Universal constraint for relaxation rates of semigroups of qubit Schwarz maps

Unital qubit Schwarz maps interpolate between positive and completely positive maps. It is shown that the relaxation rates of the qubit semigroups of unital maps enjoying the Schwarz property satisfy a universal constraint, which provides a modification of the corresponding constraint known for completely positive semigroups. As an illustration, we consider two paradigmatic qubit semigroups: Pauli dynamical maps and phase-covariant dynamics. This result has two interesting implications: it provides a universal constraint for the spectra of qubit Schwarz maps and gives rise to a necessary condition for a Schwarz qubit map to be Markovian.


Introduction
Markovian dynamical semigroups are governed by the celebrated GKLS master equation [1][2][3] where H denotes an effective system Hamiltonian, L k are noise operators, and γ k > 0 are positive transition rates (We set = 1 and use the standard notations for the commutator and anti-commutator: [A, B] := AB − BA, {A, B} := AB + BA, respectively.)It gives rise to the general representation for the generator of Markovian semigroup {Λ t = e tL } t≥0 of completely positive trace-preserving maps (CPTP) [4,5].Solutions of GKLS master equation define very good approximations to evolutions of many real systems evolution, provided the system environment interaction is sufficiently weak and there is a large enough separation of timescales for the system and environment [3,[6][7][8].The requirement for complete positivity gives rise to nontrivial constraints among relaxation rates which characterize the evolution of the system [9][10][11].They provide information on how fast the system relaxes to an asymptotic state and/or how fast it decoheres, and are expected to serve as an experimental verification of the validity of complete positivity in open quantum dynamics.The spectrum of any GKLS generator is in general complex, however, if ℓ belongs to the spectrum then so does ℓ * .Moreover, there is a leading eigenvalue ℓ 0 = 0 (the corresponding eigenvector corresponds to an invariant state of the evolution), and all remaining eigenvalues satisfy Re ℓ k ≤ 0. The corresponding relaxation rates are defined by Γ k := −Re ℓ k (for more detailed exposition of spectral properties of maps and generators cf.e.g.[12][13][14]).It should be stressed that contrary to rates γ k which depend upon a particular representation of L, the relaxation rates Γ k can be measured in the laboratory.The properties of relaxation properties of GKLS generators were studied by many authors [3,[16][17][18][19][20].For two-level systems a universal constraint was derived in [9] where there are three rates {Γ 1 , Γ 2 , Γ 3 } which satisfy Interestingly, this reproduces [2,9] the celebrated constraint between transversal Γ T and longitudinal Γ L rates: The above constraint was well tested in several experiments [3,23].For any d-level system, universal constraints were derived in [10,15].Moreover, in [11] it was conjectured that the following constraint is both the universal and the tight where now Γ = Γ 1 + . . .+ Γ d 2 −1 .This provides generalization of (2) for arbitrary (but finite) number of levels.
Here, 'universality' means that the constraint is valid for any completely positive dynamical semigroup, i.e., for any GKLS master equation.The bound (4) is tight (that is, it cannot be further improved) since for any 'd' there exists a GKLS generator for which the inequality in ( 4) is saturated.This conjecture was verified for many well known examples including unital semigroups, so called Davies semigroups derived in the weak coupling limit and many others [11] (cf. also [15]).
In this paper we analyze the properties of qubit generators leading to semigroups of maps which are not necessarily completely positive but satisfy the Schwarz inequality in the (dual) Heisenberg picture (cf.next Section).All completely positive unital maps satisfy the Schwarz inequality but the converse is generally not true.However, all unital Schwarz maps are necessarily positive.Hence, in the qubit case, Schwarz maps interpolate between positive and completely positive unital maps.Schwarz maps are widely used in many aspects of mathematical physics (cf. the recent paper [21]).In particular they are connected to several monotonicity properties (e.g.monotonicity of relative entropy) which are of great importance for quantum information theory [22].
For semigroups of positive maps the corresponding generator can still be represented via formula (1) but some of the rates γ k can be negative.In this paper we prove (cf.Theorem 4.1) that if L generates a semigroup of positive maps which satisfy (in the Heisenberg picture) the Schwarz inequality then the constraint (2) is modified to It turns out that the above bound is also tight.It is, therefore, clear that for the qubit case one has where for positive maps for completely positive maps (6b) In [12] (see also [21]), Schwarz maps were referred to as 3 2 -positive maps, albeit as a 'not too serious alternative'.It is intriguing that our results provide another evidence validating this nomenclature.In this paper we often use the term α-positive map meaning: positive (α = 1), Schwarz (α = 3/2), and completely positive (α = 2) map.Note that, the very condition (5) provides a modification of (3) as follows: which is exemplified by a phase-covariant qubit evolution (cf.Section 3).The paper is organized as follows: Section 2 provides a brief introduction to Schwarz maps and Markovian generators of semigroups of unital Schwarz maps.In Section 3 we analyze two paradigmatic qubit semigroups of Pauli and phase-covariant maps.For these two examples necessary and sufficient for the corresponding generators to generate semigroups of Schwarz maps are derived.These offer not only a simple illustration of the validity of the constraint (5) but also provide examples that satisfy the equality, thereby demonstrating the 'tightest' aspect of the constraint.In section 4 we provide the proof of (5) for arbitrary qubit semigroup enjoying the Schwarz property.As applications of our results we discuss in Section 5 necessary conditions for the spectra of unital α-positive maps and necessary conditions for α-positive maps to be Markovian.Final conclusions are collected in Section 6.Additional technical details are presented in the Appendix.

