Dependencies

For \(t\ge 0\) and \(n\in \mathbb {N}\), one has \(T_{nt}=(T_t)^n\).

Every density matrix is nonzero.

If a corner compression is invariant under \(E\), it remains invariant under every power \(E^n\).

If \(L(X)=\mu X\), then \(e^{tL}(X)=e^{t\mu }X\) for all \(t\in \mathbb {R}\).

If \(\mu \) is in the spectrum of \(L\), then \(e^{t\mu }\) is in the spectrum of \(e^{tL}\).

If \(e^{it\theta }\) is a root of unity for every \(t{\gt}0\), then \(\theta =0\).

Peripheral spectral points of the generator have vanishing real part.

The number of peripheral eigenvalues is bounded by \(\dim (M_{D}(\mathbb {C}))\).

If all powers of a peripheral eigenvalue remain peripheral, then some positive power not exceeding \(\dim (M_{D}(\mathbb {C}))\) equals \(1\).

For an irreducible channel with faithful fixed point, every power of a peripheral eigenvalue is again peripheral.

For \(t \ge 0\), setting \(s=t\) in (5) gives

where \(M = \sup _{u \in [0,t]} \| e^{uL}\| \).

For \(t \ge 0\), if \(\| e^{sL}\| \le 1\) and \(\| e^{sL'}\| \le 1\) for every \(s \in [0,t]\), then \(\| e^{tL'} - e^{tL}\| \le t\, \| L' - L\| \).

The Dyson–Phillips series \(\sum _n \widetilde{T}^{(n)}_t\) converges in operator norm for every \(t \ge 0\).

Every nonzero fixed point of an irreducible channel has nonzero trace.

One has \(\sum _{n=0}^{\infty }\widetilde{T}^{(n)}_t = e^{tL'}\) in tsum form.

For a Lindblad operator \(L_j\), the adjoint dissipator on observables is

The adjoint generator of a Lindblad form is

The adjoint kernel is the set of observables \(A\) satisfying \(L^*(A)=0\).

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is a quantum channel if it is both completely positive and trace-preserving (CPTP), following  [ Wol12 ] .

The commutant of a Lindblad form is the set of matrices commuting with \(H\), with every Lindblad operator \(L_j\), and with every adjoint \(L_j^\dagger \).

The commutator form of the Lindblad equation writes the generator as

where \([A,B] = AB - BA\). This is [ Wol12 , Equation (7.22) ] .

A dynamical semigroup \(T\) is norm-continuous if the map \(t \mapsto T_t\) is continuous in the operator norm topology. In finite dimension this coincides with strong continuity.

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is completely positive (CP) if it admits a Kraus representation: there exist operators \(\{ K_i\} _{i=0}^{r-1}\) with \(K_i \in M_{D}(\mathbb {C})\) such that, for every \(X \in M_{D}(\mathbb {C})\),

The Kraus representation also gives entrywise positivity on every positive block matrix by Theorem 2.2.1, and hence the associated completely positive map between matrix \(C^*\)-algebras in Theorem 2.2.2.

The set of density matrices in \(M_{D}(\mathbb {C})\) is

A family of linear maps \(T : \mathbb {R}\to (M_{D}(\mathbb {C}) \to _{\ell } M_{D}(\mathbb {C}))\) is a dynamical semigroup if

1. (semigroup law) \(T_{t+s} = T_t \circ T_s\) for all \(t,s \ge 0\); and

2. (initial condition) \(T_0 = \operatorname{id}\).

This is [ Wol12 , Equation (7.1) ] .

Given a generator \(L \in \operatorname{End}_{\mathbb {C}}(M_{D}(\mathbb {C}))\), the exponential semigroup is

A generator decomposition consists of a completely positive map \(\phi : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) and a matrix \(\kappa \in M_{d}(\mathbb {C})\), defining the linear map

This is [ Wol12 , Equation (7.14) ] .

For a projection \(P\), the generator preserves the compression \(PM_{D}(\mathbb {C})P\) if, for every \(X\in M_{D}(\mathbb {C})\),

Condition (4): there exists a nontrivial projector \(P\) and a Lindblad form \((H,\{ L_j\} )\) for \(L\) such that \((\mathbb {1}-P)L_jP=0\) and \((\mathbb {1}-P)\kappa P=0\) for all \(j\), where \(\kappa =iH+\frac{1}{2}\sum _jL_j^\dagger L_j\).

