19 Quantum Dynamical Semigroups
This chapter develops the theory of one-parameter semigroups of quantum channels and characterizes their generators via the GKSL (Gorini–Kossakowski–Sudarshan–Lindblad) theorem, following [ Wol12 , Chapter 7 ] .
Section 19.1 establishes definitions and the basic exponential form (Proposition 19.1.14). Section 19.2 gives the Duhamel formula and perturbation bound. Section 19.3 develops the generator theory and proves the GKSL theorem.
19.1 Dynamical semigroups and the exponential form
A family of linear maps \(T : \mathbb {R}\to (M_{D}(\mathbb {C}) \to _{\ell } M_{D}(\mathbb {C}))\) is a dynamical semigroup if
(semigroup law) \(T_{t+s} = T_t \circ T_s\) for all \(t,s \ge 0\); and
(initial condition) \(T_0 = \operatorname{id}\).
This is [ Wol12 , Equation (7.1) ] .
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.
Given a generator \(L \in \operatorname{End}_{\mathbb {C}}(M_{D}(\mathbb {C}))\), the exponential semigroup is
For all \(t,s \in \mathbb {R}\), \(e^{(t+s)L} = e^{tL} \cdot e^{sL}\).
Since \((t \cdot L)\) commutes with \((s \cdot L)\), this follows from the exponential addition formula for commuting elements in a Banach algebra.
The continuous-linear-map exponential satisfies \(e^{0L} = \mathbb {1}\).
The exponential of the zero endomorphism is the identity.
The map \(t \mapsto e^{tL}\) is continuous in the operator norm topology.
The exponential is analytic (convergence radius \(= \infty \)), hence continuous.
For all \(t \in \mathbb {R}\), one has \(\frac{d}{dt}e^{tL} = e^{tL} \cdot L\). This is [ Wol12 , Equation (7.2) ] .
Apply the chain rule to the composition \(\mathbb {R}\xrightarrow {t \mapsto t \cdot L} \operatorname{End}(M_{D}(\mathbb {C})) \xrightarrow {\exp } \operatorname{End}(M_{D}(\mathbb {C}))\).
One has \(\left.\frac{d}{dt}e^{tL}\right|_{t=0} = L\).
If \(e^{tL} = e^{tL'}\) for all \(t \ge 0\), then \(L = L'\).
Both semigroups agree on \([0,\infty )\), so their derivatives within \([0,\infty )\) at \(t=0\) agree. Since \([0,\infty )\) has unique differentials at \(0\), \(L = L'\).
The linear-map exponential semigroup and the continuous-linear-map exponential semigroup agree under the canonical identification of endomorphisms with continuous linear endomorphisms.
This is a direct unpacking of the definition of the linear-map exponential semigroup.
For every \(X \in M_{D}(\mathbb {C})\) and every \(t \in \mathbb {R}\), \(\frac{d}{dt}(e^{tL}(X)) = e^{tL}(L(X))\).
Evaluate the derivative of the continuous-linear-map semigroup at \(X\) and transport the statement through Theorem 19.1.10.
The linear-map exponential semigroup satisfies \(e^{0L} = \mathbb {1}\).
The family \(t \mapsto e^{tL}\) satisfies the semigroup law and the initial condition, hence is a dynamical semigroup.
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 ] .
Since \(T_0 = \mathbb {1}\) and \(T\) is continuous, the integral \(M_\varepsilon = \int _0^\varepsilon T_s\, ds\) is invertible for small \(\varepsilon {\gt} 0\). Using the semigroup property, \(T_t = M_\varepsilon ^{-1}(M_{t+\varepsilon } - M_t)\) is differentiable for \(t \ge 0\). ODE uniqueness then gives \(T_t = e^{tL}\) for all \(t \ge 0\).
19.2 Perturbation theory
Let \(\Delta = L' - L\). For \(t \ge 0\),
This is [ Wol12 , Lemma 7.1 ] .
