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We propose schemes for preparing W and GHZ photonic states entangled in polarization via quantum walks, where the trajectory exchanging of identical photons plus postselection induces the coupling between their polarization degrees of freedom. Being different from the polarization beam splitter-based approach, the indistinguishability of identical photons provides an alternative for coupling photons, which is more phase stable and may induce richer actions than the traditional polarization beam splitter on input photons. Our schemes demonstrate that this new coupling mechanism for photons is originated from the fact that trajectory-exchanging operation plus postselection can extract entangled states from the initially product states, and this new coupling mechanism may find considerable potential applications in both quantum communication and quantum computation.

We present a two-copy-based protocol for directly measuring the concurrence of two-photon polarization entangled mixed states (Collins–Gisin class state and the more complicated bipartite mixed entangled state—Werner class state) without quantum state tomography. The quantum circuit designed for directly measuring concurrence can be realized in an optical system. Our protocol works without the sophisticated controlled-NOT gate, which makes it much simpler than the previous ones. Because all the operations used here are local, the scheme can be used for directly measuring remote mixed entanglement too.

In 1935, Einstein, Podolsky and Rosen claimed that quantum wave function does not provide a complete description of physical reality, which is called EPR paradox1. Based on EPR paradox and hidden variable theory, Bell quantitatively analyzed and put forward the Bell inequality in a seminal paper in 19642. More precisely, the hidden variable theory is expressed in mathematics, which reveals that spatially separated quantum systems can have strong correlations. This kind of correlation is known as nonlocality, and it plays a crucial role in quantum information theory, such as nonlocal computation3. Meanwhile, Bell theory provides a significant criterion for the experimentalists to prove the validity of quantum mechanics, and the corresponding experiments verified the nonlocality property of quantum mechanics4,5,6. From John Bell’s original inequality, John Clauser, Michael Horne, Abner Shimony, and Richard Holt derived a new inequality—CHSH inequality—in a much-cited paper published in 19697. The maximum violation of CHSH inequality can reach in quantum mechanics domain, i.e. the so-called Tsirelson’s bound8, rather than the maximum value 2 in classical domain. Up to now, most of the previous Bell inequality test experiments suffer from the following three loopholes, i.e. the locality loophole (or communication loophole), the freedom-of-choice loophole, and the fair-sampling loophole (or detection loophole). The fact that the measurement choice on one subsystem may influence the outcome of the other (and vice-versa) opens the locality loophole. In a Bell test, the two users must be free to choose random measurement choices that are physically independent of one another and of any property of the particles, otherwise, there comes the freedom-of-choice loophole. The detection efficiency must be independent of the measurement settings, i.e. the sample of detected pairs provides a fair statistical sample of all the pairs. If this is not true, it opens the fair-sampling loophole (or detection loophole). The results of the Bell test experiments with any one of these three loopholes only can be accepted with some assumptions. Very recently, Bell tests that close the most significant two loopholes simultaneously have been reported9,10,11,12. Although loopholes have negative effects on Bell test, they play constructive roles in simulating post-quantum correlations whose violations of Bell inequality surpass the so-called Tsirelson’s bound. The most typical representative of this kind of correlations is the famous Popescu and Rohrlich (PR) correlation. Popescu and Rohrlich showed that it is possible to construct various causality satisfying models, where the violation of CHSH inequality can exceed the quantum mechanical bound and reach the algebraic maximal value 413. The nonlocality revealed by the violation of Bell’s inequality can be described by a correlation box shared between two parties. The boxes with the algebraic maximal violation 4 of CHSH inequality are termed PR boxes. Even though previous researches14,15,16,17 suggested that these post-quantum correlations cannot be implemented by classical or quantum systems, they can be simulated by exploiting the loopholes in a Bell test. Obvious violations of Bell inequality beyond Tsirelson’s bound caused by the fair-sampling loophole (or detection loophole) have been observed in experiments where one of the entangled photons is measured and amplified18 or re-generated18,19, or we have the knowledge of the states being measured20. The loss-induced fair-sampling loophole (or detection loophole) can lead to a violation of Bell inequality beyond Tsirelson’s bound too21,22,23,24,25. Among these studies, the fair-sampling loophole is opened by the selection of states to be measured. Actually, the fair-sampling loophole is still open if we select the results after measurement. In other words, it is still possible to simulate post-quantum correlations via post-select the measurements results. Cabello showed this possibility by simulating bipartite correlations beyond Tsirelson’s bound via appropriately post-selecting two qubits of a three-qubit GHZ state system26, and Chen et al. observed this kind of supercorrelations in optical system experimentally27. But this kind of simulation of post-quantum correlation must make use of tripartite state, which obviously limits its persuasiveness. If this kind of selection is done directly on the two subsystems in a bipartite entangled state, the effect of the result-selection induced fair sampling loophole can be shown more obviously. Marcovitch et al. showed that it can be done within “two-state vector formalism”, namely, a state described by “two-state vector formalism” can exhibit a strong violation of CHSH inequality, which can exceed Tsirelson’s bound and even reach the algebraic maximal value (4)28. Here, the measurement is done on all the samples, and the post-selection is only done after measurement, which is more in line with Bell theory than the case with the selection of the states before measurement. The “two-state vector formalism” is a new concept defined by Yakir Aharonov and Lev Vaidman, which is a complete description of a quantum system at a given time based on the results of experiments performed both before and after this time29. In addition, because the “two-state vector formalism” violates the information causality25,30, it opens the locality loophole too , which makes this kind of result-selection induced loophole a comprehensive loophole (including both locality and fair-sampling loopholes) in Bell test. Besides information causality25,30, the comprehensive loophole opened by the result selection within “two-state vector formalism” is another possible explanation of why post-quantum correlations are incompatible with quantum mechanics and seem not to exist in nature. So, in this paper, we will propose a physical scheme for simulating PR correlations by using the comprehensive loophole opened by the result selection within “two-state vector formalism”. In linear optical system, a PR correlation can be simulated by appropriately pre-selecting photon ensemble to be measured and post-selecting the measurement results. Because all the optical elements used here are very common ones, the physical scheme proposed here is feasible.