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Recently, Zhang et al. constructed one family of orthogonal product states which cannot be perfectly distinguished by local operations and classical communication (LOCC) in the $$2m\otimes 2n$$ quantum system with $$m,n\ge 2$$ (Sci Rep 6: 28864, 2016). However, it is an interesting question that what entanglement resources are necessary and/or sufficient for this task to be possible with LOCC. In this paper, we study the local distinguishability of mutually orthogonal product states with quantum entanglement as an auxiliary resource. Specifically, we put forward that the locally indistinguishable orthogonal product states in a low-dimensional system can be locally distinguished with certainty merely by utilizing an additional $$2\otimes 2$$ maximally entangled state. Then, we generalize the distinguishing method to the states in $$2m\otimes 2n$$ . These results reveal the phenomenon of less nonlocality with more entanglement. And they enable us to better understand the role of quantum entanglement in the local discrimination of quantum states and the relationship between entanglement and nonlocality.

Recently, Halder et al. (in Phys Rev Lett 122:040403, 2019) present two sets of strong nonlocality of orthogonal product states based on the local irreducibility. However, for a set of locally indistinguishable orthogonal entangled states, the remaining question is whether the states can reveal strong quantum nonlocality. Here we present a general definition of strong quantum nonlocality based on the local indistinguishability. Then, in $$2 \otimes 2 \otimes 2$$ quantum system, we show that a set of orthogonal entangled states is locally reducible but locally indistinguishable in all bipartitions, which means the states have strong nonlocality. Furthermore, we generalize the result in N-qubit quantum system with $$N\geqslant 3$$ . Finally, we also construct a class of strong nonlocality of orthogonal entangled states in $$d\otimes d\otimes \cdots \otimes d, d\geqslant 3$$ . Our results extend the concept of strong nonlocality for entangled states.

In the study of local discrimination for multipartite unitary operations, Duan et al. (Phys Rev Lett 100(2):020503, 2008) exhibited an ingenious expression: Any two different unitary operations $$U_1$$U1 and $$U_2$$U2 are perfectly distinguishable by local operations and classical communication in the single-run scenario if and only if 0 is in the local numerical range of $$U_1^\dag U_2$$U1?U2. However, how to determine when 0 is in the local numerical range remains unclear. So it is generally hard to decide the local discrimination of nonlocal unitary operations with a single run. In this paper, for two-qubit diagonal unitary matrices V and their local unitary equivalent matrices, we present a necessary and sufficient condition for determining whether the local numerical range is a convex set or not. The result can be used to easily judge the locally perfect distinguishability of any two unitary operations $$U_1$$U1 and $$U_2$$U2 satisfying $$U_1^\dag U_2=V$$U1?U2=V. Moreover, we design the corresponding protocol of local discrimination. Meanwhile, an interesting phenomenon is discovered: Under certain conditions with a single run, $$U_1$$U1 and $$U_2$$U2 such that $$U_1^\dag U_2=V$$U1?U2=V are locally distinguishable with certainty if and only if they are perfectly distinguishable by global operations.

In this paper, we focus on the global discrimination of projective measurements in which the rank of all projectors is one. Firstly, the relation between single-qubit observables and measurement---unitary operation---measurement scheme (M---U---M scheme) is studied. We show that single-qubit observables can generally not be perfectly discriminated by the M---U---M scheme, and the dimension is the most essential reason. Moreover, when we only consider that projective measurements on $$m$$m-dimensional space with $$m\ge 3$$m?3 are perfectly discriminated by the M---U---M scheme, the concrete form of the general unitary matrix is presented, which improves the previous results. Lastly, these results are applied to perfectly distinguish projective measurements with the rank of all projectors being one.

We investigate the upper bound on unambiguous discrimination by local operations and classical communication. We demonstrate that any set of linearly independent multipartite pure quantum states can be locally unambiguously discriminated if the number of states in the set is no more than $$\max \{d_{i}\}$$max{di}, where the space spanned by the set can be expressed in the irreducible form $$\otimes _{i=1}^{N}d_{i}$$?i=1Ndi and $$d_{i}$$di is the optimal local dimension of the $$i\hbox {th}$$ith party. That is, $$\max \{d_{i}\}$$max{di} is an upper bound. We also show that it is tight, namely there exists a set of $$\max \{d_{i}\}+1$$max{di}+1 states, in which at least one of the states cannot be locally unambiguously discriminated. Our result gives the reason why the multiqubit system is the only exception when any three quantum states are locally unambiguously distinguished.

In this letter, we mainly study the local indistinguishability of mutually orthogonal maximally entangled states, which are in canonical form. Firstly, we present a feasible sufficient and necessary condition for distinguishing such states by one-way local operations and classical communication (LOCC). Secondly, for the application of this condition, we exhibit one class of maximally entangled states that can be locally distinguished with certainty. Furthermore, sets of $$d-1$$d-1 indistinguishable maximally entangled states by one-way LOCC are demonstrated in $$d \otimes d$$d?d (for $$d=7, 8, 9, 10$$d=7,8,9,10). Interestingly, we discover there exist sets of $$d-2$$d-2 such states in $$d \otimes d$$d?d (for $$d=8, 9, 10$$d=8,9,10), which are not perfectly distinguishable by one-way LOCC. Finally, we conjecture that there exist $$d-1$$d-1 or fewer indistinguishable maximally entangled states in $$d \otimes d(d \ge 5)$$d?d(d?5) by one-way LOCC.