Herramienta Simulación

In any given expansion of a quantum state into component vectors, the scalar coefficients of each component vector areprobability amplitudes(cf. page 17). The interpretation of these numbers as

3.6. Modelling the Measurement of an EPR Pair ❦ 57 square roots of probabilities is due to Max Born and is fundamental to relating the mathematical formalism of quantum mechanics to the results of observations31_{. In particular, for an entangled}

state such as (cf.page 20)

jψi_[12] =c1 ju1v1i+c2 ju2v2i= p1

2 ju1v1i+ 1

p

2 ju2v2i the probability of the whole state collapsing toju1v1iisc21= p1₂

2

= 1₂, while the probability of it collapsing toju2v2iisc22= p1₂

2

= 1₂. This collapse occurs as a consequence of the first mea- surement on any of the two component particles. In the first case, the result of the measurement isju1i; in the second case the result isju2i. This much can be inferred from the definition of an

entangled state using the Dirac notation. We will now try to formalise these facts using the no- tion of conditional probability. The relevance of Bayesian probabilities in the study of “quantum behaviour” is discussed at length in the work of Caves, Fuchs and Schack32.

The probability of a particular outcome in the second measurement isconditionalon the out- come of the first measurement. Since this conditional probability is known for all the different eprstates, the outcome of the second measurement is always predictable. In order to write these probabilities, it is necessary to define appropriate events. epr pair measurement is a random process with the following events, whereψ_jis one of theeprstates_ψ

1,ψ2,ψ3,ψ4:

Meas(ψ_j, 1, 0) Particle 1 of aneprpair in state_ψis measured asj0i.

Meas(ψ_j, 1, 1) Particle 1 of aneprpair in state_ψis measured asj1i.

Meas(ψ_j, 2, 0) Particle 2 of aneprpair in state_ψis measured asj0i.

Meas(ψ_j, 2, 1) Particle 2 of aneprpair in state_ψis measured asj1i.

These events can be defined as predicates over the measurement outcomes m1(d,ψ_j) and

m2(2 d,ψ_j), which were used in the previous section:

Meas(ψ_j, 1, 0) = (m1(d,ψ_j) =0)

Meas(ψ_j, 1, 1) = (m1(d,ψ_j) =1)

Meas(ψ_j, 2, 0) = (m2(2 d,ψ_j) =0)

Meas(ψ_j, 2, 1) = (m2(2 d,ψ_j) =1)

The probabilities describing the outcomes of the first measurement are:

PrfMeas(ψ_j, 1, 0)g=PrfMeas(ψ_j, 1, 1)g=0.5

That is to say, whatever the entangled state ψ_j, the outcomes of the first measurement are

always j0i and j1i, and both outcomes are equally probable. To express the possibilities that may arise in the second measurement, we use conditional events; but the outcomes depend on whetherψ₁andψ₂are being modelled, orψ₃andψ₄.

58 ❦ Chapter 3. Model Checking Techniques PrfMeas(ψ₁, 2, 0)jMeas(ψ₁, 1, 1)g = 1 PrfMeas(ψ₂, 2, 0)jMeas(ψ₂, 1, 1)g = 1 PrfMeas(ψ₁, 2, 1)jMeas(ψ₁, 1, 0)g = 1 PrfMeas(ψ₂, 2, 1)jMeas(ψ₂, 1, 0)g = 1 However: PrfMeas(ψ₃, 2, 0)jMeas(ψ₃, 1, 0)g = 1 PrfMeas(ψ₄, 2, 0)jMeas(ψ₄, 1, 0)g = 1 PrfMeas(ψ₃, 2, 1)jMeas(ψ₃, 1, 1)g = 1 PrfMeas(ψ₄, 2, 1)jMeas(ψ₄, 1, 1)g = 1

All other conditional probabilities for theeprpair measurement problem are identically zero. Let’s see how a probabilistic transition system can be used to describe the measurement of aneprpair in one of the quantum states_ψ

3andψ4: here, the second measurement produces the

same outcome as the first.

We construct a probabilistic transition systemhS,( _!),πinitiwith five states:

s0, the initial state, in which no measurement has been performed yet;

s1, the state which arises when the first measurement producesj0i;

s2, the state which arises when the second measurement producesj0i;

s3, the state which arises when the first measurement producesj1i; and

s4, the state which arises when the second measurement producesj1i.

The initial distributionπinitensures that the only possible initial state iss0; we have

πinit(s0) =1, πinit(sk) =0 for k>0

From the initial state, the next state represents the outcome of the first measurement. There is thus a transition from the initial state to a distribution of successor statesπ0:

s0!π0 or hs0,π0i 2( !) where:

π0(s1) =0.5

π0(s3) =0.5

π0(sk) =0 fork2 f/ 1, 3g

The above transition describes the fact that the first measurement can produce eitherj0iorj1i

with equal probability.

3.6. Modelling the Measurement of an EPR Pair ❦ 59 result for the second measurement:

s1!π1 or hs1,π1i 2( !) where: π1(s2) =1 π1(sk) =0 fork6=2 s3!π3 or hs3,π3i 2( !) where: π3(s4) =1 π3(sk) =0 fork6=4

So the set of all transitions is( _!) =_fhs0,π0i,hs1,π1i,hs3,π3ig. It is also possible to construct

a simpler transition system with only three states (the initial state, a state for outcomej0i, and a state for outcomej1i) for this problem, but we will not discuss this further.