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, CP 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 CP 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.

On the relevance of weak measurements in dissipative quantum systems

“…Using N and M as meter observables enables separating the real and imaginary components of the weak value, which is often desirable in many experimental setups. In addition, they enable describing the argument of the weak value [27,28], here as a function of the dissipation time τ , in the meter phase space [29]. In section 4, we will show how equation (19) connects directly the meter measurement to the modulus and argument of the weak value.…”

“…Now, we turn our attention to the dependence of the meter shifts on the dissipation duration τ , which determines the post-selection time. For reasons that will soon become apparent, we define the polar representation of the weak value [27][28][29] by σ S,w (τ ) = |σ S,w (τ )| e i φw(τ ) , where |σ S,w (τ )| is its modulus and φ w (τ ) is its argument as a function of the dissipation duration τ . We consider that the meter initial state is the vacuum state for simplicity.…”

“…However, we see that the consequences of dissipation can always be anchored to the final state. In particular, if we set γ = 0 to cancel the effects of dissipation, we recover the expression of a standard weak value of a Pauli operator [28,29]:…”

Revisiting weak values through non-normality

“…They also contribute intrinsically to dynamical phenomena [18]. While the real and imaginary parts generate the meter shifts in typical weak measurements, the modulus and argument of weak values also hold significance [21][22][23], particularly as the argument characterizes geometric phases within the quantum state manifold. In this work, special attention will be paid to the modulus of weak values.…”