Why Quantum Mechanics Requires Complex Numbers as Props to Build Its Foundation On

Didik Nur Huda

The study of quantum mechanics has been built on a Hilbert space, the complex vector space with inner product <.|.>:VxV--> C with C is a complex vector space dan is complex numbers. Hilbert module is a generalization of Hilbert space, by changing the scalar set become ring with certain conditions so that a Hilbert module. This ring is called the number ring. Hilbert modules that will be used to build the modular foundations of quantum mechanics. Keywords : Scalar, Rings, Hilbert Module, Quantum Mechanics

Ijaz Durrani

Songsong Dai

While complex numbers form the foundational number system of quantum mechanics, the theoretical possibility of higher-dimensional hypercomplex alternatives persists. In this paper, we present a test protocol designed to distinguish between complex and quaternionic quantum mechanics. The protocol is founded upon the quantum superdense coding scheme and functions by detecting the presence of multiple distinct quantum phase gates. We also provide the corresponding quantum circuit required to perform this test.

Yìlè Yīng, Maria Ciudad Alañón, Daniel Centeno +5 more

Jarosław Adam Miszczak

In this work we demonstrate a simplified version of quantum mechanics in which the states are constructed using real numbers only. In the standard for-mulation of quantum mechanics the state is represented by positive semidefinite, normalized linear operators. In the following we focus on linearity and hermic-ity properties of density matrices as they are crucial for the properties of quantum channels. The main advantage of the introduced approach is that the simulation of the n-dimensional quantum system requires O(n2) real numbers, in contrast to the standard case where O(n4) real numbers are required. The main disadvantage is the lack of hermicity in the representation of quantum states. This leads to the occurrence of complex eigenvalues of the real-valued density matrices. During the last few years Mathematica computing system has become very popular in the area of quantum information theory and the foundations of quantum mechanics. The main reason for this is its ability to merge the symbolic and nu-