Symmetries and Monotones in Markovian Quantum Dynamics

Symmetries and monotones in Markovian quantum dynamics

Georgios Styliaris and Paolo Zanardi

Department of Physics and Astronomy, and Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089-0484, USA

What can one infer about the dynam- At the heart of such approaches lies the far- ical evolution of quantum systems just reaching idea of symmetry. The most prominent by symmetry considerations? For Marko- example is perhaps provided by Noether’s theo- vian dynamics in finite dimensions, we rem which, in the context of Lagrangian mechan- present a simple construction that assigns ics, yields a conserved quantity for each differ- to each symmetry of the generator a fam- entiable symmetry of the generator [1]. In the ily of scalar functions over quantum states quantum realm, the theorem indicates that all that are monotonic under the time evo- moments of an observable that is a symmetry of lution. The aforementioned monotones the time evolution are conserved [2]. can be utilized to identify states that are In this paper we consider open quantum sys- non-reachable from an initial state by the tems in finite dimensions evolving under Marko- time evolution and include all constraints vian dynamics [3]. We address the question: imposed by conserved quantities, provid- Given the generator of the dynamics and an ini- ing a generalization of Noether’s theorem tial state, what constraints can symmetry consid- for this class of dynamics. As a special erations impose on the set of states that are reach- case, the generator itself can be consid- able under the time evolution? ered a symmetry, resulting in non-trivial Open quantum dynamics is, in general, dissi- constraints over the time evolution, even pative. As such, the role of conserved quantities if all conserved quantities trivialize. The can be rather limited. For instance, Markovian construction utilizes tools from quantum dynamics with a unique steady state does not ad- information-geometry, mainly the theory mit any non-trivial conserved quantities [4]. For of monotone Riemannian metrics. We an- this reason, we approach the problem by instead alyze the prototypical cases of dephasing seeking to utilize symmetries to obtain mono- and Davies generators. tones, i.e., functions of the quantum state that are monotonic (in our case, non-increasing) un- 1 Introduction der the time evolution. Similarly to conserved quantities, monotones can be utilized to exclude One of the main tasks in the study of non- state transitions that are impossible under the relativistic quantum dynamical systems is pre- dynamics, i.e., states that do not belong in the dicting how quantum states and observables trajectory of a given initial state. evolve over time given some dynamical law, for An accessible discussion regarding the connec- instance, a Hamiltonian operator and the associ- tion between symmetries and conserved quanti- arXiv:1912.04939v3 [quant-ph] 27 Apr 2020 ated equations of motion. It is often the case, ties for Markovian dynamics can be found in the however, that in practice the trajectory either papers by Baumgartner and Narnhofer [5], and does not admit an explicit closed form, or even by Albert and Jiang [4]. Symmetries have also if it does, it can be complicated to draw conclu- been discussed for the closely related case of iter- sions to physical questions of interest from it. It ated quantum channels [6]. From the viewpoint is therefore important to have accessible methods of quantum resource theories [7], consequences of and tools that allow one to extract the essential symmetries in quantum dynamics have been sys- features of the evolution directly from the dynam- tematically considered in the theory of reference ical laws. frames and asymmetry [8,9]. There, one investigates the allowed state transitions under quan- Georgios Styliaris: [email protected] tum dynamics that respects a specified symme-

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 1 try1. In particular, Marvian and Spekkens estab- ties in open systems. We also briefly discuss how lished the fact that, for the case of Hamiltonian one can construct symmetries for the generator of dynamics, asymmetry monotones yield conserved the dynamics and the monotones. In section 3, quantities that can be independent of the Noether we apply the method to dephasing generators and ones [10]. Asymmetry monotones have also been in section 4 for generators of the Davies form, ex- utilized to put constraints on the evolution of tracting in both cases the qualitative features of quantum coherences by Lostaglio et al. [11]. Sym- the evolution with symmetry considerations. We metry considerations have moreover been invoked conclude in section 5. to study transport in a Markovian model of open spin chains by Buča and Prosen [12]. 2 Monotones of the evolution from Monotones generalize the concept of a conserved quantity and, as we show in our construc- symmetries of the dynamics tion, in fact one can deduce from them the full 2.1 Setting the stage set of conserved quantities. Our main result consists of a method to assign to each pair of sym- We consider quantum systems described by a metries of the generator a one-parameter family state ρ ∈ S(H) ⊆ B(H), where the Hilbert ∼ d 2 of monotones for the time evolution. As a spe- space H = C is finite dimensional . We as- cial case, the generator itself can be considered sume that the dynamics is Markovian and time- a symmetry of the dynamics, resulting in non- homogeneous, i.e., trivial constraints over the time evolution, even d if all conserved quantities trivialize. ρt = L(ρt) (1) The basic idea we invoke to obtain monotones dt of the time evolution relies on the “infinitesi- where the generator of the dynamics L, also mal version” of quantum data-processing inequal- known as the Lindbladian, is a time independent ities. Distinguishability measures D(ρ, σ) ≥ 0, superoperator and hence can be expressed in the defined over pairs of quantum states such that standard form D(ρ, ρ) = 0, are said to obey the data-processing X † 1 † inequality if they are non-increasing under the L(X) = −i [H,X] + LiXL − L Li,X i 2 i joint action of a quantum operation on both ar- i (2) guments and play a central role in quantum information theory [13, 14]. Since Markovian dynam- (see, e.g., Ref. [3] for more details). Equation (1) ics is fully characterized by its generator, we in- gives rise to a one-parameter family of time- voke an infinitesimal version of distinguishability evolution superoperators measures, connecting with monotone Riemannian metrics in the space of quantum states. Such met- Et = exp (tL) , t ≥ 0 , (3) rics have proven useful for the study of Markovian 3 quantum dynamics, as for instance for the study forming a semigroup in the space of Completely of the mixing time and convergence rates [15–17]. Positive and Trace Preserving (CPTP) maps, also The paper is organized as follows. In section 2 known as quantum channels. We define as sym- we begin by first non-rigorously presenting the metries of a Lindbladian those superoperators basic idea for establishing the correspondence be- that commute with L, i.e., tween symmetries and monotones of the evolu- Sym(L) := {M ∈ B (B (H)) | [M, L] = 0} . (4) tion. We then present the general construction, which is related to monotone Riemannian metrics We first informally present the basic idea of and, after giving a simple example, we establish how one can obtain monotones of the evolution a connection with the Noether conserved quanti- 2We use B(H) to denote the space of (bounded) linear operators. S(H) denotes the space of non-negative linear 1More specifically, given a group G, one investigates operators with unit trace. the possibility of state transitions under the class of quan- 3 0 tum operations that are symmetric with respect to a uni- I.e., E0 = I and EtEt0 = Et+t0 for all t, t ≥ 0. This tary representation of G, namely the operations such that type of quantum dynamics is also called semigroup evolu- [E, Ug] = 0 ∀g ∈ G. tion.

