Equation 10 · The Entry a Relabeling Cannot Write
What does this equation mean?
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol ρ
ρ is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Symbol B
B is an input to the expression that computes the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →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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What the article says around this equation
A coarse description reports less. It is built from a completely positive, trace-preserving map , a quantum channel sending fine states to coarse states, together with a coarse Hilbert space and a Hermitian coarse Hamiltonian on it. The simplest case is the partial trace over the bath, = [] with = ; nothing below depends on that particular choice, and a passive relabeling built from no discarding at all, treated next, is a channel too.
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