Jump histories, distinguishability, and limits of Kraus rank

The expansion in [F5] expresses the channel as a no-jump contribution plus integrals of contributions indexed by ordered times and jump labels. In the emitter model those labels include the modeled polarizations; the amplitudes act on the source basis states. The lift in [F1] changes the diagonal energies and hence the phases in the intervening propagators. This identifies a possible phase mechanism without asserting that an energy shift alone establishes distinguishability.

In the fixed-jump benchmark, [F2] supplies the exact finite-time ranks. By [F7], a threshold count can grow as previously positive weights pass the threshold even when exact rank is already constant; the threshold comparison also requires the same normalization. Under the further exit-rate and model hypotheses in [F3], the limiting channel has the stated nine source-to-ground operators. A limiting rank can differ from every finite-time rank because positive weights may tend to zero.

The record in [F4] retains both diagonal and off-diagonal information and depends on the channel and its restriction to a code. Use [F8] to specify the encoding and decoded recovery and [F9] to specify the fidelity in the Remark. The supremum defining $F_{\mathrm{o}\mathrm{p}\mathrm{t}}$ is well-defined: the fidelity values lie in $[0,1]$, and the decoder set is nonempty, since $Y\mapsto \mathrm{Tr}(Y)\mathrm{\mid }0⟩⟨0\mathrm{\mid }$ is a channel into $H_L={\mathbb{C}}^d$. Indeed, for an output orthonormal basis $\mathrm{\mid }a⟩$, the operators $K_a=\mathrm{\mid }0⟩⟨a\mathrm{\mid }$ give this map, sum to the identity in ${\sum }_a{K}_a^{†}{K}_a$, and each ancilla amplification is a sum of conjugations preserving positivity. We now prove non-determination and lack of a universal rank ordering by explicit channels.

Fix $H_L=H_P=H_O={\mathbb{C}}^2$ and $V=I$. Put $X=\mathrm{\mid }0⟩⟨1\mathrm{\mid }+\mathrm{\mid }1⟩⟨0\mathrm{\mid }$ and $Z=\mathrm{\mid }0⟩⟨0\mathrm{\mid }-\mathrm{\mid }1⟩⟨1\mathrm{\mid }$, and write ${\mathcal{U}}_U(A)=UA{U}^{†}$ for $U=I,X,Z$. These matrices are unitary; their conjugations preserve trace, and their ancilla amplifications are unitary conjugations, so they are channels by [F8]. Nonnegative convex combinations of these maps are also channels, since every amplification preserves positivity and their weights sum to one. Let $\mathrm{\mid }\mathrm{\Phi }⟩=(\mathrm{\mid }00⟩+\mathrm{\mid }11⟩)\mathrm{/}\sqrt{2}$, $\mathrm{\mid }\mathrm{\Psi }⟩=(\mathrm{\mid }01⟩+\mathrm{\mid }10⟩)\mathrm{/}\sqrt{2}$ and $\mathrm{\mid }\mathrm{\Omega }⟩=(\mathrm{\mid }00⟩-\mathrm{\mid }11⟩)\mathrm{/}\sqrt{2}$. They are orthonormal by their displayed coordinates, and the trace-one Choi matrices of ${\mathcal{U}}_I,{\mathcal{U}}_X,{\mathcal{U}}_Z$ are respectively their rank-one projectors: expand $\mathrm{\mid }\mathrm{\Phi }⟩⟨\mathrm{\Phi }\mathrm{\mid }=\frac{1}{2}{\sum }_{j,k=0}^1\mathrm{\mid }j⟩⟨k\mathrm{\mid }\otimes \mathrm{\mid }j⟩⟨k\mathrm{\mid }$ and apply the corresponding channel on the second factor. Thus ${\mathcal{U}}_I$ and ${\mathcal{U}}_X$ both have exact Kraus rank one by [F7]; with identity recovery their entanglement fidelities are $1$ and $0$ by [F9]. This proves non-determination even for a fixed code and recovery.