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 2 from symmetries of the generator. Let us con- then it is not hard to show that the expansion (8) sider a distinguishability measure, i.e., a function yields k = 2 and D(ρ, σ) over pairs of states with |hi|M(ρ)|ji|2 h (ρ) = X , (11) D(ρ, σ) ≥ 0 and D(ρ, ρ) = 0 , (5a) M √ √ 2 ij pi + pj such that it respects the data-processing inequal- P ity where we spectrally decomposed ρ = i pi |iihi|. A detailed derivation of Eq. (11) via elementray D(ρ, σ) ≥ D (E(ρ), E(σ)) (5b) methods can be found in AppendixA. The above quantity is well-known in quantum information- for all states ρ, σ and quantum channels E. Let geometry as the Wigner-Yanase metric [18], here us also define evaluated on M (ρ). sM fs(ρ) = D(ρ, e ρ) , with M ∈ Sym (L) , (6) 2.2 Monotones of the evolution and monotone Riemannian metrics which is a measure of the difference between a state ρ and its variation4 generated by the sym- In the previous section we discussed a way of metry M. obtaining monotones for a Markovian evolution The data-processing inequality, together with given a distinguishability measure and a symme- the fact that M is a symmetry of the time evo- try M of the generator, as in the example of lution, implies that Eq. (11). This raises the question of how to sys- tematically perform the expansion (8) for a suit- tL fs(ρ) ≥ fs(e ρ) , ∀t ≥ 0 , (7) ably wide class of distinguishability measures, a question that naturally leads to the consideration i.e., that fs is a monotone of the dynamics. How- of monotone Riemannian metrics. ever, to calculate fs for a finite value of the pa- To see that, consider to the case where rameter s one needs to first calculate the action of p D(ρ, σ) is a distance function over the manifold√ the exponential exp(sM) on ρ, which in general is S>0(H) of positive definite states such that D impractical. Instead, if the function fs is suitably arises from a Riemannian metric5 g. In that case, well-behaved, one can examine its expansion the expansion (8) yields k = 2 and the function h (ρ) is nothing else that (squared) length of f (ρ) = skh (ρ) + O(sk+1) , k ∈ , (8) M s M N the tangent vector M(ρ) over the tangent space where the form of hM depends on D and M. Tρ S>0(H), i.e., Since the inequality (7) is valid for all s ≥ 0, no hM(ρ) = gρ (M(ρ), M(ρ)) . (12) matter how small, it follows that also hM(ρ) is a monotone of the evolution More importantly, the monotonicity property (9) is satisfied if the metric is contractive under the hM(ρ) ≥ hM(Et (ρ)) , t ≥ 0 . (9) action of quantum channels T , namely Before proceeding to formalize the previous observation via basic tools from quantum gρ(X,X) ≥ gT (ρ)(T (X), T (X)) . (13) information-geometry, let us consider an exam- The aforementioned class of Riemannian met- ple. If the distinguishability measure is taken to rics, called monotone metrics [19–22], constitute be the relative α-Rényi entropy with α = 1/2, the quantum analog of the Fisher metric for prob- i.e., ability distributions (see also Ref. [23] for an ac- √ √ D(ρ, σ) = S 1 (ρ, σ) = −2 log Tr ρ σ (10) cessible introduction to the subject). While the 2 latter is uniquely determined from the mono- 4Notice that for exp(sM)(ρ) to be a valid quantum tonicity property under classical stochastic maps state for s ∈ [0,T ) ⊆ R≥0, the superoperator M should (up to a normalization constant), the same does obey certain constraints, as for instance that M(ρ) is hermitian and traceless. We will later generalize this con- 5I.e., the distance between two states is the length of struction to include any M ∈ Sym (L). the geodesic connecting them.

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 3 not hold for its quantum counterparts, which fact that the quantity show a rich variety6. −1 (λ) † We include for completeness an elementary dis- ρ M,N := Tr M(ρ) RN (ρ) + λLN (ρ) [M(ρ)] cussion of monotone Riemannian metrics in Ap- J K (17) pendixB. For the purposes of this work, the crucial prop- satisﬁes erty of such quantities is that they satisfy the (λ) (λ) data-processing inequality, Eq. (13). This mono- ρ M,N ≥ Et(ρ) M,N for λ, t ≥ 0. (18) J K J K tonicity property can be understood as a conse- We have shown the following. quence of a key operator inequality, ﬁrst proved in Ref. [25], which we repeat here due to its cen- Proposition 1. If M, N are symmetries of the tral importance for what follows. dynamics and λ ≥ 0, then the function

