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<p>In [Phys. Rev. A <b>62</b>, 062314 (2000)], Dür et al. proposed that three qubits can be entangled in two inequivalent ways. That is, the genuinely entangled states of three qubits are partitioned into two SLOCC equivalence classes: GHZ and W. In [M. Walter et al., Science 340, 1205, 7 June (2013)], via polytopes they gave a sufficient condition for genuinely entangled pure states and discussed SLOCC classification. In this paper, we study entanglement classification of pure states of n qubits via the basis state matrix (BSM). We propose the canonical form of BSM, which can be obtained by exchanging columns and rows of BSM. Then, we establish a necessary and sufficient condition for genuinely entangled states of n qubits via the canonical form of BSM. Thus, genuinely entangled states of n qubits can be partitioned into two families. One family includes all states whose BSM cannot be transformed into the canonical form. The states whose BSM cannot be transformed into the canonical form are always genuinely entangled, no matter what the nonzero coefficients are. GHZ and W states belong to this family. The other family includes all states for which BSM can be transformed into the canonical form, but for any canonical form of BSM, some two rows (or columns) of the corresponding coefficient matrix are not proportional. The cluster state belongs to this family.</p>

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N qubits can be entangled in two different ways

  • 期刊论文

  • 发表日期：

    2026年1月9日

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  • Dafa Li

摘要

In [Phys. Rev. A 62, 062314 (2000)], Dür et al. proposed that three qubits can be entangled in two inequivalent ways. That is, the genuinely entangled states of three qubits are partitioned into two SLOCC equivalence classes: GHZ and W. In [M. Walter et al., Science 340, 1205, 7 June (2013)], via polytopes they gave a sufficient condition for genuinely entangled pure states and discussed SLOCC classification. In this paper, we study entanglement classification of pure states of n qubits via the basis state matrix (BSM). We propose the canonical form of BSM, which can be obtained by exchanging columns and rows of BSM. Then, we establish a necessary and sufficient condition for genuinely entangled states of n qubits via the canonical form of BSM. Thus, genuinely entangled states of n qubits can be partitioned into two families. One family includes all states whose BSM cannot be transformed into the canonical form. The states whose BSM cannot be transformed into the canonical form are always genuinely entangled, no matter what the nonzero coefficients are. GHZ and W states belong to this family. The other family includes all states for which BSM can be transformed into the canonical form, but for any canonical form of BSM, some two rows (or columns) of the corresponding coefficient matrix are not proportional. The cluster state belongs to this family.