Entanglement in 2D Harmonic Oscillator system

Question

Let's assume a 2 dimensional harmonic oscillator system with the Hamiltonian $\hat{H} = \frac{1}{2} p_x^2 + \frac{1}{2} p_y^2 + \frac{1}{2} \omega_x^2 x^2 + \frac{1}{2} \omega_y^2 y^2$ with the solution of the ground state being simply the product of the the ground state of each independent mode.

$\psi_{gs}(x,y) = \psi_{x}(x) \psi_{y}(y) \propto e^{-w_xx^2} e^{-w_yy^2} $ which is clearly a separable solution.

But when we rotate our coordinate system by $\pi/4$ we, get the new coordinates to be $ x' = \frac{1}{\sqrt{2}}(x + y)$ and $y' = \frac{1}{\sqrt{2}}(x - y) $

and expressing the solution in those coordinates will lead to the solution $\psi_{gs}(x',y') \neq \psi_{x'}(x') \psi_{y'}(y') $ and hence a signature of entanglement.

How can rotating a physical system (or just someone choosing a different coordinate system) lead to it being entangled?

Answer 1

Entanglement means the system not separable in any basis (or coordinate system). It’s not always easy to show there is no transformation (such as the inverse of the one you propose) that would bring a system to an explicitly separable form.

The development of practical entanglement witnesses to show entanglement irrespective of the coordinate system or basis is still an active area of research.

Answer 2

To add to @ZeroTheHero answer: one usually talks about entanglement in a narrower sense, as a state of two or more particles, since what we have here can be described as a superposition of one-particle states (whatever is the basis).