(λ) Theorem (Lesniewski and Ruskai [25]). ρ M,N : S>0(H) → R≥0 J K h i defined in Eq. (17) is non-increasing under the Tr A† (R + λL )−1 (A) ≥ σ τ time evolution generated by L. −1 † Tr E(A) RE(σ) + λLE(τ) [E(A)] , (14) In the rest of this paper, we will mainly focus on two cases. where A ∈ B(H) is a linear operator, L (X) := σ (i) N = I, in which case for simplicity we de- σX (R (X) := Xτ) is the superoperator repre- τ note the resulting family of monotones as senting left (right) multiplication, E is a quan- h i tum channel and λ ∈ R≥0. The operators σ, τ ∈ (λ) † −1 ρ M = Tr M(ρ) (Rρ + λLρ) [M(ρ)] . B>0(H) are positive definite, assuring that the su- J K (19) peroperator inverses entering the inequality are well-defined, as well as that the resulting traces are non-negative. (ii) If the dynamics admits a full-rank stationary state ω, then N (X) = Tr (X) ω is a Let us now return to considering quantum symmetry. In that case, it is convenient to Markovian dynamics generated by some time- denote directly the fixed state ω instead of independent Lindbladian L. For our purposes, the symmetry, i.e., write the quantum channel in the inequality (14) is (λ) h † −1 i ρ M,ω = Tr M(ρ) (Rω + λLω) [M(ρ)] . specialized to the time evolution superoperator J K Et := exp (tL) for t ≥ 0. Letting ρ ∈ S>0(H) be (20) a full-rank state, we take (λ) The monotones ρ M can be also expressed in A = M(ρ) , where M ∈ Sym (L) (15) coordinates via spectrallyJ K decomposing the argu- P σ = τ = N (ρ) , where N ∈ Sym (L) (16) ment ρ = i pi |iihi|. Substituting, one gets

(λ) X 1 2 such that N (ρ) ∈ B>0(H) is positive deﬁnite for ρ M = |hi|M(ρ)|ji| . (21) J K λpi + pj all ρ ∈ S>0(H). We will refer to any such super- ij operators M and N as symmetries of the dynam- (λ) ics. Similarly, for the case of ρ M,ω one can decom- P J K The resulting inequality, due to the commuta- pose ω = i qi |iωihiω| resulting in tion relations [M, E ] = [N , E ] = 0, expresses the t t (λ) X 1 2 ρ M,ω = |hiω|M(ρ)|jωi| . (22) J K λqi + qj 6In fact, monotone metrics are also closely related ij to generalized relative entropies [24], as every monotone : Riemannian metric arises from a generalized relative en- For Hamiltonian dynamics, L(X) = KH (X) = tropy [25]. Hence the previous discussion can be general- −i [H,X], the monotones (17) are, in fact, con- ized to D corresponding to a relative entropy, as in our served. This follows because unitary channels are example (11). invertible, hence for this case Eq. (18) holds true

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 4 for t ∈ R. This forces monotones to maintain a the two linearly independent conserved quantities constant value along the orbit. of the evolution predicted by Noether’s theorem (λ) for Lindbladians, which we discuss momentarily. For every family { ρ M,N }λ, one can consider convex combinationsJ accordingK to the measure Now we consider again Eq. (19) but for the

µ(λ), namely symmetry M(X) = Kσz and λ = 1. Spectrally decomposing ρ = p P + p P and invoking Z + + − − (λ) (λ) Eq. (21), we have { ρ M,N }λ 7→ dµ(λ) ρ M,N . (23) J K J K (1) 2 From this construction one obtains valid mono- ρ K = 2 Tr (P+σzP−σz)(p+ − p−) J K σz tones, possibly admitting a convenient mathe- = 2 |h0|ρ|1i|2 , (27) matical form for an appropriate choice of the measure (for explicit expressions of measures, that is, coherences are non-increasing under the see [18]). However, notice that the resulting func- time evolution. tions do not impose any additional constraints The monotones from Eqs. (26) and (27) jointly compared to the ones from the parent functions7. limit the set of states R that are (possibly) reachable under time evolution. The aforementioned 2.3 A first example: Dephasing of a qubit set consists of all Bloch vectors lying on horizontal disk with radius equal to r0 (the radial dis- It might be useful at this stage to consider a sim- tance of the initial Bloch vector from the z-axis.) ple example to illustrate the formalism. Let us In fact, the set R of our example is as con- analyze a two-level system with dephasing dy- strained as possible, in the sense that additional namics described by the Lindbladian symmetry arguments (relying on the set Sym(L)) cannot restrict it more. To see this, first notice L(X) = −ig [σ ,X] + σ Xσ − X , g ∈ . z z z R that the set of symmetries of our dephasing Lind- (24) bladian does not depend on the value of the pa- The time evolution in terms of the Bloch vector rameter g, as long as g 6= 0. On the other hand, 1 by varying g, all points in the bulk of R can representation of the state ρ(t) = 2 (I + v · σ) is, in cylindrical coordinates, be reached by the trajectories of Eq. (25) (for any fixed initial condition). This is because g −2t rt = r0e , φt = φ0 + 2gt , zt = z0 . (25) controls the frequency of the rotational motion, which can be made arbitrarily high. Therefore no That is, the Bloch vector lies onto a horizontal more points in R can be excluded by symmetry plane evolving inwards in a spiral motion. arguments. Let us now illustrate how one can deduce the qualitative features of the evolution just by sym- 2.4 Monotones imply Noether conserved quan- metry considerations, by use of Eq. (19). Since tities the Hamiltonian H = gσz and the single Lind- blad operator L = σz are both diagonal in the We will say that an operator Y ∈ B(H) is con- σz := |0ih0| − |1ih1| eigenbasis, clearly the left served if multiplication superoperators L|0ih0| and L|1ih1| are h † i symmetries of the Lindbladian. One immediately Tr Y exp(tL)(ρ) = const ∀ρ and t ≥ 0 . gets that the populations (28)

(0) We will also refer to such functions of the state as ρ L = Tr (ρ |iihi|) , i = 0, 1 (26) J K |iihi| Noether conserved quantities, since they general- are non-increasing. However, since the evolution ize the corresponding Hamiltonian construction. is trace preserving, each of the populations is sep- It is well-known that arately conserved. Notice that these are exactly Y is conserved ⇐⇒ Y ∈ Ker L∗ (29) 7I.e., if the possibility of a transition ρ 7→ σ is ruled R (λ) R (λ) out by the inequality dµ(λ) ρ M,N < dµ(λ) σ M,N , (the adjoint is with respect to the Hilbert- then there exists a (non-zero measure)J K set of λ’s forJ K which Schmidt inner product). The claim follows by also ρ (λ) < σ (λ) . M,N M,N noticing that the above trace can be expressed J K J K

