Geometric description of modular and weak values in discrete quantum systems using the Majorana representation

We express modular and weak values of observables of three- and higher-level quantum systems in their polar form. The Majorana representation of N-level systems in terms of symmetric states of N-1 qubits provides us with a description on the Bloch sphere. With this geometric approach, we find that modular and weak values of observables of N-level quantum systems can be factored in N-1 contributions. Their modulus is determined by the product of N-1 ratios involving projection probabilities between qubits, while their argument is deduced from a sum of N-1 solid angles on the Bloch sphere. These theoretical results allow us to study the geometric origin of the quantum phase discontinuity around singularities of weak values in three-level systems. We also analyze the three-box paradox [1] from the point of view of a bipartite quantum system. In the Majorana representation of this paradox, an observer comes to opposite conclusions about the entanglement state of the particles that were successfully pre- and postselected

Interferences in quantum eraser reveal geometric phases in modular and weak values

In this letter, we present a new procedure to determine completely the complex modular values of arbitrary observables of pre- and post-selected ensembles, which works experimentally for all measurement strengths and all post-selected states. This procedure allows us to discuss the physics of modular and weak values in interferometric experiments involving a qubit meter. We determine both the modulus and the argument of the modular value for any measurement strength in a single step, by controlling simultaneously the visibility and the phase in a quantum eraser interference experiment. Modular and weak values are closely related. Using entangled qubits for the probed and meter systems, we show that the phase of the modular and weak values has a topological origin. This phase is completely defined by the intrinsic physical properties of the probed system and its time evolution. The physical significance of this phase can thus be used to evaluate the quantumness of weak values

cGMP favors the interaction between APP and BACE1 by inhibiting Rab5 GTPase activity

We previously demonstrated that cyclic guanosine monophosphate (cGMP) stimulates amyloid precursor protein (APP) and beta-secretase (BACE1) approximation in neuronal endo-lysosomal compartments, thus boosting the production of amyloid-\u3b2 (A\u3b2) peptides and enhancing synaptic plasticity and memory. Here, we further investigated the mechanism by which cGMP regulates the subcellular localization of APP and BACE1, finding that the cyclic nucleotide inhibits the activity of Rab5, a small GTPase associated with the plasma membrane and early endosomes. Accordingly, we also found that expression of a dominant-negative Rab5 mutant increases both APP-BACE1 approximation and A\u3b2 extracellular levels, therefore mimicking the effects induced by cGMP. These results reveal a functional correlation between the cGMP/A\uf062 pathway and the activity of Rab5 that may contribute to the understanding of Alzheimer\u2019s disease pathophysiology

On the relevance of weak measurements in dissipative quantum systems

We investigate the impact of dissipation on weak measurements. While weak measurements have been successful in signal amplification, dissipation can compromise their usefulness. More precisely, we show that in systems with non-degenerate eigenstates, weak values always converge to the expectation value of the measured observable as dissipation time tends to infinity, in contrast to systems with degenerate eigenstates, where the weak values can remain anomalous, i.e., outside the range of eigenvalues of the observable, even in the limit of an infinite dissipation time. In addition, we propose a method for extracting information about the dissipative dynamics of a system using weak values at short dissipation times. Specifically, we explore the amplification of the dissipation rate in a two-level system and the use of weak values to differentiate between Markovian and non-Markovian dissipative dynamics. We also find that weak measurements operating around a weak atom-cavity coupling can probe the atom dissipation through the weak value of non-Hermitian operators within the rotating-wave approximation of the weak interaction

Geometrical interpretation of the argument of weak values of general observables in N-level quantum systems

Observations in quantum weak measurements are determined by complex numbers called weak values. We present a geometrical interpretation of the argument of weak values of general Hermitian observables in $N$-dimensional quantum systems in terms of geometric phases. We formulate an arbitrary weak value in function of three real vectors on the unit sphere in $N^2-1$ dimensions, $S^{N^2-2}$. These vectors are linked to the initial and final states, and to the weakly measured observable, respectively. We express pure states in the complex projective space of $N-1$ dimensions, $\mathbb{C}\textrm{P}^{N-1}$, which has a non-trivial representation as a $2N-2$ dimensional submanifold of $S^{N^2-2}$ (a generalization of the Bloch sphere for qudits). The argument of the weak value of a projector on a pure state of an $N$-level quantum system describes a geometric phase associated to the symplectic area of the geodesic triangle spanned by the vectors representing the pre-selected state, the projector and the post-selected state in $\mathbb{C}\textrm{P}^{N-1}$. We then proceed to show that the argument of the weak value of a general observable is equivalent to the argument of an effective Bargmann invariant. Hence, we extend the geometrical interpretation of projector weak values to weak values of general observables. In particular, we consider the generators of SU($N$) given by the generalized Gell-Mann matrices. Finally, we study in detail the case of the argument of weak values of general observables in two-level systems and we illustrate weak measurements in larger dimensional systems by considering projectors on degenerate subspaces, as well as Hermitian quantum gates.Comment: 29 pages, 3 figure

Revisiting weak values through non-normality

Quantum measurement is one of the most fascinating and discussed phenomena in quantum physics, due to the impact on the system of the measurement action and the resulting interpretation issues. Scholars proposed weak measurements to amplify measured signals by exploiting a quantity called a weak value, but also to overcome philosophical difficulties related to the system perturbation induced by the measurement process. The method finds many applications and raises many philosophical questions as well, especially about the proper interpretation of the observations. In this paper, we show that any weak value can be expressed as the expectation value of a suitable non-normal operator. We propose a preliminary explanation of their anomalous and amplification behavior based on the theory of non-normal matrices and their link with non-normality: the weak value is different from an eigenvalue when the operator involved in the expectation value is non-normal. Our study paves the way for a deeper understanding of the measurement phenomenon, helps the design of experiments, and it is a call for collaboration to researchers in both fields to unravel new quantum phenomena induced by non-normality