Preliminaries: Schwarz maps and Markovian generators
Let M n denote the space of n × n complex matrices.A unital linear map Φ : for all X ∈ M n [4,5,24,25].It was shown by Kadison [26,27] that if Φ is a positive unital map then it satisfies celebrated Kadison inequality for all X † = X.However, not all positive unital maps satisfy (8).The simples example is provided by the transposition map.Actually, Kadison inequality was generalized by Choi [28,29] who proved that if Φ is positive and unital, then for any normal operator X.Moreover, it was shown [28,29] that if Φ(1l) > 0, then 2-positivity is equivalent to for all X, T ∈ M n , such that T is invertible.Actually, the above result maybe slightly generalized as follows [21]: Φ is 2-positive iff Φ(1l) ≥ 0 and is valid for all X ∈ M n , and T ≥ 0 with kerT ⊆ kerX † .In (12) T − denotes the Moore-Penrose inverse [30,31].It is, therefore, clear that if Φ(1l) = 1l, then (11) implies (8), i.e. any unital 2-positive map is a Schwarz map.However, it is known that there exist Schwarz maps which are not 2-positive [28,29] (cf.also [12]).Hence, for n = 2 Schwarz maps interpolate between positive and completely positive maps unital maps.Consider a dynamical semigroup {Λ t } t≥0 of linear positive trace-preserving maps on B(H), i.e. for any t, s ≥ 0 one has Λ t+s = Λ t • Λ s , Λ t (X) ≥ 0 for X ≥ 0, and TrΛ t (X) = TrX for any X ∈ B(H).Passing to the dual Heisenberg picture via where (X, Y ) = Tr(X † Y ) denotes the Hilbert-Schmidt inner product, one defines a semigroup {Λ ‡ t } t≥0 of linear positive unital maps on B(H), i.e.Λ ‡ t (1l) = 1l.A semigroup is uniquely defined by the corresponding generator L : B(H) → B(H) via Λ t = e tL .The map Λ t is trace-preserving if and only the corresponding generator L annihilates the trace Tr L(X) = 0 for any X ∈ B(H).Equivalently, the dual generator annihilates identity operator in B(H), i.e.L ‡ (1l) = 0. Proposition 2.1 ( [32]).Λ t = e tL is positive for all t ≥ 0 if and only if for any pair of mutually orthogonal rank-1 projectors P and Q.
Lindblad [1] provided the following condition for the generator L for which {Λ ‡ t } t≥0 is a semigroup of Schwarz maps.
for all X ∈ M n .
It is evident from ( 16) that if both L ‡ 1 and L ‡ 2 generate Schwarz semigroups, then the sum Note that a generator of a unital Hermiticity-preserving semigroup has the following representation [2]: where H = H † and Φ is a Hermiticity-preserving map.Using ( 16), one finds [1] that L ‡ gives rise to a semigroup of Schwarz maps if and only if for all X ∈ M n .Actually, it is sufficient to check (18) for traceless operators.Indeed, letting X = X 0 +a1l, with traceless X 0 , one easily checks 3 Paradigmatic qubit semigroups In this Section we analyze two paradigmatic qubit semigroups in terms of the corresponding relaxation rates Γ k (k = 1, 2, 3).Our analysis shows that the constraint ( 6) is satisfied.

