Any four orthogonal ququad-ququad maximally entangled states are locally markable

In quantum state discrimination, the observers are given a quantum system and aim to verify its state from the two or more possible target states. In the local quantum state marking as an extension of quantum state discrimination, there are N composite quantum systems and N possible orthogonal target quantum states. Distant Alice and Bob are asked to correctly mark the states of the given quantum systems via local operations and classical communication. Here we investigate the local state marking with N 4 ${\otimes}$ 4 systems, N=4, 5, 6, and 7. Therein, Alice and Bob allow for three local operations: measuring the local observable either ${\sigma}_{z}$ or ${\sigma}_{x}$ simultaneously, and entanglement swapping. It shows that, given arbitrary four 4 ${\otimes}$ 4 systems, Alice and Bob can perform the perfect local quantum state marking. In the N=5, 6 cases, they can perform perfect local state marking with specific target states. We conjecture the impossibility of the local quantum state marking given any seven target states since Alice and Bob cannot fulfill the task in the simplest case.Comment: 20 page

Statistical link between Bell nonlocality and uncertainty relations

Bell nonlocality and uncertainty relations are distinct features of quantum theory from classical physics. Bell nonlocality concerns the correlation strength among local observables on different quantum particles, whereas the uncertainty relations set the lower bound of the sum or product of the variance square of observables. Here we establish the statistical link between these two quantum characters using the Aharonov-Vaidman identity. Therein, the upper bounds of Bell-type inequalities are expressed in terms of the product of the local sum of the variance square. On the other hand, instead of evaluating local uncertainty relations, the uncertainty relations on two or more quantum systems are upper-bounded by the amount of Bell nonlocality therein.Comment: 13 page

Genuine Bell locality and nonlocality in the networks

In the literature on $K$-locality ($K\geq2$) networks, the local hidden variables are strictly distributed in the specific observers rather than the whole ones. Regarding genuine Bell locality, all local hidden variables, as classical objects that allow for perfect cloning in classical physics, should be cloned and then spread throughout the networks. More correlators are involved in the proposed linear and non-linear Bell-type inequalities, where their upper bounds are specified by the pre-determined output probability distribution. As for the quantum version, the no-clone theorem limits the broadcast of quantum correlations. To explore genuine Bell nonlocality in variant particle distributions in the networks, the Pauli operators stabilizing the two-qubit Bell states or multi-qubit Greenberger--Horne--Zeilinger states (GHZ states) play an essential role in designing the proposed linear and non-linear Bell tests and assigning the local incompatible measurements for the spatially separated observers. We prove the maximal violations of the proposed Bell-type inequalities quantum networks. In the end, how entanglement swapping replaces the joint measurements in the Bell tests is demonstrated