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Long-time RWA limitation

Example

Work on $C^{2}$ with $ℏ = 1$ . Keep the original commuting example $A = B = σ_{z}$ , $ω = 1$ , so $H (t) = (1 + 2 \cos t) σ_{z}$ and $H_{a v g} = σ_{z}$ . The exact and averaged propagators from time $0$ are

$U (t) = e^{- i (t + 2 \sin t) σ_{z}}, U_{a v g} (t) = e^{- i t σ_{z}} .$

Their norm difference is $2 ∣ \sin (\sin t) ∣$ . At $t_{n} = π / 2 + 2 π n$ it equals $2 ∣ \sin 1 ∣ > 0$ for every $n \geq 0$ . This proves persistent nondecay of the original commuting error. It alone does not disprove uniform-time smallness as the ratio $∥ B ∥ / ∣ ω ∣$ becomes small: in the commuting extension $A = σ_{z}$ , $B = β σ_{z}$ , $β \in R$ , $ω > 0$ , the exact error satisfies

$∥ U (t) - U_{a v g} (t) ∥ = 2 ∣ \sin (\frac{β}{ω} \sin ω t) ∣ \leq \frac{2 ∣ β ∣}{ω} (t \in R) .$

To prove the stronger intended general limitation, fix $ω > 0$ and define $σ_{+} = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array})$ , $σ_{-} = σ_{+}^{*}$ , $σ_{x} = σ_{+} + σ_{-}$ . For every integer $n \geq 1$ , let

$ε_{n} = ω \sqrt{\frac{1}{2 n} + \frac{1}{4 n^{2}}}, A_{n} = 0, B_{n} = ε_{n} σ_{+},$

$H_{n} (t) = e^{i ω t} B_{n} + e^{- i ω t} B_{n}^{*}, H_{a v g, n} = 0.$

These are smooth bounded self-adjoint Hamiltonians of precisely the form of Rotating-wave finite-time error. They are noncommuting in time: for suitable $t, u$ , $[H_{n} (t), H_{n} (u)] = 2 i ε_{n}^{2} \sin (ω (t - u)) σ_{z} \neq 0$ . Their oscillatory coefficient norm satisfies $∥ B_{n} ∥ / ω = ε_{n} / ω \to 0$ . Nevertheless at the growing times $τ_{n} = 2 π n / ω$ their exact propagator obeys $U_{n} (τ_{n}) = - I$ , while $U_{a v g, n} (τ_{n}) = I$ . Thus

$∥ U_{n} (τ_{n}) - U_{a v g, n} (τ_{n}) ∥ = 2 (n \geq 1) .$

This is the maximum possible norm distance between unitaries. In particular no general bound $\sup_{t \geq 0} ∥ U (t) - U_{a v g} (t) ∥ \leq f (∥ B ∥ / ω)$ with $f (r) \to 0$ as $r \to 0$ can hold for this whole class, even with $A = 0$ , fixed positive $ω$ and dimension two. This does not assert that the finite-time theorem's linear growth rate is sharp; it proves the absence of general uniform-time smallness without additional hypotheses. All assertions concern explicit mathematical models, not a laboratory error or measurement.

Verification

Given: the original commuting example, and the circular family $H_{n}$ with $n \geq 1$ , $ω > 0$ , $ε_{n}$ and $τ_{n}$ as above.

[F1] The Pauli matrices have $σ_{x}^{2} = σ_{z}^{2} = I$ and $σ_{x} σ_{z} + σ_{z} σ_{x} = 0$ , and $σ_{z} σ_{+} - σ_{+} σ_{z} = 2 σ_{+}$ ; these follow by multiplying their displayed two-by-two matrices, consistent with Finite spin matrices (e).

[F2] Direct differentiation of a displayed unitary solution with initial value $I$ verifies its propagator equation; uniqueness follows from the same $D^{*} D$ calculation used in the preceding example. No generic ordered Dyson formula, autonomous AC group theorem, or unbounded-domain assertion is needed.