A generator \(L\) has a faithful stationary state if there exists a positive definite density matrix \(\rho _0\) with \(L(\rho _0)=0\).

Condition (3): there exists a nontrivial orthogonal projector \(P\) such that \(e^{tL}(PM_{D}(\mathbb {C})P)\subseteq PM_{D}(\mathbb {C})P\) for all \(t\ge 0\).

The commutant of the Lindblad family \(\{ L_j\} '\) equals \(\mathbb {C}\cdot \mathbb {1}\), i.e. the Lindblad operators act irreducibly on \(M_{D}(\mathbb {C})\).

Condition (1): there exists a density matrix \(\rho _0\) with nontrivial kernel such that \(e^{tL}(\rho _0)=\rho _0\) for all \(t\ge 0\).

Condition (2): there exists a density matrix \(\rho _0\) with nontrivial kernel satisfying \(L(\rho _0)=0\).

A Lindblad form has traceless Kraus operators if \(\operatorname{tr}(L_j) = 0\) for every \(j\).

A linear map \(L : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) is conditionally completely positive (CCP) if it admits a generator decomposition, i.e. there exist a CP map \(\phi \) and a matrix \(\kappa \) such that \(L(\rho ) = \phi (\rho ) - \kappa \rho - \rho \kappa ^\dagger \). This is condition 1 of [ Wol12 , Proposition 7.2 ] .

A linear map \(L : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) is a GKSL generator if \(e^{tL}\) is a quantum channel (CPTP) for all \(t \ge 0\).

The span of the Lindblad operators is closed under Hermitian conjugation, i.e. it forms a \(*\)-subspace of \(M_{D}(\mathbb {C})\).

A projection is nontrivial if it is orthogonal and neither \(0\) nor \(\mathbb {1}\).

A quantum dynamical semigroup is a norm-continuous dynamical semigroup whose time slices are quantum channels for every \(t \ge 0\).

A quantum dynamical semigroup is reducible if it has a nontrivial invariant compression.

A linear map \(L : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) is trace-annihilating if \(\operatorname{tr}(L(\rho )) = 0\) for all \(\rho \in M_{d}(\mathbb {C})\). This is the infinitesimal version of trace preservation.

A generator decomposition \((\phi , \kappa )\) satisfies the trace constraint if \(\phi ^*(\mathbb {1}) = \kappa + \kappa ^\dagger \), where \(\phi ^*\) is the Hilbert–Schmidt adjoint. This is the condition in [ Wol12 , Equation (7.20) ] .

A Kossakowski form consists of \(H=H^\dagger \), a finite family of matrices \(\{ F_k\} _{k=0}^{n-1}\), and a positive semidefinite matrix \(C \in M_{n}(\mathbb {C})\), defining

In Wolf’s formula one takes \(\{ F_k\} \) to be a basis of traceless matrices; here we keep only the algebraic data needed for the conversion to Lindblad form.

The minimal number of Lindblad operators across all GKSL representations of \(L\), which equals the rank of the Kossakowski matrix in the orthonormal-basis representation.

A Lindblad form consists of a Hermitian matrix \(H = H^\dagger \) (the Hamiltonian) and a family of matrices \(\{ L_j\} _{j=0}^{r-1}\) (the Lindblad operators), defining the linear map

where \([A,B] = AB - BA\) and \(\{ A,B\} _+ = AB + BA\). This is [ Wol12 , Equation (7.21) ] .

The \(\mathbb {C}\)-linear span of the Lindblad operators \(\{ L_j\} \).

For matrices \(\rho ,\sigma \in M_{D}(\mathbb {C})\), define the trace-log expression

On the physical domain where \(\rho \) is a density matrix and \(\sigma \) is positive definite, this is the Umegaki relative entropy.

For \(u\ge 0\) and \(s{\gt}0\), the residual slice index is \(m_n=\left\lfloor nu/s\right\rfloor \).

For \(u\ge 0\) and \(s{\gt}0\), the residual slice time is the remainder \(r_n=s\, \mathrm{fract}\! \left(nu/s\right)\).

The trace norm of \(A\in M_{D}(\mathbb {C})\) is its Schatten one-norm:

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is trace-preserving if \(\operatorname{tr}(E(X)) = \operatorname{tr}(X)\) for all \(X\).

For Pauli directions \(a,b\in \{ x,y,z\} \) and real rates \(\gamma _a,\gamma _b\), define

In particular, \(\mathcal L_{a,b}(\mathbb {1})=0\).

