## 1 Introduction  
Fiber-based Mach-Zehnder interferometers (MZIs) are used to detect environmental variations, such as temperature, pressure, and mechanical stress, as well as to increase transmission rates in telecommunication systems [1; 2; 3]. They also have interesting applications in quantum communications. For instance, two-arm MZIs have been used to prepare and measure quantum states encoded by two-dimensional paths (qubits), as well as to evaluate entangled quantum systems [4; 5; 6]. For this application, the quality (or fidelity) of the qubit state prepared depends on the physical characteristics of the interferometer, such as polarization, and insertion loss. Although insertion loss and polarization are typically compensated for by using passive elements integrated into optical fibers, such as the relative phase between the optical paths, the symmetry of the optical path changes constantly and randomly due to variations in the refractive index caused mainly by mechanical disturbances and temperature changes.This variation in the optical path between different fibers produces a phase shift, which is known as phase noise. To enable quantum information applications, this phase noise has been compensated using various control methods based on Proportional-Integral-Derivative (PID) controller in two-arm MZIs, such as piezoelectric and/or fiber stretcher elements, showing successful performance in laboratory and field environments [7; 8; 9]. On the other hand, a multi-arm MZI (MAMZI) allows the preparation and measurement of high-dimensional quantum states, known as qudits. One of the most significant advantages of operating with qudits is the ability to encode more information into a single particle compared to a qubit, potentially improving the efficiency of specific quantum tasks, optimizing the execution of quantum algorithms and increasing sensor accuracy across entangled qudits by increasing sensitivity to minor parameter variations [10; 11; 12]. In quantum cryptography, qudits offer greater environmental noise tolerance and increased security, as the complexity of eavesdropping attacks grows exponentially with the dimension of the quantum system [13; 14]. Although path-encoded qudits can be prepared using a MAMZI, the constant change of the optical path of the multiple arms introduces multidimensional phase noise, making it impractical to generate and measure qudits efficiently using a single-core fiber array. Multicore optical fibers (MCFs) have been employed to mitigate the impact of this phase noise on path-encoded qudits [15; 16; 17] and entangled qudits [18]. Although the effects of phase noise are less than in a single-core fiber array, it is still a critical variable in fiber-based MAMZI systems [19]. Quantum cryptography schemes based on MAMZIs have been proposed to mitigate the impact of multidimensional phase noise by sending the quantum state through all possible paths of the interferometer, thus achieving phase and polarization self-compensation [20]; to implement noise control and encryption process using independent cores of multicore fibers (MCFs)[21]; and to improve the robustness and quality of the system using hybrid structures such as both single core fibers and MCFs within a single MAMZI [22; 23]. MAMZIs are composed of multiport beam splitters (MBS), which are feasible to develop directly in multicore fiber [22] and within photonic integrated circuits [20], which allows the scalability of systems of higher complexity (\(N>4\)). In this work, we analyze the controllability of any dimensional MAMZI built using MBS. Additionally, we propose a standardized control method that maximizes the specific output intensity of a MAMZI, thus stabilizing the multidimensional phase noise inside the optical circuit. This approach allows optimization algorithms as controllers to mitigate phase noise in any MAMZI configuration.