On geometric phases and non-normality in weak measurements of discrete quantum systems

Weak values are unbounded complex numbers describing the results of post-selected weak measurements through their real and imaginary parts. We show that their polar representation (argument and modulus) is of great interest. We connect the argument to geometric phases. For qubits, solid angles on the Bloch sphere describe the weak value argument. However, for qudits, the geometry of the state manifold is more complicated. The geodesic triangles associated to the geometric phase can then be described in multiple ways: using a generalized Bloch (hyper)sphere, using Majorana’s stellar representation on the sphere, or using a projection on a spherical octant that was developed previously to represent qutrit states. Interestingly, within Majorana’s representation, the geometric phase is described by solid angles with edges formed from great circle arcs that do not match the quantum state geodesics in this representation. To investigate the modulus, we express weak values of a discrete quantum system in terms of the average value of a non-normal operator. Using Henrici’s departure from normality to quantify non-normality, we study the correlation between the operator non-normality and anomalousness in weak values under weak value amplification conditions. Beyond the case of weak values, our results also establish a very general connection between quantum fluctuations and non-normality, providing an alternative point of view on physical bounds involving quantum uncertainties.