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 5 via the Hilbert-Schmidt inner product (hA, Bi := To complete the proof, we need to show Tr A†B for A, B ∈ B(H)) as that the constraints imposed by conserved Y ∈ B>0(H) are the same as the ones imposed by ar- h i Tr Y † exp(tL)(ρ) = hY, exp(tL)(ρ)i bitrary Y ∈ B(H). Indeed, consider some (possibly non-hermitian) Y ∈ Ker(L∗). Since L∗ ∗ = hexp(tL )(Y ), ρi . preserves hermiticity, then both the hermitian and anti-hermitian parts of Y are separately con- A consequence is that the number of linearly in- served i.e., conservation of Y amounts to conser- dependent conserved quantities equals the dimen- vation of the hermitian operators (Y +Y †)/2 and sion of the subspace spanned by the steady states (Y −Y †)/(2i). Finally notice that, although these of the evolution, a fact also noted in [4]. This ob- two operators might fail to be positive, due to servation follows from the identity dim Ker(L) = unitality of E∗ it holds that I ∈ Ker (L∗). Hence, ∗ t dim Ker(L ). In particular, if there is a unique if a hermitian Y is conserved, then also Y + aI is fixed point of the evolution then all conserved conserved, which can always be made positive for quantities trivialize, in the sense that they are large enough a. Notice that this amounts to just necessarily proportional to the trace of the (time- adding the constant a to the value of the Noether ∗ evolved) state, since always I ∈ Ker L . conserved quantity. (λ) On the other hand, the monotones ρ M im- The above demonstrate that the monotone pose non-trivial, in general, constraintsJ K on the (0) family ρ M,N imposes at least as many con- J K reachable states of the evolution, for instance by straints as conserved quantities. taking M = L (a specific example is given later in Figure 1). What is more, if a steady state ω 2.5 Symmetries of the generator and the is known (regardless whether it is unique or not), (λ) monotones then also monotones of the form ρ M,ω can be utilized, imposing additional constraints.J K We now very briefly address the question of how A natural question to be asked is whether the one can find symmetries of a Lindbladian, be- constraints imposed by the Noether conserved yond those guessed by inspection. Determining quantities are included in the monotones arising explicitly the entire set of symmetries Sym (L) of from Proposition 1 or not. The answer is affirma- a Lindbladian is usually too complicated to be of tive. practical importance. Instead, when a representation of the Lindbladian in Lindblad operators Proposition 2. If the Lindbladian admits a full- is known (i.e., a decomposition as in Eq. (2)), it rank stationary state, then for each conserved op- might be convenient to look at operators simul- erator Y there is an appropriate choice of the taneously commuting with all elements of the set symmetries M, N in the family of monotones S = {H} ∪ L ,L† . If A is the algebra gener- (0) i i i ρ M,N that implies the conservation. ated by S, then the element of the algebra gen- J K erated by {LX , RX }X∈A0 belong to Sym (L) (here Notice, however, the converse is not true; con- 0 8 (0) A denotes the commutant of the algebra ). We straints imposed by the monotones ρ M,N can- J K will use this fact to analyze dephasing generators not necessarily be inferred from conservation later in the paper. laws. Steady states ω give rise to symmetries, for in- Proof. Let us consider some conserved Y > 0. stance via maps N (X) = Tr(X)ω, as in Eq. (20). The superoperator M(ρ) = Tr (Y ρ) ω, for ω a Although Ker (L) is not in general closed with full-rank stationary state of the evolution, is a respect to operator multiplication, it was shown in Ref. [26] that it forms an Euclidean Jordan symmetry of the dynamics. Therefore ρ (0) = M,ω Algebra. More specifically, if the limit F := Tr (Y ρ)2 is non-increasing under the timeJ K evolu- limt→∞ exp (tL) I exists and is full-rank, then for tion, hence also Tr (Y ρ) is non-increasing. any A, B fixed points of the evolution it holds On the other hand, since ω is a full-rank steady (0) −1 8 state and Tr (Y ρ) > 0, also ρ ω,M = [Tr (Y ρ)] I.e., the operator subalgebra of B(H) consisting of all is a valid non-increasing monotone.J K As a result, (and only) the elements that commute with every element Tr (Y ρ) has to be a constant of the evolution. of A.

