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Equation 3 · Part 4 · How Fast Can a Horizon Learn Which Path You Took?

Symbol h_1

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)|.
h1h_1

What this part means

h1h_1 is an input to the expression that computes the quantity on the left.

Its job in the formula

h1h_1 is an input to the expression that computes the quantity on the left.

The passage around this formula

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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