Fundamental speed limits on entanglement dynamics of bipartite quantum systems

TL;DR

The paper addresses fundamental limits on how fast entanglement can be generated or degraded in bipartite quantum systems under both unitary and open dynamics. It develops speed limits by tracking the time evolution of the relative entropy of entanglement and the trace-distance entanglement, using the closest separable state and its dynamics via the closest separable map constructed from the Choi–Jamiołkowski isomorphism. The main contributions are explicit bounds on entanglement rate that separate the intrinsic evolution speed from a time-dependent CSS contribution, along with integrated minimal-time bounds for entanglement changes, and demonstrations of tightness in representative unitary and dephasing scenarios. These results provide fundamental guidance for optimizing entanglement generation and preservation in quantum technologies and can be extended to other quantum resources.

Abstract

The speed limits on entanglement are defined as the maximal rate at which entanglement can be generated or degraded in a physical process. We derive the speed limits on entanglement, using the relative entropy of entanglement and trace-distance entanglement, for unitary as well as for arbitrary quantum dynamics, where we assume that the dynamics of the closest separable state can be approximately described by the closest separable dynamics of the actual dynamics of the system. For unitary dynamics of isolated bipartite systems which are described by pure states, the rate of entanglement production is bounded by the product of fluctuations of the system's driving Hamiltonian and the surprisal operator, with an additional term reflecting the time-dependent nature of the closest separable state. Removing restrictions on the purity of the input and on the unitarity of the evolution, the two terms in the bound get suitably altered. Furthermore, we find a lower bound on the time required to generate or degrade a certain amount of entanglement by arbitrary quantum dynamics. We demonstrate the tightness of our speed limits on entanglement by considering quantum processes of practical interest.

Figures (3)

• Figure 1: Here we depict $T_{\rm ESL}$ vs $T$, as given by \ref{['equ:speeed_limit_entanglemen_unitary']}, for the unitary dynamics governed by the Hamiltonian given in Eq. \ref{['equ:hamiltonian']} with the pure input $\Psi_{0} = \sqrt{p}\ket{00}+\sqrt{1-p}\ket{11}$. We have taken $p=0$ and Hamiltonian parameters $\theta = 3.5$ and $\delta = 0.5$ (blue line), $\delta = 1.0$ (red line).
• Figure 2: Here we depict $T_{\rm ESL}$ vs $T$, as given by \ref{['equ:speeed_limit_entanglemen_unitary']}, for the unitary dynamics governed by the Hamiltonian given in Eq. \ref{['equ:hamiltonian']} with initial state $p\ketbra{\Psi^{+}}{\Psi^{+}}+(1-p)\ketbra{00}{00}.$ For $p=0,0.1$ and $0.2$ we have taken $\theta = 3.5, \delta = 0$ (blue line), $\theta = 3.5, \delta = 0.1$ (red line) and $\theta = 5.5, \delta = 0.1$ (black line), respectively.
• Figure 3: Here we depict $T_{\rm ESL}$ vs $T$, as given by \ref{['equ:speeed_limit_entanglemen']}, for the pure dephasing process. We have taken the pure input state $\Psi_{0}=\sqrt{p} \ket{00}+\sqrt{1-p}\ket{11}$, and we have taken state parameter $p=0.5,0.4$ and $0.3$ and decay rate $\gamma = 0.1$.