The standard purification LOCC operation considered in this chapter, as shown in Fig.
6.1, should be mentioned first. In each purification LOCC operation, Alice and Bob first perform local operations by operators U and U∗, which will be defined latter, respectively.
Then Alice and Bob each performs a quantum control-not operation. They then measure
U
U
U
*U
*A B
Figure 6.1: The standard purification LOCC operations including the local controlled-NOT operation, single qubit measurement, and local unitary operation in each party.
Note that the classical communication is not shown in this figure.
the target qubits in the computational basis, and if the outcomes, communicated via classical channel, coincide they keep the control pair for the next step and discard the target pair. If the outcomes do not coincide, both pairs are discarded. In the purification LOCC operation, the state to be purified needs not be of a Werner form. We express the mixed state in the Bell basis {|Φ+i, |Ψ−i , |Ψ+i, |Φ−i}:
Φ±
= 1
√2(|00i ± |11i), Ψ±
= 1
√2(|01i ± |10i), (6.1)
where |0i and |1i form the computational basis of the two-dimensional space belonging to the EPR pairs. Let {a0, b0, c0, d0} be the average initial diagonal elements of the density operator representing the mixed state before the protocol is begun with, and {ar, br, cr, dr} be the average diagonal elements of the surviving state after the r-th step. It can be shown that a purification LOCC operation in fact is relative to a nonlinear map, where the diagonal entries of the surviving state after the LOCC operation are nonlinear functions of those before the operation. Therefore the purification protocol considered in this work is composed of consecutive nonlinear maps of the Bell-diagonal elements used to transform an initial state asymptotically to a desired pure state. Suppose the state
|Φ+i hΦ+| is the desired one to be purified through the purification, we then are willing
to map step by step the initial state {a0, b0, c0, d0}, where one of the elements should be greater than 1/2, to converge to the desired attractor {1, 0, 0, 0} as the step number r is sufficiently large. But the intrinsic property of the nonlinear map reveals that the desired attractor is not the only one, as can be seen in the article of Macchiavello [131], who has given the analytical convergence in the recurrence scheme of the QPA protocol.
The interesting nonlinear behavior of the recurrence scheme in a distillation protocol is dominantly influenced by the local unitary operations operators U and U∗ applied by Alice and Bob in the purification LOCC operation. Generalized expression for U, controlled by two phases θ and φ, is given by
It is clear that distinct choices of θ and φ will lead to different destinations of the protocol.
For example, in using the original QPA protocol, Alice and Bob choose θ = φ = π/2 , i.e., they apply the operator
Bob obtain coinciding outcomes in the measurements on the target pairs (so only pr−1/2 of the pairs before the rth step is surviving after the step). Let us define the domains
Da = {a ∈ (0.5, 1]; a + b + c + d = 1}, Db = {b ∈ (0.5, 1]; a + b + c + d = 1}, Dc = {c ∈ (0.5, 1]; a + b + c + d = 1}, Dd = {d ∈ (0.5, 1]; a + b + c + d = 1}, Dab = Da∪ Db ,
Dcd = Dc ∪ Dd ,
Dabcd = Da∪ Db∪ Dc∪ Dd. (6.5)
In what follows we will consider the case that an initial mixed state to be purified is in the applicable Dabcd because any state ρ ∈ Dabcd is distillable. It has been proved [131] that, for the Oxford protocol, an initial state in the domain Dab will eventually be mapped to converge to the attractor {1, 0, 0, 0} representing the desired pure state
|Φ+i hΦ+|. While if the initial state is in the domain Dcd, then it will be mapped to approach another attractor {0, 0, 1, 0}, or the pure state |Ψ+i hΨ+|. In the end, according to Ref. [76], using the QPA protocol, Alice and Bob will regain the desired pure state from any state ρ ∈ Dabcd provided they first take efforts additional to the standard purification LOCC operations to transform the pure state |Ψ+i hΨ+|, or |Φ−i hΦ−|, into the desired state |Φ+i hΦ+| if the input state is in the domain Dcd. Meanwhile, such efforts also have meaningful implication as if the QPA is considered to be combined with the hashing protocol [74, 75] to improve its output yield. These tedious transformations cannot be avoided even when the input state is already in the domain Dab, because Alice and Bob initially do not have an idea about whether the input state is exactly in the domain Dab or Dcd. For example, if the input state has the element c0 = 0.7, then Alice and Bob should transform the state |Ψ+i hΨ+| into |Φ+i hΦ+| before the purification procedure so that the mixed state in turn will have the element a0 = 0.7.
As another example, if Alice and Bob choose θ = π/2 and φ = 0, then they have the operator
U(π/2, 0) = XH = 1
√2
1 −1 1 1
, (6.6)
where X is quantum NOT gate and H is the Hadamard transformation. Accordingly, in this case, the recurrence scheme is described by
ar = a2r−1+ c2r−1
pr−1 , br = 2br−1dr−1 pr−1 , cr = b2r−1+ d2r−1
pr−1 , dr = 2ar−1cr−1
pr−1 , for θ = π/2, φ = 0, (6.7)
where pr−1 = (ar−1+cr−1)2+(br−1+dr−1)2. It should be mentioned here that the relations (6.7) can also be resulted from the utility of Hadamard transformation only, i.e., U = H, but this transformation does not belong to the SU(2) operator defined in (6.2). Although the analytical convergency in the recurrence scheme (6.7) has not yet been proved, we find that an initial state in some domain Du ⊂ Dabcd, which is not yet defined, will be mapped to approach the periodic attractor representing a state interchanging step by step between {0.5, 0, 0, 0.5} and {0.5, 0, 0.5, 0}, while a state in the domain Dcu, where Duc∪Du = Dabcd, will be mapped to converge to the fixed attractor {1, 0, 0, 0}, as wanted.
For example, one can easily check to see that the initial state {0.1, 0.2, 0.6, 0.1} will be mapped to converge to the fixed attractor but the initial state {0.2, 0.1, 0.6, 0.1}, on the other hand, will be mapped to approach the mentioned periodic attractor. So a protocol in which the operator XH is used, unlike the QPA protocol, will not guarantee to purify pure maximally entangled pairs.