Department of Mathematics, Indian Institute of Science Education and Research Berhampur, Transit Campus, Government ITI, Berhampur 760010, Odisha, India.
S. N. Bose National Center for Basic Sciences, Block JD, Sector III, Bidhannagar, Kolkata 700098, India.
Phys Rev Lett. 2019 Feb 1;122(4):040403. doi: 10.1103/PhysRevLett.122.040403.
Quantum nonlocality is usually associated with entangled states by their violations of Bell-type inequalities. However, even unentangled systems, whose parts may have been prepared separately, can show nonlocal properties. In particular, a set of product states is said to exhibit "quantum nonlocality without entanglement" if the states are locally indistinguishable; i.e., it is not possible to optimally distinguish the states by any sequence of local operations and classical communication. Here, we present a stronger manifestation of this kind of nonlocality in multiparty systems through the notion of local irreducibility. A set of multiparty orthogonal quantum states is defined to be locally irreducible if it is not possible to locally eliminate one or more states from the set while preserving orthogonality of the postmeasurement states. Such a set, by definition, is locally indistinguishable, but we show that the converse does not always hold. We provide the first examples of orthogonal product bases on C^{d}⊗C^{d}⊗C^{d} for d=3, 4 that are locally irreducible in all bipartitions, where the construction for d=3 achieves the minimum dimension necessary for such product states to exist. The existence of such product bases implies that local implementation of a multiparty separable measurement may require entangled resources across all bipartitions.
量子非局域性通常与纠缠态相关，因为它们违反了贝尔类型不等式。然而，即使是非纠缠系统，其部分也可以单独制备，也可以表现出非局域性质。具体来说，如果一组乘积态是局部不可区分的，那么它们就被称为“没有纠缠的量子非局域性”；也就是说，不可能通过任何序列的局域操作和经典通信来最优地区分这些态。在这里，我们通过局部不可约性的概念，在多体系统中展示了这种非局域性的一种更强的表现形式。一组多体正交量子态被定义为局部不可约的，如果不可能从集合中局部地消除一个或多个态，同时保持后测量态的正交性。这样的集合，根据定义，是局部不可区分的，但我们表明，反之并不总是成立。我们提供了第一个例子，即在 C^{d}⊗C^{d}⊗C^{d}（其中 d=3,4）上的正交乘积基，在所有二分体中都是局部不可约的，其中对于 d=3 的构造实现了存在这种乘积态所需的最小维度。这样的乘积基的存在意味着多体可分离测量的局部实现可能需要在所有二分体中使用纠缠资源。