Define \(f(s) = e^{(t-s)L}e^{sL'}\). Then \(f'(s) = e^{(t-s)L}\Delta \, e^{sL'}\) by Theorem 19.1.7, and \(e^{tL'} - e^{tL} = f(t) - f(0) = \int _0^t f'(s)\, ds\), which gives (??).
For \(t \ge 0\),
where \(\Delta = L' - L\). This is [ Wol12 , Corollary 7.1 ] .
Apply the triangle inequality and submultiplicativity of the operator norm to the Duhamel integral (??).
For \(t \ge 0\), if \(\| e^{sL}\| \le 1\) and \(\| e^{sL'}\| \le 1\) for every \(s \in [0,t]\), then \(\| e^{tL'} - e^{tL}\| \le t\, \| L' - L\| \).
Apply (??) and bound both suprema by \(1\).
Assume a factorial majorant of Dyson–Phillips iterates:
Then the Dyson series is summable in operator norm.
This is the Weierstrass M-test step, using the scalar convergence of \(\sum _n a^n/n!\) and lifting via norm-bounded summability.
For \(s \in [0,t]\) and \(M = \sup _{u \in [0,t]} \| e^{uL}\| \),
Induction on \(n\). The base case uses the semigroup supremum bound. The inductive step bounds the integrand pointwise by \(M\| \Delta \| \cdot M(u\| \Delta \| M)^n/n!\) and integrates \(\int _0^s u^n\, du = s^{n+1}/(n+1)\), recovering the factorial.
The Dyson–Phillips series \(\sum _n \widetilde{T}^{(n)}_t\) converges in operator norm for every \(t \ge 0\).
The bound (??) gives a factorial majorant. Lemma 19.2.4 then applies the Weierstrass M-test.
For \(t \ge 0\), setting \(s=t\) in (??) gives
where \(M = \sup _{u \in [0,t]} \| e^{uL}\| \).
This is the special case \(s=t\) of (??).
Each Dyson iterate \(t \mapsto \widetilde{T}^{(n)}_t\) is continuous in the time parameter.
By induction on \(n\). The base case is continuity of the matrix exponential. For the inductive step, the parametric integral \(t\mapsto \int _0^t e^{(t-s)L}\Delta \, \widetilde{T}^{(n)}_s\, ds\) is continuous by the continuous-parameter primitive theorem; the pasting lemma extends this to all of \(\mathbb {R}\) since \(\widetilde{T}^{(n+1)}_t = 0\) for \(t \le 0\).
For \(s \in [0,t]\), \(M = \sup _{u \in [0,t]}\| e^{uL}\| \), and \(M' = \sup _{u \in [0,t]}\| e^{uL'}\| \),
Induction on \(N\). The base case is the semigroup supremum bound. The inductive step uses the telescoping remainder identity \(R_{N+1}(s) = \int _0^s e^{(s-u)L}\Delta \, R_N(u)\, du\) derived from the Duhamel formula (??), bounds the integrand pointwise, and integrates to recover the factorial.
For \(t \ge 0\), the Dyson–Phillips series equals the perturbed semigroup:
By (??), the partial-sum remainder \(\| e^{tL'}-\sum _{n{\lt}N}\widetilde{T}^{(n)}_t\| \le M'(t\| \Delta \| M)^N/N!\) tends to zero as \(N \to \infty \), since \(c^N/N! \to 0\) for any constant \(c\). Together with the summability of the series (Corollary 19.2.6), this identifies the sum via uniqueness of limits.
One has \(\sum _{n=0}^{\infty }\widetilde{T}^{(n)}_t = e^{tL'}\) in tsum form.
The claim follows from (??).
19.3 GKSL/Lindblad generators
This section characterizes generators of semigroups of completely positive and trace-preserving maps. The central result is the GKSL theorem (Theorem 19.3.8.5), which shows that such generators have the standard Lindblad form.
19.3.1 Generator decomposition and conditional complete positivity
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) ] .
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 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) ] .
