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

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\nu(T)=\langle h_0(T)|h_1(T)\rangle, \qquad V(T)=|\nu(T)|.

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Inputs and operationslangle h_0(T)|h_1(T)rangle, qquad V(T)=|nu(T)|

Result or conditionnu(T)

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\nu

Symbol nu

nu is part of the quantity the equation computes from the expression on the right.

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T

Symbol T

T is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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h_0

Symbol h_0

$h_0$ is an input to the expression that computes the quantity on the left.

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h_1

Symbol h_1

$h_1$ is an input to the expression that computes the quantity on the left.

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V

Symbol V

the remaining fringe visibility.

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=

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

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

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What the article says around this equation

Tracing out the field multiplies the off-diagonal element of the path density matrix by the conditional-state overlap $\nu(T)=\langle h_0(T)|h_1(T)\rangle, \qquad V(T)=|\nu(T)|$ . 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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