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Trace-norm Cesàro dephasing for a discrete Gibbs spectrum

Statement

Assume the Axiom of Choice and the exact first-clause interface of Trace-class Gibbs normalization for a semibounded self-adjoint operator: a nonzero complex Hilbert space has a finite or countable complete orthonormal energy eigenbasis $(u_{j})$ for a self-adjoint semibounded $A$ , with each energy eigenspace finite-dimensional. The supplier is a current draft whose statement and complete argument have been inspected; this theorem is conditional on that interface and does not claim its independent acceptance. Let $ℏ > 0$ , define $U_{t} u_{j} = e^{- i E_{j} t / ℏ} u_{j}$ , and extend linearly and continuously. For any positive trace-class density operator $ρ$ with $Tr ρ = 1$ , set $ρ_{t} = U_{t} ρ U_{t}^{*}$ . If $P_{E}$ denotes the projection onto the full eigenspace of energy $E$ , then the trace-norm Riemann average ${\overset{ˉ}{ρ}}_{T} : = \frac{1}{T} \int_{0}^{T} ρ_{t} d t$ converges as $T \to \infty$ to $ρ_{d i a g} : = \sum_{E} P_{E} ρ P_{E}$ in trace norm. The full degenerate energy blocks are retained. Thus expectations of every bounded observable converge to their diagonal-ensemble values. This asserts Cesàro convergence, not convergence of $ρ_{t}$ at individual late times, and is restricted to the stated discrete trace-class spectrum.

Facts & Assumptions

Given: The inspected discrete eigenbasis interface, a positive trace-one trace-class density, and the unitary phase convention.

[A1]

AC is adopted; the actual draft supplier statement Trace-class Gibbs normalization for a semibounded self-adjoint operator assumes a nonzero complex Hilbert space, self-adjoint $A$ with spectral support in $[a, \infty)$ , $β > 0$ , and positive trace-class Gibbs operator $K_{β}$ . Its first clause supplies a finite or countable complete orthonormal basis of eigenvectors $A u_{j} = E_{j} u_{j}$ , finite multiplicity for each energy, and a positive finite partition trace. Its complete proof is present as draft; the dependency is used conditionally.

[A2]

$ρ$ is positive trace class with trace one; $ℏ > 0$ is fixed and the phase is $e^{- i E_{j} t / ℏ}$ .

[F1]

Trace class operator: trace-class operators have finite trace norm and form the space on which the trace and trace norm are defined.

[F2]

Trace class is a two sided Banach operator ideal: trace class is a Banach space, and $∥ A R B ∥_{1} \leq ∥ A ∥ ∥ R ∥_{1} ∥ B ∥$ for bounded $A, B$ .

[F3]

Finite-rank orthogonal compressions converge in trace norm: if finite-rank orthogonal projections converge strongly to the identity on a separable Hilbert space, their two-sided compressions of a trace-class operator converge in trace norm.

[F4]

Trace of a positive operator is the sum of its eigenvalues and Trace is absolutely convergent and basis independent: positive trace is the sum of diagonal eigenbasis entries, and the trace is basis independent.

[F5]

Cyclicity of the trace: for trace-class $S$ and bounded $B$ , $Tr (S B) = Tr (B S)$ .

[F6]

The complex exponential by its power series, $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , and The complex exponential is entire and its complex derivative is itself: the complex exponential is continuous, has derivative itself, obeys addition, and has unit modulus on imaginary arguments. The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ evaluates its real and imaginary antiderivatives componentwise on compact real intervals.

[F7]

Real and complex inner-product spaces and their induced length: the Hilbert inner product is linear in the first argument.

Proof

Given: [A1]–[A2] and [F1]–[F7].

1.1A1A2F6F7

Each multiplier $e^{- i E_{j} t / ℏ}$ has modulus one by [F6]. Thus $U_{t}$ preserves the norm of finite linear combinations of the complete orthonormal basis; it extends by density to an isometry on the Hilbert space. The similarly defined $U_{- t}$ is its inverse, so $U_{t}$ is unitary and depends continuously on $t$ on each finite-dimensional energy span. The arguments $E_{j} t / ℏ$ are dimensionless.

2.1A2F2F5F7step 1.1

By [F2], $ρ_{t} = U_{t} ρ U_{t}^{*}$ is trace class. It is positive because $⟨ ρ_{t} x, x ⟩ = ⟨ ρ U_{t}^{*} x, U_{t}^{*} x ⟩ \geq 0$ , and cyclicity [F5] gives $Tr ρ_{t} = Tr ρ = 1$ .

