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Equation 3 · Part 6 · How Fast Can a Horizon Learn Which Path You Took?

=

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)|.
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

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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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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