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Finite Cesaro dephasing with degenerate energy blocks

Statement

Under the finite-dimensional unitary dynamics of Finite unitary dynamics and stationary Gibbs states, let $P_{E}$ be the orthogonal spectral projection of $H$ for its distinct energy $E$ . For every density matrix $ρ$ ,

$\frac{1}{T} \int_{0}^{T} U (t) ρ U (t)^{*} d t ⟶ \sum_{E} P_{E} ρ P_{E}$

in trace norm as $T \to \infty$ . The diagonal energy blocks, including all blocks within degenerate eigenspaces, remain. This is Cesaro convergence of the averaged state; pointwise convergence of $U (t) ρ U (t)^{*}$ is not asserted.

Facts & Assumptions

Given: A finite-dimensional $H$ , a Hermitian $H$ , $ℏ > 0$ , and a positive matrix $ρ$ with trace one.

[F1]

Finite unitary dynamics and stationary Gibbs states: $U (t) = e^{- i t H / ℏ}$ is unitary and has the finite-dimensional spectral dynamics of the Statement.

[F2]

Finite matrix trace-norm ideal inequalities: the finite trace norm is a norm on the matrix space, so convergence of finite sums in that norm is well defined.

Proof

Given: The finite-dimensional unitary evolution and density matrix above.

1.1givenF1algebra

Write $H = \sum_{E} E P_{E}$ by finite Hermitian spectral calculus. Then $U (t) ρ U (t)^{*} = \sum_{E, F} e^{- i (E - F) t / ℏ} P_{E} ρ P_{F}$ . There are finitely many ordered pairs $(E, F)$ , so the time average is the corresponding finite sum of averaged coefficients times fixed matrices.

2.1step 1.1F2algebra∎

If $E = F$ , the averaged scalar coefficient is $1$ . If $E \neq F$ , then $∣ T^{- 1} \int_{0}^{T} e^{- i (E - F) t / ℏ} d t ∣ = ∣ \frac{ℏ (e^{- i (E - F) T / ℏ} - 1)}{- i (E - F) T} ∣ \leq \frac{2 ℏ}{T ∣ E - F ∣} \to 0$ . Since the finite trace norm is a norm by [F2], the finite matrix sum converges in that norm to $\sum_{E} P_{E} ρ P_{E}$ . Equal-energy blocks are not removed, so degeneracies are retained.

Finite Cesaro dephasing with degenerate energy blocks

Statement

Facts & Assumptions

Proof

Depends on

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