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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? $$ \rho_A(t)=Tr_B \left( U(t)|\psi_{AB}\rangle\langle\psi_{AB}|U'(t)\right) \sim\pmatrix{x(t)\\y(t)\\z(t)}_{\text{Bloch}_A} \text{ with } \frac{x^2(t)}{a^2}+\frac{y^2(t)}{b^2}=1 $$

draks ...
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  • Does this paper on elliptical orbits in the bloch sphere help in anyway? – GaussStrife Aug 06 '20 at 16:19
  • Thanks, I'll have a closer look... – draks ... Aug 06 '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 07 '20 at 09:45
  • 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 07 '20 at 11:48
  • it is equivalent, as pointed out here: https://quantumcomputing.stackexchange.com/a/13209/5280 – draks ... Aug 07 '20 at 12:32
  • but still it looks like that the ellipse is drawn in the mixed/local&non-local $k-l$ plane, s. eq. 15. So my ellipse depends somehow on this initial state. Hmm, let's try... – draks ... Aug 07 '20 at 13:25
  • so if I understand, you fix an initial state $|\psi_{AB}\rangle$ and look for the unitaries $U(t)$ which give a state which reduced on $A$ traces the trajectory you want. That's certainly possible, but this means that each $U(t)$ maps the same initial state $|\psi_{AB}\rangle$ to the target one, so this isn't really a time evolution, as you are just sending the initial state in different output ones – glS Aug 10 '20 at 20:51

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