© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

International Journal of Advanced Computer Science and Applications

Entanglement Classification for a Three-qubit System using Special Unitary Groups, SU(2) and SU(4)

Siti Munirah Mohd, Bahari Idrus, Hishamuddin B Zainuddin, Muriati Bt. Mukhtar

Published Jan 1, 2019• 6 citations

Abstract

Entanglement is a physical phenomenon that links a pair, or a set of particles that correlates with each other, regardless of the distance between them. Recent researches conducted on entanglement are mostly focused on measurement and classification in multiqubit systems. Classification of two qubits will only distinguish the quantum state as either separable or entangled, and it can be done by measurement. Meanwhile, in a three-qubit system, it becomes more complex because of the structure of the three qubits itself. It is not sufficient to do measurement because the states are divided into three types, including fully separable state, biseparable state, and genuine entangled state. Therefore, the classification is needed to distinguish the type of states in the three-qubit system. This study aims to classify the entanglement of three-qubit pure states using a combination model of special unitary groups, SU(2) and SU(4), by changing the angle of selected parameters in SU(4) and acting on the separable pure state. The matrix represents SU(2) is 2×2 matrix while matrix for SU(4) is 4×4 matrix. Hence, the combination of SU(2) and SU(4) represent 8×8 matrix. This classification uses the von Neumann entropy and three tangle measurements to classify the class, respectively. The results of this study have indicated that the three-qubit pure state has been successfully classified into different classes, namely, A-B-C, A-BC, C-AB, GHZ, and W, with A-B-C being a fully separable state, A-BC and C-AB are biseparable states, and GHZ and W are genuine entangled states. The results show that this model can change separable pure states to other entanglement class after transformation is done.

DOI: 10.14569/ijacsa.2019.0100751

Related papers

Entanglement Classification for Three-qubit Pure Quantum System using Special Linear Group under the SLOCC Protocol

Amirul Asyraf Zhahir, Siti Munirah Mohd, Mohd Ilias M Shuhud +4 more

Quantum technology has been introduced in the IR 4.0, breeding a new era of advanced technology revolutionizing the future. Hence, understanding the key resources of quantum technology, quantum entanglement is vital. Growing interests in quantum technologies has raised comprehensive studies of quantum entanglement, especially on the entanglement classification. Special Linear group, SL(n) of multipartite entanglement classification under the SLOCC protocol is not widely studied due to its complex structure, creating a curb in developing its classification method. Therefore, this paper developed and delivered a classification method of pure multipartite, three-qubit quantum state using a combination of Special Linear group, SL(2) x SL(2) x SL(2) model operator under the SLOCC, classifying entanglement using the model operator with certain selected parameters. Further analysis was done resulting in the determination of the six subgroups, namely fully separable (A-B-C), bi-separable (A-BC, B-AC and C-AB) and genuinely entangled (W and GHZ).

International Journal of Advanced Computer Science and Applications•Jan 1, 2023•0 citations

A New Method of Classification of Pure Tripartite Quantum States

Jyoti Faujdar, Anoopa Joshi, Satyabrata Adhikari

The classification of the multipartite entanglement is an important problem in quantum information theory. We propose a class of two qubit mixed states $σ_{AB}= p|χ_{1}\rangle\langleχ_{1}|\otimesρ_{1}+(1-p)|χ_{2}\rangle\langleχ_{2}|\otimesρ_{2}$, where $|χ_{1}\rangle=α|0\rangle+β|1\rangle$, $|χ_{2}\rangle=β|0\rangle+(-1)^{n}α|1\rangle$. We have shown that the state $σ_{AB}$ represent a classical state when $n$ is odd while it represent a non-classical state when $n$ is even. The purification of the state $σ_{AB}$ is studied and found that the purification is possible if the spectral decomposition of the density matrices $ρ_{1}$ and $ρ_{2}$ represent pure states. We have established a relationship between three tangle, which measures the amount of entanglement in three qubit system and the quantity $\langleχ_{1}|χ_{2}\rangle$, which identifies whether the two qubit mixed state is classical or non-classical. The three qubit purified state is then classified as a separable or biseparable or W-type or GHZ-type state using the quantum correlation, measured by geometric discord, of its reduced two qubit density matrix.

