Nonlocal Properties of Two-Qubit Gates and Mixed States, and the Optimization of Quantum Computations
Nonlocal Properties of Two-Qubit Gates and Mixed States, and the Optimization of Quantum Computations
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DOI：
10.1023/a:1022144002391
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发表时间：
2002-08-01
影响因子：
2.5
通讯作者：
Makhlin, Yuriy
中科院分区：
文献类型：
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作者：
Makhlin, Yuriy
量子系统两部分的纠缠是一种非定域性质，不受这些部分的局域操纵的影响。它可以用局部酉变换下不变的量来描述。在这里，我们针对两个量子位的系统提出了一组不变量，它提供了非局部属性的完整描述。该集合包含密度矩阵的条目的18个真实的多项式。我们证明了两个混合态中的一个可以通过单量子比特操作转换成另一个，当且仅当这些状态的所有18个不变量的值相等。可以有效地找到相应的本地操作。没有这18个不变量中的任何一个，集合都是不完整的。类似地，双量子比特酉门的非定域纠缠特性在单量子比特操作下是不变的。给出了酉门的3个真实的多项式不变量的完整集合。我们的结果是有用的量子计算的优化，因为它们提供了一个有效的工具来验证是否以及如何使用一个基本的两个量子位门，实现了一个基本的物理操作（和任意多个单量子位门），可以执行给定的两个量子位操作。
Entanglement of two parts of a quantum system is a nonlocal property unaffected by local manipulations of these parts. It can be described by quantities invariant under local unitary transformations. Here we present, for a system of two qubits, a set of invariants which provides a complete description of nonlocal properties. The set contains 18 real polynomials of the entries of the density matrix. We prove that one of two mixed states can be transformed into the other by single-qubit operations if and only if these states have equal values of all 18 invariants. Corresponding local operations can be found efficiently. Without any of these 18 invariants the set is incomplete. Similarly, nonlocal, entangling properties of two-qubit unitary gates are invariant under single-qubit operations. We present a complete set of 3 real polynomial invariants of unitary gates. Our results are useful for optimization of quantum computations since they provide an effective tool to verify if and how a given two-qubit operation can be performed using exactly one elementary two-qubit gate, implemented by a basic physical manipulation (and arbitrarily many single-qubit gates).