The maximally mixed qubit state is \(\tau _1=\frac12\mathbb {1}_2\).

Assume a factorial majorant of Dyson–Phillips iterates:

Then the Dyson series is summable in operator norm.

Let \(L\) be a GKSL generator in Lindblad form with Hamiltonian \(H\) and Lindblad operators \(\{ L_j\} \). Define the adjoint generator \(L^*\) by (25). If \(L\) has a faithful stationary state \(\rho _0{\gt}0\) with \(L(\rho _0)=0\), then, for any \(A\), the condition \(L^*(A)=0\) implies, for every \(j\),

That is, \(\ker (L^*)\subseteq \{ H,L_j,L_j^\dagger \} '\). This is [ Wol12 , Theorem 7.2 ] .

Under the same hypotheses as Theorem 12.6.8,

That is, the adjoint kernel equals the commutant of \(\{ H,L_j,L_j^\dagger \} \). This is [ Wol12 , Theorem 7.2 ] .

If the Lindblad data of a GKSL generator are block-upper-triangular with respect to some nontrivial projector \(P\), then there exists a density matrix \(\rho _0\) with nontrivial kernel satisfying \(L(\rho _0)=0\). This is [ Wol12 , Proposition 7.6, (4)\(\Rightarrow \)(2) ] .

If \(L\) is CCP and \(P = \mathbb {1}- |\Omega \rangle \! \langle \Omega |\), then

This is [ Wol12 , Proposition 7.2 ] .

If \(L\) is CCP, then \(e^{tL}\) is CP for all \(t \ge 0\). This is [ Wol12 , Proposition 7.3 ] .

If \(L\) is Hermiticity-preserving and \(P((L\otimes \operatorname{id})(|\Omega \rangle \! \langle \Omega |))P \ge 0\), where \(P = \mathbb {1}- |\Omega \rangle \! \langle \Omega |\), then \(L\) is CCP. This is [ Wol12 , Proposition 7.2 ] .

If an observable commutes with \(H\), with every \(L_j\), and with every \(L_j^\dagger \), then it lies in \(\ker (L^*)\).

The commutator form (Definition 12.3.2.5) defines the same linear map as the standard Lindblad form (Definition 12.3.2.1).

Every norm-continuous dynamical semigroup \(T\) on \(M_{D}(\mathbb {C})\) is of the form \(T_t = e^{tL}\) for some generator \(L \in \operatorname{End}_{\mathbb {C}}(M_{D}(\mathbb {C}))\) and every \(t \ge 0\). This is [ Wol12 , Proposition 7.1 ] .

\(e^{tL}\) is CP for all \(t \ge 0\) if and only if \(L\) is CCP.

If \(e^{tL}\) is CP for all \(t \ge 0\), then \(L\) is CCP. This is [ Wol12 , Proposition 7.3 ] .

Let \(\Delta = L' - L\). For \(t \ge 0\),

This is [ Wol12 , Lemma 7.1 ] .

For \(t \ge 0\), the Dyson–Phillips series equals the perturbed semigroup:

Each Dyson iterate \(t \mapsto \widetilde{T}^{(n)}_t\) is continuous in the time parameter.

If \(E\) is trace-preserving, \(E(V)=\mu V\), and \(\operatorname{tr}(V)\ne 0\), then \(\mu =1\).

If \(t_0{\gt}0\), \(T_{t_0}\) is irreducible, and \(\sigma \) is fixed for all times, then there exists a positive time step \(u\) such that \(T_{t_0}=(T_u)^{(\dim M_{D}(\mathbb {C}))!}\), the slice \(T_u\) is a channel, \(T_u\) is irreducible, and \(T_u(\sigma )=\sigma \).

If \(T_{t_0}\) is irreducible for some \(t_0{\gt}0\), then there exists a positive reduced time step \(u\) such that \(T_u\) is primitive.

Suppose \(u\ge 0\) and \(s{\gt}0\), and that \(T_s(\delta )=\delta \) and \(T_{nu}(\delta )\to 0\) as \(n\to \infty \). Then there exists \(a\in [0,s]\) such that \(T_a(\delta )=0\).

Under the reduced-time hypotheses \(T_{t_0}=(T_u)^{(\dim M_{D}(\mathbb {C}))!}\), if \(\mu \) is a peripheral eigenvalue of \(T_u\), then \(T_u\) admits a \(\mu \)-eigenvector with nonzero trace.

