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Invariant finite truncation for lowering generators

Statement

Let $V = ⨁_{j = 0}^{J_{*}} V_{j}$ be a finite-dimensional complex inner product space expressed as an orthogonal direct sum, with $J_{*}$ a nonnegative integer. Let $H = H^{†}$ preserve each $V_{j}$ , and let a finite family of linear maps $L_{a}$ satisfy $L_{a} V_{j} \subseteq V_{j - 1}$ for every $j$ , where $V_{- 1} = {0}$ . Let $L (X) = - i [H, X] + \sum_{a} (L_{a} X L_{a}^{†} - {L_{a}^{†} L_{a}, X} / 2)$ and $N_{t} = e^{t L}$ for $t \geq 0$ .

For any integer $0 \leq J_{0} \leq J_{*}$ , put $W = ⨁_{j = 0}^{J_{0}} V_{j}$ and let $P$ be the orthogonal projection onto $W$ . The operators supported on $W$ , meaning those $X$ with $X = P X P$ , are preserved by $N_{t}$ . Under the natural inclusion of operators on $W$ into operators on $V$ , the restriction of $N_{t}$ is exactly $e^{t L_{W}}$ , where $L_{W}$ is the same GKSL formula on $W$ with $H_{W} = H ∣_{W}$ and $L_{a, W} = L_{a} ∣_{W}$ . If $W = {0}$ , this equality is understood on the zero operator space. The conclusion requires the stated lowering condition; a low-temperature heuristic alone does not establish it.

Facts & Assumptions

[F1]

In orthonormal coordinates on a nonzero finite-dimensional complex inner product space, a Hermitian $H$ and a finite family of jumps define the displayed finite GKSL generator and its channel as the operator-norm exponential series for $t \geq 0$ . Finite GKSL evolution and the QEC channel interface

[F2]

An inner product is positive definite and Hermitian; orthogonality means that the inner product of vectors from different summands is zero. Real and complex inner product spaces, with the inner product linear in the first argument

[F3]

The orthogonal projection onto a subspace sends a vector to its component in that subspace along its orthogonal complement. The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥

[F4]

The exponential series of a bounded operator on a real or complex Banach space converges in operator norm. The exponential series of a bounded operator

[F5]

The adjoint $T^{†}$ is characterized by $⟨ T u, v ⟩ = ⟨ u, T^{†} v ⟩$ ; every linear map between finite-dimensional real or complex inner product spaces has a unique such adjoint, including zero-dimensional spaces. The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V Every linear map between finite-dimensional inner product spaces has a unique adjoint

[F6]

A real or complex normed space admitting an ordered basis of finite length is Banach. Every finite-dimensional normed space is Banach

[F7]

Every linear map between normed spaces over the same scalar field is bounded if its domain admits an ordered basis of finite length. A linear map from a finite-dimensional normed space is bounded

[F8]

Every finite-dimensional real or complex inner product space has an orthonormal basis, with the empty basis in dimension zero. Every finite-dimensional real or complex inner product space has an orthonormal basis

Proof

Given: The finite orthogonal grading, Hermitian $H$ , lowering maps, cutoff $J_{0}$ , and nonnegative $t$ of the statement.

Goal: Prove support invariance and equality with the compressed exponential.

1.1givenF1F2F5F8choosealgebra

By [F8], choose an ordered orthonormal basis in each of the finitely many blocks $V_{j}$ . Concatenation in increasing block order gives an ordered orthonormal basis of $V$ whose initial segment is a basis of $W$ . All daggers exist by [F5]. If $A$ has entries $a_{i j}$ in orthonormal bases, the identity in [F5] evaluated on basis vectors says that the entry of $A^{†}$ in row $j$ , column $i$ is $\overline{a_{i j}}$ . Thus the abstract daggers are the conjugate transposes used by [F1], and on every nonzero space the abstract GKSL formula identifies with that finite matrix interface.

1.2givenF2F3F5algebra

For $u \in V_{j}$ and $v \in V_{k}$ , the adjoint identity and conjugate symmetry give $⟨ L_{a}^{†} L_{a} u, v ⟩ = ⟨ L_{a} u, L_{a} v ⟩$ . When $j \neq k$ , the latter is zero because the two images lie in distinct orthogonal summands, with the zero image allowed. Thus $L_{a}^{†} L_{a}$ preserves every $V_{j}$ . The block-preserving $H$ therefore commutes with $P$ , as does each $L_{a}^{†} L_{a}$ . Moreover $L_{a} W \subseteq W$ , so $L_{a} P = P L_{a} P$ .

2.1givenF1F3F5step 1.1step 1.2algebra

Let $ι (Y)$ denote extension by zero to $W^{⊥}$ of an operator $Y$ on $W$ . In the adapted basis from step 1.1, this has diagonal blocks $Y$ and zero. Since $L_{a} W \subseteq W$ , the lower-left block of $L_{a}$ is zero; block multiplication gives $L_{a} ι (Y) L_{a}^{†} = ι (L_{a, W} Y L_{a, W}^{†})$ . The upper-left block of $L_{a}^{†} L_{a}$ is $L_{a, W}^{†} L_{a, W}$ , and step 1.2 makes its off-diagonal blocks zero. The same block property holds for $H$ , and $H_{W}$ is Hermitian by the adjoint identity. Hence the commutator and anticommutator terms also compress exactly, proving $L (ι (Y)) = ι (L_{W} (Y))$ . Induction gives $L^{n} ι (Y) = ι (L_{W}^{n} (Y))$ for every nonnegative integer $n$ . The restricted jumps form a finite family, so the compressed formula has the finite GKSL form whenever $W$ is nonzero.

2.2givenF4F6F7step 1.1algebra

Suppose $W \neq {0}$ . Equip each of the complex spaces of operators on $V$ and on $W$ with the coordinate norm $∥ X ∥_{e n t, 1} = \sum_{i, j} ∣ X_{i j} ∣$ in its basis from step 1.1. This is a norm: definiteness and absolute homogeneity follow entrywise, and summing $∣ X_{i j} + Y_{i j} ∣ \leq ∣ X_{i j} ∣ + ∣ Y_{i j} ∣$ proves its triangle inequality. The matrix units, ordered lexicographically by $(i, j)$ , are finite ordered bases. Thus [F6] makes both normed operator spaces Banach. The displayed formulas are complex-linear, so [F7] makes $L$ and $L_{W}$ bounded. With the induced norm $∥ S ∥ = \sup_{X \neq 0} ∥ S (X) ∥_{e n t, 1} / ∥ X ∥_{e n t, 1}$ , [F4] therefore applies and gives operator-norm convergence of both exponential series.

3.1givenF1F3step 2.1step 2.2algebra∎

In these adapted coordinates $∥ ι (Y) ∥_{e n t, 1} = ∥ Y ∥_{e n t, 1}$ because extension only adds zero entries. Applying step 2.1 to finite exponential partial sums and using this isometry to pass to their limits from step 2.2 gives $N_{t} (ι (Y)) = ι (e^{t L_{W}} Y)$ . An operator is supported on $W$ exactly when it is such an extension, proving invariance and exact compressed evolution. If $W = {0}$ , the only supported operator is zero; its image under any linear map is zero, and the exponential on the zero operator space is its unique map. If also $V = {0}$ , the same convention defines the ambient exponential, so the equality still holds.

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