*Mirko Cormann¹² and Yves Caudano¹²
¹Research Centre in Physics of Matter and Radiation (PMR) ²Namur Center for Complex Systems (NaXyS) University of Namur, rue de Bruxelles 61, Namur, Belgium*
In the last decade, there has been considerable advancement in the study of weak quantum measurements. The experimental observations resulting from weak quantum measurements of pre- and post-selected sub-ensembles can usually be described using weak and modular values [1,2]. Weak values are a form of generalization of the expectation value of an observable for pre- and post-selected sub-ensembles. Taken as expectation values, weak values are very unusual, though: they can be complex numbers or outside the range of the eigenvalues of the observable. Modular values were less reported in the literature. They are related to unitary operators. In a sense, they are counterpart of the weak value for unitary operators: weak and modular values have similar expressions, but applied to observables and unitary operators, respectively.
Weak and modular values of two-level quantum states possess a geometric representation in terms of three dimensional vectors on the Bloch sphere. In this case, the complex values are most usefully expressed using their polar form (modulus and argument) rather than their cartesian representation (real and imaginary components) . The argument of the polar form has a topological origin that depends on a solid angle intercepted on the Bloch sphere during the system evolution from its initial to final state, i. e. from the pre- to the post-selected state.
In this work, we express weak and modular values of three-level and higher-level quantum systems by their polar form. By considering the Majorana representation of qudits, we describe complex weak and modular values of N-level systems using a purely geometrical approach on the Bloch sphere. We show that the modulus of these values is determined by the product of N-1 square roots of probability ratios. We find that their argument is deduced from a sum of N-1 solid angles. We use this theoretical approach to examine the well-known discontinuous effects around singularities of the weak value for a three-level system. Furthermore, we discuss the feasibility to measure experimentally the polar components of the modular value by encoding the three-level state in a bipartite qubit state. This is a preliminary step in the experimental realization of weak and modular value measurements of higher-level systems.
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 M. Cormann, M. Remy, B. Kolaric, and Y. Caudano, Phys. Rev. A 2016, in press, arXiv:1508.01353.