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 6 † that S = {H}∪ Li,Li i mutually commute with each 1 other. We will refer to this class as dephasing gen- A • B := (AF BF + BF AF ) (30) erators. Notice that this family is a generalization F 2 of the qubit example of section 2.3. 9 −1/2 −1/2 is also a fixed point , where XF := F XF . Dephasing Lindbladians are both unital and In addition, in Ref. [27] the authors constructed Hilbert-Schmidt normal, as can be inferred with a a family of maps that are bijections over the direct calculation invoking the standard form (2) subspace of fixed points of the dynamics. More of the generator. Hence by the spectral theorem specifically, the mappings can be constructed they admit a complete family of eigenoperators. with the input of an operator monotone func- The corresponding time evolution can be formally tion [28] and two full-rank steady states. The expressed via resorting to the (maximal) projec- above facts can be of use to generate more fixed tors onto the joint eigenspaces of all elements in n points, and hence symmetries, out of a set of S. Let us denote as {Pi}i=1 (n ≤ d) this com- known ones. plete family of orthogonal projectors, and also Finally, we address the question of how to con- as Pij(X) := PiXPj the corresponding family of struct monotones that themselves possess some (Hilbert-Schmidt orthogonal) projector superop- symmetry. Let us consider a unitary representa- erators. Then, for dephasing generators, the time † tion Ug of a group G such that Ug = Ug(·)Ug is a evolution can be written as symmetry of the Lindbladian for all g ∈ G. Then X λij t one can define a (left) group action on the space exp (tL) = e Pij . (34) of monotones via ij (λ) g (λ) (λ) Invoking once again the standard form of the gen- · M,N 7−→ · U M,N = · M,U N . (31) J K J K g J K g−1 erator (2), it follows that the eigenvalues satisfy ∗ From the latter, it is straightforward to construct λij = λji and λii = 0 ∀i, while Re (λij) < 0 for monotones that are invariant under the action by i 6= j. invoking the (Haar) group average, i.e., consider The last inequality expresses that fact that there are no steady states with support over any (λ) Z (λ) Q (ρ) := dµ (g) ρ . (32) of the eigenspaces Pij with i 6= j. This is true M,N Haar UgM,N J K since, for unital Lindbladians, Ker (L) coincides For instance, with A0 [31], where A is the (abelian) algebra generated by the set S. In our case, this commu- (0) h † −1i QM,ω(ρ) = Tr P M(ρ) M(ρ) ω , (33) tant is exactly given by operators in the range of P Pii. R i where P(X) := dµHaar(g)Ug(X) is the projector Qualitatively, the evolution dictates that the superoperator onto the G-invariant subspace, i.e., P diagonal elements i Pii(ρ) are conserved while for any operator X it holds U P(X) = P(X)U P g g the off-diagonal parts i Pij(ρ) (i 6= j) decay, ∀g. The projector P can be constructed explic- possibly with oscillations, which is the defining itly in terms of the irreducible representations of characteristic of dephasing. {Ug}g using standard techniques from theory of Let us now extract the same qualitative in- noiseless subsystems [29, 30]. formation from a symmetry analysis. Clearly,

[LPi , L] = 0 ∀i hence

3 Dephasing generators (0) ρ = Tr (ρPi) , ∀i (35) LPi Here we consider Lindblad generators of the gen- J K eral form (2) such that all elements of the set are non-increasing. On the other hand, the evolution is trace-preserving, therefore all Pi are, in 9 Notice that any operator X ∈ Ker (L) can be decom- fact, conserved. These are all the linearly inde- posed into (up to four) steady states by considering the 1 † 1 † pendent Noether conserved quantities. positive/negative part of 2 (X+X ) and 2i (X−X ). Each of the latter, after proper normalization, forms a steady In addition, LPi RPj is a symmetry of the evo- state. This follows from the fact that the corresponding lution since all elements in S mutually commute. quantum channel Et preserves positivity. Combining this with the fact that the evolution

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 7 is unital, hence ω = I is a ﬁxed point, we directly 1.0 get from Eq. (20) that

(0) 0.5 ρ L R ,I = Tr (PiρPjρ) (36) J K Pi Pj is non-increasing. This corresponds to the decay z 0.0 of the (i, j)-coherences of the state ρ and is a v generalization of Eq. (27).

We have shown the following general fact. -0.5 Proposition 3. Let L be a Lindbladian.

† -1.0 (i) If LP is a symmetry of L, then Tr ρtP P -1.0 -0.5 0.0 0.5 1.0 is a non-increasing function of time. vx

(ii) If L is unital and LP RQ is a symmetry, then Figure 1: Constraints imposed on the time evolution † † Tr P P ρtQ Qρt is a non-increasing func- by considering as symmetry the Lindbladian L itself. In this example L is a single qubit Davies√ generator with tion of time. + − H = σz and jump operators L1 = 2σ , L2 = σ . For P,Q orthogonal projectors, then (i) corre- We depict in Bloch coordinates contours separating the sponds to decaying population and (ii) to decay- allowed (inside) from the forbidden (outside) regions as predicted by the monotone ρ (1/2) for various ρ. The ing coherence. L monotone is symmetric underJ K rotations around the z- axis, so we only plot an x − z slice of the Bloch ball. 4 Davies generators The corresponding states ρ are marked with a point and are chosen to correspond to instances of the actual tra- 4.1 Preliminaries jectory (as projected onto√ the x−z plane) of the initially pure state v = (1/2, 0, − 3/2). A physically significant class of Lindbladians is provided by the Davies generators [32] (for a more Let us recall a few useful mathematical facts modern treatment, see also Refs. [33–35]). This regarding the detailed balance condition [33, 36]. family arises from a first-principles derivation un- First of all, since I ∈ Ker (D∗), then Eq. (37b) der the main assumption that the interaction be- implies that τ is a fixed point of the evolution. tween the system and its environment is weak The detailed balance condition can be equiva- and gives rise to steady states that are thermal. lently understood as the requirement that D∗ is Davies generators have the following properties: hermitian with respect to the (possibly degener- (i) The Hamiltonian part commutes with the ate) scalar product dissipative part, i.e., L = KH + D with † hA, Biτ := Tr τA B . (38) [KH , D] = 0 . (37a) Assuming that τ is full rank, the detailed balance (ii) It satisfies the quantum detailed balance condition Eq. (37b) can be recast in a yet another condition, i.e., there exists a state τ such form, namely D being hermitian with respect to that the scalar product h·, ·iτ −1 . The last claim follows by multiplying both from the left and the right −1 Eq. (37b) with Rτ . Let us now express the general form of the DR = R D∗ , (37b) τ τ time evolution of a Davies generators. Notice KH (τ) = 0 . (37c) that Davies generators are always normal operators, hence admit a complete family of eigenop- For the scope of this paper, we will refer to erators. This is because, by assumption, D is her- any Lindbladian that satisfies the conditions of mitian with respect to the scalar product (38) (for Eqs. (37) as a Davies generator. τ −1) and it commutes with the Hamiltonian part

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 8 (which is also anti-hermitian with respect to the 4.3 Davies generators: General remarks same scalar product). Let us formally denote as Let us make a few observations regarding mono- {P } the complete family of hermitian (with re- i i tones and Davies generators. First of all, notice spect to h·, ·i −1 ) superoperator projectors. Then τ that monotones (20) for ω = τ (the detailed bal- the general solution has the form ance ﬁxed point) and λ = 0 are directly related