Assume \(d {\gt} 0\). Given any Kraus operators \(\{ K_j\} \), there exist \(c_j \in \mathbb {C}\) such that \(K_j' = K_j+c_j\mathbb {1}\) is traceless. This is part of [ Wol12 , Proposition 7.4 ] .

The family \(t \mapsto e^{tL}\) satisfies the semigroup law and the initial condition, hence is a dynamical semigroup.

The linear-map exponential semigroup and the continuous-linear-map exponential semigroup agree under the canonical identification of endomorphisms with continuous linear endomorphisms.

The linear-map exponential semigroup satisfies \(e^{0L} = \mathbb {1}\).

For all \(t,s \in \mathbb {R}\), \(e^{(t+s)L} = e^{tL} \cdot e^{sL}\).

The map \(t \mapsto e^{tL}\) is continuous in the operator norm topology.

The continuous-linear-map exponential satisfies \(e^{0L} = \mathbb {1}\).

If \(t_0{\gt}0\), \(T_{t_0}\) is irreducible, and \(\sigma \) is its unique fixed density matrix, then \(T_u(\sigma )=\sigma \) for every \(u\ge 0\).

If \(T_{t_0}\) is irreducible with faithful fixed density \(\sigma \), then every fixed point \(V\) of \(T_{t_0}\) satisfies \(V=\operatorname{tr}(V)\sigma \).

If \(E\) is an irreducible channel with faithful fixed density \(\sigma \), then every nonzero fixed point \(V\) of \(E\) satisfies \(V=\operatorname{tr}(V)\sigma \).

If \(T_u\) is primitive and \(\sigma \) is the common fixed density, then every fixed point of any positive-time slice \(T_s\) has the form \(\tau =\operatorname{tr}(\tau )\sigma \).

Let \((H,\{ L_j\} )\) be a Lindblad form. If the algebra generated by \(\{ L_j\} \) and \(\kappa \) is the entire matrix algebra \(M_{D}(\mathbb {C})\), then the Lindblad data do not admit a block-upper-triangular decomposition.

If the semigroup preserves \(PM_{D}(\mathbb {C})P\) for all non-negative times, then the generator preserves the same compression.

If \(K_j' = K_j+c_j\mathbb {1}\) and \(\kappa '\) is adjusted per [ Wol12 , Equation (7.19) ] , then \((\phi ',\kappa ')\) and \((\phi ,\kappa )\) define the same generator.

If \(e^{tL} = e^{tL'}\) for all \(t \ge 0\), then \(L = L'\).

If \(L\) is a GKSL generator, then there exist a CP map \(\phi \) and a matrix \(\kappa \) such that

If two Lindblad forms \(F,F'\) induce the same generator and both have traceless Kraus operators, then \(\phi =\phi '\) and \(\kappa '=\kappa +i\lambda \, \mathbb {1}\) for some \(\lambda \in \mathbb {R}\).

If two Lindblad forms \(F,F'\) induce the same generator and both have traceless Kraus operators, then their drift matrices differ by an imaginary scalar: \(\kappa ' = \kappa +i\lambda \, \mathbb {1}\) for some \(\lambda \in \mathbb {R}\). This is part of [ Wol12 , Proposition 7.4(2) ] .

If two Lindblad forms \(F,F'\) induce the same generator and both have traceless Kraus operators, then their CP parts agree: \(\phi = \phi '\). This is part of [ Wol12 , Proposition 7.4(2) ] .

\(L\) is a GKSL generator if and only if \(L\) is CCP and trace-annihilating.

\(L\) is a GKSL generator if and only if

with \(H=H^\dagger \). This is [ Wol12 , Theorem 7.1 ] .

If \(L(\rho )=\phi (\rho )-\kappa \rho -\rho \kappa ^\dagger \) with \(\phi \) CP and \(\phi ^*(\mathbb {1})=\kappa +\kappa ^\dagger \), then \(L\) is a GKSL generator. This is [ Wol12 , Theorem 7.1, Equation (7.20) ] .

If a GKSL generator preserves a nontrivial compression, then its Lindblad data can be chosen block-upper-triangular with respect to that compression.

For every \(X \in M_{D}(\mathbb {C})\) and every \(t \in \mathbb {R}\), \(\frac{d}{dt}(e^{tL}(X)) = e^{tL}(L(X))\).

