according toπ holds for all testableI-local properties and not just on states satisfying cI, as in the definition.
The proof system is also enriched with an additional inference rule called Local Atomicity Rule: ifI 6=N and p does not occur inψ, φ, θ then from
`LQPn ψ∧T(pI)∧pI≤φ→pI ≤θ infer
`LQPn ψ∧I(φ)→φ≤θ
To represent quantum protocols we need to introduce a few more abbrevi- ations and axioms. For example, we need to know that the action X indeed behaves like the corresponding quantum gate. To this end we introduce a group of axioms describing the effect of the action X on the basis:
• 0i →[Xi]1i
• 1i →[Xi]0i
• +i→[Xi]+i
where iis the subsystem on which Xi acts. We introduce similar groups of
axioms for CN OT, H and Z. The Bell states are characterized by formulas
βxyij := (Z1x;X1y)ij.
Theorem 3 (Theorem 7 in [12], p.28). The proof system of LQPn is sound with respect to the class ΓCQDFn.
2.3
Expressing Quantum Protocols
We now turn to the representation of quantum algorithms. The main result concerning the correctness proofs of known quantum algorithms is
Theorem 4 ([12] and [6]). In the logic LQPn we can give a formal cor- rectness proof of the following algorithms: Teleportation, Quantum Secret Sharing, Superdense Coding, Entanglement Swapping and Logic Gate Tele- portation.
We exemplify the case of the Teleportation protocol, describing how it is coded in the logic. We use an alternative formulation of the same protocol. After the preparation of the protocol, instead of measuring in the Bell basis Alice performs the following actions.
The first move of Alice is the entanglement of the two qubits in her possession: she first perform a CNOT gate on the two qubits and successively a Hadamard gate on the first one (the one inH1). She then measures the system of both qubits in the standard basis. The rest of the protocol remains
unchanged: she communicates the result of the measurement to Bob, who performs a unitary correction on his qubits.
The program inLQPn corresponding to the Teleportation protocol can be written as
T el := [
x,y∈{0,1}
(CN OT1,2;H1; (x1∧y2)?;X3y;Z3x)
whereX0 andZ0are the identity. Notice that the program follows closely the sequential application of the quantum gates described above. The big union in the front captures the indeterminacy of the measurement outcome, and at the same time divides the program into 4 branches which are dependent on such outcome. The validity expressing the correctness of the overall procedure is
T el[q1∧β0023] =3id13[q1]
which reads as follows: the state obtained after performing the protocol T el
to a state prepared inq1∧β0023 is, with respect to the subsystem indexed by 3, equivalent to the state having subsystem 3 in stateq1. For the proof of this formula from the axioms ofLQPn we refer to [12] p. 30.
Despite these encouraging results, we can see that LQPn falls short in
one respect: its formalism cannot express probabilities. Hence this logic will not be able to represent any algorithm in which probabilities play a significant role. This suggests the need for a further improvement ofLQPn.
Part II
Drawing the connection
Chapter 3
Modal Logic for Small
Categories
In this chapter we present a procedure to obtain a Modal Logic frame from a small category. The primary aim is to have a formal tool to connect the approaches we presented in the previous two chapters. Our methodology however touches a very general issue, namely the relation between Category Theory and Modal Logic. In particular, the combination of the types given by the categorical setting with a dynamic logic formalism seems to us especially interesting, as it can be used to describe typed processes. We spend a considerable part of this chapter analyzing such Dynamic Logic with Types, abbreviatedDLT.
The chapter is structured as follows. First we introduce such a procedure in full generality. Second, we devote some pages to the study of different logics that can be used to describe Modal Logic frames arising from small categories: the dynamic logic with types DLT, the logic S4 and Hybrid Logic. Third, we elaborate on the possibility to connect the categorical structure to the logic. We conclude the chapter extending the approach to cover locally small categories.
3.1
Frames and models
Definition 23. Asmall category is a category such that the collection of objects and the collection of maps are both sets. Alocally small category is a category such that, for each pair of objectsI, J, the collection of morphisms from I toJ is a set.
In what follows we will only consider small categories, if not indicated otherwise. We will use the notationC0 and C1 to indicate the collection of objects and arrows of a categoryC, respectively.
Definition 24. Given a small category C and a functor U :C →Rel, a (C, U)-frame is a pairhW, Reli such that
• The setW is defined as
W :=[{U(I)|I ∈C0} • The setRel of relations onW is defined as
Rel:={U(f)|f ∈C1}
Notice that ifCis small then W is the union of set-many sets, and thus is a set. Similarly, as there are set-many morphisms inC,Rel will be a set. Definition 25. Call Γ the class of all (C, U)-frames, that is, the class of Modal Logic frames arising from any such pair (C, U).
Definition 26. Given a setAt of atomic proposititons, a (C, U)-model is a triplehW, Rel, Vi such that
• hW, Reliis a (C, U)-frame • V is a functionV :At→℘(W)
Definition 27. CallM(Ψ) the class of models over the Modal Logic frames in the class Ψ.