Unveiling the Dynamical Genesis of Quantum Entanglement in Linear Systems: Internal causality breaking in the reduced subsystem evolution

TL;DR

The paper develops an exact, path-integral treatment of a simple bilinear two-mode bosonic system to reveal how entanglement can dynamically arise from purely unitary evolution of the full system. By tracing out one mode and solving the coherent-state influence functional with stationary paths, it shows that internal causality breaking in the reduced subsystem drives entanglement and the emergence of statistical probabilities, even without dissipation or thermal noise. The authors derive an exact reduced master equation with time-dependent, non-unitary terms tied to the initial quantum features (squeezing) of the environment mode and establish conditions under which the subsystem stays pure (coherent) or becomes mixed due to entanglement. They further argue that this internal causality breaking provides a fundamental mechanism for the dynamical genesis of both entanglement and quantum statistics, with broad implications for understanding measurement, thermalization, and quantum technologies, and it generalizes to more complex open quantum systems.

Abstract

Utilizing the general theory of open quantum systems to investigate the exact dynamical evolution of simple bilinear systems, we discover a mechanism of the dynamical genesis of quantum entanglement. We focus in detail on the exact quantum evolution dynamics of two photonic modes (or any two bosonic modes) coupled to each other through a linear interaction, as the simplest system of open quantum systems that we have investigated in the last two decades. Such a linear coupling alone fails to produce two-mode entanglement. We also start with an initially separable pure state of the two modes. By solving exactly the quantum equation of motion without relying on the probabilistic interpretation, we find that when the initial state of one mode is different from a coherent state (a minimum uncertainty wave packet with equal variance in the conjugate quadratures that corresponds to a well-defined classically "particle"), the causality in the time-evolution of each mode is internally violated. It also leads to the emergence of quantum entanglement between the two modes. The lack of causality is the nature of statistics. We discover that it is the internal violation of causality in the reduced (subsystem) dynamical evolution that results in the emergence of entanglement and statistic probability in quantum mechanics, even though the dynamical evolution of the whole system completely obeys the deterministic Schrödinger equation. This conclusion is valid for the quantum dynamics of more complicated composite systems. It may provide the fundamental mechanism of the dynamical genesis for both the entanglement and the statistical probability within the deterministic framework of quantum mechanics, which is the longest-standing problem that has not been fully understood since the birth of quantum mechanics.

Figures (3)

• Figure 1: (Colour online) The stationary paths of the dimensionless variable $z_1(\tau)$ for mode 1 in the path integrals of Eq. (\ref{['propagating']}), determined by the equation of motion Eq. (\ref{['eom-z1']}) or its solution Eq. (\ref{['sz1']}), as a function of time $\tau$ varying from the initial time $t_0=0$ to the end (delayed-choice) time $t=10/\omega_1$ or $15/\omega_1$, respectively. The left and right panels take arbitrarily two different set of the boundaries (i.e. the initial and final states in the path integral) given by $z_{1i}=z'^*_{1i} =1i; z^*_{1f}=z'_{1f}=2$ and $z_{1i}=1i, z'^*_{1i}=2i ; z^*_{1f}=2, z'_{1f}=1+\sqrt{3}i$, respectively. The plots (a1)-(b1) are obtained with $\omega_2=2\omega_1, \gamma=2$, which shows that different later time choices produce in advance the different stationary paths, as a direct evidence of causality breaking; the plots (a2)-(b2) take $\omega_2=\omega_1, \gamma=2$ which corresponds to the two resonant modes that manifest the similar causality breaking effect; the plots (c1)-(c2) are given by $\omega_2=2 \omega_2$ but $\gamma=0$, namely no squeezing in this case so that the stationary paths reproduce precisely the classical trajectories for different later time choices, which show that the internal causality is preserved. The other parameters $V_{12}=0.5 \omega_1$ and $\alpha_2=1$.
• Figure 2: (Colour online) The dimensionless coefficient $\delta(t)$ in Eq. (\ref{['rho_1(t)m']})-(\ref{['simplifed']}) as a function of time, which characterizes the mixing degree of the reduced density matrix $\rho_1(t)$. (a) The first few terms in the expansion of Eq. (\ref{['rho_1(t) sol']}), $\delta^n(t)$ with $n=0,1, 2$, the other parameters in the calculation are $(\omega_1/\omega_2)=2$, $V_{12}= \omega_2$, and the squeezed parameter $\gamma=2$. (b-d) $\delta(t)$ for different squeezed parameter $\gamma=2, 1, 0.5$, the different coupling strength parameters $V_{12}=(2, 1, 0.5) \omega_2$, and the different frequencies setting $(\omega_1,\omega_2)= (1,1), (2, 1), (\sqrt{2},1)$, respectively.
• Figure 3: (Colour online) Schematic plots of (a) classical wave propagation given by coherent states and (b) quantum waves of entangling two modes through the beam splitting, where $\alpha_1(t) = u(t,t_0)\alpha_1+v_0(t,t_0)\alpha_2$ as shown by Eq. (\ref{['rdmps']}), and $\rho_1(t)$ is given by Eq. (\ref{['rho_1(t) sol']}). The detailed dynamics is fully solved from the deterministic quantum mechanics.