Dependencies

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

Every density matrix is nonzero.

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 \(e^{tL}\).

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 (8) 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 \(j\).

For a complex square matrix \(H\) and \(t\in \mathbb {R}\), set \(U_H(t)=e^{itH}\).

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 fixed basis \(F_1,\ldots ,F_{d^2-1}\) of the traceless matrices in \(M_{d}(\mathbb {C})\), and a positive semidefinite matrix \(C\in M_{d^2-1}(\mathbb {C})\), defining

This is [ Wol12 , Theorem 7.1, Equation (7.23) ] . Allowing an arbitrary finite family \(\{ F_k\} \) gives the corresponding algebraic form used in the factorization argument.

The infimum of the numbers of Lindblad operators across all representations of \(L\). When such a representation exists, this is the minimum; the definition is totalized to zero otherwise.

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 a map \(\alpha :M_D(\mathbb {C})\to M_D(\mathbb {C})\), a matrix \(X_0\), and an index \(a\), define \(X_{ij}=(\alpha (E_{ja})X_0)_{ia}\).

A semigroup \(T\) relaxes to a density matrix \(\rho _\infty \) if \(\rho _\infty {\gt}0\) and

for every density matrix \(\rho \). A generator \(L\) has a simple faithful kernel if its kernel consists precisely of the scalar multiples of a positive-definite density matrix.

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.

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 \(\tau _1=\frac12\mathbb {1}_2\).

The factorization \(C=(\sqrt C)^\dagger \sqrt C\) gives Lindblad operators \(L_j=\sum _k(\sqrt C)_{jk}F_k\) for the same generator. Their linear span has dimension \(\operatorname{rank}(C)\).

If every Lindblad operator belongs to a linear subspace \(V\), then the same generator has a Lindblad representation with at most \(\dim V\) operators.

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 (35). 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 12.8.8,

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 12.5.2.5) defines the same linear map as the standard Lindblad form (Definition 12.5.2.1).

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 \(t \ge 0\). Moreover, the same formula \(T_z=e^{zL}\) for \(z\in \mathbb {C}\) gives a complex parameter group, whose restriction to \(\mathbb {R}\) gives the real parameter group:

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 \(\mu =1\).

Assume \(d {\gt} 0\). Given any Kraus operators \(\{ L_i\} \), there exist \(c_i \in \mathbb {C}\) such that \(L_i' = L_i+c_i\mathbb {1}\) is traceless. Hence every Lindblad form has an equivalent one with traceless Lindblad operators; shifting preserves any linear subspace that contains \(\mathbb {1}\). This is the final assertion of [ Wol12 , Proposition 7.4(1) ] .

For complex square matrices \(X,Y\), of possibly different sizes,

In particular, for every complex square matrix \(H\) and \(t\in \mathbb {R}\),

Let \(K\) be a complex square matrix with \(K\notin \mathbb {C}\mathbb {1}\). There exists \(\varepsilon {\gt}0\) such that \(e^{itK}\notin \mathbb {C}\mathbb {1}\) whenever \(0{\lt}|t|{\lt}\varepsilon \). Hermiticity of \(K\) is not required.

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 \(V=\operatorname{tr}(V)\sigma \).

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.

Suppose \(\phi (\rho )=\sum _iL_i\rho L_i^\dagger \). If

where \(\phi '(\rho )=\sum _iL_i'\rho (L_i')^\dagger \), then \((\phi ',\kappa ')\) and \((\phi ,\kappa )\) define the same generator. This is [ Wol12 , Proposition 7.4(1), Equations (7.18)–(7.19) ] .

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

Suppose \((\phi ,\kappa )\) and \((\phi ',\kappa ')\) represent the same generator, and let \(\{ L_i\} _{i=0}^{r-1}\) and \(\{ L_i'\} _{i=0}^{r'-1}\) be traceless Kraus representations of \(\phi \) and \(\phi '\), respectively. Then

for some \(\lambda \in \mathbb {R}\). Put \(N=\max \{ r,r'\} \) and pad the smaller family with zero operators to obtain \(\{ \widetilde L_j\} _{j=0}^{N-1}\) and \(\{ \widetilde L_i'\} _{i=0}^{N-1}\). There is a unitary \(U\in M_N(\mathbb {C})\) such that

for every \(i\). This is [ Wol12 , Proposition 7.4(2) ] .

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) ] .

A linear map \(L:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) generates a continuous dynamical semigroup of quantum channels if and only if it has the Kossakowski form (23) in a fixed basis of the traceless matrices, with \(C\geq 0\). This is [ Wol12 , Theorem 7.1, Equations (7.20)–(7.23) ] .

\(L\) 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.

For every complex square matrix \(H\), the path \(U_H\) is continuous, \(U_H(0)=\mathbb {1}\), and \(U_H(s+t)=U_H(s)U_H(t)\) for all real \(s,t\). If \(H=H^\dagger \), then \(U_H(t)U_H(t)^\dagger =\mathbb {1}\), so \(U_H\) is a continuous homomorphism from \((\mathbb {R},+)\) to the unitary group.