If \((\phi , \kappa )\) satisfies \(\phi ^*(\mathbb {1}) = \kappa + \kappa ^\dagger \), then \(L(\rho ) = \phi (\rho ) - \kappa \rho - \rho \kappa ^\dagger \) is trace-annihilating.
Let \(\phi (\rho ) = \sum _i K_i\rho K_i^\dagger \). By the cyclic property of trace, \(\operatorname{tr}(L(\rho )) = \operatorname{tr}((\sum _i K_i^\dagger K_i)\rho ) - \operatorname{tr}(\kappa \rho ) - \operatorname{tr}(\kappa ^\dagger \rho ) = \operatorname{tr}((\kappa +\kappa ^\dagger )\rho ) - \operatorname{tr}(\kappa \rho )-\operatorname{tr}(\kappa ^\dagger \rho )=0\).
19.3.2 The Lindblad form
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) ] .
Every Lindblad form defines a trace-annihilating linear map.
The Hamiltonian part satisfies \(\operatorname{tr}(i[\rho ,H]) = 0\) by the cyclic property. Each dissipator term satisfies \(\operatorname{tr}(L_j\rho L_j^\dagger ) = \operatorname{tr}(L_j^\dagger L_j\rho ) = \operatorname{tr}(\rho L_j^\dagger L_j)\), so the three terms sum to zero.
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) ] .
Compute \(\kappa ^\dagger = -iH+\frac{1}{2}\sum _j L_j^\dagger L_j\) using \(H^\dagger = H\) and \((A^\dagger A)^\dagger = A^\dagger A\). Setting \(S = \sum _j L_j^\dagger L_j\), expand \(\phi (\rho )-\kappa \rho -\rho \kappa ^\dagger = \sum _j L_j\rho L_j^\dagger -iH\rho -\tfrac {1}{2}S\rho +i\rho H-\tfrac {1}{2}\rho S\), which is \(i[\rho ,H]+\sum _j(L_j\rho L_j^\dagger -\tfrac {1}{2}L_j^\dagger L_j\rho -\tfrac {1}{2}\rho L_j^\dagger L_j)\).
Every Lindblad form defines a conditionally completely positive map.
By Theorem 19.3.2.3, the Lindblad form equals a generator decomposition with \(\phi \) CP.
The commutator form of the Lindblad equation writes the generator as
where \([A,B] = AB - BA\). This is [ Wol12 , Equation (7.22) ] .
Expanding the commutator brackets \([L_j,\rho L_j^\dagger ]\) and \([L_j\rho ,L_j^\dagger ]\) in (??) yields the standard dissipator terms in (??).
19.3.3 Characterization of conditional complete positivity
If \(L\) is CCP and \(P = \mathbb {1}- |\Omega \rangle \! \langle \Omega |\), then
This is [ Wol12 , Proposition 7.2 ] .
Write \(L(\rho ) = \phi (\rho )-\kappa \rho -\rho \kappa ^\dagger \). The left- and right-multiplication terms are annihilated by \(P\), so the projected Choi matrix reduces to the projected Choi matrix of \(\phi \), which is positive.
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 ] .
Decompose the Hermitian Choi matrix as \(\tau _L = Q-|\psi \rangle \langle \Omega |-|\Omega \rangle \langle \psi |\) with \(Q \ge 0\) supported on \(|\Omega \rangle ^\perp \). The Choi–Jamiolkowski isomorphism applied to \(Q\) gives the CP map \(\phi \), and \((\kappa \otimes \mathbb {1})|\Omega \rangle = |\psi \rangle \) defines \(\kappa \).
19.3.4 Completely positive semigroups and conditionally completely positive generators
If \(e^{tL}\) is CP for all \(t \ge 0\), then \(L\) is CCP. This is [ Wol12 , Proposition 7.3 ] .
