Finite GKSL evolution and the QEC channel interface

Let $d\ge 1$ be an integer and $M_d(\mathbb{C})$ the complex $d\times d$ matrices. Fix a Hermitian matrix $H=H^{†}$ and a finite family $L_1,\dots ,L_m$ in $M_d(\mathbb{C})$, allowing $m=0$. Here $A^{†}$ denotes conjugate transpose, $[A,B]=AB-BA$, and $\{A,B\}=AB+BA$. In units with $\mathrm{\hslash }=1$, define the linear map $\mathcal{L}$ by $\mathcal{L}(X)=-i[H,X]+{\sum }_{a=1}^m(L_{a}X{L}_a^{†}-\{L_a^{†}{L}_a,X\}\mathrm{/}2)$, with the empty sum equal to zero. The units of $H$ and $L_a$ are chosen so that every term has inverse-time units.

For $t\ge 0$, define the noise channel ${\mathcal{N}}_t:M_d(\mathbb{C})\to M_d(\mathbb{C})$ by ${\mathcal{N}}_t=e^{t\mathcal{L}}:={\sum }_{n=0}^{\mathrm{\infty }}{t}^n{\mathcal{L}}^n\mathrm{/}n!$, with ${\mathcal{L}}^0$ the identity map and convergence in operator norm. A channel here means a complex-linear, completely positive and trace-preserving map: complete positivity means positivity after tensoring with the identity on every finite matrix ancilla, and trace preservation means $\mathrm{Tr}({\mathcal{N}}_t(X))=\mathrm{Tr}(X)$ for every $X$. A matrix is positive when it is Hermitian and has nonnegative quadratic form on every vector.

This is a mathematical construction. Assigning it to a physical emitter requires a separate physical assumption; no negative-time channel is asserted.

Given: The finite matrices, finite family and nonnegative time in the definition, with compatible units and $\mathrm{\hslash }=1$.

Goal: Establish that the displayed series defines a unique finite channel with the stated domain and codomain.

1.1 For $1\le i,j\le d$, let $E_{ij}$ be the matrix whose entry in row $i$, column $j$ is $1$ and whose other entries are $0$. Every matrix $X$ has the expansion $X={\sum }_{i,j=1}^d{X}_{ij}{E}_{ij}$; comparing entries makes its coefficients unique, so these matrices, ordered lexicographically by $(i,j)$, form a finite ordered complex basis. Define $\mathrm{\parallel }X{\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t},1}={\sum }_{i,j=1}^d\mathrm{\mid }{X}_{ij}\mathrm{\mid }$. This function is nonnegative, and it vanishes exactly when every entry is zero, hence exactly when $X=0$. For every complex scalar $\alpha$, $\mathrm{\parallel }\alpha X{\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t},1}=\mathrm{\mid }\alpha \mathrm{\mid }\mathrm{\parallel }X{\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t},1}$; summing $\mathrm{\mid }{X}_{ij}+Y_{ij}\mathrm{\mid }\le \mathrm{\mid }{X}_{ij}\mathrm{\mid }+\mathrm{\mid }{Y}_{ij}\mathrm{\mid }$ gives $\mathrm{\parallel }X+Y{\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t},1}\le \mathrm{\parallel }X{\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t},1}+\mathrm{\parallel }Y{\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t},1}$. Thus this is a complex norm on a space with a finite ordered basis, and [F4] makes this normed matrix space Banach. [given, F4, construct, algebra]

2.1 The displayed GKSL formula is complex-linear. Let $C={\max }_{1\le i,j\le d}\mathrm{\parallel }\mathcal{L}(E_{ij}){\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t},1}$, which is finite because $d\ge 1$ and all matrices and the jump family are finite. Linearity and the norm axioms give $\mathrm{\parallel }\mathcal{L}(X){\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t},1}\le {\sum }_{i,j=1}^d\mathrm{\mid }{X}_{ij}\mathrm{\mid }\mathrm{\parallel }\mathcal{L}(E_{ij}){\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t},1}\le C\mathrm{\parallel }X{\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t},1}$, so $\mathcal{L}$ is bounded. Applying [F5] on the Banach space from step 1.1, with the induced operator norm, gives a unique operator-norm limit for the exponential series. [given, F1, F5, step 1.1, algebra]

3.1 The hypotheses of the finite GKSL channel theorem are exactly those in the definition, including the finite family and Hermitian $H$. By [F3] the resulting exponential is a finite channel for $t\ge 0$; [F2] gives the stated meaning of that term. Hence the construction has the claimed type and is well-defined. [given, F1, F2, F3, step 2.1] ∎