arXiv:1408.1539•Aug 7, 2014•0 citations

Detection and classification of three-qubit states using l_1 norm of coherence

Anu Kumari, Satyabrata Adhikari

Entanglement is a purely quantum mechanical phenomenon and thus it has no classical analog. On the other hand, coherence is a well-known phenomenon in classical optics and in quantum mechanics. Recent research shows that quantum coherence may act as a useful resource in quantum information theory. We will employ here quantum coherence to detect and classify the entanglement property of three-qubit states. Moreover, we have shown that if any three-qubit state violates another necessary condition for the detection of a general biseparable state then the given three-qubit state cannot be a biseparable state. Since there are only three categories of states for the three-qubit system so if we detect that the state under probe is neither a separable nor a biseparable state then we can definitely conclude that the given three-qubit state is a genuine entangled state. We have illustrated our results with a few examples.

arXiv:2106.09072•Quantum Information and Computation•Apr 1, 2023•0 citations

Enhancing Quantum Information Processing – SU(2) Operator Model Development for Three-Qubit Quantum Systems Entanglement Classification

Amirul Asyraf Zhahir, Siti Munirah Mohd, Mohd Ilias M Shuhud +4 more

The study introduces the development of the operator model within the Local Unitary (LU) protocol for entanglement classification, specifically of the pure three-qubit quantum systems. Addressing the challenge of accurately distinguishing different classes of entanglement, this research aims to enhance the understanding and utilization of entangled quantum states in quantum information processing. A systematic approach was employed in the development of the model, designed to effectively distinguish different classes of entanglement. This study contributes significantly to quantum information processing, by providing valuable insights into the nature of entangled quantum states and enabling researchers to gain a better understanding, utilizing them effectively in various quantum applications. The developed operator model holds significant potential for the advancements in entanglement classification and broaderscope of quantum information processing. The model marks a notable achievement in enabling a more precise and efficient entanglement of entangled quantum systems, which is crucial for advancing various quantum technologies. Future research should extends this model to a higher-qubit and higher-dimensional quantum systems and explore its integration with several other entanglement classification protocols and Lie groups to further advance the field.

International Journal of Advanced Research in Computational Thinking and Data Science•Sep 15, 2024•0 citations

Identification of three-qubit entanglement

Ming‐Jing Zhao, Ting-Gui Zhang, Xianqing Li‐Jost +1 more

We present a way of identifying all kinds of entanglement for three-qubit pure states in terms of the expectation values of Pauli operators. The necessary and sufficient conditions to classify the fully separable, biseparable, and genuine entangled states are explicitly given. The approach can be generalized to multipartite high-dimensional cases. For three-qubit mixed states, we propose two kinds of inequalities in terms of the expectation values of complementary observables. One inequality has advantages in entanglement detection of the quantum state with positive partial transpositions, and the other is able to detect genuine entanglement. The results give an effective method for experimental entanglement identification.

arXiv:1301.4317•Physical Review A•Jan 17, 2013•28 citations

Construction of three-qubit biseparable states distinguishing kinds of entanglement in a partial separability classification

Kyung Hoon Han, Seung-Hyeok Kye

We construct seven kinds of three-qubit biseparable states to show that every class of biseparable states in the partial separability classification proposed by Szalay and K\"ok\'enyesi [S. Szalay and Z. K\"ok\'enyesi, Phys. Rev. A 86, 032341 (2012)] is nonempty.

arXiv:1812.01227•Physical Review A•Mar 4, 2019•12 citations

Entanglement is a physical phenomenon that links a pair, or a set of particles that correlates with each other, regardless of the distance between them. Recent researches conducted on entanglement are mostly focused on measurement and classification in multiqubit systems. Classification of two qubits will only distinguish the quantum state as either separable or entangled, and it can be done by measurement. Meanwhile, in a three-qubit system, it becomes more complex because of the structure of the three qubits itself. It is not sufficient to do measurement because the states are divided into three types, including fully separable state, biseparable state, and genuine entangled state. Therefore, the classification is needed to distinguish the type of states in the three-qubit system. This study aims to classify the entanglement of three-qubit pure states using a combination model of special unitary groups, SU(2) and SU(4), by changing the angle of selected parameters in SU(4) and acting on the separable pure state. The matrix represents SU(2) is 2×2 matrix while matrix for SU(4) is 4×4 matrix. Hence, the combination of SU(2) and SU(4) represent 8×8 matrix. This classification uses the von Neumann entropy and three tangle measurements to classify the class, respectively. The results of this study have indicated that the three-qubit pure state has been successfully classified into different classes, namely, A-B-C, A-BC, C-AB, GHZ, and W, with A-B-C being a fully separable state, A-BC and C-AB are biseparable states, and GHZ and W are genuine entangled states. The results show that this model can change separable pure states to other entanglement class after transformation is done.

