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**Title: **Geometrical and non–normal properties of weak values

**Authors: **Lorena Ballesteros Ferraz, Dominique Lambert, Timoteo Carletti, Riccardo Muolo,

and Yves Caudano

**Abstract: **Quantum measurements are essential to all quantum experiments and are key to developing new quantum technologies. In particular, weak measurements have attracted much interest, for both theoretical and experimental reasons. When performing a weak measurement with preselection (which consists of choosing the initial state) and post–selection (which requires a projective measurement after the weak measurement and corresponds to choosing the final state), weak values appear. In that case, the ancilla’s wavefunction is shifted in position by a quantity that is proportional to the real part of the weak value.

Its wavefunction is also shifted in momentum by a quantity that is proportional to the imaginary part of the weak value. Even though weak values are regularly studied in terms of their real and imaginary parts, we focus our analysis in terms of their modulus and argument. In this work, we have studied the geometrical interpretation of the argument of weak values of general observables in N–level systems. This argument corresponds to a geometric phase (Berry phase) that is associated to the symplectic area generated by the geodesic triangle spanned by the three normalized vectors representing the initial state, the application of the observable over the initial state and the postselected state. These vectors represent quantum pure states in complex projective space CP^(N–1) and they can be mapped to a subspace of the (𝑁² − 2)–sphere, a generalization of the Bloch sphere (the usual S² sphere, used to describe qubits).

The argument of the weak value of an N–level system observable can also be mapped to the sum of N–1 arguments of weak values of two–level projectors up to a phase that specifies the quadrant. Each of these arguments is associated to the solid angle on the Bloch sphere spanned by three vectors representing the pre–selected state, the observable and the post–selected state. To obtain this decomposition, we employ the Majorana representation to factor the weak value of the N–level system observable into the product of N–1 weak values of qubit projectors and a real constant.

Studying the modulus of the weak value provides information on the ability of weak values to

amplify minute phenomena as well as their role in improving the estimation of small physical

parameters. Weak values can be written as the expectation value of a non–normal operator.

Investigating the non–normal properties of the operator, we found a direct link with the

amplitude of the modulus of the weak value. This paves the way for a deeper understanding of these values from an energetics point of view and links the fields of weak measurements and non–normality.

The seminar will take place in **Room S08** at the Faculty of Sciences.