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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 Ndimensional quantum systems in terms of geometric phases. We formulate an arbitrary weak value in terms 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, C P N − 1 , which has a nontrivial 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 Nlevel quantum system describes a geometric phase associated to the symplectic area of the geodesic triangle spanned by the vectors representing the preselected state, the projector and the postselected state in C 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 GellMann matrices. Finally, we study in detail the case of the argument of weak values of general observables in twolevel systems and we illustrate weak measurements in larger dimensional systems by considering projectors on degenerate subspaces, as well as Hermitian quantum gates. To conclude, we discuss the interpretation and usefulness of geometric phases in weak values in connection to weak measurements.
Original language  English 

Article number  045028 
Journal  Quantum Science and Technology 
Volume  7 
Issue number  4 
DOIs  
Publication status  Published  Oct 2022 
Keywords
 Bargmann invariant
 generalized Bloch sphere
 geodesic triangle
 geometric phase
 symplectic area
 weak measurement
 weak value
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 1 Finished

WeaM: Quantum weak measurements: theoretical foundations, experimental approach, philosophical and logical interpretation
Caudano, Y. (PI), Carletti, T. (CoI), Hespel, B. (CoI) & Lambert, D. (CoI)
1/10/19 → 30/09/23
Project: Research
Student theses

Fundamental study of weak values and postselected measurements: from geometry and quantum foundations to quantum information and open systems
Ballesteros Ferraz, L. (Author)Caudano, Y. (Supervisor), Carletti, T. (CoSupervisor), Mayer, A. (President), Matzkin, A. (Jury), Martin, J. (Jury), Lambert, D. (Jury) & Milman, P. (Jury), 13 Sept 2023Student thesis: Doc types › Doctor of Sciences
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