OBJETIVO 1: CARACTERIZAR LAS PRÁCTICAS ASOCIADAS AL CULTIVO DE

For n = 2 and n = 3, the minimum required fidelity coincides with the values found for the purification of GHZ-states [58]. The reason for this coincidence is that the two- and tree-party linear cluster states are (up to local unitary operations) equal to two- and tree-qubit GHZ states, and that the cluster purification protocol is (in these two cases) equivalent to the GHZ purification protocol. 2 3 4 5 6 7 0.1 0.2 0.3 0.4 0.5 0.6 minim um initial fidelit y Numbernof parties

Figure 4.3: The required initial fidelity as a function of the number n of parties. The dotted curve is an exponential fit to the exact values (circles).

4.2

Noisy operations

In a realistic scenario, the n parties have to use some physical apparatus in order to perform the purification protocol. However, no physical apparatus can ever work perfectly, i. e. it will introduce noise by itself. In the case of

78 4. Cluster state purification

bipartite entanglement purification, the influence of noisy operations is well- understood (see Section2.4.2, Chapter3, and Refs. [17,36]). The main result is that bipartite entanglement purification still works with noisy apparatus, as long as its reliability is above a certain threshold value. However, there is a price which one has to pay: it is not possible to get perfectly entangled pairs using noisy apparatus; the fidelity will converge to a maximum fidelity

Fmax<1, which depends on the reliability of the quantum operations.

For the cluster purification protocol, one expects that the situation is quite similar. However, it is not a priori clear how the required reliability scales with the number n of parties. In particular, if the required reliability would approach unity (i. e., perfect operations) exponentially with growing

n, the cluster purification protocol would not be of any practical use for large

n. However, as we will see, the reliability threshold seems to be independent of n.

For the analysis of the purification process using noisy apparatus, we again concentrate on density operators which are diagonal in the cluster basis. On the one hand, this is not really a restriction since it is always possible to get rid of the off-diagonal elements using a twirling operation. On the other hand, it greatly simplifies the calculation; without this simplification, the number n of parties for which it is possible to perform the calculations in a given amount of time and memory would be smaller by a factor of two.

4.2.1

One-qubit white noise

In order to keep the influence of noise computable, we restricted our attention to one-qubit white noise (uncorrelated one-qubit depolarizing channel, see also Eq.2.10). If we start with an-qubit state ρa···n (which later on will be

“close” to a pure cluster state), the depolarization of qubit iwith reliability

pcan be written in the form

ρa···n→ρ0a···n =pρa···n+ 1−p 2 Ii⊗triρa···n =pρa···n+ 1−p 4 3 X l=0 σ(_li)ρa···nσl(i) ≡ D(i) p ρa···n, (4.6)

where we defined the (linear) partial depolarization super-operator Dp(i) of

4.2 Noisy operations 79 by ρa···n → n Y i=1 D_p(i)ρa···n. (4.7)

The partial depolarization super-operators are convex combinations of the actions of the Pauli operators. One can easily check that for a cluster state |(k1, . . . , ki−1, ki, ki+1, . . . kn)i, a phase flip on qubit i (σx(i)) inverts the

eigenvalue ki of the correlation operator Ki, and a spin flip on qubit i (σ(xi))

inverts the eigenvalues ki−1 and ki+1 of the adjacent correlation operators.

The Pauli operatorσy inverts thus (up to an irrelevant phase) the eigenvalues ki−1, ki, and ki+1.

If the density operator ρa···n is diagonal in the cluster basis (which we

conveniently write asρa···n=cdiag(d~)), a flip of thei-th cluster bit (ki → −ki)

changes the diagonal vector in the following way:

~

d −→σ˜(i)

x d~ (4.8)

Here, ˜σx(i) is the i-th cluster bit flip operator, which looks in the cluster

basis like the Pauli operator σx in the computational basis. Note that, in

general, it is necessary to apply the squared modulus of a unitary operation to the diagonal vector (see Section 6.3). However, the σx spin flip operation

coincides with its squared modulus.

Since each of the Pauli operators flips zero, one, two or three cluster bits, we get (in the cluster basis)

D(i) p diag (d~) = diag µµ 3p+ 1 4 I+ 1−p 4 ¡ ˜ σ(i) x + ˜σx(i−1)σ˜(xi+1)+ ˜σx(i−1)σ˜x(i)σ˜(xi+1) ¢¶_~ d ¶ ≡diag ³ D(i) p d~ ´ . (4.9) The effect of a partial depolarization, applied to alln qubits, on the diagonal vector is now given by the product Dp =

Q_n

i=1D (i)

p . For a n-qubit cluster

state, Dp is a real 2n×2n-matrix, while a general super-operator would be

described by a 22n _×22n_{-matrix. While this is already a simplification, we}

can still do better, as we will see in the next paragraphs.

For a given number of qubits n and a reliability parameter p, it is now possible to calculate Dp once and re-use it for each application of the depo-

80 4. Cluster state purification

the one-qubit depolarizing matrices Dp(i) separately, which can be effectively

be described by 8×8-matrices (for 1 < i < n) or 4×4-matrices for i = 1 ori=n. We define the components of vector d~= (dk1k2···kn) and the matrix

Dp(i) =

³

δk0i−1ki0k0i+1

ki−1kiki+1 ´

. All indices take the values −1 or 1, so Dp(i) is really a

8×8-matrix. The components of the vector d~0 _=D(i)

p d~are then given by d0_k1k2···kn = X

k0

i−1ki0k0i+1

δki0−1k0iki0+1

ki−1kiki+1dk1···ki−2ki0−1k0ik0i+1ki+2kn. (4.10)

This method reduces the n(2n₎3_{-overhead for the calculation of the matrix}

Dp, which has to be done once, and the 22n-overhead for the calculation of

the productDpd~, which has to be calculated for each application of the depo-

larizing channel, to an2n_{-overhead for each application of the depolarization.}

4.2.2

Results

For bipartite entanglement purification protocols, it is well-known that there exists a threshold value for the reliability of the operations used in the pu- rification process. If the apparatus is worse than this reliability value, then the purification process does not produce any entanglement. If the appara- tus is above the threshold, the purification process works, and one can reach entangled quantum states up to some maximum fidelity (which depends on the reliability of the apparatus). Of course, one expects that the situation is quite similar in the case of multi-partite purification protocols. However, it is crucial how the threshold reliabilitypthreshold of the noisy apparatus depends

on the number n of the parties. If the required reliability approached unity for increasing n (maybe even exponentially), the protocol would be practi- cally useless for a large number of parties. However, as it turns out, this is not the case, and the threshold reliability seems to approach a value of about 0.933 for increasingn (see Table4.1).