X λit exp (tL) = e Pi (39) to the Davies inner product i (0) ρ M,τ = hM(ρ), M(ρ)iτ −1 . (42) P J K with PiPj = δijPi and i Pi = I. In particular, the case M = I expresses the fact that 4.2 Davies generator: Single qubit (0) 1 2 ρ I,τ = Tr ρ (43) Let us consider a qubit system whose Lindbladian J K τ is described by H = σz together with two Lind- is non-increasing under the time evolution, which blad operators {aσ+, bσ−}. The unique steady is a generalization of the well-known fact that the 1 2 state τ = 2 (I + w · σ) has a corresponding Bloch purity Pur(ρ) := Tr(ρ ) is non-increasing under a2−b2 unital dynamics. Equivalently, the distance vector lying along the z-axis with wz = a2+b2 . One can easily check directly that Eqs. (37) are 2 kρt − τkτ −1 = hρt − τ, ρt − τiτ −1 (44) satisfied, hence L is indeed a valid Davies generator. Setting g = |a|2 + |b|2, the dynamical is non-increasing. evolution of the Bloch vector is, in cylindrical co- One can easily generalize the construction ordinates, of the monotone (41). Each of the projectors in Eq. (39) can be written as Pi(X) = d −gt/2 P i hY ,Xi Y , where {Y } is a com- rt = r0e , φt = φ0 + 2t , k=1 i,k τ −1 i,k i,k i,k plete orthonormal family of eigenoperators. z = (1 − e−gt)w + e−gtz . (40) t z 0 Hence, for any such operator, M(X) = Let us analyze some of its symmetries now. By hY,Xiτ −1 Y is clearly a symmetry of the dynamics, and because for these symmetries definition of the Davies generator, KH = Kσz is a 2 symmetry, therefore Eq. (27) is a valid monotone, (0) 1 † indicating that coherences can only decay. This ρ M,τ = Tr Y ρ (45) J K τ corresponds to the inward spiral motion of the −1 † trajectory. the functions Tr τ Y ρ are monotones, gen- Regarding the z-evolution of the Bloch vector eralizing Eq. (41). towards the fixed point, let us assume that τ is known to be a fixed state of the evolution (but 5 Discussion and outlook we do not explicitly assume that it is the unique fixed state). Then by inspection one can verify Identifying symmetries and their consequences in that M(X) = hσz,Xiτ −1 σz is a symmetry of the physical systems is an important task for var- evolution. Therefore ious fields of physics. The present manuscript 2 constitutes an attempt to provide a simple corre- (0) 1 −1 ρ = Tr σzρ Tr τ (41) spondence between symmetries in the generators M,τ τ J K of quantum Markovian dynamics and monotones is non-increasing, implying that also of the corresponding evolution. More specifi- −1 Tr τ σzρ is a monotone. In the Bloch cally, we presented a construction that assigns representation, the last quantity equivalently ex- to every pair of symmetries of the generator a presses the fact that |vz − wz| is non-increasing, one-parameter family of functions over quantum i.e., the z-component of the Bloch vector states that are non-increasing under the time evo- monotonically approaches the steady state. lution. Such monotonic functions can be em- Finally, we numerically investigate in Figure 1 ployed in order to identify states that are non- a example for the constraints imposed on the reachable by the evolution. The construction uti- qubit time evolution just by considering as sym- lizes powerful tools from quantum information- metry the Lindbladian itself. geometry, mainly from the theory of monotone

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 9 Riemannian metrics. Finally, we have demon- GKS–lindblad generators: II. General. Jour- strated how one can deduce from symmetries the nal of Physics A: Mathematical and Theo- qualitative features of the evolution for the proto- retical, 41(39):395303, 2008. doi:10.1088/ typical cases of dephasing and Davies generators. 1751-8113/41/39/395303. One can easily rephrase the question addressed [6] Victor V. Albert. Asymptotics of quan- in the present paper to fit in the framework of tum channels: conserved quantities, an adi- quantum resource theories. Given some fixed abatic limit, and matrix product states. Lindbladian, one can define as free operations Quantum, 3:151, 2019. doi:10.22331/ those associated with the time evolution {Et := q-2019-06-06-151. exp (tL)}t≥0. Our main result, Proposition 1, [7] Eric Chitambar and Gilad Gour. provides a family of monotones for each pair of Quantum resource theories. Rev. symmetries of the generator. However, even if all Mod. Phys., 91:025001, 2019. doi: possible symmetries are considered, the resulting 10.1103/RevModPhys.91.025001. families of monotones are not in general complete, [8] Stephen D. Bartlett, Terry Rudolph, and i.e., they do not exclude all states that are non- Robert W. Spekkens. Reference frames, su- reachable by the evolution. It remains an open perselection rules, and quantum informa- question to find a general construction for a com- tion. Rev. Mod. Phys., 79:555–609, 2007. plete family of monotones. doi:10.1103/RevModPhys.79.555. We hope that the present work demonstrates yet another way in which ideas and tools from [9] Iman Marvian and Robert W. Spekkens. quantum information theory can provide useful The theory of manipulations of pure state insights to the analysis of quantum dynamics. asymmetry: I. Basic tools, equivalence classes and single copy transformations. New Journal of Physics, 15(3):033001, 2013. doi: Acknowledgments 10.1088/1367-2630/15/3/033001. [10] Iman Marvian and Robert W. Spekkens. G.S. is thankful to Á.M. Alhambra, N. Anand, Extending Noether’s theorem by quantify- L. Campos Venuti, A. Hamma, S. Montangero, ing the asymmetry of quantum states. Na- and N.A. Rodríguez Briones for the interesting ture communications, 5:3821, 2014. doi: discussions and to Th. Apostolatos and P.J. Ioan- 10.1038/ncomms4821. nou for their teachings. P.Z. acknowledges partial support from the NSF award PHY-1819189. [11] Matteo Lostaglio, Kamil Korzekwa, and Antony Milne. Markovian evolution of quantum coherence under symmetric dynamics. References Phys. Rev. A, 96:032109, 2017. doi:10. 1103/PhysRevA.96.032109. [1] Lev D. Landau and Evgeny M. Lifshitz. [12] Berislav Buča and Tomaž Prosen. A Classical Mechanics. Addison-Wesley, 1959. note on symmetry reductions of the Lind- [2] Jun John Sakurai and Jim Napolitano. blad equation: transport in constrained Modern quantum mechanics. Cambridge open spin chains. New Journal of University Press, 2017. doi:10.1017/ Physics, 14(7):073007, 2012. doi:10.1088/ 9781108499996. 1367-2630/14/7/073007. [3] Heinz-Peter Breuer and Francesco Petruccione. The theory of open [13] Edward Witten. A mini-introduction to in- quantum systems. Oxford University formation theory. arXiv:1805.11965. Press, 2002. doi:10.1093/acprof: [14] Mark M. Wilde. Quantum information the- oso/9780199213900.001.0001. ory. Cambridge University Press, 2013. doi: [4] Victor V. Albert and Liang Jiang. Symme- 10.1017/CBO9781139525343. tries and conserved quantities in Lindblad [15] Kristan Temme, Michael James Kasto- master equations. Phys. Rev. A, 89:022118, ryano, Mary Beth Ruskai, Michael Marc 2014. doi:10.1103/PhysRevA.89.022118. Wolf, and Frank Verstraete. The χ2- [5] Bernhard Baumgartner and Heide Narn- divergence and mixing times of quantum hofer. Analysis of quantum semigroups with Markov processes. Journal of Mathemati-