For all \(t \in \mathbb {R}\), one has \(\frac{d}{dt}e^{tL} = e^{tL} \cdot L\). This is [ Wol12 , Equation (7.2) ] .

One has \(\left.\frac{d}{dt}e^{tL}\right|_{t=0} = L\).

If the Lindblad data are block-upper-triangular with respect to a nontrivial projection, then the associated semigroup preserves that compression.

If the Lindblad span is closed under Hermitian conjugation and its commutant is \(\mathbb {C}\cdot \mathbb {1}\), then the Lindblad data do not admit a block-upper-triangular decomposition.

If \(T_{t_0}\) is irreducible for some \(t_0{\gt}0\), then every positive-time slice \(T_s\) is irreducible.

If \(T_{t_0}=(T_u)^N\) and \(T_{t_0}\) is irreducible, then \(T_u\) is irreducible.

Let \(T_t=e^{tL}\) be a quantum dynamical semigroup. If \(T_{t_0}\) is irreducible for some \(t_0{\gt}0\), then \(T_t\) is primitive (and irreducible) for every \(t{\gt}0\). This is [ Wol12 , Proposition 7.5 ] .

A quantum dynamical semigroup is reducible if and only if its generator preserves some nontrivial compression.

Let \((H,\{ L_j\} )\) be a Lindblad form with \(\kappa =iH+\tfrac {1}{2}\sum _jL_j^\dagger L_j\), and let \(P\) be an orthogonal projection such that the generator preserves the compression \(PM_{D}(\mathbb {C})P\). If \((\mathbb {1}-P)L_jP=0\) for every \(j\), then \((\mathbb {1}-P)\kappa P=0\).

A linear map admits a Kossakowski form iff it admits a Lindblad form.

If the Kossakowski rank satisfies \(\operatorname{rank}(C)+d\ge d^2+1\), then no block-upper-triangular Lindblad form exists.

A Lindblad form with Hamiltonian \(H\) and operators \(\{ L_j\} \) defines the same linear map as the generator decomposition \((\phi ,\kappa )\) with \(\phi (\rho ) = \sum _j L_j\rho L_j^\dagger \) and \(\kappa = iH + \frac{1}{2}\sum _j L_j^\dagger L_j\). This is [ Wol12 , Equation (7.24) ] .

Every Lindblad form defines a conditionally completely positive map.

Every Lindblad form defines a trace-annihilating linear map.

For \(s \in [0,t]\), \(M = \sup _{u \in [0,t]}\| e^{uL}\| \), and \(M' = \sup _{u \in [0,t]}\| e^{uL'}\| \),

For \(s \in [0,t]\) and \(M = \sup _{u \in [0,t]} \| e^{uL}\| \),

If the algebra generated by \(\{ L_j\} \) and \(\kappa \) is the entire matrix algebra \(M_{D}(\mathbb {C})\), then the QDS is not reducible. This is [ Wol12 , Corollary 7.2(1) ] .

If the Lindblad span is Hermitian-closed and its commutant is trivial, then the QDS is not reducible. This is [ Wol12 , Corollary 7.2(2) ] .

If \(\operatorname{rank}(C){\gt}d^2-d\), then the QDS is not reducible. This is [ Wol12 , Corollary 7.2(3) ] .

If a GKSL generator admits no block-upper-triangular Lindblad form, then the associated quantum dynamical semigroup is not reducible.

For Pauli directions \(a,k\in \{ x,y,z\} \),

Consequently, the dissipator generated by \(\sigma _a\) is \(\mathcal D_{\sigma _a}(X)=\sigma _aX\sigma _a-X\).

If a quantum dynamical semigroup is irreducible at every positive time, then every peripheral eigenvalue of every positive-time slice equals \(1\).

Under the same hypotheses, every peripheral eigenvalue of \(T_u\) equals \(1\).

For every positive time \(t\), a quantum dynamical semigroup irreducible at every positive time has the following property: every peripheral eigenvalue \(\mu \) of \(T_t\) admits a nonzero eigenvector \(V\) and a positive integer \(p\) such that \(T_t(V)=\mu V\) and \(T_{pt}(V)=V\).

For every positive time \(t\), under the same hypotheses every peripheral eigenvalue of \(T_t\) has a nonzero eigenvector with nonzero trace.

For \(t \ge 0\),

where \(\Delta = L' - L\). This is [ Wol12 , Corollary 7.1 ] .

