Entanglement and wave function description are two of the core concepts that make quantum mechanics such a unique theory. A method to directly measure the wave function, using Weak Values, was demonstrated by Lundeen et al., Nature 474, 188(2011). However it is not applicable to a scenario of two disjoint systems, where nonlocal entanglement can be a crucial element, since that requires obtaining Weak Values of nonlocal observables. Here, for the first time, we propose a method to directly measure a nonlocal wave function of a bipartite system, using Modular Values. The method is experimentally implemented for a photon pair in a hyper-entangled state, i.e. entangled both in polarization and momentum degrees of freedom.
Quantum processes of inherent dynamical nature, such as quantum walks (QWs), defy a description in terms of an equilibrium statistical physics ensemble. Up to now, it has remained a key challenge to identify general principles behind the underlying unitary quantum dynamics. Here, we show and experimentally observe that split-step QWs admit a characterization in terms of a dynamical topological order parameter (DTOP). This integer-quantized DTOP measures, at a given time, the winding of the geometric phase accumulated by the wave-function during the QW.We observe distinct dynamical regimes in our experimentally realized QWs each of which can be attributed to a qualitatively different temporal behavior of the DTOP. Upon identifying an equivalent many-body problem, we reveal an intriguing connection between the nonanalytic changes of the DTOP in QWs and the occurrence of dynamical quantum phase transitions.
There are several findings: (i) when the noise level is low, GLLS performs well. However, it exhibits large bias and poor precision especially in k*3 and k*4 when the noise level is high. (ii) BF is a promising method with superior bias and precision properties, and is less affected by the scan duration used. (iii) The weighting factors in the nonlinear estimation methods are important: a good choice of weights can help to make the estimates more accurate and reliable. Weighting based on noisy data should be avoided. (iv) It confirms that PGA is little affected by noise, but the assumptions of PGA could induce bias. It also confirms that 60 min is not long enough to give reliable estimates of k*4 especially for the linear estimation methods LLS, TLS, and GLLS.
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