Emerging dynamics arising from coarse-grained quantum systems

The purpose of physics is to describe nature from elementary particles all the way up to cosmological objects like cluster of galaxies and black holes. Although a unified description for all this spectrum of events is desirable, this would be highly impractical. To not get lost in unnecessary details, effective descriptions are mandatory. Here we analyze the dynamics that may emerge from a full quantum description when one does not have access to all the degrees of freedom of a system. More concretely, we describe the properties of the dynamics that arise from quantum mechanics if one has access only to a coarse-grained description of the system. We obtain that the effective maps are not necessarily of Kraus form, due to correlations between accessible and nonaccessible degrees of freedom, and that the distance between two effective states may increase under the action of the effective map. We expect our framework to be useful for addressing questions such as the thermalization of closed quantum systems, as well as the description of measurements in quantum mechanics.

Figure 1

Coarse-graining-induced dynamics. Schematic diagram representing the different levels of description connected by a coarse graining. Given an initial state of the system, with density operator ${\psi }_0$, its evolution, ${\mathfrak{U}}_t$, and a coarse-graining map ${\mathrm{\Lambda }}_{\text{CG}}$, we want to determine the induced dynamics ${\mathrm{\Gamma }}_t$ and its properties, such that ${\mathrm{\Gamma }}_t\circ {\mathrm{\Lambda }}_{\text{CG}}\left({\psi }_0\right)={\mathrm{\Lambda }}_{\text{CG}}\circ {\mathfrak{U}}_t\left({\psi }_0\right)$.

Figure 2

Operational interpretation of a CPTP linear map $\mathrm{\Lambda }$.

Figure 3

The distinct levels and dynamics induced by the coarse-graining ${\mathrm{\Lambda }}_{\text{CG}}$.

Figure 4

Effective evolution as seen by a blurred-saturated detector. The plot shows an oscillatory behavior for the purity of the effective state. The inset shows the effective state trajectory in the Bloch sphere. The underlying dynamics is determined by the Hamiltonian $H=\hslash J{\sigma }_z\otimes {\sigma }_z$, and we set $J=1\mathrm{rad}/\mathrm{s}$.

Figure 5

Simple example of fixing ${\rho }_0$ and changing the effective map. For the case where we fix the unitary mapping as the $SWAP$, i.e., $U_t|ij\rangle =SWAP|ij\rangle =|ji\rangle$, and the coarse graining as the usual partial trace on the second component, we see that different underlying states generate different effective maps. The fact that the emergent maps cannot be the same is clear because if that was the case the same input would lead to two different outputs.

Figure 6

The $\gamma$ space, and the effects of changing the underlying state. The intersection between the hyperplanes defines the effective state. A non-point-like region reflects the fact that many underlying states lead to the same effective state. (a) The top panel shows a change in the underlying state that does not change the effective state ${\rho }_0$. This change may, nevertheless, have impact on the effective map. (b) Changing the underlying state such that its Bloch vector $\vec{\gamma }$ moves normally to the hyperplanes changes only the effective state preserving the effective map. The region obtained by such normal displacement of the Bloch vector defines the domain of an effective map.

Figure 7

Recovering strict complete positivity for the effective map. One way to obtain a family of CP effective maps is to require the map ${\mathcal{N}}_{t_k}={\mathrm{\Lambda }}_{CG}\circ {\mathfrak{U}}_{t_k}$ to be CP divisible.

Figure 8

Distance increase for the effective dynamics. In the above plots, $\left|\right|{\rho }_t-{\rho }_t^{\prime }{\left|\right|}_1, \left|\right|{\psi }_0-{\psi }_0^{\prime }{\left|\right|}_1$, and $\left|\right|{\rho }_0-{\rho }_0^{\prime }{\left|\right|}_1$ are represented, respectively by the blue continuous line, red dashed line, and the black dot-dashed line. Contrary to the usual contractive property of CPTP linear maps, on the effective level, the distance between two states undergoing the same process may increase. This increase is, however, upper bounded by the distance between the underlying states (red dashed line). (a) The underlying interaction is dictated by the Hamiltonian $H=\hslash J{\sigma }_z\otimes {\sigma }_z$. We see that the distance oscillates, increasing for some time intervals. Nevertheless, in this case, we always have $\left|\right|{\rho }_0-{\rho }_0^{\prime }{\left|\right|}_1\ge \left|\right|{\rho }_t-{\rho }_t^{\prime }{\left|\right|}_1$. (b) The underlying evolution is dictated by the Hamiltonian $H=\hslash J{\sigma }_z\otimes {\sigma }_z+\hslash \mathrm{\Omega }({\sigma }_x\otimes \mathbb{1}+\mathbb{1}\otimes {\sigma }_x)$, and we set $J=1\mathrm{rad}/\mathrm{s}$ and $\mathrm{\Omega }=3\mathrm{rad}/\mathrm{s}$. In this case, we see that $\left|\right|{\rho }_t-{\rho }_t^{\prime }{\left|\right|}_1$ can even go beyond $\left|\right|{\rho }_0-{\rho }_0^{\prime }{\left|\right|}_1$.