1.1F1F2algebra

The original displayed diagonal exponentials differentiate to $i U^{'} = (1 + 2 \cos t) σ_{z} U$ , $i U_{a v g}^{'} = σ_{z} U_{a v g}$ , with initial value $I$ . They are therefore the required propagators by [F2]. Left multiplication of their difference by $e^{i t σ_{z}}$ leaves its norm unchanged and gives diagonal entries $e^{\mp 2 i \sin t} - 1$ , whose common modulus is $2 ∣ \sin (\sin t) ∣$ . Substitution at $π / 2 + 2 π n$ proves the original positive nondecaying subsequence.

1.2F1F2algebra

Define $W (t) = \exp (i ω t σ_{z} / 2)$ and $G_{n} = ε_{n} σ_{x} + (ω / 2) σ_{z}$ . The diagonal formula for $W$ gives $W σ_{+} W^{*} = e^{i ω t} σ_{+}$ and $W σ_{-} W^{*} = e^{- i ω t} σ_{-}$ . Hence $H_{n} (t) = ε_{n} W (t) σ_{x} W (t)^{*}$ . Put $V_{n} (t) = W (t) e^{- i t G_{n}}$ . Both factors are unitary, $V_{n} (0) = I$ , and differentiating gives $i V_{n}^{'} = - (ω / 2) σ_{z} V_{n} + W G_{n} e^{- i t G_{n}} = ε_{n} W σ_{x} W^{*} V_{n} = H_{n} V_{n}$ , because $W$ commutes with $σ_{z}$ . By [F2], $U_{n} = V_{n}$ . The sign of $W$ and the positive $ω σ_{z} / 2$ term in $G_{n}$ are essential in this cancellation.

2.1step 1.1algebra

For the commuting extension with real $β$ , integrate its diagonal scalar entries to obtain $U (t) = \exp [- i (t + 2 β \sin (ω t) / ω) σ_{z}]$ . The same diagonal norm calculation gives $2 ∣ \sin ((β / ω) \sin ω t) ∣ \leq 2 ∣ β ∣ / ω$ . Thus the original commuting nondecay example admits a time-uniform small-ratio estimate; the stronger failure requires the separate noncommuting construction.

2.2F1step 1.2algebra

By [F1], $G_{n}^{2} = (ε_{n}^{2} + ω^{2} / 4) I = ν_{n}^{2} I$ , where $ν_{n} = (ω / 2) (1 + 1 / n) > 0$ from the exact definition of $ε_{n}$ . Splitting the absolutely convergent matrix exponential into even and odd powers gives $e^{- i t G_{n}} = \cos (ν_{n} t) I - i \sin (ν_{n} t) G_{n} / ν_{n}$ . At $τ_{n} = 2 π n / ω$ , the angles are $ω τ_{n} / 2 = π n$ and $ν_{n} τ_{n} = π (n + 1)$ . Therefore $W (τ_{n}) = (- 1)^{n} I$ and $e^{- i τ_{n} G_{n}} = (- 1)^{n + 1} I$ , so $U_{n} (τ_{n}) = - I$ . Since $A_{n} = 0$ , $U_{a v g, n} (t) = I$ for every $t$ , and the norm difference at $τ_{n}$ is $∥ - 2 I ∥ = 2$ .

3.1step 2.2algebra∎

The matrix $σ_{+}$ sends $(x, y)$ to $(y, 0)$ , so its operator norm is exactly $1$ ; hence $∥ B_{n} ∥ / ω = \sqrt{1 / (2 n) + 1 / (4 n^{2})} \to 0$ . Also $τ_{n} \to \infty$ for fixed $ω$ . For noncommutativity, the products $σ_{+} σ_{-} = diag (1, 0)$ and $σ_{-} σ_{+} = diag (0, 1)$ give $[H_{n} (t), H_{n} (u)] = ε_{n}^{2} (e^{i ω (t - u)} - e^{- i ω (t - u)}) σ_{z} = 2 i ε_{n}^{2} \sin (ω (t - u)) σ_{z}$ . The proposed general function $f (r) \to 0$ would give $2 \leq f (∥ B_{n} ∥ / ω) \to 0$ , a contradiction. This proves the full intended no-general-uniform-smallness conclusion while retaining the distinct commuting nondecay example.

Long-time RWA limitation

Example

Verification

Depends on

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