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Entanglement detection and classification of different kinds of entangled states in quantum many-body systems have always been a key topic in quantum information and quantum computation. In this work, we investigate the entanglement detection and classification of three special entangled states: 4-qubit GHZ state, 4-qubit $ W\overline{W} $ state, and 4-qubit SGT state, which cannot be distinguished by the general quantum Fisher information (QFI) under the usual local operations. By utilizing the experimentally mature and controllable one-axis twisting model, accompanied by the optimized rotation and adjustable interaction strength, we successfully classify the three states by QFI. Besides, we have also studied the effects of four types of environmental noises on the entanglement detection, i.e., bit-flip channel, amplitude-damping channel, phase-damping channel, and depolarizing channel. The results show that under the local operation, the change of the QFI from the 4-qubit GHZ state with respect to the decoherence parameter p in four noise channels is significantly different from those from the $ W\overline{W} $ state and SGT state, and it can be distinguished. However, the QFI from the $ W\overline{W} $ state and the SGT state exhibit the same variations and cannot be classified. In the one-axis twisting model, the variation curves of the QFI from the three states under the four noise channels are mutually distinct and can be clearly observed. It should be noted that, in the bit-flip channel, the QFI of the $ W\overline{W} $ state and the SGT state overlaps in the middle region ($ p\approx0.5 $), failing to be classified. Our work provides a new way to realize the entanglement detection and classification in quantum many-body systems, which will contribute to the future research in quantum science and technology.
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Keywords:
- entanglement detection and classification /
- quantum Fisher information /
- one-axis twisting model /
- GHZ state /
- $W\overline{W} $ state /
- SGT state
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图 1 (color online) 4量子比特GHZ态、$ W\overline{W} $态和SGT态在单轴旋转模型下的量子Fisher信息. 黑色圆点, 蓝色三角形, 红色方块依次表示GHZ态, $ W\overline{W} $态和SGT态的量子Fisher信息随相互作用参数γ的变化结果. 其中, (a)表示彼此独立优化旋转方向n下的结果, 算式(25); (b)表示单轴旋转为x轴的结果, 算式(26); (c)表示单轴旋转为y轴的结果, 算式(27); (d)单轴旋转为z轴的结果, 算式(29)
Figure 1. The QFI of the 4-qubit GHZ state, $ W\overline{W} $ state and SGT state under the one-axis twisting model. Black dots, blue triangles and red squares represent the QFI of the GHZ state, $ W\overline{W} $, and SGT state with respect to the interaction parameter γ. (a) shows the results under the condition that the rotation directions n are optimized independently, Eq. (25); (b) shows the results when the rotation is along the x-axis, Eq. (26); (c) shows the results when the rotation is along the y-axis, Eq. (27); (d) shows the results when the rotation is along the z-axis, Eq. (29).
图 2 (color online) 在局域操作下, 4量子比特GHZ态、$ W\overline{W} $态和SGT态在比特翻转, 相位阻尼, 振幅阻尼和去极化通道下的量子Fisher信息. 黑色圆点, 蓝色三角形, 红色方块依次表示GHZ态, $ W\overline{W} $态和SGT态的量子Fisher信息随退相干参数p的变化结果. 其中, (a)表示量子态在比特翻转通道下的结果; (b)表示量子态在相位阻尼通道中的结果; (c)表示在振幅阻尼通道中的结果; (d)表示在去极化通道下的结果
Figure 2. The QFI of the 4-qubit GHZ state, $ W\overline{W} $ state and SGT state under the bit-flip, phase damping, amplitude damping, and depolarizing channels with local operations. Black dots, blue triangles and red squares represent the QFI of the GHZ state, $ W\overline{W} $, and SGT state with respect to the decoherence parameter p. (a) shows the results under the bit-flip channel; (b) shows the results under the phase damping channel; (c) shows the results under the amplitude damping channel; (d) shows the results under the depolarizing channel.
图 3 (color online) 在单轴旋转模型下($ \gamma=2 $), 4量子比特GHZ态、$ W\overline{W} $态和SGT态在比特翻转, 相位阻尼, 振幅阻尼和去极化通道下的量子Fisher信息. 黑色圆点, 蓝色三角形, 红色方块依次表示GHZ态, $ W\overline{W} $态和SGT态的量子Fisher信息随退相干参数p的变化结果. 其中, (a)表示量子态在比特翻转通道下的结果; (b)表示量子态在相位阻尼通道中的结果; (c)表示在振幅阻尼通道中的结果; (d)表示在去极化通道下的结果
Figure 3. The QFI of the 4-qubit GHZ state, $ W\overline{W} $ state and SGT state under the bit-flip, phase damping, amplitude damping, and depolarizing channels in the one-axis twisting model ($ \gamma=2 $). Black dots, blue triangles and red squares represent the QFI of the GHZ state, $ W\overline{W} $, and SGT state with respect to the decoherence parameter p. (a) shows the results under the bit-flip channel; (b) shows the results under the phase damping channel; (c) shows the results under the amplitude damping channel; (d) shows the results under the depolarizing channel.
图 4 (color online) 在单轴旋转模型下($ \gamma=5 $), 4量子比特GHZ态、$ W\overline{W} $态和SGT态在比特翻转, 相位阻尼, 振幅阻尼和去极化通道下的量子Fisher信息. 黑色圆点, 蓝色三角形, 红色方块依次表示GHZ态, $ W\overline{W} $态和SGT态的量子Fisher信息随退相干参数p的变化结果. 其中, (a)表示量子态在比特翻转通道下的结果; (b)表示量子态在相位阻尼通道中的结果; (c)表示在振幅阻尼通道中的结果; (d)表示在去极化通道下的结果
Figure 4. The QFI of the 4-qubit GHZ state, $ W\overline{W} $ state and SGT state under the bit-flip, phase damping, amplitude damping, and depolarizing channels in the one-axis twisting model ($ \gamma=5 $). Black dots, blue triangles and red squares represent the QFI of the GHZ state, $ W\overline{W} $, and SGT state with respect to the decoherence parameter p. (a) shows the results under the bit-flip channel; (b) shows the results under the phase damping channel; (c) shows the results under the amplitude damping channel; (d) shows the results under the depolarizing channel.
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