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ν(T)=⟨h0(T)∣h1(T)⟩,V(T)=∣ν(T)∣\nu(T)=\langle h_0(T)|h_1(T)\rangle, \qquad V(T)=|\nu(T)|

Why this formula appears here

Tracing out the field multiplies the off-diagonal element of the path density matrix by the conditional-state overlap ν(T)=⟨h0(T)∣h1(T)⟩,V(T)=∣ν(T)∣\nu(T)=\langle h_0(T)|h_1(T)\rangle, \qquad V(T)=|\nu(T)|. Under balanced arms and otherwise ideal readout, V is the remaining fringe visibility. That statement belongs to ordinary complementarity, not black-hole mysticism. Englert derived a quantitative visibility–which-way bound for a two-path interferometer, and atom-interferometer experiments later measured visibility and distinguishability independently in agreement with the relation [ 11 , 12 ] .

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ν(T)=⟨h0(T)∣h1(T)⟩,V(T)=∣ν(T)∣.\nu(T)=\langle h_0(T)|h_1(T)\rangle, \qquad V(T)=|\nu(T)|.

Equation 3 · Evolutionary Physics

How Fast Can a Horizon Learn Which Path You Took?

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Tracing out the field multiplies the off-diagonal element of the path density matrix by the conditional-state overlap ν(T)=⟨h0(T)∣h1(T)⟩,V(T)=∣ν(T)∣\nu(T)=\langle h_0(T)|h_1(T)\rangle, \qquad V(T)=|\nu(T)|. Under balanced arms and otherwise ideal readout, V is the remaining fringe visibility. That statement belongs to ordinary complementarity, not black-hole mysticism. Englert derived a quantitative visibility–which-way bound for a two-path interferometer, and atom-interferometer experiments later measured visibility and distinguishability independently in agreement with the relation [ 11 , 12 ] .

Meanings in this article

  • VV: the remaining fringe visibility.
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