Universal equivalence between divisibility and information flow ...

Universal equivalence between divisibility and information flow notions of quantum Markovianity ´ Dariusz Chru´sci´ nski1 and Angel Rivas2, 3 1

arXiv:1710.06771v1 [quant-ph] 18 Oct 2017

Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudzi¸adzka 5/7, 87–100 Toru´ n, Poland 2 Departamento de F´ısica Te´ orica I, Facultad de Ciencias F´ısicas, Universidad Complutense, 28040 Madrid, Spain. 3 CCS-Center for Computational Simulation, Campus de Montegancedo UPM, 28660 Boadilla del Monte, Madrid, Spain We prove the equivalence between CP-divisibility and the lack of information backflow for an arbitrary dynamical map. This equivalence is universal, that is, does not depend upon the invertibility of the map. Our approach reconciles various attempts where the invertibility was essential (divisibility and/or distinguishability) or inessential (guessing probability). Interestingly this result sheds new light into the structure of the time-local generators giving rise to CP-divisible evolutions. We show that if the map is not invertible then positivity of dissipation/decoherence rates is no longer necessary for CP-divisibility. PACS numbers: 03.65.Yz, 03.65.Ta, 42.50.Lc

Introduction.— Recently, the notion of non-Markovian quantum evolution received considerable attention (see recent review papers [1–3]). Quantum systems interacting with an environment [4, 5] are of increasing relevance due to rapidly developing modern quantum technologies like quantum communication or quantum computation [6]. It turns out that recent experimental techniques allow one to go beyond standard Markovian approximations (weak coupling and/or neglecting memory effects caused by the interaction with environment). Therefore, there is a need for the exploration of non-Markovian regime, and characterizing the genuine properties of this kind of evolution. To this end, two main approaches which turned out to very influential are based on the concept of CPdivisibility [7] and information flow [8]. A dynamical map Λt is family of completely positive (CP) and tracepreserving (TP) maps acting on space B(H) of bounded operators on the Hilbert space H. We say that Λt is divisible if it can be decomposed as Λt = Vt,s Λs ,

(1)

where Vt,s : B(H) → B(H) for any t ≥ s. Note that since Λt is TP the map Vt,s is necessarily TP on the range of Λs but needs not be trace-preserving on the entire B(H). However, if Vt,s is TP on the B(H), one calls Λt P-divisible if Vt,s is also a positive map on the entire B(H), and CP-divisible if Vt,s is CP on the entire B(H) [9]. Acording to [7] the evolution is considered Markovian iff the corresponding dynamical map Λt is CP-divisible. A second idea is based on a physical feature of the system-reservoir interaction. It is claimed [8] that the phenomenon of reservoir memory effects may be associated with an information backflow, that is, for any pair of density operators ρ1 and ρ2 one can define the information flow σ(ρ1 , ρ2 ; t) =

d ||Λt ρ1 − Λt ρ2 ||1 , dt

(2)

where ||A||1 denotes the trace norm of A. Following [8] Markovian evolution is characterized by σ(ρ1 , ρ2 ; t) ≤ 0. Whenever σ(ρ1 , ρ2 ; t) > 0 one calls it information backflow meaning that the information flows from the environment back to the system. In this case the evolution displays nontrivial memory effects and it is evidently nonMarkovian. Interestingly, both P- and CP-divisible maps have a clear mathematical characterization [10]. Theorem 1 Let us assume that Λt is an invertible dynamical map. Then Λt is P-divisible iff d ||Λt X||1 ≤ 0, dt

(3)

for any Hermitian X ∈ B(H). It is CP-divisible iff d e 1 ≤ 0, ||(1l ⊗ Λt )X|| dt

(4)

e ∈ B(H ⊗ H). for any Hermitian X

Actually, the condition σ(ρ1 , ρ2 ; t) ≤ 0 is slightly weaker version of (3): one takes X = ρ1 − ρ2 which means that X is Hermitian but traceless. In [10] we proposed how to reconcile P-divisibility with information flow by noticing that any Hermitian operator X can be interpreted (up to some multiplicative constant) as a so-called Helstrom matrix [11] X = p1 ρ1 − p2 ρ2 with p1 + p2 = 1. It characterizes the error probability of discriminating between states ρ1 or ρ2 with prior probabilities p1 and p2 , respectively [10]. The relation between divisibility and information flow was recently reconsidered by Bylicka et al. [12]. They proved Theorem 2 Let us assume that Λt is an invertible dynamical map. Then Λt is CP-divisible iff d ||(1d+1 ⊗ Λt )(e ρ1 − ρe2 )||1 ≤ 0, dt

(5)

2 for any pair of density operators ρe1 , ρe2 in B(H′ ⊗ H) with dim(H′ ) − 1 = dim(H) = d.

