This paper first employs the mathematical framework of quantum mechanics, including Hilbert space theory and the superposition principle of quantum states, to analyze the processing methods and transmission processes of quantum information under second-order conditions. It delves into the core role of quantum gates as orthogonal operators and their correspondence with information transmission results through matrix multiplication. Subsequently, it investigates quantum information transmission and dense encoding techniques based on controllable non-qubit gates (CNOT) at different orders. The CNOT gate is represented by a second-order tensor product matrix: when the control qubit is
∣1>, it performs a Pauli-X operation on the target qubit (matrix: [
[0,1], [1, 0]]), otherwise remaining unchanged. By defining the CNOT matrix, Hadamard matrix, maximum entangled state and corresponding multi-order unitary operations (described using elements of the
SU(n) group for higher-order quantum system transformations), this study combines density matrix representations of quantum states with quantum measurement theories (such as the mathematical description of projection measurements). It reveals the mechanistic principles underlying quantum information transmission and decryption processes. Through computational verification of quantum state fidelity and transmission success rates across different orders, the effectiveness of cross-order information transmission is demonstrated.
An expanded theory of information transmission and processing has emerged over the past few years, encompassing the processing and transmission of intact quantum states, the interaction of quantum and classical information, the quantitative theory of entanglement, and the use of quantum information processing to speed up certain classical computations. We review this field, concentrating on the parallels with and differences from classical information theory, which is now best seen as a part of the new theory.