Remote Sampling with Applications to General Entanglement Simulation

. 2019 Jan 19;21(1):92. doi: 10.3390/e21010092

Algorithm 2 Modified rejection algorithm—Protocol for the leader

Input: Value of

t_{0}

1:
Compute $p_{x} (t_{0})$ for each $x \in X$

{The leader needs information from the custodians in order to compute these approximations}
2:
Compute C and $q = {(q_{x})}_{x \in X}$ as per Equations (1) and (2)
3:
Sample X according to q
4:
Sample U uniformly on $[0, 1]$
5:
for $t = t_{0}$ ∞ do
6:
if $U C q_{X} \leq p_{X} (t) - 2^{- t}$ then
7:
return X {X is accepted}
8:
else if $U C q_{X} > p_{X} (t) + 2^{- t}$ then
9:
go to line 3 {X is rejected}
10:
else
11:
    Continue the for loop

    {We cannot decide whether to accept or reject because $- 2^{- t} < U C q_{x} - p_{x} (t) \leq 2^{- t}$ ;

    communication may be required in order for the leader to compute $p_{X} (t + 1)$ ;

    it could be that bits previously communicated to compute $p_{x} (t)$ can be reused.}
12:
end if
13:
end for