Reduced Density Matrix Equation of Motion to describe an Ellipse - Quantum Computing Stack Exchange most recent 30 from quantumcomputing.stackexchange.com 2022-01-20T17:56:50Z https://quantumcomputing.stackexchange.com/feeds/question/13196 https://creativecommons.org/licenses/by-sa/4.0/rdf https://quantumcomputing.stackexchange.com/q/13196 3 Reduced Density Matrix Equation of Motion to describe an Ellipse draks ... https://quantumcomputing.stackexchange.com/users/5280 2020-08-06T15:10:48Z 2020-08-10T16:21:17Z <p>Given a pure two qubit state <span class="math-container">$|\psi_{AB}\rangle$</span>. If we trace out system <span class="math-container">$B$</span>, the remaining density matrix <span class="math-container">$\rho_A = Tr_B|\psi_{AB}\rangle\langle\psi_{AB}|$</span>, can be represented as a point lying anywhere on or inside a Blochsphere.</p> <p>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.</p> <p>How do have to steer (by applying timevarying unitaries <span class="math-container">$U(t)$</span>) the composite system <span class="math-container">$|\psi_{AB}\rangle$</span>, such that the resulting trajectory on or inside the Bloch sphere of system <span class="math-container">$A$</span> is an ellipse? <span class="math-container">$$\rho_A(t)=Tr_B \left( U(t)|\psi_{AB}\rangle\langle\psi_{AB}|U'(t)\right) \sim\pmatrix{x(t)\\y(t)\\z(t)}_{\text{Bloch}_A} \text{ with } \frac{x^2(t)}{a^2}+\frac{y^2(t)}{b^2}=1$$</span></p>