II. MARCO TEORICO
2.10 MECANISMO DE FRAGMENTACIÓN DE LA ROCA
The transition relation−→α is defined on configurations, and therefore we must first consider an equivalence on configurations before extending this to an equivalence on processes. We combine the notion of branching bisimulation, which is a weak relation that preserves branching structure, with matching for probabilistic transi- tions. The resultingprobabilistic branching bisimulation is similar to the equivalence defined by Lalire [2006], however a significant difference is in our treatment of non- determinism. For the purposes of simplification, non-deterministic branching was modelled as equiprobable choice by Lalire. However, a significant drawback of this approach is that it is not preserved by parallel composition.
A common method to account for non-determinism is the use of schedulers or
adversaries, which assign probabilities to executions. Equivalence is then based on the existence of specific schedulers or adversaries, and this approach was used by Feng et al. [2007]. To avoid the added complexity of introducing schedulers, we follow a similar approach to Lalire [2006], and define a functionµ, which assigns a probability to each transition. However, we maintain a distinction between non-deterministic and probabilistic branching, and use a function that is preserved by parallel composition. Our choice of function is based on the bisimulation by Trˇcka and Georgievska [2008], which assigns probability 1 to all non-deterministic transitions. The CQP transition system fits into the alternating class of probabilistic transition systems, because all probabilistic configurations result in a probabilistic transition to a non-probabilistic configuration. As a result, this probabilistic function is simpler than a corresponding function for a non-alternating system, which would have to account for sequences of consecutive probabilistic transitions.
LetSbe the set of configurations. The relations−→and p induce a partitioning ofS into non-deterministic configurationsSn and probabilistic configurationsSp: Let Sp ={s∈ S | ∃p∈(0,1],∃t ∈ S, s p t}; and let Sn =S \ Sp. By this definition a
configuration with no transitions belongs toSn.
We now define the probabilistic function µ:S × S →[0,1] in the style of Trˇcka and Georgievska [2008]: µ(s, t) = p, ifs p t 1, ifs=tands∈ Sn 0, otherwise.
For a setD⊂ S we defineµ(s, D) =P
t∈Dµ(s, t).
In the bisimulation, we want to compare the reduced density matrices of qubits in output actions, therefore we extend the concepts of density matrix and reduced density matrix to configurations.
Definition 3.2 (Density Matrix of Configurations). Let σ = [epqe7→ |ψi] and s = (σ;ω;P). Then • ρ(σ) =|ψihψ|, • ρqe(σ) = tr e p(|ψihψ|), • ρ(s) =ρ(σ), and • ρqe(s) =ρqe(σ).
Before defining probabilistic branching bisimulation, we introduce some notation that will be used in the remainder of this thesis. Let −→τ + denote zero or one τ
transitions, let =⇒ denote zero or more τ transitions, and let =α⇒be equivalent to =⇒−→α =⇒.
The following definition is based on the standard definition of branching bisimu- lation [van Glabbeek and Weijland 1996] with additional conditions for probabilistic configurations, using the function µ, and for matching quantum information. We require the relationRto be an equivalence relation (instead of a symmetric relation that is normally sufficient) in order to define the equivalence classesD∈ S/R. Definition 3.3 (Probabilistic Branching Bisimulation). Let s, t be configurations. An equivalence relation Ris aprobabilistic branching bisimulation on configurations if whenever (s, t)∈ Rthe following conditions are satisfied.
I. Ifs∈ Sn and s−→τ s0 then there existst0, t00 such thatt=⇒t0−→τ +t00 where a) (s, t0)∈ R, and
b) (s0, t00)∈ R.
II. Ifsc![ev,qe]
−→ s0 then∃t0, t00such thatt=⇒t0c![ev,er]
a) (s, t0)∈ R, b) (s0, t00)∈ R, c) ρeq(s0) =ρer(t00). III. Ifsc?[ev,qe]
−→ s0 then∃t0, t00 such thatt=⇒t0c?[ev,re]
−→ t00 where a) (s, t0)∈ R,
b) (s0, t00)∈ R, c) |ev|=|ue|, and d) ρqe(s0) =ρer(t00).
