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I would like to apply a General Lorentz Boost to some Multi-partite Quantum State.

I have read several papers (like this) on the theory of boosting quantum states, but I have a hard time applying this theory to concrete examples.

Let us take a $|\Phi^+\rangle$ Bell State as an example, and apply a general Lorentz Boost $$ \Lambda=\left[\begin{array}{cccc}{\gamma} & {-\gamma \beta_{x}} & {-\gamma \beta_{y}} & {-\gamma \beta_{z}} \\ {-\gamma \beta_{x}} & {1+(\gamma-1) \frac{\beta_{x}^{2}}{\beta^{2}}} & {(\gamma-1) \frac{\beta_{x} \beta_{y}}{\beta^{2}}} & {(\gamma-1) \frac{\beta_{x} \beta_{z}}{\beta^{2}}} \\ {-\gamma \beta_{y}} & {(\gamma-1) \frac{\beta_{y} \beta_{x}}{\beta^{2}}} & {1+(\gamma-1) \frac{\beta_{y}^{2}}{\beta^{2}}} & {(\gamma-1) \frac{\beta_{y} \beta_{z}}{\beta^{2}}} \\ {-\gamma \beta_{z}} & {(\gamma-1) \frac{\beta_{z} \beta_{x}}{\beta^{2}}} & {(\gamma-1) \frac{\beta_{z} \beta_{y}}{\beta^{2}}} & {1+(\gamma-1) \frac{\beta_{z}^{2}}{\beta^{2}}}\end{array}\right] $$ to this state.

Now, as I understand, we represent this Lorentz Boost as some unitary $U(\Lambda)$ in our Hilbert Space, in order to be able to boost our quantum state:$$|\Phi^{+'}\rangle=U(\Lambda)|\Phi^+\rangle$$

Unfortunately, I have found no paper that detailes just how exactly this unitary is found, they all simply state that it must always exist.

So, how would I find $U(\Lambda)$ that boosts some quantum state – like $|\Phi^+\rangle$ – from some inertial frame of reference $S$ to another $S'$?

Thanks!

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  • $\begingroup$ I'm not sure that this makes sense. Something like a "Bell state" represents quantum information at a rather abstract level. The qubits might represent any kind of degree of freedom of any kind of physical system (polarisation or position of a photon, atomic structure of some atom, state of an electron, you name it). How a coordinate transformation (relativistic or not) acts on such a state will fully depend on the actual physical state that is being represented. $\endgroup$ – glS Aug 28 at 19:55

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