The robustness of Bell's inequality (in CHSH form) violation by entangled state in the simultaneous presence of colored and white noise in the system is considered. A twophoton polarization state is modeled by twoparameter density matrix. Setting parameter values one can vary the relative fraction of pure entangled Bell's state as well as white and colored noise fractions. Bell's operator-parameter dependence analysis is made. Computational results are compared with experimental data [quant-ph/0511265] and with results computed using a oneparameter density matrix [doi: 10.1103/PhysRevA.72.052112], which one can get as a special case of the model considered in this work.
The robustness of Bell's inequality (in CHSH form) violation by entangled state in the simultaneous presence of colored and white noise in the system is considered. A twophoton polarization state is modeled by twoparameter density matrix. Setting parameter values one can vary the relative fraction of pure entangled Bell's state as well as white and colored noise fractions. Bell's operator-parameter dependence analysis is made. Computational results are compared with experimental data [quant-ph/0511265] and with results computed using a oneparameter density matrix [doi: 10.1103/PhysRevA.72.052112], which one can get as a special case of the model considered in this work.
We experimentally demonstrate that violations of Bell's inequalities for two-photon polarization-entangled states with colored noise are extremely robust, whereas this is not the case for states with white noise. Controlling the amount of noise by using the timing compensation scheme introduced by Kim et al. [Phys. Rev. A 67, 010301(R) (2003)], we have observed violations even for states with very high noise, in excellent agrement with the predictions of Cabello et al. [Phys. Rev. A 72, 052112 (2005)].
Contrary to the usual assumption that the experimental preparation of pure entangled states can be described by mixed states due to white noise, a more realistic description for polarization-entangled states produced by parametric down-conversion is that they are mixed states due to decoherence in a preferred polarization basis. This distinction between white and colored noise is crucial when we look for maximal violations of Bell's inequalities for two-qubit and two-qutrit entangled states. We find that violations of Bell's inequalities with realistic noise for polarization-entangled photons are extremely robust for colored noise, whereas this is not the case for white noise. In addition, we study the difference between white and colored noise for maximal violations of Bell's inequalities for three- and four-qubit entangled states.
It is well known that the violation of Bell's inequality in the form given by Clauser, Horne, Shimony, and Holt (CHSH) in two-qubit systems requires entanglement, but not vice versa, i.e., there are entangled states which do not violate the CHSH inequality. Here we compare some standard entanglement measures with violations of the CHSH inequality (as given by the Horodecki measure) for two-qubit states generated by Monte Carlo simulations. We describe states that have extremal entanglement according to the negativity, concurrence, and relative entropy of entanglement for a given value of the CHSH violation. We explicitly find these extremal states by applying the generalized method of Lagrange multipliers based on the Karush-Kuhn-Tucker conditions. The found minimal and maximal states define the range of entanglement accessible for any two-qubit states that violate the CHSH inequality by the same amount. We also find extremal states for the concurrence versus negativity by considering only such states which do not violate the CHSH inequality. Furthermore, we describe an experimentally efficient linear-optical method to determine the highest Horodecki degree of the CHSH violation for arbitrary mixed states of two polarization qubits. By assuming to have access simultaneously to two copies of the states, our method requires only six discrete measurement settings instead of nine settings, which are usually considered.
Loophole-free Bell tests for quantum nonlocality and long-distance secure communication require photodetection efficiencies beyond a threshold ${\ensuremath{\eta}}_{\mathrm{crit}}$ that depends on the Bell inequality and the noise affecting the entangled state received by the distant parties. Most calculations of ${\ensuremath{\eta}}_{\mathrm{crit}}$ assume that the noise is random and can be modeled as white noise. However, most sources suffer from colored noise. Indeed, since entangled states are usually created as a superposition of two possible deexcitation paths, a partial distinguishability between the two processes leads to the appearance of colored noise in the generated state. Recently, there was a proposal for a loophole-free Bell test [A. Cabello and F. Sciarrino, Phys. Rev. X 2, 021010 (2012)], where a specific colored noise appears as a consequence of the precertification of the photon's presence through single-photon spontaneous parametric down-conversion. Here we obtain ${\ensuremath{\eta}}_{\mathrm{crit}}$, the optimal quantum states, and the local settings for a loophole-free Bell test as a function of the amount of colored noise. We consider three bipartite Bell inequalities with n dichotomic settings: Clauser-Horne-Shimony-Holt (n=2), I3322 (n=3), and A5 (n=4), both for the case of symmetric efficiencies, corresponding to photon-photon Bell tests, and for the totally asymmetric case, corresponding to atom-photon Bell tests. Remarkably, in all these cases, ${\ensuremath{\eta}}_{\mathrm{crit}}$ is robust against the colored noise. The present analysis can find application in any test of Bell inequalities in which the dominant noise is of the colored type.
Bell’s inequality plays an important role in quantum mechanics. Many experiments have been made to test the Bell’s inequality by using various bipartite entangled sources. However, the maximal violation of the Bell inequality (i.e., the Cirel’son’s limit) has not been reached yet, due to various practically-existing imperfections (e.g., entangled source is not the pure maximally-entangled state, and the measurement settings used are not the best ones). In this paper, we investigate experimentally an approach to enhance the violation degree of Bell’s inequality with the commercialized entangled photons by optimizing the measurement settings for the experimentally-reconstructed density matrix of the entangled photons. It is shown that the measurement setting related to the pure entangled state can not get maximally violation generally. However, upon the density matrix reconstructed by tomographic technique, optimal measurement settings can be found. With these optimal settings, we consequently found that the violation degree of the Bell’s inequality can be increased. This suggests a possible approach to optimize the test of Bell’s inequality.
The existence of incompatible measurements is a fundamental phenomenon having no explanation in classical physics. Intuitively, one considers given measurements to be incompatible within a framework of a physical theory, if their simultaneous implementation on a single physical device is prohibited by the theory itself. In the mathematical language of quantum theory, measurements are described by POVMs (positive operator valued measures), and given POVMs are by definition incompatible if they cannot be obtained via coarse-graining from a single common POVM; this notion generalizes noncommutativity of projective measurements. In quantum theory, incompatibility can be regarded as a resource necessary for manifesting phenomena such as Clauser-Horne-Shimony-Holt (CHSH) Bell inequality violations or Einstein-Podolsky-Rosen (EPR) steering which do not have classical explanation. We define operational ways of quantifying this resource via the amount of added classical noise needed to render the measurements compatible, i.e., useless as a resource. In analogy to entanglement measures, we generalize this idea by introducing the concept of incompatibility measure, which is monotone in local operations. In this paper, we restrict our consideration to binary measurements, which are already sufficient to explicitly demonstrate nontrivial features of the theory. In particular, we construct a family of incompatibility monotones operationally quantifying violations of certain scaled versions of the CHSH Bell inequality, prove that they can be computed via a semidefinite program, and show how the noise-based quantities arise as special cases. We also determine maximal violations of the new inequalities, demonstrating how Tsirelson's bound appears as a special case. The resource aspect is further motivated by simple quantum protocols where our incompatibility monotones appear as relevant figures of merit.