Logics from Quantum Information

Richard J. Greechie, Stanley Gudder

David Ellerman

Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences, and distinguishability, and is formalized using the distinctions (`dits') of a partition (a pair of points distinguished by the partition). All the definitions of simple, joint, conditional, and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this paper is to give the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., qudits of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates (cohered together in the pure superposition state being measured) that are distinguished by the measurement (decohered in the post-measurement mixed state). Both the classical and quantum versions of logical entropy have simple interpretations as "two-draw" probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states.

Richard J. Greechie, Stanley Gudder

Alexandru Baltag, Sonja Smets

This paper provides an overview of quantum dynamic logics, showing how they have been designed and illustrating how these logics can be applied to verify the correctness of quantum protocols. Similar to the advantages of using dynamic logics to reason about the flow of classical information, the quantum analogues of these logics are tailored to the task of reasoning about the flow of quantum information. We present our logical systems in a modular way, starting with the qualitative logic of quantum measurements and unitary evolutions in single quantum systems, which can already express non-classical effects, e.g., the state-changing interference induced by quantum tests, their non-commutativity, etc. We then move on to logics for compound quantum systems that can capture the non-local features of quantum information: separability, entanglement, correlated measurements, Bell states, etc. We then briefly summarize the logic of quantum probabilities and sketch some applications to quantum protocols.

María Luisa Dalla Chiara, Roberto Giuntini