Equation 25 · The Entry a Relabeling Cannot Write
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Symbol ρ
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is a real number in units of energy for every fine state : how much the coarse description, evaluated through pulled back along , disagrees with the fine description’s own , for that specific state. Because expectation values determine a Hermitian operator uniquely — two Hermitian operators with the same trace against every density matrix are the same operator — the strongest statement follows at once: = 0 for every fine state if and only if = 0 as an operator identity, that is, if and only if = exactly. Energy is preserved for every possible input under that one algebraic condition, never as a statistical…
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is a real number in units of energy for every fine state : how much the coarse description, evaluated through pulled back along , disagrees with the fine description’s own , for that specific state. Because expectation values determine a Hermitian operator uniquely — two Hermitian operators with the same trace against every density matrix are the same operator — the strongest statement follows at once: = 0 for every fine state if and only if = 0 as an operator identity, that is, if and only if = exactly. Energy is preserved for every possible input under that one algebraic condition, never as a statistical tendency and never approximately.
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