Partial trace ( $Tr_{B}$ )

The partial trace over

B

is the linear operation characterized by

Tr [M_{A} Tr_{B} (X_{A B})] = Tr [(M_{A} \otimes I_{B}) X_{A B}] .

Applied to a density operator, it produces a reduced density matrix.

Table of contents
- Partial trace positivity from product vectors Partial trace
- Haar twirling conditional expectation Partial trace

Partial trace positivity from product vectors ( $⟨ u, Tr_{B} (ρ) u ⟩ = \sum_{b} ⟨ u \otimes e_{b}, ρ (u \otimes e_{b})⟩$ )

The partial trace of a positive operator is positive by this sum of nonnegative expectations in an orthonormal basis of the traced factor. Its trace equals the original trace, using the product orthonormal basis. Basis independence follows from

Tr [(Tr_{B} ρ) A] = Tr [ρ (A \otimes I)]

, which uniquely determines the reduced operator. Thus reductions of density matrices remain density matrices. For a positive trace-class operator the same argument uses convergent nonnegative basis sums.

Haar twirling conditional expectation ( $E_{A} (O) = d_{B}^{- 1} Tr_{B} (O) \otimes I_{B}$ )

For an operator on a tensor-product Hilbert space, averaging unitary conjugations on subsystem

B

with normalized Haar measure gives

E_{A} (O) = (Tr_{B} O / d_{B}) \otimes I_{B}

. It fixes precisely the operators acting trivially on

B

and is contractive in operator norm. Writing

O - U O U^{†} = [O, U] U^{†}

converts localization error into a commutator bound.

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The concept of the partial trace arises in the context of quantum mechanics and quantum information theory, particularly when dealing with composite quantum systems. It is a mathematical operation used to obtain the reduced density matrix of a subsystem from the density matrix of a larger composite system. Let's break it down further: ### Quantum States and Density Matrices In quantum mechanics, a system can be described by a density matrix, which encodes the statistical state of the system.

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