Equation 41 · The Entry a Relabeling Cannot Write

What does this equation mean?

\mathrm{Tr}[W\rho W^\dagger O_c] = \mathrm{Tr}[\rho W^\dagger O_c W]

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 W^dagger O_c W]

Result or conditionTr[Wρ W^dagger O_c]

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

W

Symbol W

the dual of conjugation by.

Understand this part →

\rho

Symbol ρ

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

Understand this part →

W^\dagger

Symbol W^dagger

$W^d$ agger is part of the quantity the equation computes from the expression on the right.

Understand this part →

O_c

Symbol O_c

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

Understand this part →

=

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

Understand this part →

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.

Understand this part →

How to interpret it

Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

The dual of conjugation by W is conjugation by $W^\dagger$ : from the defining pairing, $\mathrm{Tr}$ [W $\rho$ $W^\dagger$ $O_c$ ] = $\mathrm{Tr}$ [ $\rho$ $W^\dagger$ $O_c$ W] holds for every $\rho$ , so $\Lambda^\dagger(O_c)$ = $W^\dagger$ $O_c$ W . Substituting,

Read the equation in its article →

For background, read the article’s source list.

Return to The Entry a Relabeling Cannot Write

See this formula across 1 published context →

Browse the mathematical compendium →