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Bisimulation for Quantum Processes

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Published:01 December 2012Publication History
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Abstract

Quantum cryptographic systems have been commercially available, with a striking advantage over classical systems that their security and ability to detect the presence of eavesdropping are provable based on the principles of quantum mechanics. On the other hand, quantum protocol designers may commit more faults than classical protocol designers since human intuition is poorly adapted to the quantum world. To offer formal techniques for modeling and verification of quantum protocols, several quantum extensions of process algebra have been proposed. An important issue in quantum process algebra is to discover a quantum generalization of bisimulation preserved by various process constructs, in particular, parallel composition, where one of the major differences between classical and quantum systems, namely quantum entanglement, is present. Quite a few versions of bisimulation have been defined for quantum processes in the literature, but in the best case they are only proved to be preserved by parallel composition of purely quantum processes where no classical communication is involved.

Many quantum cryptographic protocols, however, employ the LOCC (Local Operations and Classical Communication) scheme, where classical communication must be explicitly specified. So, a notion of bisimulation preserved by parallel composition in the circumstance of both classical and quantum communication is crucial for process algebra approach to verification of quantum cryptographic protocols. In this article we introduce novel notions of strong bisimulation and weak bisimulation for quantum processes, and prove that they are congruent with respect to various process algebra combinators including parallel composition even when both classical and quantum communication are present. We also establish some basic algebraic laws for these bisimulations. In particular, we show the uniqueness of the solutions to recursive equations of quantum processes, which proves useful in verifying complex quantum protocols. To capture the idea that a quantum process approximately implements its specification, and provide techniques and tools for approximate reasoning, a quantified version of strong bisimulation, which defines for each pair of quantum processes a bisimulation-based distance characterizing the extent to which they are strongly bisimilar, is also introduced.

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  1. Bisimulation for Quantum Processes

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          Wolfgang Schreiner

          Quantum cryptography enables provably secure communication based on the principles of quantum mechanics [1]. However, research is hampered by the lack of good formalisms for modeling and verifying quantum protocols. One possible approach is based on extensions of process algebras, such as Milner's calculus of communicating systems (CCS), which are used to model and analyze communication and concurrency in classical systems. This paper presents a quantum process algebra, qCCS, that extends CCS by primitives for quantum communication, operator application, and measurement. After defining the syntax and operational semantics of qCCS and demonstrating its use by modeling several quantum protocols, the presentation proceeds along the lines of CCS: first, a notion of strong bisimulation of quantum processes is introduced, and the corresponding equivalence relation is shown to be a congruence, that is, preserved by all the combinators of qCCS. Then, a notion of weak bisimulation is defined that does not discriminate between internal actions. While the corresponding equivalence itself is not preserved by summation, a notion of process equality can be derived that is again a congruence. Furthermore, a notion of approximate strong bisimulation is introduced that captures the inherent imprecision of quantum processes and is more suitable in practice. While the quantum aspects of the work are hard to digest for the layman, the reader familiar with process algebra will enjoy the correspondence of concepts known from a classical setting with those in the quantum world. An envisioned extension of the work to model and analyze security properties (analogous to the Spi calculus) promises interesting future results. Online Computing Reviews Service

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