If, in a specified branch, the additional averaging hypothesis in [F6] holds, averaging the relative phases multiplies each input coherence by the corresponding Gram entry. The conclusion is conditional on that branch model. It does not by itself establish the article's comparison of threshold-crossing times. The reported role of successive jump-time profiles and the earlier onset for larger lift are retained as the article's model interpretation.

For $0<p\le 1\mathrm{/}2$, put ${\mathcal{N}}_p=(1-p){\mathcal{U}}_I+p{\mathcal{U}}_X$. Its trace-one Choi matrix is $(1-p)\mathrm{\mid }\mathrm{\Phi }⟩⟨\mathrm{\Phi }\mathrm{\mid }+p\mathrm{\mid }\mathrm{\Psi }⟩⟨\mathrm{\Psi }\mathrm{\mid }$, so its exact rank is two and its identity-recovery fidelity is $1-p$. For any decoded recovery $\mathcal{R}$, put $Q_{\mathcal{R}}=(\mathrm{i}\mathrm{d}\otimes \mathcal{R})(\mathrm{\mid }\mathrm{\Phi }⟩⟨\mathrm{\Phi }\mathrm{\mid })$, a positive trace-one matrix by [F9], and set $a=⟨\mathrm{\Phi }\mathrm{\mid }{Q}_{\mathcal{R}}\mathrm{\mid }\mathrm{\Phi }⟩$, $b=⟨\mathrm{\Psi }\mathrm{\mid }{Q}_{\mathcal{R}}\mathrm{\mid }\mathrm{\Psi }⟩$. These are nonnegative and $a+b\le 1$, by completing the two orthonormal vectors to a basis and summing the nonnegative diagonal entries of $Q_{\mathcal{R}}$. Since $(I\otimes X)\mathrm{\mid }\mathrm{\Phi }⟩=(X\otimes I)\mathrm{\mid }\mathrm{\Phi }⟩=\mathrm{\mid }\mathrm{\Psi }⟩$, factorwise tensor action gives $(\mathrm{i}\mathrm{d}\otimes \mathcal{R})(\mathrm{\mid }\mathrm{\Psi }⟩⟨\mathrm{\Psi }\mathrm{\mid })=(X\otimes I)Q_{\mathcal{R}}(X\otimes I)$. Consequently $F_e(\mathcal{R}\circ {\mathcal{N}}_p)=(1-p)a+pb\le (1-p)(a+b)\le 1-p$. Identity recovery attains the bound, so $F_{\mathrm{o}\mathrm{p}\mathrm{t}}({\mathcal{N}}_p,I)=1-p$. In particular $p=1\mathrm{/}4$ and $p=1\mathrm{/}2$ give the same rank two but different optimized fidelities $3\mathrm{/}4$ and $1\mathrm{/}2$.

Finally put $\mathcal{M}=\frac{3}{4}{\mathcal{U}}_I+\frac{1}{8}{\mathcal{U}}_X+\frac{1}{8}{\mathcal{U}}_Z$. Its trace-one Choi matrix is the corresponding positive combination of the three orthogonal projectors in step 1.4, so its exact rank is three and its identity-recovery fidelity is $3\mathrm{/}4$. Hence $F_{\mathrm{o}\mathrm{p}\mathrm{t}}(\mathcal{M},I)\ge 3\mathrm{/}4>F_{\mathrm{o}\mathrm{p}\mathrm{t}}({\mathcal{N}}_{1\mathrm{/}2},I)=1\mathrm{/}2$: increasing rank from two to three can accompany improved fixed-recovery and optimized fidelity. Conversely, increasing rank from ${\mathcal{U}}_I$ to ${\mathcal{N}}_{1\mathrm{/}2}$ lowers both fidelities from one to $1\mathrm{/}2$, since fidelity is at most one and identity recovery attains one for ${\mathcal{U}}_I$. Thus rank comparisons alone impose neither a universal increase nor a universal decrease in these fidelities. These counterexamples establish the general limitation of a rank metric; the article's proposed connection between distinguishability and degradation in its emitter model remains a mechanism interpretation, not a deduction of such a universal ordering.