Pre-Schmidt Mode Parameters in Quantum Entanglement
TAMIL SELVAM MURUGAYA, Nurisya Mohd Shah, Sharifah Kartini Said Husain Β· ASM Science Journal Β· 2024
We wrote this paper to study in detail the mathematical methods used in deriving the parameters ππβ²1, ππβ²2, π₯π₯β²1, π₯π₯β²2, ππ, πΎπΎ and ππ2ππ in the research papers by Han et al. (1999), Makarov (2018a), Han et al. (1993), and Han et al. (1995) to find the Schmidt modes π¬π¬ππ in quantum entanglement. Here, we have analysed and developed a thorough calculation in exploring the rationales behind the existence of these parameters in two and three-coupled harmonic oscillators. Various mathematical approaches were applied in the study, ranging from polynomials and linear algebra to trigonometry and the Pythagorean theorem. We found the parameters πΎπΎ and ππ2ππ using the matricesβ determinants and eigenvalues. With these rationales in deriving the parameters πΎπΎ and ππ2ππ in the research papers for the two-coupled harmonic oscillators, we have formulated similar parameters for three-coupled harmonic oscillators as the conclusion of our study.