From \((e^{tL}\otimes \operatorname{id})(|\Omega \rangle \! \langle \Omega |) \ge 0\), expand at infinitesimal \(t\) and project onto \(P\) to get \(P(L\otimes \operatorname{id})(|\Omega \rangle \! \langle \Omega |)P \ge 0\). Then Theorem 19.3.3.2 gives CCP.
If \(L\) is CCP, then \(e^{tL}\) is CP for all \(t \ge 0\). This is [ Wol12 , Proposition 7.3 ] .
Approximate \(e^{tL}\) by the completely positive Euler steps
Each step is CP, hence so are its powers. These powers converge in operator norm to \(e^{tL}\), and the cone of CP maps is closed.
\(e^{tL}\) is CP for all \(t \ge 0\) if and only if \(L\) is CCP.
19.3.5 Freedom in generator representation
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 ] .
Set \(c_j = -\operatorname{tr}(K_j)/d\).
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.
Direct algebraic expansion.
19.3.6 Uniqueness of the traceless Lindblad form
A Lindblad form has traceless Kraus operators if \(\operatorname{tr}(L_j) = 0\) for every \(j\).
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) ] .
Compare projected Choi matrices and use Choi injectivity.
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) ] .
After identifying \(\phi = \phi '\) (Theorem 19.3.6.2), the generator equality forces \(\Delta \kappa \cdot \rho +\rho \cdot (\Delta \kappa )^\dagger =0\) for all \(\rho \). Setting \(\rho =\mathbb {1}\) gives \(\Delta \kappa +(\Delta \kappa )^\dagger =0\) (skew-Hermiticity), which converts the equation to \([\Delta \kappa ,\rho ]=0\) for all \(\rho \). The scalar commutant lemma gives \(\Delta \kappa =c\cdot \mathbb {1}\), and skew-Hermiticity forces \(c=i\lambda \) for some \(\lambda \in \mathbb {R}\).
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 \(\lambda \in \mathbb {R}\).
19.3.7 Trace-annihilation and trace preservation
If \(L\) is trace-annihilating, then \(e^{tL}\) is trace-preserving for every \(t \in \mathbb {R}\).
For fixed \(\rho \), the function \(f(t)=\operatorname{tr}(e^{tL}(\rho ))\) has derivative \(f'(t)=\operatorname{tr}(L(e^{tL}(\rho )))=0\), so \(f\) is constant and \(\operatorname{tr}(e^{tL}(\rho ))=f(t)=f(0)=\operatorname{tr}(\rho )\).
If \(e^{tL}\) is trace-preserving for every \(t \ge 0\), then \(L\) is trace-annihilating.
Differentiate the identity \(\operatorname{tr}(e^{tL}(\rho ))=\operatorname{tr}(\rho )\) at \(t=0\) from the right.
19.3.8 The GKSL/Lindblad theorem
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 \(t \ge 0\).
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 \(L\) is a GKSL generator, then there exist a CP map \(\phi \) and a matrix \(\kappa \) such that
\(L\) 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 ] .
Forward: apply Theorem 19.3.8.3 to write \(L(\rho )=\phi (\rho )-\kappa \rho -\rho \kappa ^\dagger \) with trace constraint. Then choose traceless Kraus operators by Theorem 19.3.5.1 and rewrite \(\kappa \) as \(iH+\frac{1}{2}\phi ^*(\mathbb {1})\); Theorem 19.3.2.3 gives the Lindblad formula (??). Reverse: Theorems 19.3.2.4 and 19.3.2.2 show that every Lindblad form satisfies the right-hand side of Theorem 19.3.8.4.
19.3.9 Kossakowski matrix form
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.
A linear map admits a Kossakowski form iff it admits a Lindblad form.
Forward: diagonalize \(C=B^\dagger B\) and set \(L_j=\sum _k B_{jk}F_k\). Reverse: keep the same Hamiltonian and family \(F_k=L_k\), and take \(C=\mathbb {1}\).
19.4 Dissipation generated by two Pauli matrices
For Pauli directions \(a,k\in \{ x,y,z\} \),
Consequently, the dissipator generated by \(\sigma _a\) is \(\mathcal D_{\sigma _a}(X)=\sigma _aX\sigma _a-X\).