IV. Ifs∈ Sp thenµ(s, D) =µ(t, D) for all classesD∈ S/R. Naturally this leads on to the following definition of bisimilarity.
Definition 3.4(Probabilistic Branching Bisimilarity). Letsandtbe configurations. Thens andt areprobabilistic branching bisimilar, denoteds-tif and only if there exists a probabilistic branching bisimulationRsuch that (s, t)∈ R.
Our aim is to define an equivalence for processes, hence we now define bisimilarity for processes based on bisimilarity for configurations. In particular, equivalence for processes should be independent of the quantum state because, unlike a configuration, a process has no quantum state associated with it. The following definition identifies processes that produce bisimilar executions, given any initial quantum state.
Definition 3.5(Probabilistic Branching Bisimilarity of Processes). LetP andQbe processes. P andQareprobabilistic branching bisimilar, denotedP -Q, if and only
if for allσ, (σ;∅;P)-(σ;∅;Q).
Example 3.1. LetP andQbe processes defined by
P =c?[x].{x∗=Z}.{x∗=X}.d![x].0 Q=c?[x].{x∗=iY}.d![x].0
Then P -Q since the identityZX=iY ensures the state of xupon output will be the same in each process.
We now prove that probabilistic branching bisimilarity of processes is an equiv- alence relation. This doesn’t follow directly from the definition, however it is an important result if we are to use the relation for equational reasoning. A similar proof of transitivity is given by Lalire [2006].
Lemma 3.16. If Ris a probabilistic branching bisimulation and sRt, and s=⇒s0
Proof. Ifs=⇒s0 then there exists a sequence of configurationss1, . . . , sn such that
s−→τ s1
τ
−→ · · ·−→τ sn =s0. The proof is by induction onn.
The base case isn= 1. Thus ifs−→τ s1then there exist configurationst0, t00such
that t =⇒ t0 −→τ + t00 where (s, t0) ∈ R and (s1, t00)∈ R. Equivalently t =⇒t00 as
required.
For the inductive step, assume the Lemma holds for n, i.e. if s =⇒ sn there exists a configuration t0 such that t =⇒ t0 and (sn, t0) ∈ R. If sn
τ
−→ sn+1 then
there exist configurationst00, t000 such that t0 =⇒t00 −→τ +t000 where (sn, t00)∈ R and (sn+1, t000)∈ R.
Lemma 3.17. Probabilistic branching bisimilarity is an equivalence relation. Proof. We show that probabilistic branching bisimilarity is reflexive, symmetric, and transitive. The result follows from these properties.
Reflexivity: Let RI be the identity relation ((s, t)∈ RI if and only ifs=t). It is clear thatRI is a probabilistic branching bisimulation, hence-is reflexive.
Symmetry: This follows directly from the symmetry and existence property of the corresponding bisimulation relations. Let P and Q be two processes such that
P - Q. By the definition of - there exists a bisimulation RS such that for all ∀σ.((σ;∅;P)RS(σ;∅;Q)). SinceRSis symmetric we have that∀σ.((σ;∅;Q)RS(σ;∅;P)), henceQ-P.
Transitivity: This does not follow directly from the transitivity of bisimulation. We require the composition of distinct bisimulations to be a bisimulation. LetP,Q
andR be processes such thatP-QandQ-R. We now show thatP -R.
From the definition of bisimilarity we know that there exist bisimulationsR1 and
R2 such that for all quantum statesσ, (σ;∅;P)R1(σ;∅;Q) and (σ;∅;Q)R2(σ;∅;R).
LetRC denote the composition of relationsR1◦ R2.
Then letRT denote the symmetric and transitive closure ofRC, thus
sRTu⇒ ∃t1, . . . , tn |s=t0R1t1R2t2· · ·tn−2R1tn−1R2tn =u
We now show that RT is a probabilistic branching bisimulation by induction on n. The base case isn= 0 therefore (s, u)∈ RT ⇒s=u.
For the inductive case assume that RT is a probabilistic branching bisimula- tion such that (s, tn) ∈ RT. Therefore there exists t1, . . . , tn−1 such that s =
t0R1t1R2t2· · ·tn−2R1tn−1R2tn.
