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Project supported in part by the National Natural Science Foundation of China (Grant Nos. 11275118 and U1330201).
The original formula of Bell inequality (BI) in terms of two-spin singlet has to be modified for the entangled-state with parallel spin polarization. Based on classical statistics of the particle-number correlation, we prove in this paper an extended BI, which is valid for two-spin entangled states with both parallel and antiparallel polarizations. The BI and its violation can be formulated in a unified formalism based on the spin coherent-state quantum probability statistics with the state-density operator, which is separated to the local and non-local parts. The local part gives rise to the BI, while the violation is a direct result of the non-local quantum interference between two components of entangled state. The Bell measuring outcome correlation denoted by PB is always less than or at most equal to one for the local realistic model (
The non-locality is the most striking characteristic of quantum mechanics beyond our intuition of space and time in the classical field theory.[1–3] It has no classical counterpart and therefore has been receiving continuously theoretical attention ever since the birth of quantum mechanics. The two-particle entangled-state as a typical example of non-locality was originally considered by Einstein–Podolsky–Rosen to question the complicity of quantum mechanics. It has become the essential ingredients in quantum information and computation.[4–6] The quantum nonlocal correlation by local measurements on distant parts of a quantum system is a consequence of entanglement, which is incompatible with local hidden variable models.[7] This was discovered by Bell, who further established a theorem known as Bell inequality (BI)[7] to provide a possibility of quantitative test for non-local correlations, which lead necessarily to the violation of BI. The overwhelming experimental evidence[8–13] for the violation of BI in some entangled-states invalidates local realistic interpretations of quantum mechanics. Various extensions of the original BI have proposed from both theoretical and experimental viewpoints.[14–21] The nonlocality has been also justified undoubtedly in various aspects.[14,15] Soon after the pioneer work of Bell, Clauser–Horne–Shimony–Holt (CHSH) formulated a modified form of the inequality,[22] which is more suitable for the quantitative test and therefore attracts most attentions of experiments. An alternative inequality for the local realistic model was formulated by Wigner[23–25] known as Wigner inequality (WI), which needs measurements of particle number probabilities only along one direction of spin-polarization. It is assumed that the joint probability distributions for measuring outcomes satisfy the locality condition[26] in the underlying stochastic hidden variable space. The experimental evidence strongly supports the quantum non-locality, however the underlying physical-principle is obscure.[27] Various aspects relating to the initial debate remain to be fully understood.
In order to have a better understanding of the underlying physics we in previous publications[28,29] formulated the BI and its violation in a unified formalism by means of the spin coherent-state quantum probability statistics along with the assumption of measurement-outcome-independence. The density operator of a bipartite entangled-state can be separated into the local and non-local parts,[29] with which the measuring outcome correlation is then evaluated by the quantum probability statistics in the spin coherent-state base vectors. The local part of density operator gives rise to the BI, while its violation is a direct result of non-local correlations of entangled states.[29] We predicted a spin parity effect[28] in the violation of BI, which is violated by the entangled-states of half-integer but not the integer spins. It was moreover demonstrated that the violation is seen to be an effect of Berry phase induced by relative-reversal measurements of two spins.
The original formula of BI is actually valid for arbitrary two-spin entangled-state with antiparallel polarization[30] beyond the singlet state. However, it has to be modified by the change of a sign for the parallel spin polarization.[29] It is an interesting question whether or not a unified inequality exists for both antiparallel and parallel spin-polarizations. It is the main goal of the present paper to establish an extended BI valid for two kinds of entangled states. Following the recent work for the maximum violation[31] of WI, the maximum violation bound of the BI is also obtained to demonstrate a fact that the BI with three-direction measurements is equally convenient as CHSH inequality for the experimental test. A loophole-free experimental verification of the violation of CHSH inequality was reported recently by means of electronic spin associated with a single nitrogen-vacancy defect center in a diamond chip[32,33] and also for the two-photon entangled states with mutually perpendicular polarizations.[34] The formalism and results in the present paper are also suitable for the entangled photon pairs.
In our formalism the Bell-type inequalities and their violation are formulated in a unified manner by means of the spin coherent-state quantum probability statistics.[28,29] We begin with an arbitrary two-spin entangled state with antiparallel polarization
![](cpb_27_10_100303/cpb_27_10_100303_ieqn3.gif)
We assume to measure two spins independently along two arbitrary directions, say
![](cpb_27_10_100303/cpb_27_10_100303_ieqn7.gif)
![](cpb_27_10_100303/cpb_27_10_100303_ieqn8.gif)
![](cpb_27_10_100303/cpb_27_10_100303_ieqn9.gif)
In the Wigner formalism[31] the particle-number correlation probability is considered instead of the spin measuring-outcome correlation. The quantity defined by
The extended BI for both parallel and antiparallel polarizations is obviously
In Appendix A we specifically present a simple proof of the validity of the extended BI, in terms of the classical statistics with the particle-number correlation probabilities in the Wigner formulation.[23–25] An interesting question is to find the maximum violation bound, which is useful for the experimental verification.
By means of the spin coherent-state quantum probability statistics we can obtain quantum correlation probability P(a,b) for the two-spin entangled state with antiparallel polarization. The entire (quantum) correlation-probability including the non-local parts becomes
For the measuring directions with polar and azimuthal angles θa = θb = θc = π/2 and ϕa = π/2, ϕb = 0, ϕc = π, the QBCP equation (
![](cpb_27_10_100303/cpb_27_10_100303_ieqn10.gif)
The violation value depends not only on the entangled-state parameters ξ, η in our parametrization but also on the three directions of measurements.
Particularly for the two-spin singlet state
![](cpb_27_10_100303/cpb_27_10_100303_ieqn11.gif)
![](cpb_27_10_100303/cpb_27_10_100303_ieqn12.gif)
For the two-spin entangled state with parallel polarization[29]
![](cpb_27_10_100303/cpb_27_10_100303_ieqn13.gif)
The maximum violation of BI can be realized from Eq. (
In conclusion the value of QBCP is restricted by −1 ≤ PB ≤ 2 for two-spin entangled states with both parallel and antiparallel polarizations. The extended BI is violated if 1 < PB.
Polarization-entangled photon pairs play an important role in various quantum information experiments instead of the two-spin entangled state. We reformulate the extended BI and maximum violation in terms of our formalism.
Two perpendicular polarization states of a single photon may be denoted by |ex⟩ and |ey⟩ in our framework, where we have assumed that the polarization plane is perpendicular to the z axis. The entangled state of a photon pair with mutually perpendicular polarizations can be represented as
![](cpb_27_10_100303/cpb_27_10_100303_ieqn14.gif)
![](cpb_27_10_100303/cpb_27_10_100303_ieqn15.gif)
![](cpb_27_10_100303/cpb_27_10_100303_ieqn16.gif)
The state of entangled photon pairs with mutually parallel polarizations is
![](cpb_27_10_100303/cpb_27_10_100303_ieqn17.gif)
In the entangled state of equal polarizations
![](cpb_27_10_100303/cpb_27_10_100303_ieqn18.gif)
The BI and its violation are formulated in a unified way by the spin coherent-state quantum probability statistics, in which the state density operator is separated to the local and nonlocal parts. The BI is a direct result of local model, while the nonlocal part from the coherent interference of two components of the entangled state leads to the violation. The original BI, which was derived from the two-spin singlet, is extended to a unified form valid for the general entangled state with both antiparallel and parallel polarizations. Up to date the experimental test[32–34] of the inequality violation are mainly focused on the CHSH form, which provides a qualitative bound of the violation. The maximum violation value
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