Syndrome flags and logical-state-independent environmental records

Let $V:H_L\to H_P$ be a finite-dimensional isometric encoding and let $\mathcal{N}:\mathcal{L}(H_P)\to \mathcal{L}(H_O)$ be a quantum channel, as in Code restriction and decoded channel types. A density operator is a positive semidefinite operator of trace one. Fix a finite Kraus representation of $\mathcal{N}$ and its associated complementary channel ${\mathcal{N}}^c$ as in Complementary error records in a Kraus basis. The logical environmental record of a logical density operator $\rho$ is ${\mathcal{N}}^c(V\rho V^{†})$. It is logical-state independent when there is one density operator $\sigma$ such that ${\mathcal{N}}^c(V\rho V^{†})=\sigma$ for every logical density operator $\rho$.

A syndrome instrument on the record is a finite instrument $\{{\mathcal{I}}_a{\}}_{a\in \mathrm{\Omega }}$ whose input is the environment of this chosen complementary channel. A syndrome flag is its classical outcome label $a$, with probability $p_a(\rho )=\mathrm{tr}{\mathcal{I}}_a({\mathcal{N}}^c(V\rho V^{†}))$. The syndrome distribution is logical-state independent when every $p_a(\rho )$ is constant over logical density operators. The instrument may retain a quantum output; for $p_a(\rho )>0$, its conditional state is ${\mathcal{I}}_a({\mathcal{N}}^c(V\rho V^{†}))\mathrm{/}{p}_a(\rho )$, and no conditional state is assigned when the probability is zero, as in Finite instruments and conditional states.

An erasure flag is an explicitly specified, distinguishable outcome indicating that a designated data branch has been erased. In the finite-dimensional erasure channel of Finite-dimensional quantum erasure channel, the output space is $H_L\oplus \mathbb{C}e$, with $e$ a unit vector orthogonal to the data summand; on a density operator the output is $(1-ϵ)J\rho J^{†}+ϵ\mathrm{\mid }e⟩⟨e\mathrm{\mid }$, where $J$ includes the data summand and $0\le ϵ\le 1$. The orthogonal flag occurs with probability $ϵ$, while the unflagged branch retains $\rho$. A flag alone does not specify either the branch probability or the retained logical channel.

The complementary-record supplier constructs a channel for the chosen finite Kraus representation, so its output on each encoded density operator is a density operator. Equality to one fixed $\sigma$ is therefore a well-defined condition on the full environmental operator. The instrument supplier defines the outcome probabilities and conditional states, including the zero-probability convention. If the full record is independent of $\rho$, applying each fixed instrument map and taking its trace makes the syndrome distribution independent of $\rho$ as well.

The converse need not hold. Take a two-dimensional logical space and identity encoding, and the channel with Kraus operators $E_0=\mathrm{\mid }0⟩⟨0\mathrm{\mid }$ and $E_1=\mathrm{\mid }0⟩⟨1\mathrm{\mid }$. Its chosen dilation sends $\mathrm{\mid }\psi ⟩$ to $\mathrm{\mid }0⟩\otimes \mathrm{\mid }\psi ⟩$, so its full environmental record is $\rho$. Use the one-outcome instrument whose sole map is the identity on that environment. Its only outcome has probability one for every input, but its retained conditional state is $\rho$, which varies with the logical state. Thus an input-independent syndrome distribution does not imply an input-independent full record.

For the erasure channel, orthogonality of the two output summands makes the flag distinguishable from the data branch. Their traces on a density operator are respectively $ϵ$ and $1-ϵ$; when the latter is nonzero, normalizing that branch gives $J\rho J^{†}$. These statements use the specified erasure-channel formula and do not assert that an arbitrary flagged process has that formula.