A semigroup of Pauli maps
Consider the following generator where γ k (k = 1, 2, 3) are real numbers and σ k (k = 1, 2, 3) are the Pauli matrices (we use here the standard identification: Note, that L ‡ = L and hence Λ ‡ t = Λ t .Self-dual maps are necessarily unital.One finds for the spectrum: L(1l) = 0, together with with , and λ 3 = −(γ 1 + γ 2 ).Hence, the corresponding relaxation rates read It is well known that (20) gives rise to a semigroup of positive maps if and only if all Γ k ≥ 0. Indeed, in terms of the corresponding Bloch vector r = (x 1 , x 2 , x 3 ), with x k = Tr(ρσ k ), the evolution of a density operator corresponds to x k (t) = e −Γ k t and hence r(t) stays in Bloch ball if and only if Γ k ≥ 0. It shows that it is not necessary that all γ k are positive.Note, however, that at most only single γ k can be negative.Taking for example γ 1 = γ 2 = γ and γ 3 = −γ, one obtains Γ 1 = Γ 2 = 0 and Γ 3 = 2γ.Complete positivity is much more demanding and it requires all γ k ≥ 0 which is equivalent to the following relations between relaxation rates where {i, j, k} are mutually different.The above relation can be compactly rewritten as follows where To check whether a positive map Λ ‡ t satisfies Schwarz inequality one has to analyze (16).Assuming that γ 1 , γ 2 ≥ 0, and taking X = |0 1| (where |i (i = 0, 1) denotes the normalized eigenvectors of σ 3 , with eigenvalues 1, −1, respectively) one finds which implies the following necessary condition Observe, that in terms of relaxation rates condition (26) can be rewritten as follows which immediately implies Γ ≥ 3 2 Γ 3 and hence (6) holds since Γ 3 = max k Γ k .Interestingly, for a semigroup of Pauli maps the very constraint (6) may be equivalently rewritten as follows: for completely positive maps (28) where {i, j, k} are all different.
To check that the bound ( 6) is tight we still need to construct a qubit Schwarz semigroup that saturates (6).Recall, that ( 26) is necessary but not sufficient for the Schwarz inequality to hold.Assuming that γ 3 < 0 the following sufficient condition was derived in [35] (In the Appendix A we provide an independent proof of this result.)Note, that when γ 1 = γ 2 both conditions ( 26) and ( 29) coincide.Hence, taking γ 1 = γ 2 = −2γ 3 provides an example of a Schwarz semigroup that achieves the equality in (6).

A semigroup of phase-covariant maps
A linear map Φ : M 2 → M 2 is called to be a phase-covariant if where U ϕ = e −iϕσz , and ϕ is an arbitrary (real) phase.The most general phase-covariant generator has the following form where ω, γ + , γ − , γ z are real numbers and σ ± = 1 2 (σ x ±iσ y ).The above generator gives rise to a semigroup of CPTP maps if and only if γ ± ≥ 0 and γ z ≥ 0. Let us analyze when the corresponding semigroup Λ t consists of positive maps and when Λ ‡ t satisfies Schwarz inequality.Note, that ρ ss = p 0 |0 0| + p 1 |1 1|, with p 0 = γ + /(γ + + γ − ) and p 1 = γ − /(γ + + γ − ), defines a stationary state: L(ρ ss ) = 0.The spectral properties of L are characterized as follows: where the transversal Γ T and longitudinal Γ L relaxation rates are given by With the same reasoning as in Pauli semigroup, it is clear that positivity of Γ T and Γ L is necessary for positivity of Λ t .However, contrary to the semigroups of Pauli maps this condition is not sufficient.