Let \(D{\gt}0\) and let \(\alpha _t:M_D(\mathbb {C})\to M_D(\mathbb {C})\) be a family for which \(t\mapsto \alpha _t(M)\) is differentiable at zero for each \(M\). Suppose \(\alpha _0(M)=M\), and for every real \(t\) there exists a unitary \(V_t\) with \(\alpha _t(M)=V_tMV_t^\dagger \) for every \(M\). Then there exists \(H=H^\dagger \) such that

No regularity of the matrices \(V_t\) is required.

If \(T_{t_0}\) is irreducible for some \(t_0{\gt}0\), then every positive-time slice \(T_s\) 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 an algebraic Kossakowski form if and only if it admits a Lindblad form.

For a generator written in Wolf’s traceless-basis form, the minimum number of Lindblad operators equals \(\operatorname{rank}(C)\).

Two sets of Kraus operators \(\{ K_j\} \) and \(\{ \widetilde{K}_\ell \} \) represent the same completely positive map if and only if there is a unitary \(U\) with \(K_j=\sum _\ell U_{j\ell }\widetilde{K}_\ell \), where the smaller set is padded with zero operators; equivalently, after padding, the families are related by an isometry \(V\) with \(V^\dagger V=\mathbb {1}\). This is item 4 of [ Wol12 , Theorem 2.1 ] .

If the Kossakowski matrix satisfies \(\operatorname{rank}(C){\gt}d^2-d\), 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.

If \(\alpha (M)=YMY^{-1}\) with \(Y\) invertible, then \(X=(Y^{-1}X_0)_{aa}Y\) and \(\alpha (M)X=XM\) for every matrix \(M\).

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 a GKSL generator \(L\), the semigroup is not reducible if and only if \(e^{t_0L}\) is irreducible for some \(t_0{\gt}0\). Consequently, a non-reducible semigroup relaxes to a faithful stationary density matrix \(\rho _\infty \), and \(\rho _\infty \) spans \(\ker L\).

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 \(1\).

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 \(T_{pt}(V)=V\).

Let \(D\geq 1\), and let \(E\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be an irreducible quantum channel. Then every peripheral eigenvalue of \(E\) is a root of unity.

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

For \(t \ge 0\) and every operator norm induced by a norm on the finite-dimensional state space,

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 \(\operatorname{tr}(X)=0\).

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 ] .

Let \(t\ne 0\). If \(e^{tL}(X)=X\) but \(L(X)\ne 0\), then \(L\) has a nonzero eigenvalue \(\mu \) satisfying \(e^{t\mu }=1\).

Every eigenvalue \(\lambda \) of \(e^{tL}\) has the form \(\lambda =e^{t\mu }\) for an eigenvalue \(\mu \) of \(L\).

More generally, if \(\alpha \) is irrational and both \(e^{i\theta }\) and \(e^{i\alpha \theta }\) are roots of unity, then \(\theta =0\). In particular, if both \(e^{i\theta }\) and \(e^{i\pi \theta }\) are roots of unity, then \(\theta =0\).

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

For every finite-dimensional generator \(L\), there is a \(\delta {\gt}0\) such that, for all \(0{\lt}t{\lt}\delta \),

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.

Let \(T_t=e^{tL}:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) be a dynamical semigroup of trace-preserving completely positive maps, with generator represented as in Theorem 7.1. There is a positive-definite stationary density matrix \(\rho _\infty \) such that

for every density matrix \(\rho \), if one of the following holds:

1. the algebra generated by the Lindblad operators \(\{ L_j\} \) and \(\kappa \) is all of \(M_{d}(\mathbb {C})\);

2. the linear span of \(\{ L_j\} \) is Hermitian and has commutant \(\mathbb {C}\mathbb {1}\);

3. the Kossakowski matrix in a fixed traceless basis satisfies \(\operatorname{rank}(C){\gt}d^2-d\).

Moreover, the same \(\rho _\infty \) spans \(\ker L\). This is [ Wol12 , Corollary 7.2 ] .

Let \(T_t=e^{tL}\), \(t\in \mathbb {R}_+\), be a norm-continuous dynamical semigroup of trace-preserving completely positive maps. The following statements are equivalent:

1. There is a \(t_0{\gt}0\) such that \(T_{t_0}\) is irreducible.

2. \(T_t\) is irreducible for every \(t{\gt}0\).

3. \(T_t\) is primitive for every \(t{\gt}0\).

4. There is a \(\rho _\infty {\gt}0\) such that Eq. 31 holds for every density matrix \(\rho \).

5. \(L\) has a one-dimensional kernel consisting of multiples of a positive-definite density matrix \(\rho _\infty \).

This is [ Wol12 , Proposition 7.5 ] .

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) ] .