## 1 Introduction  
Visible light communication (VLC) serves as a beneficial complement to radio frequency (RF) communication [1], offering the potential to address the scarcity of spectrum resources in RF communication and thereby alleviate pressure on wireless communication networks. Recently, VLC has gained immense traction because of its low cost, low power consumption, and license-free indoor applications in the context of sixth-generation (6G) communication systems [2]. In VLC systems, indoor wireless data are transferred over the optical spectrum using intensity modulation/direct detection (IM/DD) [3]. Light-emitting diodes (LEDs) serve dual purposes for both lighting and data transmission, while photodetectors (PDs) are used to detect the transmitted data [4]. Currently, to enhance the performance of indoor communication systems, numerous LEDs and PDs are deployed within indoor environments, forming multiple-input multiple-output (MIMO) VLC systems.  
Photon-counting technology exhibits excellent capabilities in detecting optical signals. When the communication system operates at low power [5], particularly in power-constrained IoT applications, this advantage becomes even more prominent. In such scenarios, traditional optical communication receivers struggle to operate effectively in the presence of extremely weak light. However, the photon-counting receiver utilizes the principle of photon counting and high-sensitivity detectors to achieve reliable optical signal reception and decoding at extremely low optical power levels. We delve into a more microscopic perspective, considering light as being composed of discrete particles called photons.  
In the traditional field of optical communications, scholars primarily focus on communication system models affected by additive white Gaussian noises (AWGNs). In contrast, photon counting introduces quantum effects due to the discrete nature of light signals, resulting in Poisson shot noises [5], which are signal-dependent. Typically, the photon-counting process can be characterized by two independent parameters [6]--the background radiation and the expected average signal value--and we refer to it as a Poisson counting process (PCP) [5]. This process exhibits non-linear characteristics, posing significant challenges to the analysis and optimization of photon-counting communication systems.  
The achievable rate is a critical metric in communication systems, as it directly impacts the efficiency and performance of communication. This metric is equally important and worth researching in photon-counting VLC systems. In MIMO systems, alleviating multi-user interference (MUI) and enhancing achievable rates is typically achieved by using precoding techniques at the transmitter. However, this is not straightforward for photon-counting MIMO VLC systems, as the inherent correlation between the signal and noise in the PCP leads to a fundamental change in the expression of the rate, rendering the classic Shannon formula inapplicable to photon-counting systems. Moreover, this signal-dependent shot noise exhibits non-linear characteristics, posing significant challenges to the analysis and optimization of photon-counting communication systems. This also implies that the mature precoding techniques in MIMO VLC systems in AWGN channels are not applicable in photon-counting systems.