Proposition 3.1 ( [36]
). L defined by (31) gives rise to a semigroup of positive maps if and only if γ ± ≥ 0 and This result was already proved in [36] where the authors used quantum version of the Sinkhorn theorem.Here we propose a simple proof based on the defining relation (14).Let P = |ψ ψ| and ( It is therefore clear that only γ z can be negative.Consider now an arbitrary normalized vector |ψ = Now, it is evident that Tr[QL(P )Q] ≥ 0 if and only if 2 is always non-negative.✷ In the Appendix B we also show that γ ± ≥ 0 and condition (34) are equivalent to for all X † = X ∈ M 2 , which, combined with Proposition 2.2, provides another proof of Proposition 3.1.Finally, the generator giving rise to a phase-covariant semigroup of positive maps can be characterized in terms of relaxation rates Γ T and Γ L and the parameter δ := γ + − γ − .One has and hence a phase covariant generator L gives rise to semigroup of positive maps if and only if and Remark 1. Note, that for δ = 0 one has γ + = γ − = γ and the dissipative part of (31) reduces to the Pauli generator with γ 1 = γ 2 = γ and γ 3 = 2γ z .In this case conditions (39), (40) reduce to Γ L ≥ 0 and Γ T ≥ 0.
Consider now the dual generator Proposition 3.2.The dual generator L ‡ (see (41)) gives rise to a semigroup of Schwarz maps Λ ‡ t if and only if γ ± ≥ 0 together with Proof: taking Similarly, for Hence, using Proposition 2.3, one finds that conditions γ ± ≥ 0 and ( 42) are necessary in order to generate a semigroup of Schwarz maps.To show that they are also sufficient let γ z =: −γ, with γ > 0. Clearly, if γ z ≥ 0, then L ‡ generates completely positive unital and hence Schwarz semigroup.Introducing Let us observe that L ‡ can be decomposed as where and Now, conditions (29) implies that L ‡ 1 generates a semigroup of Pauli Schwarz maps while L ‡ 2 generates a semigroup of completely positive and hence Schwarz maps.It is, therefore, clear that L ‡ 1 + L ‡ 2 generates a semigroup of Schwarz maps.
Corollary 3.2.If L ‡ generates a semigroup of Schwarz maps, then Proof, indeed, conditions (42) imply The proof for Γ T is straightforward due to positivity of Γ T and Γ L .Hence, it is shown that condition (6) holds for phase-covariant Schwarz semigroups.✷ It is interesting to observe that the inequality Γ L ≤ 2  3 Γ results in a modification of the well-known relation ( 3) to (7) between longitudinal and transverse relaxation rates.Recently, the qubit dynamical maps were analyzed in terms of the corresponding relaxation rates in [38].Both positivity and complete positivity is analyzed.Note, however, that authors of [38] use different notation.

