Identifying weak values with intrinsic dynamical properties in modal theories

Abstract

The so-called eigenvalue-eigenstate link states that no property can be associated to a quantum system unless it is in an eigenstate of the corresponding operator. This precludes the assignation of properties to unmeasured quantum systems in general. This arbitrary limitation of orthodox quantum mechanics generates many puzzling situations such as for example the impossibility to uniquely define a work distribution, an essential building block of quantum thermodynamics. Alternatively, modal theories (e.g., Bohmian mechanics) provide an ontology that always allows one to define intrinsic properties, i.e., properties of quantum systems that are detached from any possible measuring context. We prove here that Aharonov, Albert, and Vaidman's notion of a weak value can always be identified with an intrinsic dynamical property of a quantum system defined in a certain modal theory. Furthermore, the fact that weak values are experimentally accessible (as an ensemble average of weak measurements which are postselected by a strong measurement) strengthens the idea that understanding the intrinsic (unperturbed) dynamics of quantum systems is possible and useful in a given modal theory. As examples of the physical soundness of these intrinsic properties, we discuss three intrinsic Bohmian properties, viz., the dwell time, the work distribution, and the quantum noise at high frequencies.

Uncertainty Relations in the Madelung Picture

“…There is at least a formal similarity between the definition of our Madelung quantities and the weak values, but, as pointed out in Section 2.1 , there are ontological differences involved that we do not intend to discuss in detail in this paper. However, we would like to refer interested readers to a recent paper by Pandey et al [ 26 ], in particular, where an approach that is formally close to ours is discussed and also shows connections between weak values and modal theories such as Bohmian mechanics.…”

“…Indeed, the essence of a theory is not reduced merely to a collection of mathematical objects and equations, but the ontology also plays an important role, e.g., in the way these objects and equations have to be understood and interpreted. In this sense, even though our Madelung quantities match the structure of the weak values [ 26 ], the ontology is totally different. Indeed, in the Madelung picture, contrary to the weak value formulation, there is no measurement process under consideration, nor are there preselected and postselected states [ 27 ].…”