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Phys.Rev.A 105.052210(2022) According to von Neumann, the global Hamiltonian of whole universe must be Hermitian in order to keep the eigenvalues real and to construct a self-consistent quantum theory. In addition to the open system approach by introducing environmental degrees of freedom to a small system, a global Hermitian Hamiltonian can also be generated through the dilation from a small Hilbert space. For example, a local non-Hermitian๐๐ฃ -๐ผ๐๐ถ๐ถ๐ฎ๐ฝ๐ป๐ฒ๐ฌ ๐ผ๐๐ผ๐ฝ๐ฎ๐ถ ๐ฌ๐ช๐ท ๐ซ๐ฎ ๐ผ๐ฒ๐ถ๐พ๐ต๐ช๐ฝ๐ฎ๐ญ ๐๐ฒ๐ฝ๐ฑ ๐ช ๐ฐ๐ต๐ธ๐ซ๐ช๐ต ๐๐ฎ๐ป๐ถ๐ฒ๐ฝ๐ฒ๐ช๐ท ๐ธ๐ท๐ฎ ๐ซ๐ ๐ฝ๐ฑ๐ฎ ๐๐ช๐ฒ๐ถ๐ช๐ป๐ด ๐ญ๐ฒ๐ต๐ช๐ฝ๐ฒ๐ธ๐ท. ๐ฆ๐ฑ๐ฎ๐ท ๐ผ๐ฑ๐ช๐ป๐ฎ๐ญ ๐ซ๐ ๐๐ต๐ฒ๐ฌ๐ฎ ๐ช๐ท๐ญ ๐๐ธ๐ซ, ๐ฝ๐ฑ๐ฎ ๐ฒ๐ท๐ฝ๐ฎ๐ป๐ท๐ช๐ต ๐ท๐ธ๐ท๐ต๐ธ๐ฌ๐ช๐ต๐ฒ๐ฝ๐ ๐ฒ๐ท ๐ผ๐พ๐ฌ๐ฑ ๐ญ๐ฒ๐ต๐ช๐ฝ๐ฎ๐ญ ๐๐ฎ๐ป๐ถ๐ฒ๐ฝ๐ฒ๐ช๐ท ๐ผ๐๐ผ๐ฝ๐ฎ๐ถ๐ผ ๐ฒ๐ผ ๐ป๐ฎ๐ฟ๐ฎ๐ช๐ต๐ฎ๐ญ ๐ป๐ฎ๐ฌ๐ฎ๐ท๐ฝ๐ต๐, ๐ซ๐พ๐ฝ ๐ธ๐ท๐ต๐ ๐๐ฒ๐ฝ๐ฑ ๐ช ๐ฝ๐๐ธ-๐ฏ๐ธ๐ต๐ญ ๐ผ๐ฝ๐ป๐พ๐ฌ๐ฝ๐พ๐ป๐ฎ. ๐๐ท ๐ฝ๐ฑ๐ฒ๐ผ ๐น๐ช๐น๐ฎ๐ป, ๐๐ฎ ๐ฎ๐๐ฝ๐ฎ๐ท๐ญ ๐ผ๐พ๐ฌ๐ฑ ๐ช ๐ญ๐ฒ๐ผ๐ฌ๐พ๐ผ๐ผ๐ฒ๐ธ๐ท ๐ฝ๐ธ ๐ฝ๐ฑ๐ฎ ๐ฐ๐ฎ๐ท๐ฎ๐ป๐ช๐ต๐ฒ๐๐ฎ๐ญ ๐ฌ๐ช๐ผ๐ฎ ๐๐ฑ๐ฎ๐ท ๐ฝ๐ฑ๐ฎ ๐ฝ๐๐ธ-๐ฏ๐ธ๐ต๐ญ ๐ผ๐ฝ๐ป๐พ๐ฌ๐ฝ๐พ๐ป๐ฎ ๐ซ๐ป๐ฎ๐ช๐ด๐ผ. ๐ฃ๐ฑ๐ฎ ๐ฒ๐ท๐ฝ๐ฎ๐ป๐ท๐ช๐ต ๐ท๐ธ๐ท๐ต๐ธ๐ฌ๐ช๐ต๐ฒ๐ฝ๐ ๐ฒ๐ผ ๐ญ๐ฒ๐ผ๐ฌ๐พ๐ผ๐ผ๐ฎ๐ญ ๐๐ฒ๐ฝ๐ฑ ๐ญ๐ฒ๐ฏ๐ฏ๐ฎ๐ป๐ฎ๐ท๐ฝ ๐ฌ๐ธ๐ป๐ป๐ฎ๐ต๐ช๐ฝ๐ฒ๐ธ๐ท ๐น๐ฒ๐ฌ๐ฝ๐พ๐ป๐ฎ๐ผ ๐ช๐ท๐ญ ๐ฝ๐ฑ๐ฎ ๐ฌ๐ธ๐ป๐ป๐ฎ๐ผ๐น๐ธ๐ท๐ญ๐ฒ๐ท๐ฐ ๐ฌ๐ธ๐ป๐ป๐ฎ๐ต๐ช๐ฝ๐ฒ๐ธ๐ท ๐ซ๐ธ๐พ๐ท๐ญ๐ผ. ๐๐พ๐ป ๐ป๐ฎ๐ผ๐พ๐ต๐ฝ๐ผ ๐น๐ป๐ธ๐ฟ๐ฒ๐ญ๐ฎ ๐ช ๐ญ๐ฎ๐ฟ๐ฒ๐ฌ๐ฎ-๐ฒ๐ท๐ญ๐ฎ๐น๐ฎ๐ท๐ญ๐ฎ๐ท๐ฝ ๐ฝ๐ฎ๐ผ๐ฝ ๐ธ๐ท ๐ฝ๐ฑ๐ฎ ๐ป๐ฎ๐ต๐ฒ๐ช๐ซ๐ฒ๐ต๐ฒ๐ฝ๐ ๐ธ๐ฏ ๐ฝ๐ฑ๐ฎ ๐ผ๐ฒ๐ถ๐พ๐ต๐ช๐ฝ๐ฒ๐ธ๐ท ๐ฒ๐ท ๐ฝ๐ฑ๐ฎ ๐ฐ๐ต๐ธ๐ซ๐ช๐ต ๐๐ฎ๐ป๐ถ๐ฒ๐ฝ๐ฒ๐ฌ๐ฒ๐ฝ๐.</๐น>