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 10 cal Physics, 51(12):122201, 2010. doi:10. maps. Master’s thesis, Technical University 1063/1.3511335. of Munich, 2013. [16] Michael J. Kastoryano, David Reeb, and [27] Jaroslav Novotn`y, Jiří Maryška, and Igor Michael M. Wolf. A cutoff phenomenon Jex. Quantum Markov processes: From at- for quantum Markov chains. Journal tractor structure to explicit forms of asymp- of Physics A: Mathematical and Theoret- totic states. The European Physical Journal ical, 45(7):075307, 2012. doi:10.1088/ Plus, 133(8):310, 2018. doi:10.1140/epjp/ 1751-8113/45/7/075307. i2018-12109-8. [17] Michael J. Kastoryano and Kristan Temme. [28] Rajendra Bhatia. Matrix analysis, volume Quantum logarithmic Sobolev inequalities 169. Springer-Verlag, 2013. doi:10.1007/ and rapid mixing. Journal of Mathematical 978-1-4612-0653-8. Physics, 54(5):052202, 2013. doi:10.1063/ [29] Emanuel Knill, Raymond Laflamme, and 1.4804995. Lorenza Viola. Theory of quantum error [18] Frank Hansen. Metric adjusted skew in- correction for general noise. Phys. Rev. formation. Proceedings of the National Lett., 84:2525–2528, 2000. doi:10.1103/ Academy of Sciences, 105(29):9909–9916, PhysRevLett.84.2525. 2008. doi:10.1073/pnas.0803323105. [30] Paolo Zanardi. Stabilizing quantum infor- [19] E. A. Morozova and N. N. Chentsov. Markov mation. Phys. Rev. A, 63:012301, 2000. invariant geometry on manifolds of states. doi:10.1103/PhysRevA.63.012301. Journal of Soviet Mathematics, 56(5):2648– [31] David W. Kribs. Quantum channels, 2669, 1991. doi:10.1007/BF01095975. wavelets, dilations and representations of [20] Dénes Petz. Monotone metrics on matrix On. Proceedings of the Edinburgh Mathe- spaces. Linear Algebra and its Applica- matical Society, 46(2):421–433, 2003. doi: tions, 244:81 – 96, 1996. doi:10.1016/ 10.1017/S0013091501000980. 0024-3795(94)00211-8. [32] E. Brian Davies. Markovian master equa- [21] Dénes Petz and Csaba Sudár. Geometries tions. Communications in mathematical of quantum states. Journal of Mathemat- Physics, 39(2):91–110, 1974. doi:10.1007/ ical Physics, 37(6):2662–2673, 1996. doi: BF01608389. 10.1063/1.531535. [33] Robert Alicki and Karl Lendi. Quantum [22] Dénes Petz and Mary Beth Ruskai. Con- dynamical semigroups and applications, vol- traction of generalized relative entropy un- ume 717. Springer-Verlag, 2007. doi:10. der stochastic mappings on matrices. Infi- 1007/3-540-70861-8. nite Dimensional Analysis, Quantum Proba- bility and Related Topics, 1(01):83–89, 1998. [34] Wojciech Roga, Mark Fannes, and Karol doi:10.1142/S0219025798000077. Życzkowski. Davies maps for qubits [23] Ingemar Bengtsson and Karol Życzkowski. and qutrits. Reports on Mathematical Geometry of quantum states: An intro- Physics, 66(3):311–329, 2010. doi:10.1016/ duction to quantum entanglement. Cam- S0034-4877(11)00003-6. bridge University Press, 2017. doi:10. [35] Angel Rivas and Susana F. Huelga. Open 1017/CBO9780511535048. quantum systems. Springer-Verlag, 2012. [24] Dénes Petz. Quasi-entropies for finite quan- doi:10.1007/978-3-642-23354-8. tum systems. Reports on mathematical [36] Lorenzo Campos Venuti and Paolo Za- physics, 23(1):57–65, 1986. doi:10.1016/ nardi. Dynamical response theory for driven- 0034-4877(86)90067-4. dissipative quantum systems. Phys. Rev. A, [25] Andrew Lesniewski and Mary Beth Ruskai. 93:032101, 2016. doi:10.1103/PhysRevA. Monotone Riemannian metrics and rela- 93.032101. tive entropy on noncommutative probabil- [37] L. Lorne Campbell. An extended Čencov ity spaces. Journal of Mathematical Physics, characterization of the information metric. 40(11):5702–5724, 1999. doi:10.1063/1. Proceedings of the American Mathematical 533053. Society, 98(1):135–141, 1986. doi:10.1090/ [26] Martin Idel. On the structure of positive S0002-9939-1986-0848890-5.

