# Reduced Density Matrix Equation of Motion to describe an Ellipse

Given a pure two qubit state $$|\psi_{AB}\rangle$$. If we trace out system $$B$$, the remaining density matrix $$\rho_A = Tr_B|\psi_{AB}\rangle\langle\psi_{AB}|$$, can be represented as a point lying anywhere on or inside a Blochsphere.

When you're on the Bloch sphere you have a separable state; when you're in the center, your state is maximally entangled. So by entanglement you can affect the distance from the center.

How do have to steer (by applying timevarying unitaries $$U(t)$$) the composite system $$|\psi_{AB}\rangle$$, such that the resulting trajectory on or inside the Bloch sphere of system $$A$$ is an ellipse? $$\rho_A(t)=Tr_B \left( U(t)|\psi_{AB}\rangle\langle\psi_{AB}|U'(t)\right) \sim\pmatrix{x(t)\\y(t)\\z(t)}_{\text{Bloch}_A} \text{ with } \frac{x^2(t)}{a^2}+\frac{y^2(t)}{b^2}=1$$

• Does this paper on elliptical orbits in the bloch sphere help in anyway? – GaussStrife Aug 6 '20 at 16:19
• Thanks, I'll have a closer look... – draks ... Aug 6 '20 at 16:30
• ... it looks to me, that the paper does not adress the fact that I trace out the system B. What do you think? – draks ... Aug 7 '20 at 9:45
• Good point. Could the unitary applied in the paper to trace the ellipse on the surface simply not be extended to both systems via taking the tensor of the individual unitary, so that it works on the individual subsystems? From Fig.1 in that paper, the wording suggests that the ellipses they achieve on both the pure and mixed state case is what is generated from a unitary on the overall 2 qubit system, and then once the trace is taken, the ellipses is on both of their bloch spheres. – GaussStrife Aug 7 '20 at 11:48
• it is equivalent, as pointed out here: quantumcomputing.stackexchange.com/a/13209/5280 – draks ... Aug 7 '20 at 12:32