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

Abstract

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.