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 11 A Derivation of Eq. (11)

We include below an explicit derivation of Eq. (11), i.e., we calculate the expansion of the 1 -Rényi √ √ 2 entropy S 1 (ρ0, ρs) = −2 log Tr ρ0 ρs for ρs = exp (sM)(ρ0) to the ﬁrst non-vanishing order in s, 2 assuming that the state ρ is full-rank. √ 0 Setting χs = ρs, we have √ √ s2 Tr ( ρ ρ ) = hχ , χ i = 1 + s hχ , χ˙ i + hχ , χ¨ i + Os3 . (46) 0 s 0 s 0 0 2 0 0 2 Moreover, from ρs = χs one ﬁnds

ρ˙s = χsχ˙ s +χ ˙ sχs = (Lχs + Rχs )χ ˙ s := Aχs (χ ˙ s) , hence also

−1 χ˙ s = Aχs (ρ ˙s) , dA−1 χ¨ = A−1 (¨ρ ) + χs (ρ ˙ ) . s χs s ds s Using the above expressions we can calculate the terms in the expansion (46). We have 1 1 hχ , χ˙ i = hA (I), A−1 (ρ ˙ )i = Tr (ρ ˙ ) = 0 s s 2 χs χs s 2 s hence the term linear in s does not contribute. The quadratic term has two parts, dA−1 hχ , χ¨ i = hχ , A−1 (¨ρ )i + hχ , χs (ρ ˙ )i s s s χs s s ds s but one of them vanishes 1 hχ , A−1 (¨ρ )i = Tr (¨ρ ) = 0 . s χs s 2 s To calculate the remaining term notice that

2 −1 2 Aχs = Aχ0 + s (Lχ˙ 0 + Rχ˙ 0 ) + O s = I + sAχ˙ 0 Aχ0 Aχ0 + O s and therefore −1 −1 −1 −1 2 Aχs = Aχ0 I + sAχ˙ 0 Aχ0 + O s , which gives

dA−1 χs = −A−1A A−1 . ds χ0 χ˙ 0 χ0 s=0 Finally, this equation implies 1 h i hχ , χ¨ i = − Tr [A (χ ˙ )] = − Tr (χ ˙ )2 0 0 2 χ˙ 0 0 0 which completes the expansions √ √ s2 h i Tr ( ρ ρ ) = 1 − Tr (χ ˙ )2 0 s 2 0 and hence 2 h 2i 3 S 1 (ρ0, ρs) = s Tr (χ ˙ 0) + O s . 2 −1 The form (11) follows by using once again the fact that χ˙ 0 = Aχ (ρ ˙0) and via expressing ρ0 = P 0 i pi |iihi|.

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 12 B Monotone Riemannian metrics and Eq. (14)

Here we recall for completeness some well-known facts about monotone Riemannian metrics and discuss how the key inequality (14), obtained by Lesniewski and Ruskai in [25], is related to the monotonicity property Eq. (13). Monotone Riemannian metrics over S>0(H), whose defining property is the data-processing inequality Eq. (13), were first classified completely by Petz [20, 21], who extended an earlier result by Morozova and Chentsov [19]. In short, every such metric admits the form

−1 gρ(X,Y ) = hX, Kρ (Y )i (47a) for X,Y ∈ Tρ S>0(H) (i.e., are hermitian and traceless operators) and

−1 Kρ := k LρRρ Rρ (47b) where k : R≥0 → R is an operator monotone function that satisﬁes k(t) = tk t−1 ∀t > 0 . (47c)

It might be instructive at this point to check that the bilinear form gρ indeed defines a Riemannian metric. Operator monotone functions are always analytic [28], hence the bilinear form (47a) is smooth −1 with respect to ρ. Equation (47c) guarantees that Kρ preserves hermiticity, making gρ real valued. −1 Since ρ is a full-rank state, Kρ is always well-defined and positive definite, implying that the bilinear form (47a) is non-degenerate, and therefore gρ is indeed a Riemannian metric. P One can easily express gρ(X,X) explicitly by invoking the spectral decomposition ρ = i pi |iihi|, which yields

p −1 g (X,X) = X p k j |hi|X|ji|2 . (48) ρ i p ij i

Notice that, for all valid k, the diagonal part i = j corresponds to the Fisher metric (up to proportion- ality constant). Consequently one recovers the classical analogue result [37] over the (d−1)-dimensional probability simplex stating that if a Riemannian metric gp satisﬁes the monotonicity property

gp(x, x) ≥ gT (p)(T (x),T (x)) (49) under the action of stochastic matrices T then it is necessarily proportional to the Fisher metric, i.e.,

δij (gp)ij ∝ . (50) pi A consequence of the richness of the monotone metrics in the quantum regime is the presence of the (λ) free parameter λ in the monotones ρ M,N introduced in the main text. Let us finally recall how inequalityJ K (14) implies the monotonicity property Eq. (13). In [25], the authors showed that for the class of functions k defined above satisfying k(1) = 1 (which amount to a −1 mere normalization condition), superoperators Kρ admit the integral representation Z ∞ −1 −1 −1 Kρ = [λRρ + Lρ] + [Rρ + λLρ] Ng(λ)dλ (51) 0 where Ng(λ)dλ is a singular measure (the detailed definition of can be found in [25, 15]). The important point is that, due to the above representation, the monotonicity of the metric (13) follows from the monotonicity of the integrand in Eq. (51); this is what is shown in the theorem of Eq. (14). In fact, notice that the theorem is slightly more general; it applies to any operator (i.e., not necessarily hermitian and traceless), a freedom that we take advantage of in the main text.

Accepted in Quantum 2020-04-27, click title to verify. Published under CC-BY 4.0. 13