Konstruksi Keadaan 3-Qubit Menggunakan Transformasi Matriks SU(2) dan SU(4)

Al Hasany, Alief Hisyam (2026) Konstruksi Keadaan 3-Qubit Menggunakan Transformasi Matriks SU(2) dan SU(4). Other thesis, Institut Teknologi Sepuluh Nopember.

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Abstract

Keterbelitan kuantum merupakan salah satu fenomena fundamental dalam mekanika kuantum yang menjadi sumber daya utama dalam komputasi kuantum, kriptografi kuantum, dan komunikasi kuantum. Pada sistem multipartit, khususnya sistem tiga qubit, struktur keterbelitan menjadi lebih kompleks dibandingkan sistem bipartit karena memungkinkan munculnya keadaan separabel, biseparabel, maupun terbelit tripartit. Penelitian ini bertujuan untuk merumuskan representasi matematis transformasi keadaan kuantum pada sistem tiga qubit menggunakan kombinasi Grup Lie SU(2) dan SU(4), menganalisis mekanisme pembentukan keterbelitan yang dihasilkan, serta mengklasifikasikan keadaan berdasarkan determinan matriks densitas tereduksi 1-qubit. Operator transformasi dikonstruksi dengan menerapkan syarat grup uniter khusus (special unitary group), yaitu U†U = I dan det(U) = 1, pada matriks kompleks simbolik dengan memanfaatkan kombinasi struktur generator Grup Lie dari SU(2) hingga SU(4). Operator SU(2) digunakan untuk membangkitkan superposisi lokal, sedangkan operator SU(4) diterapkan pada pasangan qubit untuk menghasilkan korelasi nonlokal. Evolusi keadaan dianalisis dari keadaan awal |000⟩ melalui variasi parameter sudut pada generator non-diagonal λ₂, λ₅, dan λ₁₀. Klasifikasi keterbelitan dilakukan menggunakan determinan matriks densitas tereduksi 1-qubit yang diperoleh melalui operasi partial trace. Hasil penelitian menunjukkan bahwa struktur generator yang berbeda menghasilkan pola keterbelitan yang berbeda. Variasi parameter pada generator non-diagonal mampu menghasilkan transisi dari keadaan separabel menuju keadaan biseparabel dan terbelit tripartit. Berdasarkan analisis determinan matriks densitas tereduksi 1-qubit, variasi U₂₄U₄₂ menghasilkan pola jenis keterbelitan yang berbeda dengan pola keterbelitan pada variasi U₄₂U₂₄, dengan hasil keadaan yang didominasi keterbelitan sejati W daripada keadaan GHZ. Hasil tersebut menunjukkan bahwa struktur aljabar generator dan konfigurasi parameter transformasi memengaruhi pembentukan keterbelitan pada sistem tiga qubit.
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Quantum entanglement is one of the fundamental phenomena in quantum mechanics and serves as a key resource for quantum computing, quantum cryptography, and quantum communication. In multipartite systems, particularly three-qubit systems, the structure of entanglement is more complex than that of bipartite systems because it allows the existence of fully separable, biseparable, and genuinely tripartite entangled states. This study aims to formulate a mathematical representation of quantum state transformations in three-qubit systems using a combination of the Lie groups SU(2) and SU(4), to analyze the resulting entanglement generation mechanism, and to classify the resulting quantum states based on the determinants of the reduced one-qubit density matrices. The transformation operators are constructed by imposing the special unitary group conditions, namely U†U = I and det(U) = 1, on symbolic complex matrices while exploiting the Lie group generator structures of SU(2) and SU(4). The SU(2) operators are employed to generate local superposition, whereas the SU(4) operators are applied to pairs of qubits to produce nonlocal correlations. The quantum state evolution is analyzed from the initial state |000⟩ by varying the rotation parameters associated with the non-diagonal generators λ₂, λ₅, and λ₁₀. Entanglement classification is performed using the determinants of the reduced one-qubit density matrices obtained through the partial trace operation. The results show that different generator structures produce distinct entanglement patterns. Variations in the parameters of the non-diagonal generators induce transitions from fully separable states to biseparable and genuinely tripartite entangled states. Based on the determinant analysis of the reduced one-qubit density matrices, the transformation sequence U₂₄U₄₂ produces an entanglement pattern that differs from that generated by U₄₂U₂₄, with the resulting states predominantly belonging to the genuine W entanglement class rather than the GHZ class. These findings demonstrate that the algebraic structure of the generators and the configuration of the transformation parameters significantly influence the formation of quantum correlations in three-qubit systems.

Item Type:	Thesis (Other)
Uncontrolled Keywords:	Grup Lie, SU(2), SU(3), SU(4), keterbelitan kuantum, Lie Groups, quantum entanglement
Subjects:	Q Science Q Science > QA Mathematics > QA159 Algebra Q Science > QA Mathematics > QA184 Algebra, Linear Q Science > QA Mathematics > QA401 Mathematical models. Q Science > QC Physics > QC174.17.Q38 Quantum teleportation
Divisions:	Faculty of Science and Data Analytics (SCIENTICS) > Physics > 45201-(S1) Undergraduate Thesis
Depositing User:	Alief Hisyam Al Hasany Mr
Date Deposited:	03 Aug 2026 04:11
Last Modified:	03 Aug 2026 04:11
URI:	http://repository.its.ac.id/id/eprint/138587

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