The fact that not all quantum observables are jointly measurable is one of the major differences between quantum and classical theory. In the former, non-commuting observables can only be simultaneously measured with limited precision. We report on an experimental implementation of such a simultaneous measurement of two non-commuting observables based on the framework of weak von Neumann measurements. We use a photonic setup where the polarisation degree of freedom acts as the system and the two components of the transversal position correspond to the pointers of the measurement apparatuses. In addition, the theory shows that these weak measurements demonstrate a counter-intuitive non-monotonicity: weaker measurements can potentially reveal more information than stronger ones.
The fact that not all quantum observables are jointly measurable is one of the major differences between quantum and classical theory. In the former, non-commuting observables can only be simultaneously measured with limited precision. We report on an experimental implementation of such a simultaneous measurement of two non-commuting observables based on the framework of weak von Neumann measurements. We use a photonic setup where the polarisation degree of freedom acts as the system and the two components of the transversal position correspond to the pointers of the measurement apparatuses. In addition, the theory shows that these weak measurements demonstrate a counter-intuitive non-monotonicity: weaker measurements can potentially reveal more information than stronger ones.
In an EPR setup, simultaneous measurement of two non-commuting observables is considered at station A and, possibly after a time delay, at station B . At each station one apparatus measures the z -component of a spin, while a second one measures the x -component of the same spin simultaneously. The dynamics of the measurements is solved explicitly. It is shown that no violations of Bell inequalities occur in Bell-like correlation experiments, and that, in fact, experiments of this type are subject to even stronger constraints. This is caused by the mutual influence of the two apparatuses coupled to the same spin, an effect which occurs at both stations.
In contrast to a projective quantum measurement, in a weak measurement the system is only weakly perturbed while only partial information on the measured observable is obtained. A simultaneous measurement of non-commuting observables cannot be projective, however the strongest possible such measurement can be defined as providing their values at the smallest uncertainty limit. Starting with the Arthurs and Kelly (AK) protocol for such measurement of position and momentum, we derive a systematic extension to a corresponding weak measurement along three steps: First, a plausible form of the weak measurement operator analogous to the Gaussian Kraus operator, often used to model a weak measurement of a single observable, is obtained by projecting a naïve extension (valid for commuting observable) onto the corresponding Gabor space. Second, we show that the so obtained set of measurement operators satisfies the normalization condition for the probability to obtain given values of the position and momentum in the weak measurement operation, namely that this set constitutes a positive operator valued measure (POVM) in the position-momentum space. Finally, we show that the so-obtained measurement operator corresponds to a generalization of the AK measurement protocol in which the initial detector wavefunctions is suitable broadened.
Incompatible observables underlie pillars of quantum physics such as contextuality and entanglement. The Heisenberg uncertainty principle is a fundamental limitation on the measurement of the product of incompatible observables, a 'joint' measurement. However, recently a method using weak measurement has experimentally demonstrated joint measurement. This method [Lundeen, J. S., and Bamber, C. Phys. Rev. Lett. 108, 070402, 2012] delivers the standard expectation value of the product of observables, even if they are incompatible. A drawback of this method is that it requires coupling each observable to a distinct degree of freedom (DOF), i.e., a disjoint Hilbert space. Typically, this 'read-out' system is an unused internal DOF of the measured particle. Unfortunately, one quickly runs out of internal DOFs, which limits the number of observables and types of measurements one can make. To address this limitation, we propose and experimentally demonstrate a technique to perform a joint weak-measurement of two incompatible observables using only one DOF as a read-out system. We apply our scheme to directly measure the density matrix of photon polarization states.
Non-locality is one of the most characteristic trait of Quantum Mechanics, and it is linked to the conceptually counter-intuitive ability of some objects (when adequately prepared) to instantaneously know about each other’s state, no matter how much space-separated they are.
At the development stage of Quantum Mechanics, non-locality had been an argument of intense debate among scientists due to its consequences that seemed to contradict physicists common sense.
The scientific debate on the argument faced a turning point with the publication in 1964 of the landmark paper by J. S. Bell On the Einstein-Podolsky-Rosen Paradox, which ended the discussion whether or not non-locality was a true aspect of physical reality, raising it to one of the fundamental aspect of our
comprehension of the physical world.
The scientific interest for non-locality has recently substantially increased due to its striking applications in Quantum Information Theory.
The major innovative traits of Quantum Information Theory, compared to Classical Information Theory, arise in fact from the usage of non-locality as a resource for information manipulation and transmission.
The aim of this thesis is to investigate experimentally the non-local properties of a quantum system.
This work is one of the first to prove that it is possible to share the non-local correlations of a two-photon entangled singlet state among three observers.
The sharing of non-local correlations was possible through the employment of a weak measurement scheme, used to entangle the polarization degree of freedom of photons with a path degree of freedom used as ancilla for the weak measurement.
In this thesis we briefly review all the theoretical tools necessary to understand the work (Quantum Mechanics, Optics, Quantum Optics) and we discuss in depth all the aspects of our work.
We demonstrate theoretically how non-locality sharing is achievable, showing that it is possible to simultaneously violate two CHSH inequalities among three observers sharing a two-photon singlet state.
We describe precisely our setup and its implementation.
We finally present the obtained results, which confirm our theoretical model and expectations.
The achievement of double violation and the realization of a simple weak measurement scheme may have important applications for Quantum Random Number Generation or for Quantum Key Distribution exploiting weak measurements.
Weak measurement is a potentially powerful tool for studying post-selected quantum systems. The original approach to weak measurement requires a different von Neumann interaction between the quantum system and measurement pointer for weak values of every different observable. When studying weak values of properties of more than one particle, which is necessary for studying entanglement, these interactions can range from difficult to impossible to implement experimentally. For example, the measurement of properties on a pair of optical photons requires a strong Kerr-like nonlinearity at the two-photon level. In a recent work, we described a new method for extracting weak values of operators that can be expressed as products of two local, single-particle operators by studying the correlations in experimentally feasible local, single-particle measurements. In the present work, that theoretical approach is generalized so that weak values of products of N local, single-particle operators can be extracted from the correlations in N single-particle measurements.
Sequential weak measurements of non-commuting observables are not only fundamentally interesting in terms of quantum measurement but also show potential in various applications. Previously reported methods, however, can only make limited sequential weak measurements experimentally. In this article, we propose the realization of sequential measurements of non-commuting Pauli observables and experimentally demonstrate for the first time the measurement of sequential weak values of three non-commuting Pauli observables using genuine single photons.