Tensor C*-algebra on Two Qubit Spin-1/2 System

DOI

10.2991/978-2-38476-283-5_31How to use a DOI?

Keywords

Commutation Relation; Pauli Matrices; Tensor Product

Abstract

Spin-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  σ 1 = 0 1 1 0 , σ 2 = 0 - i i 0 , and σ 3 = 1 0 0 - 1 . By inserting the identity matrix  I , we can construct a C*-algebra C ∗ I , σ 1 , σ 2 , σ 3 with  I , σ 1 , σ 2 , σ 3 ⨂ I , σ 1 , σ 2 , σ 3 = 16 = E ij : i , j = 1 , 2 , 3 , 4 , 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 , σ 1 , σ 2 , σ 3 ⨂ C ∗ I , σ 1 , σ 2 , σ 3 ≅ M 4 C . This means, that for any observable in the composite of two spin-1/2 systems, by self-adjointness of Pauli matrices, it can be expressed by the field reduction from C to R in span I , σ 1 , σ 2 , σ 3 ⨂ I , σ 1 , σ 2 , σ 3 where the structure itself is non-associative algebra. Furthermore, given a Hamiltonian H t = σ 1 ⨂ σ 1 + J sin ω t σ 3 ⨂ I + J sin ω t + π 2 I ⨂ σ 3 and the initial state which is a linear combination of Bell basis, by using the integration factor method and the commutation fact of the exponential operator, we got the following time-dependent state ψ t ⟩ = 1 2 e it ħ i J ħ ω cos ω t i J ħ ω sin ω t ν i 5 - i i = 1 4 with  ν ij = δ i + j 5 - 1 maks i , j , where δ ij is Kronecker delta.

Copyright

© 2024 The Author(s)

Open Access

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TY  - CONF
AU  - Rivani Adistia Dewi
AU  - Imam Nugraha Albania
AU  - Rizky Rosjanuardi
AU  - Sumanang Muhtar Gozali
PY  - 2024
DA  - 2024/09/03
TI  - Tensor C*-algebra on Two Qubit Spin-1/2 System
BT  - Proceedings of the  9th Mathematics, Science, and Computer Science Education International Seminar (MSCEIS 2023)
PB  - Atlantis Press
SP  - 333
EP  - 338
SN  - 2352-5398
UR  - https://doi.org/10.2991/978-2-38476-283-5_31
DO  - 10.2991/978-2-38476-283-5_31
ID  - Dewi2024
ER  -