Nakata Yoshifumi, Murao Mio
Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan.
Japan Science and Technology Agency (JST), Precursory Research for Embryonic Science and Technology (PRESTO), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan.
Entropy (Basel). 2020 Jun 19;22(6):684. doi: 10.3390/e22060684.
When a quantum pure state is drawn uniformly at random from a Hilbert space, the state is typically highly entangled. This property of a random state is known as generic entanglement of quantum states and has been long investigated from many perspectives, ranging from the black hole science to quantum information science. In this paper, we address the question of how symmetry of quantum states changes the properties of generic entanglement. More specifically, we study bipartite entanglement entropy of a quantum state that is drawn uniformly at random from an invariant subspace of a given symmetry. We first extend the well-known concentration formula to the one applicable to any subspace and then show that 1. quantum states in the subspaces associated with an axial symmetry are still highly entangled, though it is less than that of the quantum states without symmetry, 2. quantum states associated with the permutation symmetry are significantly less entangled, and 3. quantum states with translation symmetry are as entangled as the generic one. We also numerically investigate the phase-transition behavior of the distribution of generic entanglement, which indicates that the phase transition seems to still exist even when random states have symmetry.
当一个量子纯态从希尔伯特空间中随机均匀抽取时，该态通常具有高度纠缠性。随机态的这一性质被称为量子态的一般纠缠性，并且长期以来从许多角度进行了研究，从黑洞科学到量子信息科学。在本文中，我们探讨量子态的对称性如何改变一般纠缠性的性质这一问题。更具体地说，我们研究从给定对称性的不变子空间中随机均匀抽取的量子态的二分纠缠熵。我们首先将著名的集中公式扩展到适用于任何子空间的公式，然后表明：1. 与轴对称相关的子空间中的量子态仍然高度纠缠，尽管比无对称性的量子态的纠缠程度要低；2. 与置换对称性相关的量子态的纠缠程度显著更低；3. 具有平移对称性的量子态与一般量子态的纠缠程度相同。我们还通过数值研究了一般纠缠分布的相变行为，这表明即使随机态具有对称性，相变似乎仍然存在。
