Jonathan SteinbergPhD student
See also arxiv
Jonathan Steinberg, H. Chau Nguyen, Matthias Kleinmann
Minimal scheme for certifying three-outcome qubit measurements in the prepare-and-measure scenario
The number of outcomes is a defining property of a quantum measurement, in particular, if the measurement cannot be decomposed into simpler measurements with fewer outcomes. Importantly, the number of outcomes of a quantum measurement can be irreducibly higher than the dimension of the system. The certification of this property is possible in a semi-device-independent way either based on a Bell-like scenario or by utilizing the simpler prepare-and-measure scenario. Here we show that in the latter scenario the minimal scheme for a certifying an irreducible three-outcome qubit measurement requires three state preparations and only two measurements and we provide experimentally feasible examples for this minimal certification scheme. We also discuss the dimension assumption characteristic to the semi-device-independent approach and to which extend it can be mitigated.
Extensions and Restrictions of Generalized Probabilistic Theories
Springer BestMaster, To appear soon (2021)
Generalized probabilistic theories (GPTs) allow us to write quantum theory in a purely operational language and enable us to formulate other, vastly different theories. As it turns out, there is no canonical way to integrate the notion of subsystems within the framework of convex operational theories. Sections can be seen as generalization of subsystems and describe situations where not all possible observables can be implemented. We discuss the mathematical foundations of GPTs using the language of Archimedean order unit spaces and investigate the algebraic nature of sections. This includes an analysis of the category theoretic structure and the transformation properties of the state space. Since the Hilbert space formulation of quantum mechanics uses tensor products to describe subsystems, we show how one can interpret the tensor product as a special type of a section. In addition we apply this concept to quantum theory and compare it with the formulation in the algebraic approach. Afterwards we give a complete characterization of low dimensional sections of arbitrary quantum systems using the theory of matrix pencils. In addition, we combine the notion of sections with the dynamics in a GPT and consider the implications for quantum theory. As an application we introduce Spekkens' toy model, a hidden variable theory which mimics many of the features of quantum theory. We show that this model cannot be obtained as a section of a qubit system, but emerges naturally as a section of the Kadison representation what can be interpreted as an eight-level quantum system.
We revisit the formulation of quantum mechanics over the quaternions and investigate the dynamical structure within this framework. Similar to standard complex quantum mechanics, time evolution is then mediated by a unitary which can be written as the exponential of the generator of time shifts. By imposing physical assumptions on the correspondence between the energy observable and the generator of time shifts, we prove that quaternionic quantum theory admits a time evolution for systems with a quaternionic dimension of at most two. Applying the same strategy to standard complex quantum theory, we reproduce that the correspondence dictated by the Schr\"odinger equation is the only possible choice, up to a shift of the global phase.