Direct multiplication of the three Pauli matrices proves the identities; substituting \(\sigma _a^\dagger =\sigma _a\) and \(\sigma _a^2=\mathbb {1}\) into the Lindblad dissipator gives the final formula.
For Pauli directions \(a,b\in \{ x,y,z\} \) and real rates \(\gamma _a,\gamma _b\), define
In particular, \(\mathcal L_{a,b}(\mathbb {1})=0\).
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 \(-2(\gamma _a+\gamma _b)\).
Apply Theorem 19.4.1 separately to the two dissipators in (??) and collect the two scalar coefficients.
For Pauli directions \(a,b\in \{ x,y,z\} \), real rates \(\gamma _a,\gamma _b\), and all \(X,Y\in M_{2}(\mathbb {C})\),
Cyclicity of the trace makes each map \(X\mapsto \sigma _aX\sigma _a-X\) symmetric for the Hilbert–Schmidt pairing. Real linear combinations preserve this symmetry.
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 \(X\in \mathbb {C}\mathbb {1}\).
Expand \(X\) in the basis \(\{ \mathbb {1},\sigma _x,\sigma _y,\sigma _z\} \). The three Pauli coefficients vanish because the three eigenvalues in (??) are nonzero, while the identity coefficient is unrestricted.
The maximally mixed qubit state is \(\tau _1=\frac12\mathbb {1}_2\).
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.
The defining formula gives \(\mathcal L_{a,b}(\tau _1)=0\). Theorem 19.4.5 identifies its kernel, and exponentiating a kernel vector leaves it fixed. Conversely, a vector fixed for every non-negative time lies in the generator kernel.
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.
Entropy decay and Pinsker first give
The time hypothesis (??) implies \(e^{-\gamma _*t/2}\le N^{-2p}/(2N\log 2)\), so the squared trace norm is at most \(N^{-2p}\). Both sides are non-negative, and taking square roots gives the result.
19.5 Primitivity and irreducibility of QDS
19.5.1 Auxiliary spectral and semigroup lemmas
A quantum dynamical semigroup is a norm-continuous dynamical semigroup whose time slices are quantum channels for every \(t \ge 0\).
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.
If all eigenvalues satisfy \(|\lambda |{\lt}1\), then the spectral radius is \({\lt}1\).
If \(E\) is an irreducible primitive channel with faithful fixed density \(\sigma \), then \(E^n(X)\to 0\) for every matrix \(X\) with \(\operatorname{tr}(X)=0\).
Decompose \(E^n\) into the fixed-point projection and its complementary part. On the complement, every eigenvalue has modulus \({\lt}1\), so the complementary powers tend to \(0\); the trace-zero hypothesis removes the fixed-point component.
If \(E\) is an irreducible channel with faithful fixed density \(\sigma \), then every nonzero fixed point \(V\) of \(E\) satisfies \(V=\operatorname{tr}(V)\sigma \).
Subtract the trace multiple \(\operatorname{tr}(V)\sigma \) and apply the vanishing theorem for trace-zero fixed points of an irreducible channel.
Every nonzero fixed point of an irreducible channel has nonzero trace.
Otherwise the preceding theorem would force the fixed point to vanish.
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 \(T_{pt}(V)=V\).
Peripheral powers remain peripheral for an irreducible channel with faithful fixed point. The bounded-order lemma then gives \(\mu ^p=1\), and the semigroup power identity turns the eigenvector equation for \(T_t\) into a fixed-point equation for \(T_{pt}\).
For every positive time \(t\), under the same hypotheses every peripheral eigenvalue of \(T_t\) has a nonzero eigenvector with nonzero trace.
Apply Theorem 19.5.1.16 and then use the previous corollary at the irreducible time \(pt\).
If \(E\) is trace-preserving, \(E(V)=\mu V\), and \(\operatorname{tr}(V)\ne 0\), then \(\mu =1\).
