Manuscript En09-A@1 · En09-A.tex · proposition · eafec9de

Proposition: Commensurate-spectrum realization

An exact finite stationary realization · lines 685–763 · proof included or linked

En09-A@1#prop:spectral-realization, En09-A@1#eq:clock-spectrum, En09-A@1#eq:spectral-calibration, En09-A@1#eq:fourier-clock, En09-A@1#eq:stationary-encoding, En09-A@1#eq:history-encoding, En09-A@1#eq:spectral-recorded-channel

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Let $d = 2^{m}$ , $ω > 0$ , and let ${Π_{a}}_{a = 0}^{d - 1}$ be mutually orthogonal projections on $H_{S}$ with $\sum_{a} Π_{a} = I_{S}$ ; zero projections are allowed. Set

H_{S} = ℏ ω a = 0 \sum^{d - 1} a Π_{a} .

Choose a clock energy basis ${∣ e_{a} ⟩}_{a = 0}^{d - 1}$ and define

H_{C} Δ ∣ τ_{k} ⟩ W ∣ ϕ ⟩ = - ℏ ω a = 0 \sum^{d - 1} a ∣ e_{a} ⟩ ⟨ e_{a} ∣, = \frac{2 π}{d ω}, t_{k} = k Δ, = \frac{1}{d} a = 0 \sum^{d - 1} e^{2 π iak / d} ∣ e_{a} ⟩, = a = 0 \sum^{d - 1} ∣ e_{a} ⟩ \otimes Π_{a} ∣ ϕ ⟩ .

Then $W : H_{S} \to H_{C} \otimes H_{S}$ is an isometry, $K W = 0$ , and the clock basis is covariant at the readings $t_{k}$ . Moreover,

W = \frac{1}{d} k = 0 \sum^{d - 1} ∣ τ_{k} ⟩ \otimes U_{S} (t_{k}),

and, for every operator $X$ ,

M_{C} (W X W^{†}) = \frac{1}{d} k = 0 \sum^{d - 1} ∣ k ⟩ ⟨ k ∣_{R} \otimes U_{S} (t_{k}) X U_{S} (t_{k})^{†} .

All identities hold with every reference.

Proof. Orthogonality and completeness of the projections imply

W^{†} W = a \sum Π_{a} = I_{S} .

On the term $∣ e_{a} ⟩ \otimes Π_{a} ∣ ϕ ⟩$ , the clock and system energies sum to zero, so $K W = 0$ .

The finite Fourier identity

\frac{1}{d} a = 0 \sum^{d - 1} e^{2 π ia (k - ℓ) / d} = δ_{k ℓ}

proves orthonormality of ${∣ τ_{k} ⟩}$ . The sign in (eq:clock-spectrum) gives

U_{C} (t_{k}) ∣ τ_{0} ⟩ = ∣ τ_{k} ⟩,

hence covariance.

Expanding the right side of (eq:history-encoding), using

U_{S} (t_{k}) = a \sum e^{- 2 π iak / d} Π_{a},

and summing over $k$ , leaves precisely (eq:stationary-encoding). Applying $D_{k}$ to (eq:history-encoding) gives $D_{k} W = d^{- 1/2} U_{S} (t_{k})$ . Substitution in (eq:readout) proves (eq:spectral-recorded-channel). Tensoring the operator identities with an identity proves the reference versions.

Displayed formulas in this passage

En09-A@1#eq:clock-spectrum · line 695
$H_{C} Δ ∣ τ_{k} ⟩ W ∣ ϕ ⟩ = - ℏ ω a = 0 \sum^{d - 1} a ∣ e_{a} ⟩ ⟨ e_{a} ∣, = \frac{2 π}{d ω}, t_{k} = k Δ, = \frac{1}{d} a = 0 \sum^{d - 1} e^{2 π iak / d} ∣ e_{a} ⟩, = a = 0 \sum^{d - 1} ∣ e_{a} ⟩ \otimes Π_{a} ∣ ϕ ⟩ .$
En09-A@1#eq:spectral-calibration · line 695
$H_{C} Δ ∣ τ_{k} ⟩ W ∣ ϕ ⟩ = - ℏ ω a = 0 \sum^{d - 1} a ∣ e_{a} ⟩ ⟨ e_{a} ∣, = \frac{2 π}{d ω}, t_{k} = k Δ, = \frac{1}{d} a = 0 \sum^{d - 1} e^{2 π iak / d} ∣ e_{a} ⟩, = a = 0 \sum^{d - 1} ∣ e_{a} ⟩ \otimes Π_{a} ∣ ϕ ⟩ .$
En09-A@1#eq:fourier-clock · line 695
$H_{C} Δ ∣ τ_{k} ⟩ W ∣ ϕ ⟩ = - ℏ ω a = 0 \sum^{d - 1} a ∣ e_{a} ⟩ ⟨ e_{a} ∣, = \frac{2 π}{d ω}, t_{k} = k Δ, = \frac{1}{d} a = 0 \sum^{d - 1} e^{2 π iak / d} ∣ e_{a} ⟩, = a = 0 \sum^{d - 1} ∣ e_{a} ⟩ \otimes Π_{a} ∣ ϕ ⟩ .$
En09-A@1#eq:stationary-encoding · line 695
$H_{C} Δ ∣ τ_{k} ⟩ W ∣ ϕ ⟩ = - ℏ ω a = 0 \sum^{d - 1} a ∣ e_{a} ⟩ ⟨ e_{a} ∣, = \frac{2 π}{d ω}, t_{k} = k Δ, = \frac{1}{d} a = 0 \sum^{d - 1} e^{2 π iak / d} ∣ e_{a} ⟩, = a = 0 \sum^{d - 1} ∣ e_{a} ⟩ \otimes Π_{a} ∣ ϕ ⟩ .$
En09-A@1#eq:history-encoding · line 714
$W = \frac{1}{d} k = 0 \sum^{d - 1} ∣ τ_{k} ⟩ \otimes U_{S} (t_{k}),$
En09-A@1#eq:spectral-recorded-channel · line 722
$M_{C} (W X W^{†}) = \frac{1}{d} k = 0 \sum^{d - 1} ∣ k ⟩ ⟨ k ∣_{R} \otimes U_{S} (t_{k}) X U_{S} (t_{k})^{†} .$
unlabeled · line 691
$H_{S} = ℏ ω a = 0 \sum^{d - 1} a Π_{a} .$
unlabeled · line 734
$W^{†} W = a \sum Π_{a} = I_{S} .$
unlabeled · line 741
$\frac{1}{d} a = 0 \sum^{d - 1} e^{2 π ia (k - ℓ) / d} = δ_{k ℓ}$
unlabeled · line 748
$U_{C} (t_{k}) ∣ τ_{0} ⟩ = ∣ τ_{k} ⟩,$
unlabeled · line 754
$U_{S} (t_{k}) = a \sum e^{- 2 π iak / d} Π_{a},$

Proposition: Commensurate-spectrum realization

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