The Quantum Logic Formalism

Gary M. Hardegree

Thomas A. Brody

Kenji Tokuo

Abstract A modal logic based on quantum logic is formalized in its simplest possible form. Specifically, a relational semantics and a sequent calculus are provided, and the soundness and the completeness theorems connecting both notions are demonstrated. This framework is intended to serve as a basis for formalizing various modal logics over quantum logic, such as quantum alethic logic, quantum temporal logic, quantum epistemic logic, and quantum dynamic logic.

Thomas A. Brody

Olivier Brunet

In this work, we present a logical formalism for reasoning about quantum systems in finite dimension. Contrary to the usual approach in quantum logic, our formalism is based classical first-order logic, which allows us to use the tools of model theory in our study. In particular, we show that our formal theory is complete, meaning that it entirely determines the behaviour of quantum systems. Moreover, we provide a characterization of the models of our formal theory, thus providing new insights in the study of hidden variable models of quantum theory.

Canh Minh Do, Tsubasa Takagi, Kazuhiro Ogata

While constructing practical quantum computers by big companies remains a challenge, the application of quantum communication and cryptography has made remarkable progress. Therefore, it is crucial to verify quantum protocols before they can be trusted in safety and security-critical applications. We have proposed Basic Dynamic Quantum Logic (BDQL) to formalize and verify sequential models of quantum protocols with a support tool developed in Maude. However, BDQL does not support concurrency in its formalization. This article introduces Concurrent Dynamic Quantum Logic (CDQL), an extension of BDQL, to formalize and verify concurrent models of quantum protocols. CDQL is more expressive than BDQL by considering concurrent behavior and communication among participants in quantum protocols. Since CDQL is a conservative extension of BDQL, we extend the syntax of BDQL to CDQL and make a transformation from CDQL to BDQL without interrupting the semantics of BDQL. We also extend the implementation of BDQL to support CDQL, making a new support tool in Maude. The new support tool is equipped with a lazy rewriting strategy to make the verification process significantly faster. Several quantum communication protocols are successfully formalized and verified in BDQL/CDQL, demonstrating the effectiveness of our automated approach and tool in verifying quantum protocols.