Take traces in \(E(V)=\mu V\) and use trace preservation.
If a quantum dynamical semigroup is irreducible at every positive time, then every peripheral eigenvalue of every positive-time slice equals \(1\).
If a quantum dynamical semigroup is irreducible at every positive time, then every positive-time slice is primitive.
The unique faithful fixed density gives the one-dimensional peripheral spectrum criterion, and Theorem 19.5.1.19 collapses the peripheral set to \(\{ 1\} \).
Starting from one irreducible time \(t_0\), we construct a positive time step \(u=t_0/N\) with \(T_{t_0}=(T_u)^N\), prove peripheral eigenvalue collapse at time \(u\), and deduce primitivity of \(T_u\).
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 \(u\ge 0\).
By semigroup commutativity, \(T_{t_0}(T_u(\sigma ))=T_u(T_{t_0}(\sigma ))=T_u(\sigma )\), and uniqueness of the fixed density at time \(t_0\) identifies \(T_u(\sigma )\) with \(\sigma \).
If \(T_{t_0}\) is irreducible with faithful fixed density \(\sigma \), then every fixed point \(V\) of \(T_{t_0}\) satisfies \(V=\operatorname{tr}(V)\sigma \).
This is the fixed-point trace-scalar theorem for the irreducible channel \(T_{t_0}\).
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 \(r_n=s\, \mathrm{fract}\! \left(nu/s\right)\).
For \(u\ge 0\) and \(s{\gt}0\), one has \(r_n\in [0,s]\) and \(nu=m_ns+r_n\).
This is the floor-plus-fraction decomposition of \(nu/s\), multiplied by \(s\).
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 \(T_a(\delta )=0\).
The residual slice times lie in the compact interval \([0,s]\), so a subsequence converges to some \(a\in [0,s]\). The residual decomposition rewrites \(T_{nu}(\delta )\) in terms of \(T_{r_n}(\delta )\), and continuity of the semigroup identifies the limit with \(T_a(\delta )\).
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.
Under the same hypotheses, every peripheral eigenvalue of \(T_u\) equals \(1\).
Under the same hypotheses, \(T_u\) is primitive.
If \(T_{t_0}=(T_u)^N\) and \(T_{t_0}\) is irreducible, then \(T_u\) is irreducible.
Any invariant compression for \(T_u\) remains invariant under powers, hence also for \(T_{t_0}\); irreducibility of \(T_{t_0}\) rules this out.
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 \(T_u(\sigma )=\sigma \).
Take \(u=t_0/N\) with \(N=(\dim M_{D}(\mathbb {C}))!\). The semigroup power identity gives the factorization of \(T_{t_0}\), channelity comes from the semigroup hypotheses, and Theorem 19.5.1.32 gives irreducibility of \(T_u\).
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 \(\tau =\operatorname{tr}(\tau )\sigma \).
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.
If \(T_{t_0}\) is irreducible for some \(t_0{\gt}0\), then every positive-time slice \(T_s\) is irreducible.
Let \(\sigma \) be the unique faithful fixed density of \(T_{t_0}\). By Theorem 19.5.1.23, it is fixed for all times, and Theorem 19.5.1.24 makes the fixed-point space of \(T_{t_0}\) one-dimensional. Choose a primitive reduced time step by Theorem 19.5.1.35; then the trace-scalar description of fixed points from Theorem 19.5.1.34 gives the uniqueness criterion for irreducibility at every positive time.
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 ] .
For a quantum dynamical semigroup \(T_t=e^{tL}\),
This is [ Wol12 , Proposition 7.5 ] .
Forward: apply Theorem 19.5.1.37. Reverse: take \(t_0=1\) (or any positive time); the right-hand side already includes irreducibility of \(T_t\) for every \(t{\gt}0\).
19.6 Kernel of the adjoint Liouvillian
A generator \(L\) has a faithful stationary state if there exists a positive definite density matrix \(\rho _0\) with \(L(\rho _0)=0\).
