Optimal phase-corrected four-level recovery

The model postulates a finite-dimensional channel with code and decay spaces as stated, rate $\gamma \ge 0$, and real ${\omega }_0,\mathrm{\Delta }$. Four-level spontaneous-decay model

On code inputs, the channel has a null branch of fixed weight $1-p$ supported on the code and a decay branch of fixed weight $p$ supported on the orthogonal decay space. For $p>0$, identifying the latter with the logical basis gives the coherence channel ${\mathcal{D}}_{\chi }$: diagonal entries are unchanged and the $01$ entry is multiplied by $\chi$. The null branch is the identity after this identification. Exact four-level Kraus operators Conditional decoded state and decay coherence

For $p>0$, $\chi \mathrm{\ne }0$, $r\le 1$, and the displayed $D_{+}$ cancels the phase of the decay coherence. At $p=0$ the null branch alone is present. Explicit four-level transpose decoder

For a qubit channel $\mathcal{R}$, unitary $U$, and $Z=\mathrm{diag}(1,-1)$, the maximally mixed-input entanglement fidelities satisfy $F_e(\mathcal{R}\circ {\mathrm{Ad}}_U)+F_e(\mathcal{R}\circ {\mathrm{Ad}}_{ZU})\le 1$. Here ${\mathrm{Ad}}_U(X)=UX{U}^{†}$. Pauli-complementary qubit recovery fidelity inequality

The coherence channel ${\mathcal{D}}_{\lambda }$ has $F_e=(1+\mathrm{Re}\lambda )\mathrm{/}2$ for $\mathrm{\mid }\lambda \mathrm{\mid }\le 1$, and its composition and fixed-weight branch averages act as stated by the coherence-channel interface. Qubit coherence channels and fidelity functionals

For fixed input-independent weights on orthogonal output blocks, the maximum decoder fidelity equals the weighted sum of the separate branch maxima, and a project-then-decode channel attains it. Recovery optimization on orthogonal flagged branches

For fixed $\gamma >0,\mathrm{\Delta }$, the fraction $(1-r)\mathrm{/}2$ is ${\mathrm{\Delta }}^2{t}^2\mathrm{/}48+O(t^3)$ as $t\downarrow 0$. Decay Kraus weights and photon overlap

Bras, kets, outer products, conjugate-transpose adjoints and rectangular matrix products use the finite ordered-orthonormal-basis conventions. Finite operator and ket conventions

For a qubit channel $\mathcal{E}$, entanglement fidelity at $I\mathrm{/}2$ is $F_e(\mathcal{E})=⟨{\mathrm{\Phi }}_2\mathrm{\mid }(\mathrm{i}\mathrm{d}\otimes \mathcal{E})(\mathrm{\mid }{\mathrm{\Phi }}_2⟩⟨{\mathrm{\Phi }}_2\mathrm{\mid })\mathrm{\mid }{\mathrm{\Phi }}_2⟩$, where $\mathrm{\mid }{\mathrm{\Phi }}_2⟩=(\mathrm{\mid }00⟩+\mathrm{\mid }11⟩)\mathrm{/}\sqrt{2}$. This overlap is linear in the channel and belongs to $[0,1]$ for channels. Entanglement fidelity on a finite logical space

At fixed scalar parameters, the exponential series has a controlled Taylor remainder, and multiplication of scalar expansions has the controlled remainder obtained by truncating their polynomial product. Finite exponential and smooth quotient remainder control

For finite-dimensional complex inner-product spaces $H,K$, a quantum channel is a complex-linear, completely positive, trace-preserving map $\mathrm{\Phi }:\mathcal{L}(H)\to \mathcal{L}(K)$. Trace preservation is required on every input operator. Finite-dimensional quantum channel