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Zeitschrift für Naturforschung A

Construction of Two-Ququart Quantum Entanglement by Using Magnetic Resonance Selective Pulse Sequences

Mikail Doğuş Karakaş, Azmi Gençten

Published Jul 25, 2018• 1 citation
Publisher

Abstract

Abstract Ad-dimensional unit of information in quantum computing is called a qudit. Ford= 4 there exist four magnetic quantum numbers of spin-3/2. These four levels can be called ququarts. Then, for the SI (S= 3/2,I= 3/2) spin system, 16 two-ququart states are obtained. In this study, first, two-ququart entangled states are constructed by using matrix representation of Hadamard and CNOT logic gates. Two-ququart entangled states are also constructed by using magnetic resonance selective pulse sequences of Hadamard and CNOT logic gates. Then, a generalised expression is obtained for the transformation of two-qudit entangled states between each other. This expression is applied for two-ququart entangled states.

DOI
10.1515/zna-2017-0441

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Abstract Ad-dimensional unit of information in quantum computing is called a qudit. Ford= 4 there exist four magnetic quantum numbers of spin-3/2. These four levels can be called ququarts. Then, for the SI (S= 3/2,I= 3/2) spin system, 16 two-ququart states are obtained. In this study, first, two-ququart entangled states are constructed by using matrix representation of Hadamard and CNOT logic gates. Two-ququart entangled states are also constructed by using magnetic resonance selective pulse sequences of Hadamard and CNOT logic gates. Then, a generalised expression is obtained for the transformation of two-qudit entangled states between each other. This expression is applied for two-ququart entangled states.
Quantum entanglement is essential for some applications of quantum information processing such as quantum cryptography, quantum teleportation and superdence coding. A qubit is a two level quantum system and four two-qubit entangled states called Bell states can be easily obtained for two-qubit states. A qutrit is a three level quantum system and Zeeman levels of spin-1 electron or nucleus can be referred as qutrit. For SI (S=1, I=1) spin system there exist nine two-qutrit states. So nine two-qutrit entangled states can be obtained by using the Hadamard and CNOT logic gates. In this study by considering N+@C60 molecule as SI (S=1, I=1) spin system, two-qutrit entangled states are also obtained by using the magnetic resonance selective pulse sequences of Hadamard and CNOT logic gates. Then it is shown that these entangled states can be transformed into each other by the suggested transformation operators.
In quantum information processing, spin-3/2 electron or nuclear spin states are known as two-qubit states. For SI (S = 3/2, I = 1/2) spin system, there are eight three-qubit states. In this study, first, three-qubit CNOT logic gates are obtained. Then three-qubit entangled states are obtained by using the matrix representation of Hadamard and three-qubit CNOT logic gates. By considering single 31P@C60 molecule as SI (S = 3/2, I = 1/2) spin system, three-qubit entangled states are also obtained using the magnetic resonance pulse sequences of Hadamard and CNOT logic gates.
A d (d≥3) dimensional unit of information in quantum information is called a qudit. For d=6, there exist six magnetic quantum numbers of spin-5/2. For SI (S=5/2, I=5/2) spin system, 36 two qudit states are determined. First, two qudit CNOT quantum logic gates are obtained for this spin system. Then 36 two qudit entangled states are constructed by using Hadamard (for d=6) and CNOT quantum logic gates. In this study, Mn@C60 molecule is considered as SI (S=5/2, I=5/2) spin system. Possible applications of this spin system in quantum information are also discussed.
In NMR quantum computing, spin states of spin-1/2 nuclei are called qubits. Quantum logic gates are represented by unitary matrices. As a universal gate, controlled-NOT (CNOT) is a two-qubit gate. For the IS (I = 1/2 and S = 1/2) spin system, two-qubit CNOT gate is represented by a 4 × 4 matrix. SWAP logic gate, which exchanges two quantum states, is constructed by CNOT gates. In this study, first, four-qubit CNOT gates are constructed for the IS (I = 3/2, S = 3/2) spin system. Then, by using these CNOT gates, a four-qubit SWAP logic gate is found. As an application and verification, an obtained SWAP logic gate is applied to the matrix representation of product operators for the IS (I = 3/2, S = 3/2) spin system. SWAP logic gate can also be presented by an NMR pulse sequence. By using the product operator theory, the pulse sequence of the SWAP logic gate is applied to product operators of the IS (I = 3/2, S = 3/2) spin system. The expected exchange results are obtained for both the matrix representation and the pulse sequence of SWAP logic gate.
In this research, a quantum system consists of three qubits is considered; it has eight states (|000˃, |001˃, |010˃, |011˃, |100˃, |101˃, |110˃ and |111˃).Each of these states is passed through a quantum circuit made of a sequence of quantum gates.The circuit applies X-gate on the first qubit from the left, then using CNOT-gate on the first and second qubits and using the Hadamard-gate on the first qubit.Finally, Hadamard-gate is applied on the third qubit.When the state |000˃ is passed as input through the proposed quantum circuit then the output includes four entanglement states; for the other input states the output will be another non-repeatable entanglement states.So, for using all states in quantum circuit the result will be eight equations each one consists of four entanglement states; which called ML-Quadripartite.