Issues Implementing Pure Function

Question

Thanks in advance for any assistance you may be able to provide. I'm new to Mathematica and running into issues creating a usable function from the code seen below. For context, all of this code works as intended, outside of the implementation of the pure function itself (see last 2-3 lines):

ClearAll["Global`*"]

n = 2;

Do[θpsi[i] = RandomReal[π/2], {i, 1, (2^n) - 1}];
Do[ϕpsi[i]  =   RandomReal[2 π], {i, 1, (2^n) - 1}];

For[i = 0, i <= 2^n, i++,
 Which[
  i == 1, ψ[i] =  {Cos[θpsi[i]]};,
  i != 1 && 
   i != 2^n, ψ[
     i] = {Product[Sin[θpsi[j]], {j, 1, i - 1}]*
      Cos[θpsi[i]]*E^(I*ϕpsi[i - 1])};,
  i == 2^n , ψ[
     i] = {Product[Sin[θpsi[j]], {j, 1, i - 1}]*E^(
      I*ϕpsi[i - 1])};
  ](*Which*)
 ](*For*)

ψ = Array[ψ, 2^n]  

ϕState[
   i_] := {{Cos[Subscript[θ, 
     i]]}, {Sin[Subscript[θ, i]]*E^(I*Subscript[ϕ, i])}};

kronk = Fold[KroneckerProduct];
seperableStates = Table[ϕState[i], {i, 1, n}];
Φ = kronk[seperableStates];

x = (ConjugateTranspose[ψ].Φ)[[1, 1]]

(* Generate objective function *)
f = Function[{θ1, θ2, ϕ1, ϕ2}, x];

f[1, 2, 3, 4]
     

My goal is to take the result of x = (ConjugateTranspose[ψ].Φ)[[1, 1]] and turn it into a user-friendly function that can be fed inputs, as seen on the last two lines.

Am I on the right track here? If not, how can I improve upon this code? I'm still trying to figure out Mathematica in general (recent convert from MATLAB), so any general feedback would be appreciated as well.

Cheers!

Answer 1

The problem with your definition of f is x is an expression that uses 2 subscripted symbols θ and ϕ, but f is a function of 4 symbols θ1, ... . Here is a definition of f that works (for n=2)

f = x /. {Subscript[θ, 1] -> #1, Subscript[θ, 2] -> #2, 
     Subscript[ϕ, 1] -> #3, Subscript[ϕ, 2] -> #4} &;

f[1, 2, 3, 4]

This definition of f can be generalized for other values of n as follows:

Clear[g]
g = Function@Evaluate[x /. Flatten@
      Table[{Subscript[θ, k] -> Slot[k], 
        Subscript[ϕ, k] -> Slot[n + k]}, {k, n}]];

g[1, 2, 3, 4]

By the way, anyone who uses subscripts should be aware of what is said about them at these links: well-behaved indexed variables, sidenote on using Subscript, and avoid using subscripts

Answer 2

In addition to the issue of subscripts, which I recommend avoiding, there is an issue of substitution within a Function as this requires the parameter Symbols to be literally present in the body:

x = a + b * c;

f = Function[{a,b,c}, x];

f[1, 2, 3]

a + b c      (* substitution did not occur *)

This construct using a Block analog addresses both points:

SetAttributes[ssFuntion, HoldAll]

ssFuntion[p : {__Subscript}, body_] :=
 Internal`LocalizedBlock[p, p = {##}; body] &

Usage:

f = ssFuntion[{Subscript[θ, 1], Subscript[θ, 2], Subscript[ϕ, 1], Subscript[ϕ, 2]}, x];

f[1, 2, 3, 4]

0.306819 + 0.138836 I

Reference:

• Expressions containing globally undefined symbols inside a function where they are defined
• What is the purpose of Internal`LocalizedBlock?