Lindbladian exponentiation and Choi Kraus extraction

PDF p. 17: Numerical method, final sentence

The inputs are a finite nonzero physical Hilbert space $H_P$ with an ordered orthonormal basis $(e_i{)}_{i=1}^d$, a Hermitian Hamiltonian $H$, a finite family of jump operators $L_a$ on $H_P$, and one or more evolution times $t\ge 0$. For the article's benchmark, use the finite emitter, downward jumps, equal-rate rigid-rotor parameters and phenomenological diagonal lift specified by the two model postulates, with all their physical, spectral and choice hypotheses. The lift is an assumed model term, not an observed measurement. Define the Lindbladian, a linear map on operators, by $\mathcal{L}(X)=-i[H,X]+{\sum }_a(L_{a}X{L}_a^{†}-\{L_a^{†}{L}_a,X\}\mathrm{/}2)$, in units with $\mathrm{\hslash }=1$; $[A,B]=AB-BA$ and $\{A,B\}=AB+BA$.

For each input time, form the full Liouville matrix $L$ of $\mathcal{L}$ in the ordered matrix-unit basis $E_{ij}=\mathrm{\mid }{e}_i⟩⟨e_j\mathrm{\mid }$ and calculate $S=e^{tL}$. Construct the input-first, unnormalized Choi matrix $J$ by the index reshuffle $J_{(i,k),(j,l)}=S_{(k,l),(i,j)}$. Diagonalize $J$ and, for each positive eigenvalue ${\lambda }_k$ with a chosen orthonormal eigenvector $v_k$, reshape $\sqrt{{\lambda }_k}{v}_k$ into $K_k$ using ${\sum }_i{e}_i\otimes K_k{e}_i=\sqrt{{\lambda }_k}{v}_k$. Output the resulting canonical Kraus operators and their weights $w_k=\mathrm{Tr}(K_k^{†}{K}_k)={\lambda }_k$. The Choi matrix is unnormalized: its trace is $d$. The operators may depend on the eigenbasis chosen in a degenerate eigenspace.

For a supplied isometric code $V:{\mathbb{C}}^{d_L}\to H_P$, with $d_L\ge 1$, optionally form the quantum-error-correction matrix $M_{[\mu ,k],[\nu ,l]}=⟨\mu \mathrm{\mid }{V}^{†}{K}_k^{†}{K}_{l}V\mathrm{\mid }\nu ⟩$. Output the transpose-recovery entanglement fidelity at the maximally mixed logical input as $d_L^{-2}\mathrm{\parallel }{\mathrm{Tr}}_L\sqrt{M}{\mathrm{\parallel }}_F^2$: ${\mathrm{Tr}}_L$ contracts the logical indices and $\mathrm{\parallel }B{\mathrm{\parallel }}_F^2={\sum }_{k,l}\mathrm{\mid }{B}_{kl}{\mathrm{\mid }}^2$. Entanglement fidelity is the overlap of a maximally entangled logical-reference state with its image after the encoded channel and support-completed transpose recovery.

The exact Kraus rank is the number of positive Choi eigenvalues. If a numerical threshold is used, instead report the strict count of estimated eigenvalues above the specified positive threshold, the Choi normalization, the estimates and treatment of negative estimates; record any certified eigenvalue error bound, or state that it is unknown. A threshold count alone does not determine exact rank. The instructions describe the exact target operations; a floating-point implementation produces approximations and requires its own error assessment.

The article uses the full matrix for its small systems even though, when the Hamiltonian is diagonal in $J_z$ and each jump has a definite $J_z$ weight, operator sectors spanned by matrix units with fixed magnetic-label difference are invariant and can be evolved separately. This item neither performs the calculation nor asserts a measured result, numerical convergence, or replication of the article's figures.

Given: The finite matrices, ordered basis, nonnegative times and, when fidelity is requested, the isometric code specified above.

Goal: Explain why the exact operations define the stated channel representation and optional fidelity.

1.1 By [F1], the exponential defines a channel. Its Liouville matrix and Choi matrix have the stated relationship by [F2]. Applying [F3] to that finite channel gives the Kraus family, its weights and exact rank; choosing an eigenbasis resolves the nonuniqueness within repeated eigenspaces. [given, F1, F2, F3]

2.1 For the optional code, the hypotheses of [F4] hold for the channel and Kraus family from step 1.1, giving the displayed fidelity even in the singular case. The distinction between exact and thresholded counts and the conditional status of any error bounds are as in [F5]. The full-matrix instruction is well defined independently of a sector decomposition; [F6] describes the alternative only when its hypotheses hold. Thus the outputs are defined under the stated inputs, without asserting numerical accuracy or a physical observation. [given, F4, F5, F6, step 1.1] ∎