General qubit semigroup of Schwarz maps
Any generator for a qubit trace-preserving semigroup can be represented in the basis of Pauli matrices as follows [2,3,37] with Hermitian 3×3 matrix C ij , and Hermitian H.The evolution generated by ( 50) is completely positive if and only if C ij is positive definite.Observe, that the matrix C can be decomposed as C = S +iA, where S is real symmetric and A is real antisymmetric.Now, let O be an orthogonal matrix which diagonalizes S, i.e. S = ODO T , where D is diagonal.Defining C := O T CO one finds which shows that off-diagonal elements of C are purely imaginary, as O T AO remains antisymmetric.
Following [2] it is convenient to parameterize C as follows with g := (g 1 , g 2 , g 3 ), a := (a 1 , a 2 , a 3 ) ∈ R 3 and g = g 1 + g 2 + g 3 .Note, that g = Tr C = Tr C = Tr D. Finally, one arrives at the following form Now, defining Note, that Hermitian matrices F i satisfies the same algebra as Pauli matrices, i.e.
where F := (F 1 , F 2 , F 3 ).The GKLS master equation ρt = L(ρ t ) may be rewritten as the following equation for the evolution of the corresponding Bloch vector where c := 4a and the matrix where Note, that the spectrum of L apart from ℓ 0 = 0 coincides with the spectrum of '−G', i.e. three relaxation rates Γ k are real parts of eigenvalues of G.
Remark 2. For two paradigmatic qubit semigroups considered in the previous section we have For a semigroup of Pauli maps one finds g k = 4Γ k and clearly h k = 0 since H = 0.For the phasecovariant case one finds g 1 = g 2 = Γ T /4, g 3 = Γ L /4, and h 3 = ω/2, that is, The spectrum of G reads: Γ L , Γ T ± iω.Hence, for both semigroups the parameters g k essentially recover the corresponding relaxation rates.Note, however, that in general it is not true.It happens if and only if G is diagonal, which corresponds to H = 0, or there exists an axial symmetry: for example g 1 = g 2 and h 1 = h 2 = 0.These two cases correspond exactly to semigroups of Pauli maps and phase-covariant maps, respectively.Proof: the above result was already proved in [9] for α = 2 corresponding to completely positive scenario.Here we provide a proof for any α ∈ [1,2].There are two separate cases: either exactly a single eigenvalue is real or all three are real.
Given a unital α-positive map Φ : M 2 → M 2 one may ask is it possible to find L such that Φ = e L (following [42,43] one calls such map Markovian).Note, that in this case eigenvalues λ k of Φ reads λ k = e ℓ k , where ℓ k are eigenvalues of L. Hence, if Γ k = −Reℓ k satisfy (6), then and hence for any k = 1, 2, 3.The above condition provides a universal constraint for the spectrum of qubit Markovian unital α-positive map.In particular for the Pauli map one obtains which for α = 2 (complete positivity) was independently derived in [11,44,45].For α = 1 it is trivially satisfied.However, for α = 3/2 it provides a necessary condition for the spectra of qubit Markovian unital Schwarz map:

Conclusions
In this paper we have proved that relaxation rates for any qubit Schwarz semigroup satisfy the constraint (5), thereby completing the universal constraints (6) for qubit semigroups with respect to positive, Schwarz, and completely positive maps.This general result is illustrated by two paradigmatic qubit evolution: semigroups of Pauli maps and phase-covariant maps.For these semigroups it is simply possible to derive the conditions for the corresponding generator in terms of relaxation rates Γ k and then check validity of (5).It should be stressed that the bounds (6) are tight.Indeed, consider a semigroup of Pauli maps: let Hence Γ 3 = Γ saturates the bound for α = 1.
Schwarz maps play important role in mathematical physics [21] and we hope that presented analysis contributes in a nontrivial way to the discussion on the structure of Schwarz qubit maps.Interestingly, our results strongly support a proposal to call unital Schwarz map as 3  2 -positive (see also recent paper [21]).Finally, one may pose a natural question what happen to (5) beyond qubit scenario.Actually, we conjecture that the following constraint is satisfied for any semigroup of Schwarz maps of d-level quantum system.The clarification of this conjecture is postpone for the future research.

A Appendix A
In this appendix, we demonstrate that the conditions specified in (29) provide sufficient criteria for semigroups of Schwarz Pauli maps: 20) gives rise to a semigroup of Schwarz Pauli maps.
This was originally observed in [35], but the proof presented here is independent.Indeed, we demonstrate it as a corollary of a more general result, Proposition (A.2) below, which is of interest in its own right.
To show the sufficiency, assume γ ± ≥ 0 and √ γ + γ − + 2γ z ≥ 0. As mentioned above, it is enough to show the positivity of M for all X of the form (91).If z = 0, one has However, this is shown to be positive through condition (34) by employing the Arithmetic Mean-Geometric Mean Inequality.In the case where z = 0, we may assume z = 1 without loss of generality.Now, the positivity of TrM = 4(γ + +γ − )+2x 2 (γ + +γ − +4γ z ) follows from the conditions (again using the AM-GM Inequality).Moreover, the positivity of detM also follows by observing that equation b 2 − 4ac in (93) is negative.Since M ≥ 0 if and only if TrM ≥ 0 and detM ≥ 0, this completes the proof.

Corollary 3 . 1 .
A phase covariant generator L ‡ gives rise to semigroup of Schwarz maps if and only if