Finding a unitary transformation between two quantum states.

[Qubix231]

I have to find a unitary transformation that takes me from one quantum state to another (or if there is such a transformation), given the two quantum states in matrix form. The matrices are huge (smallest is 16x16) , so doing it on paper is not an option. Does anyone know how I can do this in Mathematica?

 

[JorisL]

Do you need the explicit transformation? Otherwise you could prove it exists and use it as a general operator.

If you do need it, try to get it to work for smaller examples.

A last question have you googled? Because I suppose if it's got a built in solution you would find loads of info.
If not please do so, good lookup skills are one of the most important things you'll ever learn.

 

[Qubix231]

I don't necessarily need to see the matrix form of the unitary, I just have to prove that it does exist, i.e. that my two states are unitarily equivalent. And yep, I did google.

 

[JorisL]

What is your system?
Because in the cases I encountered you can use the strong mathematical formalism to show this.
It can be non-trivial but once you find the solution you often smack yourself in the head.

 

[Qubix231]

3 to 6 qubits. (forgot to mention initially that the smallest is 8x8, not 16x16).

 

[JorisL]

It has been a while since I've worked with this. Which is why I took so long to answer.
However you'll need to determine what the Hilbert space is.

That's where my ideas get shaky.
I will refrain from giving information that is likely to have grave errors in it. That way you don't have to unlearn faulty information.

I noticed a mentor to move this to the QM forum where you'll get quality answers.

 

[Qubix231]

Thanks JorisL. So if it is of any help, the two matrices are here:

http://pastebin.com/s3B1T0HD

I want to see if there is a unitary (up to some approximation anyway) , that takes me from one matrix to the other. Anyone know how to do this in Mathematica? or Numpy Python ?

 

[Strilanc]

If the states are pure, then an operation that transforms between them is just $M_{b \leftarrow a} = I + \left| b \right\rangle \left\langle a \right|$. To make it unitary just pick some arbitrary other basis vectors to complete the $a$ basis and $b$ basis and add a mapping between them in as well.

If the states are mixed, I guess you'd take the schmidt decomposition then map each schmidt basis vector in $a$ across the basis vectors in $b$ so the coefficients end up matching... but if $b$ is more pure than $a$ then there's likely an obstacle that prevents it from working.