Proof: If Λt is divisible and X ∈ Ker(Λs ), then

So comparing (4) with (5) one enlarges the dimension of ancilla d → d+1 but uses only equal probabilities p1 = p2 like in the original approach to the information flow [8]. In the proof of Theorems 1 and 2 the assumption of invertibility was essential. Note that any open system dynamics can be written as Λt ρ = TrE [U (t, 0)ρ ⊗ ωE U † (t, 0)] where ωE is a fixed state of the environment, and according to the postulates of the quantum mechanics U (t, s) is a unitary evolution family which satisfies the Schr¨odinger Equation and so it is continuous and differentiable. Since the partial trace is continuous but noninvertible, a dynamical map Λt is always a continuous map, however it can be non-invertible. Interestingly, Buscemi and Datta [13] analyzed information backflow defined in terms of the guessing probability of discriminating an ensemble of states {ρi } (i = 1, 2, . . .) acting on H ⊗ H with prior probabilities pi . It was shown [13] that an evolution is CP-divisible if and only if the guessing probability decreases for any ensemble of states. In this approach invertibility of the map plays no role and hence this approach is universal. However, the price one pays is the use of ensembles containing arbitrary number of states ρi which makes the whole approach hardly implementable. Anyway, this result posses an important question whether the assumption of invertibility in Theorem 1 may be removed. In this Letter we show that indeed the assumption of invertibility may be removed (both in Theorem 1 and 2). This result sheds a new light into time-local master equations

and hence X ∈ Ker(Λt ). Suppose now that (8) is satisfied. To show that Λt is divisible we provide a construction for Vt,s . This construction is highly non-unique: if Y ∈ Im(Λs ), i.e. there exists X such Λs X = Y , we define Vt,s Y = Λt X. Suppose now that Y ∈ / Im(Λs ) and let Πs : B(H) → Im(Λs ) be a projector onto Im(Λs ), that is, Πs Πs = Πs is an identity operation on Im(Λs ). Define

d Λt = Lt Λt , Λt=0 = 1l. dt

(6)

One usually says that the corresponding solution Λt is CP-divisible if the two-point propagator Vt,s = T e

Rt s

Lτ dτ

,

(7)

is CPTP for any t ≥ s, and hence one concludes that Lt is time-dependent GKLS generator. However, it turns out to be true only for invertible maps. In this Letter we show how to generalize this result for arbitrary dynamical maps. Divisible maps.— Interestingly, the property of divisibility is fully characterized by the following Proposition 1 A dynamical map Λt is divisible if and only if Ker(Λt ) ⊇ Ker(Λs ), for any t > s.

(8)

Λt X = Vt,s (Λs X) = Vt,s 0 = 0,

Vt,s Y := Λt X,

(9)

where X is a arbitrary element such that Πs Y = Λs X. Note that this is a well-defined construction. Indeed, if Λs X = Λs X ′ = Πs Y , then our construction implies Λt X = Λt X ′ for t > s. Specifically, ∆ = X − X ′ ∈ Ker(Λs ) and hence due to (8) one has ∆ ∈ Ker(Λt ) which implies Λt ∆ = Λt X − Λt X ′ = 0. It should be stressed, however, that Vt,s needs not be TP due to the fact that the projector Πs needs not be TP.  Remark 1 If Λt is invertible, then it is always divisible due to Vt,s = Λt Λ−1 s . In this case condition (8) is trivially satisfied: Ker(Λt ) = Ker(Λs ) = 0. Actually, there is a simple sufficient condition for divisibility Proposition 2 If the dynamical map Λt satisfies d ||Λt X||1 ≤ 0, dt

(10)

for all Hermitian X ∈ B(H), then Λt is divisible. Proof: Suppose that (10) is satisfied but Λt is not divisible, that is, there exists X such that Λs X = 0 but Λt X 6= 0 (t > s). This shows ||Λt X||1 > 0 = ||Λs X||1 and hence ||Λt X||1 does not monotonically decrease.  Remark 2 Clearly, the above condition is sufficient but not necessary since any invertible Λt is divisible even if it does not satisfy (10). Arbitrary dynamical maps.— Now we prove the central result which provide generalization of Theorems 1 and 2 for arbitrary, that is, not necessarily invertible, dynamical map. Theorem 3 A dynamical map Λt is P-divisible iff d ||Λt X||1 ≤ 0, dt

(11)

for any Hermitian X ∈ B(H). It is CP-divisible iff d e 1 ≤ 0, ||(1l ⊗ Λt )X|| dt e ∈ B(H ⊗ H). for any Hermitian X

(12)

