Since the ˆ P α E E depend only on positions in E , while the ˆ P α S S depend only on positions in S , ˆ P α S S ˆ P α E E = ˆ P α E E ˆ P α S S , for all α$_{S}$ and α$_{E}$ . Therefore, it is possible to conceive in general of joint eigenstates of the ˆ P α E E and ˆ P α S S with uncorrelated outcomes such that and Further, we can choose these states so that the | s $^{α}$, e$^{α}$〉 0 with correlated outcomes are just the states that appear in Eq. (34). Consider now the unitary operator This operator has no effect on the outcomes for the environment relative to the ˆ P α E , but it swaps the β and γ outcomes for the system: Note that the resulting state represents a different “universe” than what would be ex- pected from normal interaction based on the initial state | Ψ$_{I}$ 〉 $_{0}$. It can be effectively consid- ered to be the result of the same basic interaction, but proceeding from a different initial state So, a priori , one cannot assume that the intrinsic properties of the universe represented by the swapped state will be the same as those of the universe represented by the original state. However, suppose that ψ β S = ψ γ $_{S}$. Then we can apply a unitary “counterswapping” operator for the environment, which swaps the β and γ outcomes of the environment, but leaves the outcomes of the system unchanged. Swapping first system outcomes and then environment outcomes gives 23 ˆ U ( β ↔ γ ) S ˆ U ( β ↔ γ ) E ˆ G$_{V}$ | Ψ$_{I}$ 〉 0 = ψ β S | e $^{γ}$, s$^{γ}$〉 0 + ψ γ S | e $^{β}$, s$^{β}$〉 0 + ∑ α = β,γ ψ α S | s $^{α}$, e$^{α}$〉 0 . ˆ U ( β ↔ γ ) E ≡ ∑ ζ ( | s $^{ζ}$, e$^{β}$〉 0 0 〈 s $^{ζ}$, e$^{γ}$| + | s $^{ζ}$, e$^{γ}$〉 0 0 〈 s $^{ζ}$, e$^{β}$| + ∑ α = β,γ | s $^{ζ}$, e$^{α}$〉 0 0 〈 s $^{ζ}$, e$^{α}$| ) , | Ψ ′ I 〉 0 = ˆ G − 1 V ˆ U ( β ↔ γ ) S ˆ G$_{V}$ | Ψ$_{I}$ 〉 0 . ˆ U ( β ↔ γ ) S ˆ S$_{V}$ | Ψ$_{I}$ 〉 0 = ψ β S | s $^{γ}$, e$^{β}$〉 0 + ψ γ S | s $^{β}$, e$^{γ}$〉 0 + ∑ α = β,γ ψ α S | s $^{α}$, e$^{α}$〉 0 . ˆ U ( β ↔ γ ) S ≡ ∑ ζ ( | s $^{β}$, e$^{ζ}$〉 0 0 〈 s $^{γ}$, e$^{ζ}$| + | s $^{γ}$, e$^{ζ}$〉 0 0 〈 s $^{β}$, e$^{ζ}$| + ∑ α = β,γ | s $^{α}$, e$^{ζ}$〉 0 0 〈 s $^{α}$, e$^{ζ}$| ) . ˆ P α E | s α S , e α E 〉 0 = δ$_{αα}$$_{E}$ | s α S , e α E 〉 0 ˆ P α S | s α S , e α E 〉 0 = δ$_{αα}$$_{S}$ | s α S , e α E 〉 0