Consider a configuration tn+1 such that (tn, tn+1) ∈ R1. The same argument
applies respectively if (tn, tn+1)∈ R2. We consider the four conditions of probabilistic
branching bisimulation in turn:
1. Ifs−→τ s0then byRT there exist configurationstn0 , t00nsuch thattn=⇒t0n τ −→+
such that tn+1 =⇒ t0n+1 and (t0n, tn0+1) ∈ R1. If t0n 6= t00n then there exist configurations t00n+1, t000n+1 such thatt0n+1=⇒t00n+1
τ
−→+t000n+1 where (t0n, t00n+1)∈
R1 and (t00n, t000n+1) ∈ R1. Therefore sRTt0nR1t00n+1 and s0RTt00nR1t000n+1. The
symmetric property is proved in an identical manner. 2. Ifsc![eu]
−→s0then byRT there exist configurationst0n, t00nsuch thattn=⇒t0n c![ev]
−→t00n
where (s, t0n) ∈ RT and (s0, t00n) ∈ RT. By Lemma 3.16 there exists t0n+1
such that tn+1 =⇒ t0n+1 and (t0n, t0n+1) ∈ R1. Therefore there exist configu-
rations t00n+1, t000n+1 such thattn0+1 =⇒t00n+1c![we]
−→t000n+1 where (tn0, t00n+1)∈ R1 and
(t00n, t000n+1) ∈ R1. Therefore sRTt0nR1t00n+1, s0RTt00nR1t000n+1 and ρu = ρv =ρw.
The symmetric property is proved in an identical manner. 3. An identical argument applies to input actions.
4. For the probability function µR we must show that µ(s, D) = µ(u, D) for all D∈ S/RT.
First we consider the relationship between the equivalence classes ofR1,R2and
RT, denoted by{Ai}i∈I, {Bj}j∈J, and {Ck}k∈K respectively. For two states
s, t ∈ Ai for some i ∈ I we have sR1tR2t using the reflexivity of R2, thus
sRTt. As a result it must be the case that for eachiandk, eitherAi⊆Ck or
Ai∩Ck =∅. Similarly we find that for eachjandkit is the case thatBj⊆Ck or Bj ∩Ck = ∅. Furthermore, for each state s ∈ Ck there exist i ∈ I, j ∈ J such that s∈Ai ands∈Bj hence each equivalence class of RT is partitioned by some subset {Ai}i∈Ik of the equivalence classes of R1, and separately by a
subset{Bj}j∈Jk of the equivalence classes ofR2. We can thus say for everyCk:
Ck = [ i∈Ik Ai= [ j∈Jk Bj
Therefore, for some s then µ(s, Ck) = Pi∈Ikµ(s, Ai) =
P
j∈Jkµ(s, Bj). This follows from the definition of µ for both non-deterministic and probabilistic states. Additionally, from the defintion ofRT we know that for everys, u∈Ck there is a sequencet1, . . . , tmof states inCksuch thatsR1t1R2t2R1· · · R1tmR2u.
By induction on m: the base case (m= 0) is trivial. Assume that µ(s, Ck) =
µ(tm, Ck) and (tm, tm+1)∈ R1. Thenµ(tm+1, Ck) =Pi∈Ikµ(tm+1, Ai). Since R1 is a bisimulation, for eachi∈Ik,µ(tm+1, Ai) =µ(tm, Ai), hence it follows that µ(tm+1, Ck) =µ(tm, Ck) =µ(s, Ck).
We have now shown that the relationRT is a bisimulation, therefore we have for all
Teleport = (qbity, z)({z∗=H}.{z, y∗=CNot}.(νe:b[Int,Int])(Alice kBob))
Alice = c?[x:Qbit].{x, z∗=CNot}.{x∗=H}.e![measurez,measurex].0
Bob = e?[r:Int, s:Int].{y∗=Xr}.{y∗=Zs}.d![y].0
QChannel = c?[x:Qbit].d![x].0
Figure 3.8. Quantum teleportation modelled in CQP (Teleport) and its specification process (QChannel).