Divisibility of dynamical maps: Schrödinger vs. Heisenberg picture

Abstract

Divisibility of dynamical maps is a central notion in the study of quantum non-Markovianity, providing a natural framework to characterize memory effects via time-local master equations. In this work, we generalize the notion of divisibility of quantum dynamical maps from the Schrödinger to the Heisenberg picture. While the two pictures are equivalent at the level of physical predictions, we show that the divisibility properties of the corresponding dual maps are, in general, not equivalent. This inequivalence originates from the distinction between left and right generators of time-local master equations, which interchange roles under duality. We demonstrate that Schrödinger and Heisenberg divisibility are distinct concepts by constructing explicit dynamics divisible only in one picture. Furthermore, we introduce a quantifier for the violation of Heisenberg P-divisibility, analogous to the trace-distance-based measure of non-Markovianity, and provide it with an operational interpretation in terms of the guessing probability between effects. Our results show that Heisenberg divisibility is an independent witness of memory effects and highlight the need to consider both pictures when characterizing non-Markovian quantum dynamics.

Figures (5)

• Figure 1: Left: guessing between states (Schrödinger divisibility): Alice prepares either $\rho$ or $\sigma$ and Bob has to guess which state was prepared. Right: guessing between effects (Heisenberg divisibility): Alice can measure either $E$ or $F$ and Bob has to guess which effect was measured.
• Figure 2: Phase covariant dynamics. Dynamics of $D_\infty(\Phi_t^*[X], \Phi^*_t[Y])$ for different initial effects $X$, $Y$. Inset: rates $\gamma_+$ (red solid), $\gamma_-$ (red dashed), $\xi_+$ (blue solid), and $\xi_-$ (blue dashed) corresponding to Eq. \ref{['eq:ph_cov_counterexample']}. At $t\approx1.9$ (vertical dashed line), $\xi_+$ becomes negative and, accordingly, $D_\infty$ behaves non-monotonically, thus witnessing violations of Heisenberg P-divisibility.
• Figure 3: Off diagonal elements of $L(t)$ and $R(t)$. Solid: $\ell_{1,2}$\ref{['eq:l_class']}, \ref{['eq:l_class_2']}, dashed: $r_{1,2}$\ref{['eq:r_class']}, \ref{['eq:r_class_2']}. Left panel corresponds to \ref{['ab1']}, right panel to \ref{['ab2']}.
• Figure 4: Representation of the time evolution of the Bloch sphere under the maps $\Lambda^{(1,2)}$ of Eq. \ref{['eq:OD_and_DO']}, in units $t_1=1$. For $t\le1$, $\Lambda^{(1)}=\Lambda^{(2)}$. At later times, $\Lambda^{(1)}$ is just an orthogonal rotation of the ellipsoid giving the evolution of the Bloch sphere, while $\Lambda^{(2)}$ does not change the image of the ellipsoid, with the image of the pure states moving along its surface.
• Figure 5: Dynamics of $\Phi_t$ corresponding to $\Lambda^{(1)}$ of Eq. \ref{['eq:OD_and_DO']}. Left panel: time evolution of $D_\infty\left(\Phi^*_t\left[{\ket{+_y}\bra{+_y}}\right], \Phi^*_t\left[{\ket{-_y}\bra{-_y}}\right]\right)$ and of $D_1\left(\Phi_t\left[{\ket{+_y}\bra{+_y}}\right], \Phi_t\left[{\ket{-_y}\bra{-_y}}\right]\right)$, where $\ket{\pm_y}$ are the eigenstates of $\sigma_y$. Since the dynamics is Schrödinger CP-divisible but Heisenberg P-indivisible, $D_1$ is monotonic while $D_\infty$ is not. Inset: dynamical functions $\lambda(t)$ and $\beta(t)$ of Eqs. \ref{['eq:D_dephasing']}, \ref{['eq:O_dephasing']}. Right panel: dynamics of incompatibility $I_0\left(\Phi^*_t\left[{\ket{+_y}\bra{+_y}}\right], \Phi^*_t\left[{\ket{+_x}\bra{+_x}}\right]\right)$ (for this particular dynamics, $I_0=I_{\text{steer}}$) and sharpness $\Sigma\left(\Phi^*_t\left[{\ket{+_y}\bra{+_y}}\right]\right)$ of Eqs. \ref{['eq:incop_p']}, \ref{['eq:incop_steering']}, and \ref{['eq:sharpness']}. $\ket{+_x}$ is the eigenstate of $\sigma_x$. Notice that the dynamics presents a full revival in sharpness and only a partial revival in incompatibility. The vertical lines represent the time at which Heisenberg P-divisibility is broken.

Theorems & Definitions (3)

• Proposition 1
• Example 1
• Corollary 1