Tensor C*-Algebra on Two Qubit Spin-1/2 System
Rivani Adistia Dewi (a), Imam Nugraha Albania (b*), Rizky Rosjanuardi (c), Sumanang Muhtar Gozali (d)

Mathematics Study Program, Universitas Pendidikan Indonesia

Abstract

Spin-\frac{1}{2} system is one of the most important phenomenon in quantum mechanics. Such a system is known to be expressed as Pauli matrices, that is \sigma_1=\begin{bmatrix}0 & 1 \\1 & 0 \end{bmatrix}, \sigma_2=\begin{bmatrix}0 & -i \\i & 0 \end{bmatrix}, and \sigma_3=\begin{bmatrix}1 & 0 \\0 & -1 \end{bmatrix}. By inserting the identity matrix I, we can construct a C*-algebra C^{*}({I,\sigma_1,\sigma_2,\sigma_3}) with \mid{I,\sigma_1,\sigma_2,\sigma_3}\otimes{I,\sigma_1,\sigma_2,\sigma_3}\mid=16=\mid{E_ij:i,j=1,2,3,4}\mid, where E_ij is the canonical basis of C*-algebra M_4 (C). In this study, by using the linear independence concept, we got C^{*}({I,\sigma_1,\sigma_2,\sigma_3})\otimes C^{*}({I,\sigma_1,\sigma_2,\sigma_3})\cong M_4 (C). This means, that for any observable in the composite of two spin-\frac{1}{2} systems, it can be expressed by the field reduction from C to R in span⁡{I,\sigma_1,\sigma_2,\sigma_3}\otimes {I,\sigma_1,\sigma_2,\sigma_3} where the structure itself is non-associative algebra. Furthermore, given a Hamiltonian
H(t)=(\sigma_1\otimes\sigma_1)+J\sin(\omega t)(\sigma_3\otimes I)+J⁡\sin(\omega t+\frac{\pi}{2})(I\otimes\sigma_3)
and the initial state which is a superposition (linear combination) of Bell basis, by using the integration factor method and the commutation fact of the exponential operator, we got the time-dependent state
\mid\psi(t)\rangle=\frac{1}{2}(e^{\frac{it}{\hslash}\frac{iJ}{\hslash\omega}\cos(\omega t)\frac{iJ}{\hslash\omega}\sin(\omega t)\nu_{i(5-i)}})_{i=1}^4,
with \nu_{ij}=\delta_{(i+j)5}(-1)^{\max{i,j}}, where \delta_{ij} is Kronecker delta.

Keywords: Pauli matrices, tensor product, commutation relation

Topic: Mathematic