We present a study of the evolution of entanglement entropy of matter and geometry in quantum cosmology. For a variety of initial quantum states of the Universe, and with evolution defined with respect to a relational time, we show numerically that (i) entanglement entropy increases rapidly at very early times, and subsequently saturates to a constant non-zero value, and (ii) that the saturation value of this entropy is a linear function of the energy associated to the quantum state: $S_{\text{ent}}^\psi = \gamma \langle \hat{H} \rangle_\psi$. These results suggest a remnant of quantum entanglement in the macroscopic Universe from the era of the Big Bang, independent of the initial state, and a "First Law" associated with matter-gravity entanglement entropy in quantum gravity.

中文翻译：

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量子宇宙学中的物质几何纠缠

我们对量子宇宙学中物质和几何的纠缠熵的演化进行了研究。对于宇宙的各种初始量子态，以及相对于相关时间定义的演化，我们从数值上表明（i）纠缠熵在很早的时候迅速增加，随后饱和到一个恒定的非零值，和(ii) 该熵的饱和值是与量子态相关的能量的线性函数：$S_{\text{ent}}^\psi = \gamma \langle \hat{H} \rangle_\psi$ . 这些结果表明，从大爆炸时代开始，宏观宇宙中存在量子纠缠的残余，独立于初始状态，以及与量子引力中的物质-重力纠缠熵相关的“第一定律”。