The notion of mutual unbiasedness for coarse-grained measurements of quantum continuous variable systems is considered. It is shown that while the procedure of "standard" coarse graining breaks the mutual unbiasedness between conjugate variables, this desired feature can be theoretically established and experimentally observed in periodic coarse graining. We illustrate our results in an optics experiment implementing Fraunhofer diffraction through a periodic diffraction grating, finding excellent agreement with the derived theory. Our results are an important step in developing a formal connection between discrete and continuous variable quantum mechanics.Introduction. The ability to measure a system in an infinite number of non-commuting bases distinguishes the quantum world from classical physics. Wave-particle duality and more generally the complementarity principle are directly rooted in this feature of quantum mechanics. Though one can measure a quantum system in several distinct bases, uncertainty relations limit the amount of information that can be obtained. It is well known that projection onto an eigenstate of one basis reduces the information that can be obtained through or inferred about subsequent measurement in a different basis. The information is minimum for mutually unbiased bases (MUBs), for which all outcomes of the second measurement are equally likely, so that total uncertainty is always substantial (the sharpest uncertainty relations [1]) and most insensitive to input states [2]. MUBs play an important role in complementarity [3], quantum cryptography [4] and quantum tomography [5,6], are useful for certifying quantum randomness [7], and for detecting quantum correlations such as entanglement [8][9][10] and steering [11][12][13][14][15][16][17][18].

We present an experimental proof-of-principle for the generation and detection of pure two-qubit states which have been encoded in degrees of freedom that are common to both classical-light beams and single photons. Our protocol requires performing polarization tomography on a single qubit from a qubit pair. The degree of entanglement in the qubit pair is measured by concurrence, which can be directly extracted from intensity measurements-or photon counting-entering single-qubit polarization tomography. Entangled qubit pairs are basic units in schemes devised to implement quantum information processes such as quantum communication, quantum cryptography, etc., as well as in schemes designed to address foundational issues of quantum mechanics. The exploitation of entanglement is one of the most challenging goals of quantum information technologies. There are good reasons to believe that entanglement plays a key role in the advantage that quantum circuits would have over classical circuits [1]. Entanglement is however difficult to characterize experimentally. So-called entanglement witnesses are state specific, in the sense that they are tailored to detecting some types of entanglement while they are blind to others. Alternatively, one can rely on entanglement measures, which are designed to be state independent. A prominent example is concurrence, which is defined for any pure, bipartite state Φ as C(Φ) = | Φ|(σ y ⊗ σ y)|Φ * |, where σ y is the Pauli matrix and |Φ * the complex-conjugate of |Φ in the computational basis of the tensor-product space to which Φ belongs. Now, confronted with this measure, the experimentalist sees no obvious way to implement it directly in the laboratory. To begin with, complex conjugation is an unphysical operation, because it does not conserve positive-definiteness. Thus, the only way to obtain C(Φ) from measurements seems to be by means of full tomographic determination of state Φ, which is experimentally demanding and prone to inaccuracies. The evaluation of C(Φ), which nonlinearly depends on the parameters fixing Φ, can then be too inaccurate. Back in 2005, Mintert et al. [2] found a way out of the

We report experiments in which the Bell parameter S that enters the Clauser-Horne-Shimony-Holt inequality:. In our experiments, we used two spatially separated optical beams, the electric fields of which were correlated to one another. The amount of correlation was quantified by the spectral degree of coherence , h a b ( ). This quantity measures the correlation between fields that exist at two distant locations and whose respective polarizations are given in terms of angles α and β, which can be set independently from one another. Such a correlation qualifies for the construction of the Bell parameter S. By changing the amount of field correlation, we could cover a range that goes from S. Our experimental findings should provide useful material for the ongoing, theoretical discussions about the quantum-classical border.

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