Refinement and extension of QC-DMRG based on recent quantum information concepts

我提出的研究计划的主要目标是解释DMRG在量子化学（QC-DMRG）中的惊人成功，最终目的是克服其主要局限性。正如我计划以一种简洁的方式展示的那样，在约束费米子系统中，能量最小化和费米子交换对称性之间的强烈冲突迫使粒子和轨道/模式纠缠减少。因此，类似于适当的晶格系统，局部结构出现在QC-DMRG的基础上的人工一维晶格建立从适当的空间轨道。为了详细阐述这些一般性想法，以下目标发挥着至关重要的作用。首先，对于相同费米子的系统，粒子和模式/轨道纠缠的坚实基础应提供。基于费米子纠缠测度的合理公理，指出了一般测度的缺陷：由于波函数的反对称性而产生的人为“关联”不再对粒子纠缠有贡献。此外，模式/轨道纠缠测量在量子化学中的应用不得违反数宇称超选择规则，这反映了自然界从不混合偶数和奇数费米子数态的事实。为了简化它们对实际费米子系统的适用性，我们需要为这些测度构造边界和证明。其次，我们需要通过约化密度矩阵泛函理论（RDMFT）构造交换力的形式，将能量最小化和交换对称性之间的矛盾具体化。为此，我将证明在一个建设性的方式，费米子交换对称性表现在RDMFT在一个有效的“潜力”的形式。第三，在全面的QC-DMRG研究中，将系统地验证量子化学和谐波陷阱系统的粒子和轨道/模式纠缠在基态中比在一般态中显著减少。应验证缠结减少与交换力强度之间的预期强关系。为了克服QC-DMRG（恢复动态相关性）的主要限制，将利用更一般的张量网络ansatzes，更高的虚拟轨道应合并到超级站点和新的见解，从交换力和RDMFT将用于系统地改善QC-DMRG基础的“晶格站点”的选择。 总之，这将最终为QC-DMRG-黑盒计算铺平道路。

It is the main goal of my proposed research programme to explain the surprising success of DMRG in quantum chemistry (QC-DMRG) with the ultimate aim of overcoming its main limitations. As I plan to show in a concise way, it is the strong conflict between energy minimization and fermionic exchange symmetry in systems of confined fermions which enforces a reduction of particle and orbital/mode entanglement. Hence, similar to proper lattice systems, a local structure emerges in QC-DMRG on the underlying artificial one-dimensional lattice built from appropriate spatial orbitals. To elaborate on these general ideas, the following objectives are playing a crucial role. First, for systems of identical fermions, a solid foundation for particle and for mode/orbital entanglement shall be provided. Based on plausible axioms for fermionic entanglement measures, the flaws of common measures shall be sorted out: The artificial "correlations" due to the antisymmetry of the wave function shall no longer contribute to the particle entanglement. Furthermore, the application of mode/orbital entanglement measures in quantum chemistry must not violate the number parity superselection rule, reflecting the fact that nature never mixes even and odd fermion number states. To simplify their applicability to realistic fermionic systems, bounds and witnesses shall be constructed for those measures.Second, the conflict between energy minimization and fermionic exchange symmetry shall be concretized in the form of an exchange force constructed via reduced density matrix functional theory (RDMFT). For this, I will prove in a constructive way that the fermionic exchange symmetry manifests itself in RDMFT in the form of an effective "potential". The exchange force will then be introduced as the derivative of that potential with respect to the natural occupation numbers.Third, in a comprehensive QC-DMRG study, it shall be systematically verified for quantum chemical and harmonic trap systems that particle and orbital/mode entanglement are both significantly reduced in ground states compared to generic states. The expected strong relation between the reduction of entanglement and the strength of the exchange force shall be verified. To overcome the main limitation of QC-DMRG (recovering dynamic correlations), more general tensor network ansatzes will be exploited, higher virtual orbitals shall be merged to supersites and new insights from the exchange force and RDMFT will be used to improve systematically the choice of the "lattice sites" underlying QC-DMRG. Altogether, this shall eventually pave the way for QC-DMRG-blackbox calculations.