the atoms reaches its maximum value at the critical point of the quantum phase transition where the system is most chaotic. Noticeable change in the Shannon information entropy near or at the critical point of the quantum phase transition is also observed. In addition, the quantum phase transition may be observed not only in the ground state mean photon number and the ground state atomic inversion as shown previously, but also in fluctuations of these two quantities in the ground state, especially in the atomic inversion fluctuation.

Based on modern quantum measurement theory, we use Zurek's "triple model" to study, from the viewpoint of quantum information theory, the wave and particle nature of a photon in a symmetric Mach-Zehnder interferometer. In the process of quantum measurement, the state of both the system and the detector is not an entangled state but a correlated state. We find that the information gain about the photon is related to the correlations (including classical and quantum correlations) between the photon and the detector. We also derive the relationship between the information gain and the fringe visibility. We find that the classical correlations remain consistent with the path distinguishability and can be used to describe the particle-like property of the photon. Quantum correlations are not exactly the same as fringe visibility, but both can represent the quantum coherence of the photon. Finally, we provide an analytical expression for quantum correlations of one type of two-qubit separable states.

We study the fringe visibility and the distinguishability of a general Mach-Zehnder interferometer with an asymmetric beam splitter. Both the fringe visibility V and the distinguishability D are affected by the input state of the particle characterized by the Bloch vectorand the second asymmetric beam splitter characterized by paramter β. For the total system is initially in a pure state, it is found that the fringe visibility reaches the upper bound and the distinguishability reaches the lower bound when cos β = −S x . The fringe visibility obtain the maximum only if S x = 0 and β = π/2 when the input particle is initially in a mixed state. The complementary relationship V 2 + D 2 ≤ 1 is proved in a general Mach-Zehnder interferometer with an asymmetric beam splitter, and the conditions for the equality are also presented.

We study the fringe visibility and the which-path information (WPI) of a general Mach–Zehnder interferometer with an asymmetric beam splitter (BS). A minimum error measurement in the detector is used to extract the WPI. Both the fringe visibility V and the WPI Ipath are affected by the initial state of the photon and the second asymmetric BS. The condition in which the WPI takes the maximum is obtained. The complementarity relationship V 2 + I path 2 ≤ 1 is found, and the conditions for equality are also presented.