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OPTIMAL ENSEMBLE FOR GHZ-TYPE MIXED STATES

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FACTORIAL

## Function that calculates the factorial of a number ## Usage : f=factorial(n) function f=factorial(n) ## Initialize the output f=1; ## Check whether the input is correct if ( (n<0) || (rem(n,1)~=0) ) printf("n cannot be a negative number. Exiting...\n"); return endif for num=1:n f*=num; endfor endfunction
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REALISTIC AND NON-REALISTIC STRINGS

## A 'realistic', 'non-elastic' string, which responses to any ## bending and has stifness. This script takes in the previous ## and the present profiles and iterates to find ## the profile in the next time step. The ratio 'r' is not 1 ## like in the 'non-realistic' string since the speed of the wave ## always less than the speed of the string it should be less than 1 ## for best and most stable solution ##constants dx=1e-2 ## Spatial increment (m) L=2 ## Length of the string (m) M=L/dx ## Dimensionless partition E=1e-4 ## Dimensionless stiffnes function ynext=propagate_stiff(ynow,yprev,r) ## Quick and dirty way to fix boundary conditions -- for each step ## they are the same as the previous step. ynext=ynow; ynow(1)=ynow(2)=0; ##Entering the loop for i=3:length(ynow)-1 ## boundary condition ynow(length(ynow)-1)=ynow(length(ynow)-2)=0; ## Divide the ynext with many terms into three parts for easiness ynext(i)=(2−(2*r^2)−(6*E*(r^2)*...
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NEWTON’S METHOD FOR MINIMUM

##Newton's Method to find ##the minimum of the function F(x)=(x-2)^4-9 ##with the initial guess xmin=1.0 ##Constants and initializations xmin=[]; ##The empty array of x that minimizes the F(x) xmin(1)=1.0; ##Initial value of the xmin Fmin=[]; ##Minimum values of F(x) x=0.0:0.1:4.0; ##Only for plotting purposes F=[]; ##Our examined Function evaluated on x-space Fp=[]; ##First derivative of F(x) wrt x Fpp=[]; ##Second derivative o F(x) wrt x NSteps=50; ##Step number of iteration ##Algorithm for n=1:NSteps Fmin(n)=(xmin(n)-2)^4-9; Fp(n)=4*(xmin(n)-2)^3; Fpp(n)=12*(xmin(n)-2)^2; xmin(n+1)=xmin(n)-Fp(n)/Fpp(n); Fmin(n+1)=(xmin(n+1)-2)^4-9; endfor printf("x*, at which F(x) is minimum, is %1.6f\n",xmin(n+1)) printf("Minimum of F(x) is %1.6f\n",Fmin(n+1)) F=(x-2).^4-9; subplot(2,1,1) plot(x,F) title('Newton^,s Method-F(x) vs x'); xlabel('x'); ylabel('F(x)'); text(2,-7,'\downarrow') text(1.7,-5.6,'(xmin,Fm...
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