xxxi Symbols Tr(O) trace of the operator direct product ⊕ direct sum for all. 3 there is at least one...such that a∈X the element a pertains to the set X XCY X is a proper subset of Y a is sufficient condition of b inclusive disjunction(OR) conjunction(AND) a→b a maps to b tends to. 10),11) arbitrary basis for a two-level system.qubits 11),12）,13),140 set of eigenstates of a path observable 10)=10,0,0) vacuum state arbitrary basis for a two-level system eigenstate of the spin observable (in the z-direction) 1) state of horizontal polarization state of vertical polarization state of 135 polarization Ie.)e living-and dead-cat states,respectively 小-= commutator anti comm ator Poisson brackets a= partial derivatives = with j=x.y.xxxi Symbols Tr(Oˆ) trace of the operator Oˆ ⊗ direct product ⊕ direct sum ∀ for all . . . ∃ there is at least one . . . such that a ∈ X the element a pertains to the set X X ⊂ Y X is a proper subset of Y a ⇒ b a is sufficient condition of b ∨ inclusive disjunction (OR) ∧ conjunction (AND) a → b a maps to b → tends to . . . |0, |1 arbitrary basis for a two-level system, qubits |1, |2, |3, |4 set of eigenstates of a path observable |0 = |0, 0, 0 vacuum state |↑, |↓ arbitrary basis for a two-level system, eigenstates of the spin observable (in the z-direction) |↔ state of horizontal polarization | state of vertical polarization |! state of 45◦ polarization |" state of 135◦ polarization |* c , |+ c living- and dead-cat states, respectively [·, ··]− = [·, ··] commutator [·, ··]+ anticommutator {·, ··} Poisson brackets ∂t = ∂ ∂t partial derivatives ∂ j = ∂ ∂ j , with j = x, y,z