Normal Forms and Tensor Ranks of Pure States of Four Qubits
Normal Forms and Tensor Ranks of Pure States of Four Qubits
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发表时间：
2006-12
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通讯作者：
Oleg Chterental;D. Djoković
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作者：
Oleg Chterental;D. Djoković
我们考察了F.Verstraete等人对四个量子比特的(非归一化)纯态的SLOCC分类。对其基本结果进行了严格的证明，并进行了必要的修正。我们用不变量理论解决了行列式1和量子比特置换的SLOCC变换下纯态的等价性问题。作为副产品，我们给出了F_4型Weyl群不变量的一组新的生成元。我们完成了由J.-L.Brylinski发起的4量子位纯态张量秩的确定。因此，我们得到了一个计算这些秩次的简单算法。我们还得到了至多3的纯秩态的一个非常简单的分类。
We examine the SLOCC classification of the (non-normalized) pure states of four qubits obtained by F. Verstraete et al. The rigorous proofs of their basic results are provided and necessary corrections implemented. We use Invariant Theory to solve the problem of equivalence of pure states under SLOCC transformations of determinant 1 and qubit permutations. As a byproduct, we produce a new set of generators for the invariants of the Weyl group of type F_4. We complete the determination of the tensor ranks of 4-qubit pure states initiated by J.-L. Brylinski. As a result we obtain a simple algorithm for computing these ranks. We obtain also a very simple classification of pure states of rank at most 3.