密度算子LU-等价不变量的完备集
A Complete Set of Invariants for LU-Equivalence of Density Operators
摘要：
我们证明了两个混合量子态的密度算符在同一个局域幺正轨道上当且仅当它们在某个诺特环中的多项式不变量上一致，且其度界已知.这隐式地给出了局部酉等价的有限完备不变量集。这是通过证明密度算子的局部酉等价于局部${\rm GL}$等价，然后使用代数几何和几何不变理论的技术来实现的。我们还对SLOCC多项式不变量进行了分类，并给出了$n$ -量子比特纯态情形下不变环生成元的度界。当然，众所周知，多项式不变量不是SLOCC的不变量的完整集合。
英文摘要：
We show that two density operators of mixed quantum states are in the same local unitary orbit if and only if they agree on polynomial invariants in a certain Noetherian ring for which degree bounds are known in the literature. This implicitly gives a finite complete set of invariants for local unitary equivalence. This is done by showing that local unitary equivalence of density operators is equivalent to local ${\rm GL}$ equivalence and then using techniques from algebraic geometry and geometric invariant theory. We also classify the SLOCC polynomial invariants and give a degree bound for generators of the invariant ring in the case of $n$-qubit pure states. Of course it is well known that polynomial invariants are not a complete set of invariants for SLOCC.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
