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This section describes the postselection step similar to Refs. [34, 35]. Usually, the key is not directly extracted from the state ρXY E, because the data might be only weakly correlated. Alice and Bob typically postselect on highly correlated data before proceeding with the protocol. A typical example is the basis sifting. In many QKD protocols Alice encodes the values of the key into quantum states in various bases, and Bob randomly chooses to measure in one of those bases. If they measured in the same basis, their data is typically strongly correlated, but if they measure in different bases, the data is only weakly correlated. In this case, they need to be able to identify where their basis choice matched, so that they can keep those signals with matching basis, and discard the others. Another possible postselection is to discard those events, where Bob did not record a detection event, because the signal got lost.

Let us first examine the classical version of the postselection step, which starts with the ccq state ρXY E. Alice and Bob calculate to each measurement outcome x and y some values f (x) = v and f (y) = w, and announcing v and w publicly. Typically, the announcements v and w do not reveal any information about the key. Based on the announcements, Alice and Bob decide if they want to keep the data or discard it (filtering). For example, they only keep data with matching announcements, v = w_{≡ u. In the case of sifting, v and w} plays the role of a basis announcement.

By identifying the values v and w, Alice and Bob effectively partition their original POVMs MA and MB into subsets mvA = {FAx : f (x) = v} and mwB = {F

y

B : f (y) = w}, each containing the POVM elements labeled by the value v or w of the announcement.

3.5.1

Quantum description of postselection

The quantum version of the postselection procedure is described by a two-step process: in the first step, the announcement is described by a coarse-grained measurement, represented by a quantum map _{E, with the classical outcomes u or “discard”. In the second step, a} refined measurement yields the final data.

The quantum map _{E is described by the Kraus operators KA}u _{⊗ KB}u that are acting on the state ρAB. Since only events with w = v ≡ u are kept, the Kraus operators come in pairs with the same index u. There is also a Kraus operator corresponding to the discarded events. The Kraus operators with index u satisfyP

uK u† AK u A⊗ K u† BK u B ≤ 1, and are related to the POVM elements in the sets mu

A and muB by the rule KAu =

q_P mu AF x A and K_Bu =qP mu B F y

B. For each Kraus operator, the outcome u is announced to all parties and stored in three classical registers ¯A, ¯B and ¯E held by Alice, Bob and Eve, respectively. The action of the quantum map on ρAB results thus in a state

E(ρAB) = X

u

p(u)_Ku[ρAB]⊗ |uihu|_{A ¯}¯_{B ¯E}, (3.23) with normalized conditional states

Ku_[ρ AB] = KAu ⊗ K u BρAB(KAu ⊗ K u B) † /˜p(u) (3.24) ˜ p(u) = tr_{KAu _{⊗ KB}uρAB(KAu ⊗ K u B) † }. (3.25)

each appearing with probability p(u) = _pp(u)˜

kept. The probability that the state ρAB is kept

during the postselection is pkept =P_up(u).˜

The announcement and filtering step is followed by the refined measurement. Each Ku_[ρ

AB] is measured with respect to a new (normalized) joint POVM MuAB ={F x,u

A ⊗F

y,u

B :

F_Ax,u _{∈ M}u_A, F_By,u_{∈ M}u_{B} conditioned on u. The POVMs}

Mu_A=_{FAx,u_{} = {(KA}u)−1F_Ax(K_Au†)−1 : F_Ax _{∈ m}u_A}, (3.26) Mu_B =_{FBy,u_{} = {(KB}u)−1F_By(K_Bu†)−1 : F_By _{∈ m}u_B}, (3.27) are constructed by renormalizing the sets mu

A and muB. The inverses (KAu)

−1 _{and (K}u B)

−1 are pseudo-inverses, which means they are only defined on the non-zero subspace of K_Au and Ku

B. The new measurement guarantees that measuring Ku[ρAB] with respect to MuAB results in the same distribution as measuring ρAB with respect to MA⊗ MB followed by classical postselection.

3.5.2

Key rate formula with postselection

We choose to calculate the key rate for each coinciding announcement u independently (alternatively, one could also merge the data from the different subsets u, and extract a key from the joint data string). Thus, Eve is given the purification _|Ψu_{i of the state} Ku_[ρ

AB]. If we denote the state after the measurement by ρuXY E, the key rate including postselection is ¯ r(_E(ρAB)) = X u p(u)r(ρu_{XY E}). (3.28)

Each individual rate r(ρu

XY E) = I(Ku[ρAB], MuAB)−χ(Ku[ρAB], MuA) is calculated according Eq. (3.10). We denote in the following the mutual information and Holevo quantity including postselection by ¯ I(_E(ρAB)) := X u p(u) I(_Ku[ρAB], MuAB), (3.29) ¯ χ(_E(ρAB)) := X u p(u) χ(_Ku[ρAB], MuA). (3.30)