Quantum technology has been introduced in the IR 4.0, breeding a new era of advanced technology revolutionizing the future. Hence, understanding the key resources of quantum technology, quantum entanglement is vital. Growing interests in quantum technologies has raised comprehensive studies of quantum entanglement, especially on the entanglement classification. Special Linear group, SL(n) of multipartite entanglement classification under the SLOCC protocol is not widely studied due to its complex structure, creating a curb in developing its classification method. Therefore, this paper developed and delivered a classification method of pure multipartite, three-qubit quantum state using a combination of Special Linear group, SL(2) x SL(2) x SL(2) model operator under the SLOCC, classifying entanglement using the model operator with certain selected parameters. Further analysis was done resulting in the determination of the six subgroups, namely fully separable (A-B-C), bi-separable (A-BC, B-AC and C-AB) and genuinely entangled (W and GHZ).

The classification of the multipartite entanglement is an important problem in quantum information theory. We propose a class of two qubit mixed states

σ_{AB}= p|χ_{1}\rangle\langleχ_{1}|\otimesρ_{1}+(1-p)|χ_{2}\rangle\langleχ_{2}|\otimesρ_{2}

, where

|χ_{1}\rangle=α|0\rangle+β|1\rangle

,

|χ_{2}\rangle=β|0\rangle+(-1)^{n}α|1\rangle

. We have shown that the state

σ_{AB}

represent a classical state when

n

is odd while it represent a non-classical state when

n

is even. The purification of the state

σ_{AB}

is studied and found that the purification is possible if the spectral decomposition of the density matrices

n

and

ρ_{2}

represent pure states. We have established a relationship between three tangle, which measures the amount of entanglement in three qubit system and the quantity

\langleχ_{1}|χ_{2}\rangle

, which identifies whether the two qubit mixed state is classical or non-classical. The three qubit purified state is then classified as a separable or biseparable or W-type or GHZ-type state using the quantum correlation, measured by geometric discord, of its reduced two qubit density matrix.

Entanglement is a purely quantum mechanical phenomenon and thus it has no classical analog. On the other hand, coherence is a well-known phenomenon in classical optics and in quantum mechanics. Recent research shows that quantum coherence may act as a useful resource in quantum information theory. We will employ here quantum coherence to detect and classify the entanglement property of three-qubit states. Moreover, we have shown that if any three-qubit state violates another necessary condition for the detection of a general biseparable state then the given three-qubit state cannot be a biseparable state. Since there are only three categories of states for the three-qubit system so if we detect that the state under probe is neither a separable nor a biseparable state then we can definitely conclude that the given three-qubit state is a genuine entangled state. We have illustrated our results with a few examples.

The study introduces the development of the operator model within the Local Unitary (LU) protocol for entanglement classification, specifically of the pure three-qubit quantum systems. Addressing the challenge of accurately distinguishing different classes of entanglement, this research aims to enhance the understanding and utilization of entangled quantum states in quantum information processing. A systematic approach was employed in the development of the model, designed to effectively distinguish different classes of entanglement. This study contributes significantly to quantum information processing, by providing valuable insights into the nature of entangled quantum states and enabling researchers to gain a better understanding, utilizing them effectively in various quantum applications. The developed operator model holds significant potential for the advancements in entanglement classification and broaderscope of quantum information processing. The model marks a notable achievement in enabling a more precise and efficient entanglement of entangled quantum systems, which is crucial for advancing various quantum technologies. Future research should extends this model to a higher-qubit and higher-dimensional quantum systems and explore its integration with several other entanglement classification protocols and Lie groups to further advance the field.

We present a way of identifying all kinds of entanglement for three-qubit pure states in terms of the expectation values of Pauli operators. The necessary and sufficient conditions to classify the fully separable, biseparable, and genuine entangled states are explicitly given. The approach can be generalized to multipartite high-dimensional cases. For three-qubit mixed states, we propose two kinds of inequalities in terms of the expectation values of complementary observables. One inequality has advantages in entanglement detection of the quantum state with positive partial transpositions, and the other is able to detect genuine entanglement. The results give an effective method for experimental entanglement identification.