2.2A1A2F2F3step 1.1

Enumerate the distinct energy eigenspaces and let $P_{N}$ project onto the first $N$ of them, with $P_{N} = I$ after the list is exhausted if it is finite. Their union spans the eigenbasis, so $P_{N} \to I$ strongly. The Hilbert space is separable by the countable complete basis in [A1]. Therefore [F3] gives $∥ P_{N} ρ P_{N} - ρ ∥_{1} \to 0$ . Unitary invariance of trace norm follows from [F2] in both directions: $∥ U_{t} R U_{t}^{*} ∥_{1} \leq ∥ R ∥_{1}$ and $R = U_{t}^{*} (U_{t} R U_{t}^{*}) U_{t}$ gives the reverse inequality. Consequently $∥ U_{t} P_{N} ρ P_{N} U_{t}^{*} - ρ_{t} ∥_{1} = ∥ P_{N} ρ P_{N} - ρ ∥_{1}$ uniformly in $t$ .

3.1F2step 2.2

For fixed $N$ , the matrix of $U_{t} P_{N} ρ P_{N} U_{t}^{*}$ on its finite-dimensional range has continuous entries, hence is continuous in trace norm. Step 2.2 makes $ρ_{t}$ a uniform trace-norm limit on compact time intervals of these continuous maps; therefore $t \mapsto ρ_{t}$ is trace-norm continuous. On $[0, T]$ it is uniformly continuous. Riemann sums for a uniformly continuous Banach-valued function are Cauchy: two sums are compared on a common refinement and their difference is bounded by $T$ times the modulus of continuity at the larger mesh. Completeness in [F2] supplies the trace-class integral and its usual norm bound.

4.1A1A2F6F7step 3.1

In the eigenbasis, the $(i, j)$ matrix entry of $U_{t} P_{N} ρ P_{N} U_{t}^{*}$ is $⟨ ρ u_{j}, u_{i} ⟩ e^{i (E_{j} - E_{i}) t / ℏ}$ . Its time-average factor is $1$ if $E_{i} = E_{j}$ ; if $E_{i} \neq E_{j}$ , direct integration using [F6] gives absolute value at most $2 ℏ / (T ∣ E_{j} - E_{i} ∣)$ , which tends to zero. Since the compression is finite dimensional, entrywise convergence implies trace-norm convergence. Hence the finite compressed average tends to $\sum_{E among the first N} P_{E} ρ P_{E}$ .

5.1A1A2F1F4step 4.1

Each block $P_{E} ρ P_{E}$ is positive trace class. The sum of their traces is $\sum_{E} \sum_{u_{j} \in ran P_{E}} ⟨ ρ u_{j}, u_{j} ⟩ = Tr ρ = 1$ by [F4] and the complete basis in [A1]. Therefore their finite partial sums are Cauchy in trace norm: for a positive finite block tail its trace norm equals its trace, and the scalar trace tail tends to zero. Their limit $ρ_{d i a g}$ exists in trace norm, is positive, and has trace one.

6.1F2step 2.2step 3.1step 4.1step 5.1

Given $ϵ > 0$ , choose $N$ so the compression error in step 2.2 and the diagonal tail in step 5.1 are each below $ϵ / 3$ . The norm bound for the Riemann integral in step 3.1 makes the average of the compression error at most $ϵ / 3$ , uniformly in $T$ . For this fixed $N$ , step 4.1 makes the finite compressed average differ from its finite diagonal limit by less than $ϵ / 3$ for all sufficiently large $T$ . The triangle inequality then gives $∥ {\overset{ˉ}{ρ}}_{T} - ρ_{d i a g} ∥_{1} < ϵ$ .

7.1A2F2F4step 6.1∎

For bounded observable $B$ , the trace-class ideal and trace estimates in [F2] and [F4] give $∣ Tr (({\overset{ˉ}{ρ}}_{T} - ρ_{d i a g}) B) ∣ \leq ∥ {\overset{ˉ}{ρ}}_{T} - ρ_{d i a g} ∥_{1} ∥ B ∥$ . Step 6.1 sends the right-hand side to zero. The conclusion is only convergence of time-averaged states in the stated trace norm.

Depends on

Dependency tree · 2 levels

Sources

Daniel Arovas, Lecture Notes on Thermodynamics and Statistical Mechanics