Mapping the Schrödinger picture of open quantum dynamics

For systems described by finite matrices, an affine form is developed for the maps that describe evolution of density matrices for a quantum system that interacts with another. This is established directly from the Heisenberg picture. It separates elements that depend only on the dynamics from those that depend on the state of the two systems. While the equivalent linear map is generally not completely positive, the homogeneous part of the affine maps is, and is shown to be composed of multiplication operations that come simply from the Hamiltonian for the larger system. The inhomogeneous part is shown to be zero if and only if the map does not increase the trace of the square of any density matrix. Properties are worked out in detail for two-qubit examples.

Figure 1

(Color online) The compatibility domain for $⟨{\Xi }_1⟩$, $⟨{\Xi }_2⟩$, $⟨{\Xi }_3⟩$, $⟨{\Sigma }_1{\Xi }_2⟩$, $⟨{\Sigma }_1{\Xi }_3⟩$, $⟨{\Sigma }_2{\Xi }_1⟩$, $⟨{\Sigma }_2{\Xi }_3⟩$, $⟨{\Sigma }_3{\Xi }_1⟩$, and $⟨{\Sigma }_3{\Xi }_2⟩$ all equal to $\frac{1}{4}$. (A) The whole compatibility domain inside the unit sphere. (B) The section of the domain in the $({\Sigma }_1,{\Sigma }_2)$ plane. The sections of the domain in the $({\Sigma }_1,{\Sigma }_3)$ and $({\Sigma }_2,{\Sigma }_3)$ planes are identical.

Figure 2

(Color online) (A) Sections in the $({\Sigma }_1,{\Sigma }_2)$ plane of the compatibility domain (dotted region) and the positivity domain (thick curve) for $⟨{\Xi }_1⟩$, $⟨{\Xi }_2⟩$, $⟨{\Xi }_3⟩$, $⟨{\Sigma }_1{\Xi }_2⟩$, $⟨{\Sigma }_1{\Xi }_3⟩$, $⟨{\Sigma }_2{\Xi }_1⟩$, $⟨{\Sigma }_2{\Xi }_3⟩$, $⟨{\Sigma }_3{\Xi }_1⟩$, and $⟨{\Sigma }_3{\Xi }_2⟩$ all equal to $\frac{1}{4}$ and ${\gamma }_1=2\sqrt{5}$, ${\gamma }_2=2\sqrt{3}$, ${\gamma }_3=2\sqrt{2}$. (B) Sections of where the map takes the compatibility domain, the positivity domain and the unit sphere (thick curve plus the dashed curve). The dotted circle is the section of the unit sphere.

Figure 3

(Color online) The compatibility domain for $⟨{\Xi }_1⟩$ and $⟨{\Sigma }_3{\Xi }_1⟩$ equal to $1∕\sqrt{3}$ and $⟨{\Xi }_2⟩$, $⟨{\Xi }_3⟩$, $⟨{\Sigma }_1{\Xi }_2⟩$, $⟨{\Sigma }_1{\Xi }_3⟩$, $⟨{\Sigma }_2{\Xi }_1⟩$, $⟨{\Sigma }_2{\Xi }_3⟩$, $⟨{\Sigma }_3{\Xi }_2⟩$ all equal to zero. (A) The whole compatibility domain inside the unit sphere. (B) The section of the domain in the $({\Sigma }_1,{\Sigma }_3)$ plane. (C) The section of the domain in the $({\Sigma }_2,{\Sigma }_3)$ plane. The compatibility domain does not intersect the $({\Sigma }_1,{\Sigma }_2)$ plane.