A Maximizing Two Functions with Constraints in Phase Covariant Cloning Machine

[elere]

hello, in the task of finding the optimal phase covariant cloning machine, i have to maximize two functions of six variables :f1=a.C+b.D and f2=a.B+c.D , they are many constraints, but I've already used them to get to those expressions in the first place, the variables are real scalars and vary between -1 and 1. of course I'm not searching for numerical values for the six variables, in fact I've already got a transformation (set of variables) which i know is optimal, all I'm looking for is a way to prove that it is actually optimal and i know there can be others sets of variables which provide optimality.

 

[mfb]

The functions are clearly maximized if all variables become 1. You can also flip the signs of a, B and C together, or the signs of b, c, and D together, or both groups together, without changing the result. Proof: looked at the expression. But you can also use some inequalities to prove that those solutions maximize both f1 and f2 in a formal way.

 

[elere]

The functions are clearly maximized if all variables become 1..

my fault, the 6 variables along with two others that I've managed to simplify in the expressions verifies a orthonormalization conditition :
a^2+b^2+c^2+d^2=1 and aA+bB+cC+dD=0 . and here are the ramaining constraints : Ac+Bd=Ab+Cd=0 and ac+AC+bd+BD=ab+AB+cd+CD=0. the set that i want to proof it's optimality and which verify the above conditions is {a=1,b=c=d=0, A=D=0}.

 

[mfb]

Okay, so a=B=C=1, and everything else zero? Then f1=f2=1.

From looking at it: a=-b=C=D=1/sqrt(2) and everything else zero lead to f1=1 as well, but then f2=0. How exactly do you want to maximize the functions? The sum of both? Both individually? Something else?

There are 6 constraints on 8 variables, unless you need a mathematical proof most algorithms should quickly find all local maxima.

 

[elere]

well, B and C aren't equal to one, they are parameters which satisfy A^2+B^2=1 (first constraint). and for the way to maximize the functions, well i guess in anyway, as the two variables B and C (which can be seen as Cos and Sin) are not specified, the symmetrical case correspond to maximizing the Sum and minimizing the difference. to be specific i need to show that this set of variables with the constraint A^2+B^2=1, verify the optimality condition (df1=0, df2=0), so i was thinking that maybe i could write this conditions in a more explicit form, as a function of the variables, I've tried the la grange multipliers, but i'v got confuse with putting the second optimality condition (df2=0) as a constraint for the first one.

 

[mfb]

A^2+B^2=1 (first constraint)

Where does that come from now?

 

[elere]

Where does that come from now?

i'm really sorry, I've forget one costraint, the constraint is A^2+B^2+C^2+D^2=1, and since A=D=0 ...

 

[mfb]

Then you have 7 constraints on 8 parameters. That is a one-dimensional problem, fixing one parameter fixes all others (up to some discrete changes like flipping signs).

 

[elere]

yes it is, i managed (finally) to write the lagrange's equations, which are satisfied for the specific set of variables I've mentioned ! thanks for your time and you help