For a Lindblad operator \(L_j\), the adjoint dissipator on observables is
The adjoint generator of a Lindblad form is
For every density matrix \(\rho \) and every observable \(A\),
Expand the Schrödinger and Heisenberg formulas and apply cyclicity of the trace term by term.
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 adjoint kernel is the set of observables \(A\) satisfying \(L^*(A)=0\).
If an observable commutes with \(H\), with every \(L_j\), and with every \(L_j^\dagger \), then it lies in \(\ker (L^*)\).
The Hamiltonian commutator term vanishes, and each adjoint dissipator \(\mathcal{D}_{L_j}^*(A)\) is zero under the commutation hypotheses.
Let \(L\) be a GKSL generator in Lindblad form with Hamiltonian \(H\) and Lindblad operators \(\{ L_j\} \). Define the adjoint generator \(L^*\) by (??). 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 ] .
From \(L^*(A)=0\) and \(L^*(A^\dagger )=0\), compute \(L^*(A^\dagger A)=\sum _j[A,L_j]^\dagger [A,L_j]\). Faithfulness of \(\rho _0\) and \(\operatorname{tr}(\rho _0L^*(A^\dagger A))=0\) force \([A,L_j]=0\) for all \(j\). Similarly, \([A,L_j^\dagger ]=0\), and then \(L^*(A)=0\) directly gives \([A,H]=0\).
The inclusion \(\supseteq \) is the easy direction (commutant elements are in the kernel by a direct calculation). The reverse inclusion \(\subseteq \) is Theorem 19.6.8.
19.7 Reducibility of quantum dynamical semigroups
The four equivalent reducibility conditions of [ Wol12 , Proposition 7.6 ] are as follows.
A projection is nontrivial if it is orthogonal and neither \(0\) nor \(\mathbb {1}\).
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 \(L(\rho _0)=0\).
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 \(t\ge 0\).
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 \(\kappa =iH+\frac{1}{2}\sum _jL_j^\dagger L_j\).
For a projection \(P\), the generator preserves the compression \(PM_{D}(\mathbb {C})P\) if, for every \(X\in M_{D}(\mathbb {C})\),
A quantum dynamical semigroup is reducible if it has a nontrivial invariant compression.
If the semigroup preserves \(PM_{D}(\mathbb {C})P\) for all non-negative times, then the generator preserves the same compression.
Differentiate the identity \(Pe^{tL}(PXP)P=e^{tL}(PXP)\) at \(t=0\).
If the generator preserves \(PM_{D}(\mathbb {C})P\), then so does the exponential semigroup \(e^{tL}\) for every \(t\ge 0\).
Apply the compression map termwise to the exponential series. Every iterate of \(L\) preserves the compression, so the whole series does as well.
A quantum dynamical semigroup is reducible if and only if its generator preserves some nontrivial compression.
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) ] .
The equivalence \(L(\rho )=0\iff e^{tL}(\rho )=\rho \) for all \(t\ge 0\) follows from the generator–semigroup correspondence: differentiate at \(t=0\) for the forward direction, and exponentiate for the reverse.
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) ] .
Let \(\rho _0\) be a rank-deficient fixed density. Let \(P\) be the support projection of \(\rho _0\). Since \(\rho _0\) is rank-deficient, \(P\neq \mathbb {1}\); since \(\rho _0\neq 0\), \(P\neq 0\). Channel positivity and trace preservation force \(e^{tL}\) to preserve \(PM_{D}(\mathbb {C})P\).
If a GKSL generator preserves a nontrivial compression, then its Lindblad data can be chosen block-upper-triangular with respect to that compression.
First pass from semigroup invariance to generator invariance by Theorem 19.7.8. The generator-level vanishing condition forces \((\mathbb {1}-P)L_jP=0\) and \((\mathbb {1}-P)\kappa P=0\) through the sum-of-squares identity.
