Passive subspace and subsystem protection interface

Let $H_P$, $H_L$ and $H_G$ be nonzero finite-dimensional complex inner-product spaces, and let $V:H_L\otimes H_G\to H_P$ be an isometry. Write $\mathcal{L}(H)$ for the linear operators on $H$, and ${\mathrm{Ad}}_V(X)=VX{V}^{\ast }$. Let $\mathfrak{N}$ be a specified family of channels from $\mathcal{L}(H_P)$ to itself. The image of $V$ is the specified block; its first and second tensor factors are called the logical and gauge subsystems, respectively.

The logical subsystem is noiseless for $\mathfrak{N}$ in this interface if, for every $\mathcal{N}\in \mathfrak{N}$, there is a channel ${\mathrm{\Phi }}_{\mathcal{N}}:\mathcal{L}(H_G)\to \mathcal{L}(H_G)$ such that $\mathcal{N}\circ {\mathrm{Ad}}_V={\mathrm{Ad}}_V\circ ({\mathrm{id}}_{\mathcal{L}(H_L)}\otimes {\mathrm{\Phi }}_{\mathcal{N}})$ on all of $\mathcal{L}(H_L\otimes H_G)$. Thus the block is invariant in the sense that every encoded operator has output supported on the block, while the logical factor has identity action and the gauge factor may evolve.

When $\dim H_G=1$, this condition defines a decoherence-free subspace for the family: the induced action on operators of the encoded subspace is the identity. A separate, explicitly tracked-unitary convention permits a specified unitary $U_{\mathcal{N}}$ on $H_L$ for each $\mathcal{N}$ and replaces the identity factor by ${\mathrm{Ad}}_{U_{\mathcal{N}}}$; it is not the identity-action condition unless that induced conjugation is the identity. A change in a noise map, including one described as a symmetry perturbation, must satisfy the relevant displayed factorization to retain the corresponding property. Energy-level degeneracy alone is not part of this definition.

The tensor construction in [F1] gives the domain of the isometry in [F3] and a product basis. The finite-dimensional adjoint existence in [F4] makes $V^{\ast }:H_P\to H_L\otimes H_G$ well-defined, so $X\mapsto VX{V}^{\ast }$ has the stated operator-space domain and codomain. On elementary operator tensors the expression $(\mathrm{id}\otimes {\mathrm{\Phi }}_{\mathcal{N}})(A\otimes B)=A\otimes {\mathrm{\Phi }}_{\mathcal{N}}(B)$ extends linearly and uniquely: [F5] constructs the tensor map on $\mathcal{L}(H_L)\otimes \mathcal{L}(H_G)$, and [F6] identifies that space with $\mathcal{L}(H_L\otimes H_G)$ by its product matrix-unit basis. Both sides of the defining equality are therefore linear maps with the same domain and codomain. For a one-dimensional gauge space every operator is a scalar multiple of its identity, and trace preservation in [F2] forces its channel to be the identity; the stated subspace specialization follows. The tracked-unitary version is a separately specified convention, with ${\mathrm{Ad}}_U(A)=UA{U}^{\ast }$ using the unitary in [F3] and its adjoint from [F4].

The source describes a noiseless factor through interaction operators that act on its cofactor. The definition above states the channel-level interface used here and does not assert that an arbitrary energy degeneracy supplies its factorization.