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Equation 62 · The Entry a Relabeling Cannot Write

What does this equation mean?

ΔEΛ(ρ)=Tr[ρ(IS⊗HB+HSB)]\Delta E_\Lambda(\rho) = \mathrm{Tr}[\rho(I_S \otimes H_B + H_{SB})]

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Inputs and operationsTr[ρ(I_S otimes H_B + H_SB)]
Result or conditionΔ E_Lambda(ρ)
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

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.

Read it piece by piece

ΔEΛ\Delta E_\Lambda

Symbol Δ E_Lambda

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

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ρ\rho

Symbol ρ

ρ 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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ISI_S

Symbol I_S

ISI_S is one of the signed contributions combined to compute the quantity on the left.

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HBH_B

Symbol H_B

HBH_B is one of the signed contributions combined to compute the quantity on the left.

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HSBH_{SB}

Symbol H_SB

HSH_SB is one of the signed contributions combined to compute the quantity on the left.

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=

=

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

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addition

addition

Add the term after the plus sign to the term or group before it.

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change

change

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

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subscript

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

an exact operator identity, not an approximation valid in some limit: the ledger for a system-only description of a system-plus-bath is precisely the bath’s own energy plus the coupling energy between the two, nothing more and nothing less. Δ\Delta EΛ(ρ)E_\Lambda(\rho) = Tr\mathrm{Tr}[ρ(IS⊗HB+HSB)\rho(I_S \otimes H_B + H_{SB})] for any joint state ρ\rho .

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Sources cited in the article section

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