Editing Certified infinite randomness expansion (section)

==Pseudocode==
'''Input''': <math>t</math>, <math>l</math>, <math>C</math>, <math>k</math>, <math>N</math>

'''Output''': <math>u</math>

* <math>u_0\leftarrow t</math>
* <math>i\leftarrow1</math> to <math>k</math>:
** <math>u_i\leftarrow\textrm{clusterExpansion}(D_{i\mod2},u_{i-1},l,C,N)</math>
* <math>u\leftarrow u_k</math>

With the following subroutines defined:



'''clusterExpansion'''

'''Input''': <math>D</math>, <math>t</math>, <math>l</math>, <math>C</math>, <math>N</math>

'''Output''': <math>u</math>

* <math>u\leftarrow VVExpansion(\{D_1,D_2\},t,l,C)</math>
* <math>u\leftarrow RUVExpansion(\{D_3,D_4\},u, N)</math>



'''VVExpansion'''

'''Input''': <math>D</math>, <math>t</math>, <math>l</math>, <math>C</math>

'''Output''':  <math>u</math>
* split <math>t</math> evenly into <math>(t^{(1)},t^{(2)})</math>
* <math>\kappa\leftarrow\lceil10\log^2l\rceil</math>
* <math>m\leftarrow\lceil Cl\log^2l\rceil</math>
* initialise array <math>R,r</math> of length <math>m</math>
* For <math>i\leftarrow1</math> to <math>m</math>:
** set <math>R[i]=True</math> with probability <math>1/l</math> (seed with <math>t^{(1)}</math>)
* For <math>i\leftarrow1</math> to <math>m</math> do initialise array <math>r_i</math> of length <math>\kappa</math>
** If <math>R[i]</math>:
*** For <math>j\leftarrow1</math> to <math>\kappa</math>:
**** prepare state <math>|\Psi^+\rangle</math> and share across devices <math>D_1</math> and <math>D_2</math>
**** <math>a_j\leftarrow</math> measurement results from device <math>D_1</math> in basis <math>A_{bases}[0]</math>
**** <math>b_j\leftarrow</math> measurement results from device <math>D_2</math> in basis <math>B_{bases}[0]</math>
**** If <math>a\neq b</math>:
***** <math>\textbf{abort}</math>
**** <math>r_i[j]\leftarrow(a_j,b_j)</math>
** Else:
*** <math>d\leftarrow0</math>
*** <math>x_i\leftarrow</math> draw next random bit from <math>t^{(1)}</math>
*** <math>y_i\leftarrow</math> draw next random bit from <math>t^{(1)}</math>
*** set device <math>D_1</math> to <math>A_{bases}[x_i]</math>
*** set device <math>D_2</math> to <math>A_{bases}[0],B_{bases}[0]\}[y_i]</math>
*** For <math>j\leftarrow1</math> to <math>\kappa</math>:
**** prepare state <math>|\Psi^+\rangle</math> and share across devices <math>D_1</math> and <math>D_2</math>
**** <math>a_j\leftarrow</math> measurement results from device <math>D_1</math> in set basis <math>A_{bases}[0]</math>
**** <math>b_j\leftarrow</math> measurement results from device <math>D_2</math> in set basis <math>B_{bases}[0]</math>
**** <math>r_i[j]\leftarrow(a_j,b_j)</math>
**** <math>d\leftarrow d+(a_j\oplus b_j)/\kappa</math>
*** If <math>x_j=0</math> and <math>y_j=0</math> and <math>d\neq0</math>:
**** <math>\textbf{abort}</math>
*** If <math>y_j=1</math> and <math>d>0.16</math>:
**** <math>\textbf{abort}</math>
*** If <math>x_j=0</math> and <math>y_j=0</math> and (<math>d<0.49</math> or <math>d>0.51</math>):
**** <math>\textbf{abort}</math>
** <math>r[i]\leftarrow r_i</math>
* flatten <math>r</math> into array of bits
* <math>u\leftarrow \textrm{Ext}\Big(r, t^{(2)}, \exp\big(2C|t^{(2)}|^{1/3}\big)\Big)</math>


'''RUVExpansion'''

'''Input''': <math>D</math>, <math>t</math>, <math>N</math> 

'''Output''': <math>u</math>
* split <math>t</math> evenly into <math>(t^{(1)},t^{(2)})</math>
* <math>n\leftarrow\big\lfloor\frac{|t^{(1)}|}{2}\big\rfloor</math>
* initialise arrays <math>r</math>, <math>s</math> of length <math>n</math>
* <math>w\leftarrow0</math>
* For <math>i\leftarrow1</math> to <math>n</math>:
** prepare state <math>|\Psi^+\rangle</math> and share across devices <math>D_1</math> and <math>D_2</math>
** <math>x_i\leftarrow t^{(1)}_i</math>
** <math>y_i\leftarrow t^{(1)}_{i+1}</math>
** <math>a_i\leftarrow</math> measurement result from device <math>D_1</math> in basis <math>A_{bases}[x_i]</math>
** <math>b_i\leftarrow</math> measurement result from device <math>D_2</math> in basis <math>B_{bases}[y_i]</math>
** <math>s[i]\leftarrow(x_i,y_i)</math>
** <math>r[i]\leftarrow(a_i,b_i)</math>
** If <math>x_i\wedge y_i = a_i\otimes b_i</math>:
*** <math>w\leftarrow w+1</math>
* If <math>w < n\cos^2(\pi/8)-\frac{1}{2\sqrt{2}}\sqrt{n\log{n}}</math>:
** \textbf{abort}
* <math>\gamma_1\leftarrow</math> random number in range <math>\{0...n/N-1\}</math> (seed using <math>t^{(2)}</math>)
* <math>\gamma_2\leftarrow</math> random number in range <math>\{1...\sqrt{N}-1\}</math> (seed using <math>t^{(2)}</math>)
* initialise array <math>u</math> of length <math>\sqrt{N}</math>
* For <math>i\leftarrow0</math> to <math>\sqrt{N}</math>:
** <math>u[i]\leftarrow r[\gamma_1n/N+\gamma_2\sqrt{N}+i][0]</math>