# Unitary Transformations for States with Same Entanglement [duplicate]

$$\newcommand{\Ket}[1]{\left|#1\right>}$$ I know this has been asked before in another context (How to construct local unitary transformations mapping a pure state to another with the same entanglement?), but a similar issue dealing with the same paper (https://arxiv.org/pdf/quant-ph/9811053.pdf) caused me to ask this question.

If $$\Ket{\psi'} \sim \Ket{\psi''}$$, then $$\Ket{\psi''} = (U_A\otimes U_B) \Ket{\psi'}$$ for some local unitary transformations $$U_A$$ and $$U_B$$. Here, $$\Ket{x} \sim \Ket{y}$$ iff $$\rho_x$$ and $$\rho_y$$ have the same spectrum of eigenvalues.

The question is: How does one construct local unitary transforms to map the states $$\Ket{\psi'''_2} \rightarrow \Ket{\psi'''_1}$$, and $$\Ket{\psi'''_1} \rightarrow \Ket{\psi^4}$$, where:

$$\Ket{\psi'''_1} = \cos(\delta)\Ket{00} + \sin(\delta)\Ket{1} \big[ \cos(\gamma)\Ket{0} + \sin(\gamma)\Ket{1} \big] \\ \Ket{\psi'''_2} = \sin(\delta)\Ket{00} + \cos(\delta)\Ket{1} \big[ \cos(\gamma)\Ket{0} + \sin(\gamma)\Ket{1} \big] \\ \Ket{\psi^4} = \sqrt{\lambda_+} \Ket{00} + \sqrt{\lambda_-} \Ket{11} \\$$

Note that $$\lambda_{\pm} = \big(1\pm \sqrt{1-\sin^2(2\delta)\sin^2(\gamma)}\big)/2$$, and the Schmidt coefficients of all states are the same. I have tried solving this problem using both Mathematica and by hand (following the method of this answer), but both were yet unfruitful. Any help would be much appreciated.

EDIT: Answer can be found here

• I don't understand the difference between this and the linked question
– glS
Aug 4 '21 at 10:21
• The issue I was having was that the expressions obtained using that method were quite unwieldy, and I wasn't able to obtain a closed-form analytical solution given the conditions of orthonormality. Aug 4 '21 at 16:33
• There are many things you can try: trigonometry, numerics, guessing, a symbolic calculation package other than Mathematica, doing some special cases first (e.g. $\gamma=0$) and then you can always resort to reshaping the states from $4\times 1$ matrices to $2\times 2$ matrices to transform the task of finding the Schmidt decomposition into the task of finding the singular value decomposition. Hope this helps! :-) Aug 4 '21 at 17:16