From its very beginning, quantum theory has been revealing extraordinary and counter-intuitive phenomena, such as wave-particle duality, Schrödinger cats and quantum non-locality. Another paradoxical phenomenon found within the framework of quantum mechanics is the ‘quantum Cheshire Cat’: if a quantum system is subject to a certain pre- and postselection, it can behave as if a particle and its property are spatially separated. It has been suggested to employ weak measurements in order to explore the Cheshire Cat’s nature. Here we report an experiment in which we send neutrons through a perfect silicon crystal interferometer and perform weak measurements to probe the location of the particle and its magnetic moment. The experimental results suggest that the system behaves as if the neutrons go through one beam path, while their magnetic moment travels along the other.

A novel method was recently proposed and experimentally realized for characterizing a quantum state by directly measuring its complex probability amplitudes in a particular basis using so-called weak values. Recently Vallone and Dequal showed theoretically that weak measurements are not a necessary condition to determine the weak value [Phys. Rev. Lett. 116, 040502 (2016)]. Here we report a measurement scheme used in a matter-wave interferometric experiment in which the neutron path system's quantum state was characterized via direct measurements using both strong and weak interactions. Experimental evidence is given that strong interactions outperform weak ones. Our results are not limited to neutron interferometry, but can be used in a wide range of quantum systems. [17,18]. One can also take a pragmatic approach and simply treat the weak value as a complex number that is accessible by experiment, as done in direct state characterization [19,20] to determine complex quantum state probability amplitudes in a particular basis. The weak value of observable of a quantum system is given bywhere |ψ i and |ψ f are the initial (preselected) and final (postselected) system states respectively. To determine  w a probe system, which serves as a measurement apparatus, has to be coupled to the observed system, leading to an entanglement between them. In the usual weak measurement approach only minimally disturbing interactions, between the quantum system and the measurement apparatus are regarded. However, as was recently pointed out theoretically [21,22], the weakness of the interaction is not a necessary condition to obtain the weak value. Furthermore it was shown that strong measurements give a better direct measurement of the quantum wave function using the weak value.

Weak values of the spin operatorŜz of massive particles, more precisely neutrons, have been experimentally determined by applying a novel measurement scheme. This is achieved by coupling the neutron's spin weakly to its spatial degree of freedom in a single-neutron interferometer setup. The real and imaginary parts as well as the modulus of the weak value are obtained by a systematical variation of pre-and post-selected ensembles, which enables to study the complex properties of spin weak values.The meaning of weak values has been an issue of heated debates ever since the concept was introduced by Aharonov, Albert and Vaidman (AAV) in 1988 [1]. Unlike in the von Neumann measurement approach, where the outcome of the measurement is an eigenvalue of the observable and the premeasurement state collapses into the corresponding eigenstate [2], the obtained values, so called weak values, may lie far outside the range of the observable's eigenvalues. A weak measurement of an ob-servable invokes three steps: (i) preparation of an initial quantum state |ψ i (pre-selection), (ii) a unitary coupling of with a dynamical variable of a probe system, via a coupling Hamiltonian, with the coupling being weak so that the system is minimally disturbed. (iii) postselection of the final quantum state |ψ f , by performing a standard projective measurement of another observablê B of the quantum system. Finally, the probe system is read out, yielding the weak value given byExperimental realizations of the procedure proposed by AAV have been performed using various optical setups [3-9]. More recently experiments addressing average trajectories of single photons in a two-slit interferometer [10], direct measurement of the quantum wavefunction [11], or violation of Heisenberg's measurementdisturbance relationship [12-14] have been performed. Measurements of weak values in at least partially nonphotonic schemes are extremely recent, involving transmons in superconducting circuits [15] and spontaneous emission of photons from atoms [16]. Moreover, most experiments have either measured purely real or purely imaginary weak values, not complex quantities. Quantum paradoxes such as the three-box problem [17], Hardy's paradox [18,19] or the quantum cheshire cat [20,21] have been demonstrated by using weak values.In this paper we report the full determination of the weak value of the Pauli spin operatorσ z , i.e., its real and imaginary part, as well as its absolute value (for simplicityσ z is the relevant observable rather thanŜ z = /2σ z ). The real and imaginary contributions of the weak value of σ z are measured using matter wave interferometry with neutrons. Our experiment illuminates a peculiarity of the weak value, namely that it is a complex number in general. Purely imaginary weak values have been utilized to observe amplification effects in weak-measurement-based quantum metrology [8,22,23]. However, the physical account of the imaginary part of the weak value is still under active discussion [24][25][26]. Therefore a systematic invest...

Previous experimental tests of quantum contextuality based on the Bell-Kochen-Specker (BKS) theorem have demonstrated that not all observables among a given set can be assigned noncontextual eigenvalue predictions, but have never identified which specific observables must fail such assignment. We now remedy this shortcoming by showing that BKS contextuality can be confined to particular observables by pre-and postselection, resulting in anomalous weak values that we measure using modern neutron interferometry. We construct a confined contextuality witness from weak values, which we measure experimentally to obtain a 5σ average violation of the noncontextual bound, with one contributing term violating an independent bound by more than 99σ. This weakly measured confined BKS contextuality also confirms the quantum pigeonhole effect, wherein eigenvalue assignments to contextual observables apparently violate the classical pigeonhole principle.

For precise measurements with polarised neutrons high efficient spin-manipulation is required. We developed several neutron optical elements suitable for a new sophisticated setup, i.e., DC spin-turners and Larmor-accelerators which diminish thermal disturbances and depolarisation considerably. The gain in performance is exploited demonstrating violation of a Bell-like inequality for a spin-path entangled single-neutron state. The obtained value of S=2.365(13), which is much higher than previous measurements by neutron interferometry, is 280.25emσ above the limit of S=2 predicted by contextual hidden variable theories. The new setup is more flexible referring to state preparation and analysis, therefore new, more precise measurements can be carried out.