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Rotation covariance and operator-weight sectors

Statement

Let $J_{z} = diag (m_{1}, \dots, m_{d})$ have real diagonal entries, and let $H = H^{†}$ be diagonal in the same basis. Let $L_{1}, \dots, L_{r}$ be matrices satisfying $[J_{z}, L_{a}] = q_{a} L_{a}$ , where each $q_{a}$ is real. Define the finite GKSL generator $L (X) = - i [H, X] + \sum_{a} (L_{a} X L_{a}^{†} - {L_{a}^{†} L_{a}, X} / 2)$ and $R_{θ} = e^{- i θ J_{z}}$ for real $θ$ . Then $R_{θ} L_{a} R_{θ}^{†} = e^{- i q_{a} θ} L_{a}$ , and $L$ commutes with the map $X \mapsto R_{θ} X R_{θ}^{†}$ .

For each real difference $ω$ , the operator-weight sector $W_{ω} = span {E_{j k} : m_{j} - m_{k} = ω}$ is invariant under $L$ . Here $E_{j k}$ is the matrix unit with a single $1$ in row $j$ , column $k$ . In particular, the same conclusion applies to the standard finite-spin diagonal matrix $J_{z}$ with magnetic labels $m = - s, - s + 1, \dots, s$ . If the weight convention is instead $[J_{z}, L_{a}] = - q_{a} L_{a}$ , the jump phase is $e^{+ i q_{a} θ}$ .

Statement references

Finite GKSL evolution and the QEC channel interface

Liouville matrix and Choi reshuffling conventions

Finite spin matrices

Finite-dimensional matrix exponential

Facts & Assumptions

[F1]

The finite GKSL interface defines the displayed generator for Hermitian $H$ and a finite family of matrices, with $[A, B] = A B - B A$ , ${A, B} = A B + B A$ and $†$ denoting conjugate transpose. Finite GKSL evolution and the QEC channel interface

[F2]

In an ordered finite basis, matrix units form a basis for the matrix space, so every matrix has a unique expansion in them. Liouville matrix and Choi reshuffling conventions

[F3]

For $s \in \frac{1}{2} N$ , the standard finite-spin matrix $J_{z}$ is diagonal with real entries $m = - s, - s + 1, \dots, s$ . Finite spin matrices

[F4]

For a finite complex matrix $A$ and complex scalar $z$ , the matrix exponential is $e^{z A} = \sum_{n = 0}^{\infty} z^{n} A^{n} / n!$ , with $A^{0} = I$ and absolute convergence in operator norm. Finite-dimensional matrix exponential

[F5]

If a finite complex matrix $A$ satisfies $A^{†} = - A$ , then $e^{t A}$ is unitary for every real $t$ . A skew-Hermitian matrix exponential is unitary

Proof

Given: The matrices, real weights and real parameter in the statement.

Goal: Prove the jump phases, covariance and invariance of each stated sector.

1.1givenF1F4F5algebra

By [F4], the exponential series for the diagonal matrix gives $R_{θ} = diag (e^{- i θ m_{1}}, \dots, e^{- i θ m_{d}})$ . Since $(- i J_{z})^{†} = i J_{z} = - (- i J_{z})$ , [F5] with $A = - i J_{z}$ and $t = θ$ gives $R_{θ}^{†} = R_{θ}^{- 1}$ . The $(j, k)$ entry of $[J_{z}, L_{a}] = q_{a} L_{a}$ is $(m_{j} - m_{k} - q_{a}) (L_{a})_{j k} = 0$ . Consequently every nonzero entry of $L_{a}$ has $m_{j} - m_{k} = q_{a}$ , and entrywise multiplication gives $R_{θ} L_{a} R_{θ}^{†} = e^{- i q_{a} θ} L_{a}$ . Replacing $q_{a}$ by $- q_{a}$ gives the stated alternative phase.

2.1givenF1step 1.1algebra

Write $C_{θ} (X) = R_{θ} X R_{θ}^{†}$ . Because $H$ and $R_{θ}$ are diagonal, $C_{θ} (H) = H$ . Conjugation by the unitary $R_{θ}$ preserves products and adjoints. The phases from step 1.1 therefore cancel in both $C_{θ} (L_{a}) C_{θ} (X) C_{θ} (L_{a})^{†}$ and $C_{θ} (L_{a})^{†} C_{θ} (L_{a})$ . Applying this to each term of the generator yields $C_{θ} (L (X)) = L (C_{θ} (X))$ .

3.1givenF2F3step 1.1step 2.1algebra∎

For $E_{j k}$ , direct multiplication gives $C_{θ} (E_{j k}) = e^{- i θ (m_{j} - m_{k})} E_{j k}$ . If $X \in W_{ω}$ , linearity and step 2.1 imply $C_{θ} (L (X)) = e^{- i θ ω} L (X)$ for every real $θ$ . Expand $L (X) = \sum_{j, k} c_{j k} E_{j k}$ using [F2]. Comparing the $(j, k)$ entries for all $θ$ shows $c_{j k} = 0$ whenever $m_{j} - m_{k} \neq ω$ , since the two real-frequency exponentials then differ for some $θ$ . Hence $L (X) \in W_{ω}$ . The finite-spin instance follows from [F3].

Depends on

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