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Finding a relationship between density matrices: Solve error

Posted 5 months ago
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I am writing my equation according to scenario 1 and scenario 2 and then I am trying to find a relationship between rhof2* and *rhof (or rhoM and rhom1m2) enter image description here

Now, according to scenario 1, m1 and 2 are matrices which have the dimension 2*2 From scenario 1, I first write the rho_ m2 and I put rho_ m2 in the rho_f and I got this result :

rho_f1={{p_x*(p_h*rho00 + 2.0*p_h*rho11 + rho00*(1 - p_h)) + 
p_x*(2.0*p_h*rho00 + 1.0*p_h*rho11 + 1.0*rho11*(1 - p_h)) + 
p_x*(2.0*p_h*rho00 + p_h*rho11 + rho11*(1 - p_h)) + (1 - 
p_x)*(p_h*rho00 + 2.0*p_h*rho11 + rho00*(1 - p_h)), 
p_x*(p_h*rho01 - rho01*(1 - p_h)) + 
p_x*(-p_h*rho10 + rho10*(1 - p_h)) + 
p_x*(1.0*p_h*rho10 - 1.0*rho10*(1 - p_h)) + (1 - p_x)*(-p_h*rho01 +
rho01*(1 - p_h))}, {p_x*(-p_h*rho01 + rho01*(1 - p_h)) + 
p_x*(1.0*p_h*rho01 - 1.0*rho01*(1 - p_h)) + 
p_x*(p_h*rho10 - rho10*(1 - p_h)) + (1 - p_x)*(-p_h*rho10 + 
rho10*(1 - p_h)), 
p_x*(p_h*rho00 + 2.0*p_h*rho11 + rho00*(1 - p_h)) + 
p_x*(1.0*p_h*rho00 + 2.0*p_h*rho11 + 1.0*rho00*(1 - p_h)) + 
p_x*(2.0*p_h*rho00 + p_h*rho11 + rho11*(1 - p_h)) + (1 - 
p_x)*(2.0*p_h*rho00 + p_h*rho11 + rho11*(1 - p_h))}}

Here p_ h is a coefficient which comes from equation of rho_ m2 and p_ x is another coefficient which comes from equation rho_f. From scenario 2, I have this equation:

rho_f2 = {{g00 . p_g + 2 . g11 . p_g + g11 . (1 - p_g), -g01 . p_g + 
    g10 . (1 - p_g)}, {g01 . (1 - p_g) - g10 . p_g, 
   2. g00 . p_g + g00 . (1 - p_g) + g11 . p_g}} 

and p_g is a coeffieicent

Now I want to find a functionality/a relationship between rho and g. I want to write the g matrix in terms of rho matrix.

I tried this code(and all the combination of this code) but I am having errors:

Solve[rho_f1 . rho_f2 == {{g00, g01, g10, g11}}, Flatten[rho_f1]]

In the output it is written that:

Solve::ivar: (rho00 (1-(p:Blank[<<1>>]))+rho00 ph+2. rho11 ph) (1-px)+(rho11 (1-(p:Blank[<<1>>]))+2. rho00 ph+rho11 ph) px+(1. rho11 (1-(p:Blank[<<1>>]))+2. rho00 ph+1. rho11 ph) px+(rho00 (1-(p:Blank[<<1>>]))+rho00 ph+2. rho11 ph) px is not a valid variable.

P.S I am not even sure the solve function is the right way to do it so if you have any other suggestion, will be glad

POSTED BY: Gozde USTUN
Posted 5 months ago

Ok, the problem has been solved

  1. I was confused by Wolfram and Mathematica syntax. I thought they are all under wolfram and the syntax should be same. For instance i is not a complex i but a variable in Mathematica. SO I changed the i by esc ii
  2. I was confused by matrix multiplication and multiply a matrix with a coefficient. To multiply a matrix with a coefficient we should use '*', and for matrix multiplication we should use '.'

Thanks.

POSTED BY: Gozde USTUN
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