Quantum Foundations

the no-cloning theorem is an essential result in quantum information on top of which many quantum cryptography protocols are built. In this talk we examine the cloning question in the context of classical mechanics/Hamiltonian mechanics. We find the answer is quite subtle: whether a mechanical system can be cloned depends on the topological structure of its phase space. In particular, for a system to be clonable, its phase space must be contractible. This means certain systems (e.g. particle moving on a line) is clonable, while others (e.g. the simple pendulum) cannot be cloned. We explain the idea of the proof, which uses tools from algebraic topology (homotopy groups and Whitehead’s theorem). Finally we discuss the physical interpretations of this result: how do we reconcile this theorem with the experience that generally speaking, classical information is clonable? Can we use this no-cloning theorem to build secure communication protocols in classical systems instead of quantum ones?

To justify the existence of measurements that can not be performed jointly on quantum systems, Heisenberg put forward a heuristic argument, involving the famous gamma-ray microscope Gedankenexperiment, based on the existence of measurements that irreversibly alter the physical system on which they act. Today, the impossibility of jointly measuring some physical quantities, termed measurement incompatibility, and irreversible disturbance, namely the existence of operations that irreversibly alter the system on which they act, are understood to be distinct but related features of quantum mechanics. In our work, we formally characterized the relationship between these two properties, showing that measurement incompatibility implies irreversible disturbance, though the converse is false. The counterexamples are two toy theories: Minimal Classical Theory and Minimal Strongly Causal Bilocal Classical Theory. These two are distinct as counterexamples because the latter allows for classical conditioning. Our research followed an operational approach exploiting the framework of Operational Probabilistic Theories. In particular, it required the development of two new classes of operational theories: Minimal Operational Probabilistic Theories and Minimal Strongly Causal Operational Probabilistic Theories. These theories are characterized by a restricted set of dynamics, limited to the minimal set consistent with the set of states. In Minimal Strongly Causal Operational Probabilistic Theories, classical conditioning is also allowed.