If the Lindblad data are block-upper-triangular with respect to a nontrivial projection, then the associated semigroup preserves that compression.
The block-upper-triangular relations imply generator-level compression invariance; then apply Theorem 19.7.9.
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) ] .
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) ] .
The block-upper-triangular structure implies \(L\) preserves the compression \(PM_{D}(\mathbb {C})P\). Therefore the semigroup \(e^{tL}\) also preserves \(PM_{D}(\mathbb {C})P\) for all \(t\ge 0\). By the Brouwer fixed-point theorem, \(e^{(1/m)L}\) has a fixed density matrix \(\rho _m\) supported in \(PM_{D}(\mathbb {C})P\) for each \(m\). Passing to a subsequential limit gives a density matrix \(\rho _0\) with \(P\rho _0P=\rho _0\) and \(L(\rho _0)=0\). Since \(P\neq \mathbb {1}\), the matrix \(\rho _0\) has nontrivial kernel.
For a GKSL generator \(L:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), the four conditions
rank-deficient fixed density,
rank-deficient kernel element,
invariant compression, and
block-upper-triangular Lindblad form
are equivalent. This is [ Wol12 , Proposition 7.6 ] .
The cycle (1)\(\leftrightarrow \)(2), (1)\(\to \)(3), (3)\(\leftrightarrow \)(4), and (4)\(\to \)(2) closes the equivalence chain.
19.7.1 Sufficient conditions for non-reducibility
The \(\mathbb {C}\)-linear span of the Lindblad operators \(\{ L_j\} \).
The span of the Lindblad operators is closed under Hermitian conjugation, i.e. it forms a \(*\)-subspace of \(M_{D}(\mathbb {C})\).
The commutant of the Lindblad family \(\{ L_j\} '\) equals \(\mathbb {C}\cdot \mathbb {1}\), i.e. the Lindblad operators act irreducibly on \(M_{D}(\mathbb {C})\).
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.
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 \((\mathbb {1}-P)\kappa P=0\).
From \(L=\varphi -\kappa (\cdot )-(\cdot )\kappa ^\dagger \) and \((\mathbb {1}-P)L(P)=0\): the \((\mathbb {1}-P)\varphi (P)\) term vanishes because each \((\mathbb {1}-P)L_jP=0\), and the \((\mathbb {1}-P)(P\kappa ^\dagger )\) term vanishes because \((\mathbb {1}-P)P=0\). What remains is \((\mathbb {1}-P)\kappa P=0\).
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.
Assume for contradiction that a block-upper-triangular decomposition exists with nontrivial projection \(P\). By Theorem 19.7.1.5, \((\mathbb {1}-P)\kappa P=0\). Induction on elements of the subalgebra shows \((\mathbb {1}-P)AP=0\) for every generator. Since the generated algebra is all of \(M_{D}(\mathbb {C})\), every matrix satisfies \((\mathbb {1}-P)AP=0\), forcing \(P=0\) or \(P=\mathbb {1}\)—a contradiction.
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.
Assume for contradiction that a block-upper-triangular decomposition exists with nontrivial projection \(P\). The block vanishing \((\mathbb {1}-P)L_jP=0\) and Hermitian closure give \(PL_j(\mathbb {1}-P)=0\) as well, so \([P,L_j]=0\) for all \(j\). But then \(P\) lies in the commutant of the Lindblad span, contradicting triviality.
If the Kossakowski rank satisfies \(\operatorname{rank}(C)+d\ge d^2+1\), then no block-upper-triangular Lindblad form exists.
A block-upper-triangular traceless Lindblad family lies in a proper subspace whose dimension is at most \(d^2-d\). The formal rank minimization argument then contradicts the large-rank bound.
If a GKSL generator admits no block-upper-triangular Lindblad form, then the associated quantum dynamical semigroup is not reducible.
A reducible semigroup would have an invariant compression, and Theorem 19.7.13 would then produce a block-upper-triangular Lindblad form.
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) ] .