If \(E\) is an irreducible primitive channel with faithful fixed density \(\sigma \), then \(E^n(X)\to 0\) for every matrix \(X\) with \(\operatorname{tr}(X)=0\).

Under the same hypotheses, \(T_u\) is primitive.

If a quantum dynamical semigroup is irreducible at every positive time, then every positive-time slice is primitive.

For a quantum dynamical semigroup \(T_t=e^{tL}\),

This is [ Wol12 , Proposition 7.5 ] .

For \(u\ge 0\) and \(s{\gt}0\), one has \(r_n\in [0,s]\) and \(nu=m_ns+r_n\).

If the generator preserves \(PM_{D}(\mathbb {C})P\), then so does the exponential semigroup \(e^{tL}\) for every \(t\ge 0\).

If \(L\) is trace-annihilating, then \(e^{tL}\) is trace-preserving for every \(t \in \mathbb {R}\).

If \(e^{tL}\) is trace-preserving for every \(t \ge 0\), then \(L\) is trace-annihilating.

For every density matrix \(\rho \) and every observable \(A\),

If \((\phi , \kappa )\) satisfies \(\phi ^*(\mathbb {1}) = \kappa + \kappa ^\dagger \), then \(L(\rho ) = \phi (\rho ) - \kappa \rho - \rho \kappa ^\dagger \) is trace-annihilating.

Let \(N\in \mathbb {N}\) with \(N\ge 1\), let \(p\ge 0\) and \(\gamma _*{\gt}0\), and let \(\rho _t=\Phi _t(\rho )\) for matrices \(\rho ,\omega \in M_{2^N}(\mathbb {C})\). Assume at the time under consideration that

If

then \(\lVert \rho _t-\omega \rVert _1\le N^{-p}\). The complete modified logarithmic Sobolev estimate motivating the first hypothesis is [ GR22 , Theorems 1.1 and 3.3 ] ; that estimate, its tensorization, and quantum Pinsker remain hypotheses here.

Suppose \(a\ne b\) and \(\gamma _a,\gamma _b{\gt}0\). For \(X\in M_{2}(\mathbb {C})\), \(\mathcal L_{a,b}(X)=0\) if and only if \(X\in \mathbb {C}\mathbb {1}\).

For Pauli directions \(a,b,k\in \{ x,y,z\} \) and real rates \(\gamma _a,\gamma _b\),

Hence, when \(a\ne b\), the eigenvalues on \(\sigma _a\), \(\sigma _b\), and the remaining Pauli matrix are respectively \(-2\gamma _b\), \(-2\gamma _a\), and \(-2(\gamma _a+\gamma _b)\).

For Pauli directions \(a,b\in \{ x,y,z\} \), real rates \(\gamma _a,\gamma _b\), and all \(X,Y\in M_{2}(\mathbb {C})\),

Suppose \(a\ne b\) and \(\gamma _a,\gamma _b{\gt}0\). The state \(\tau _1\) is positive definite and belongs to \(\mathcal{D}_2\). Furthermore, \(\mathcal L_{a,b}(\tau _1)=0\), the kernel of \(\mathcal L_{a,b}\) is the one-dimensional space \(\mathbb {C}\tau _1\), and \(e^{t\mathcal L_{a,b}}(\tau _1)=\tau _1\) for every \(t\ge 0\). More generally, for \(X\in M_{2}(\mathbb {C})\),

The same equivalence holds for any autonomous semigroup \(T_t\) satisfying \(T_t=e^{t\mathcal L_{a,b}}\) at every non-negative time.

For a GKSL generator \(L:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), the four conditions

1. rank-deficient fixed density,

2. rank-deficient kernel element,

3. invariant compression, and

4. block-upper-triangular Lindblad form

are equivalent. This is [ Wol12 , Proposition 7.6 ] .

A rank-deficient density matrix is a fixed point of \(e^{tL}\) for all \(t\ge 0\) if and only if it lies in \(\ker L\). This is [ Wol12 , Proposition 7.6, (1)\(\Leftrightarrow \)(2) ] .

If \(L\) is a GKSL generator and there exists a rank-deficient fixed density matrix, then the semigroup preserves a nontrivial compression. This is [ Wol12 , Proposition 7.6, (1)\(\Rightarrow \)(3) ] .

For a GKSL generator \(L\), the semigroup preserves a nontrivial compression if and only if the Lindblad data can be chosen block-upper-triangular. This is [ Wol12 , Proposition 7.6, (3)\(\Leftrightarrow \)(4) ] .