Bohr’s complementarity principle is one of the fundamental principles in quantum mechanics. The wave-particle duality is often used to reflect Bohr’s complementarity principle, so the wave-particle duality has also attracted considerable attention. As research progresses, it has been revealed that the fundamental reason for a particle’s particle-like property lies in its quantum entanglement with the which-path detector. However, current research on complementarity based on the entanglement perspective is limited to the symmetric Mach-Zehnder interferometer model with a balanced beam splitter splitting ratio. To overcome the limitation of fixed splitting ratios in conventional balanced beam splitters, this work achieves continuous modulation of both fringe visibility and quantum entanglement by employing an asymmetric Mach-Zehnder interferometer with a continuously tunable beam-splitting ratio. This design provides an additional degree of freedom for manipulation, thus enabling a more comprehensive exploration of the complementary relationship between fringe visibility and quantum entanglement.

In this work, the wave-particle duality of a particle is investigated with an asymmetric Mach-Zehnder interferometer. Fringe visibility and quantum entanglement are employed to quantify the wave-like and particle-like properties of the particle, respectively. In order to detect the particle passing through the apparatus, a particle detector is placed at one of the output ports of the interferometer. Subsequently, the probability of detecting the particle can be obtained, and the fringe visibility V of the quantized wave-like property can be calculated. A which-path detector is placed on one of the paths of the asymmetric Mach-Zehnder interferometer. When the particle passes through the path with the which-path detector, it will be entangled with the which-path detector. The quantum entanglement, represented by the concurrence C, is used to quantify the particle-like property.

In conclusion, we find that the concurrence depends on both the input state of the particle governed by the Bloch vector \boldsymbolS=\S_x,S_y,S_z\ and the second asymmetric beam splitter characterized by parameter β. Here, S_x represents the component of the particle’s initial state on the \sigma_x, and β determines the reflection or transmission coefficient of the second beam splitter. When \beta=\textπ/2, the reflection and transmission coefficients of the second beam splitter are equal. It is found that the concurrence reaches the upper bound when S_x=-\cos\beta. This demonstrates that the quantum entanglement between the particle and the which-path detector is maximized when the parameter β of the asymmetric beam splitter is adjusted to achieve a specific matching relationship with the particle’s initial state parameter S_x, thereby achieving effective modulation of the particle-like and wave-like properties. This paper also presents the complementary relation between fringe visibility and concurrence, which is expressed as V^2+C^2 \leqslant 1. The equals sign holds when the particle is initially in the pure state and S_x=-\cos\beta. This work provides a theoretical basis for researchers to further explore the wave-like and particle-like properties of quantum systems flexibly.