3 Proof: If Λt is P-divisible then (11) clearly follows as Vt,s are TP contractions on B(H) [14, 15]. If (11) is satisfied then Λt is divisible because Proposition 2 above, and 1 d kΛt Xk1 = lim+ [kΛt+ǫ Xk1 − kΛt Xk1 ] dt ǫ→0 ǫ 1 = lim [kVt+ǫ,t (Λt X)k1 − kΛt Xk1 ] ≤ 0, ǫ→0+ ǫ (13) so Vs+ǫ,s for small ǫ is a TP contraction on Im(Λs ). Then, Vt,s is also a TP contraction on Im(Λs ) because it can be obtained by infinitesimal composition of {Vsi +ǫ,si }t>si ≥s . However, the action of Vt,s out of Im(Λs ) is not defined, so we need to prove that there exists at least one TP and contractive Vt,s . Since Λt=0 = 1 continuity implies that there exists some small ǫ such that Λǫ is invertible. Let us take t = s the smaller time instant where the dynamics becomes non-invertible, i.e. [Λt ]t 0 = ||(1d+1 ⊗ Λs )(e ρ1 − ρe2 )||1 and hence the inequality (19) is violated. The divisibility of Λt allows for the same are ′ = ρe1 − ρe2 to conclude gument as in Eq. (13) with X that 1d+1 ⊗ Vt,s is a TP contraction on the subspace of traceless operators in Im(1d+1 ⊗ Λs ). This is sufficient to prove that 1 ⊗ Vt,s is a TP contraction on Im(1 ⊗ Λs ) [16]. In order to show it, suppose Ye ∈ Im(1 ⊗ Λs ), e for some X e ∈ B(H ⊗ H). i. e. Ye = (1 ⊗ Λs )(X) Then, similarly to [12], we consider the traceless operator e − Tr(X)|d e + 1ihd + 1| ⊗ ρS ∈ B(H′ ⊗ H). Here ρS ∆ := X is a density operator on B(H) and {|ii}d+1 i=1 is a basis of H′ , being {|ii}di=1 is a basis of H. Since (1d+1 ⊗Λs )(∆) is also traceless, we use that 1d+1 ⊗ Vt,s is a TP contraction on the subspace of traceless operators in Im(1d+1 ⊗ Λs ) to obtain k(1d+1 ⊗ Vt,s Λs )(∆)k1 ≤ k(1d+1 ⊗ Λs )(∆)k1 .

(20)

e and |d + 1ihd + 1| ⊗ ρS have orthogonal Note that X e + 1i = 0 because X e ∈ B(H ⊗ H). Due to support as X|d the fact that the identity over the ancilla does not change e and |d + 1ihd + this orthogonality, the transformed of X 1| ⊗ ρS by 1d+1 ⊗ Λt have also orthogonal supports and we find e k(1d+1 ⊗ Vt,s Λs )(∆)k1 = k(1 ⊗ Vt,s )(Ye )k1 + |Tr(X)|, e k(1d+1 ⊗ Λs )(∆)k1 = kYe k1 + |Tr(X)|,

4 where we have used that Λt is a CPTP map. These results combined with inequality (20) yield k1d ⊗ Vt,s (Ye )k1 ≤ kYe k1 ,

(21)

which proves that 1 ⊗ Vt,s a is TP contraction on Im(1 ⊗ Λs ), so that Eq. (12) is satisfied. Therefore, the result follows by Theorem 3.  Examples. — now we provide a few simple examples of not invertible but CP-divisible dynamical maps. Example 1 (Amplitude damping channel) The dynamics of a single amplitude-damped qubit is governed by a single function G(t) which depends on the form of the reservoir spectral density J(ω) [4]:   |G(t)|2 ρ11 G(t)ρ12 , (22) Λt ρ = G∗ (t)ρ21 (1 − |G(t)|2 )ρ11 + ρ22 This evolution is generated by the following time-local generator Lt ρ = −

is(t) 1 [σ+ σ− , ρ] + γ(t)(σ− ρσ+ − {σ+ σ− , ρ}), 2 2

where σ± are the spin lowering and rising operators to˙ ˙ G(t) gether with s(t) = −2Im G(t) G(t) , and γ(t) = −2Re G(t) . Now, the dynamical map is invertible whenever G(t) 6= 0. Suppose now that G(t∗ ) = 0 and G(t) 6= 0 for t < t∗ (note that G(0) = 1). The image of Λt∗ is just proportional to the ground state P0 = σ− σ+ , and so a CPTP projector onto Im(Λt∗ ) reads Πt∗ X = P0 TrX.

(23)

It is, therefore clear that Λt is divisible if and only if G(t) = 0 for t ≥ t∗ . Hence, finally, the map Λt is CPdivisible iff it is divisible and γ(t) ≥ 0 for t < t∗ . Note that γ(t) for t ≥ t∗ is arbitrary. The only constraint is G(t) = 0. Hence, positivity of γ(t) is sufficient but not necessary for CP-divisibility. It is necessary only if γ(t) is finite for finite times, that is, the generator Lt is regular and the map Λt is invertible. Note that divisibility means that if the system relaxed to the ground state (at time t∗ ) it stays in that state forever. In addition, CPdivisibility means that the relaxation to the ground state d was monotonic dt |G(t)| ≤ 0. Example 2 (Relaxation to the equilibrium state) Consider the dynamical map Λt ρ = [1 − F (t)]ρ + F (t)ωTrρ,

(24)

where ω is an equilibrium density operator satisfying Λt ω = ω. Λ0 = 1l implies F (0) = 0 and if 0 ≤ F (t) ≤ 1 the map is CPTP, being invertible for F (t) < 1. Suppose that for some time t∗ one has F (t∗ ) = 1, i.e. Λt∗ ρ = ωtrρ

is not invertible. The map Λt is divisible iff Λt = Λt∗ for t ≥ t∗ , or, equivalently, iff F (t) = 1 for t ≥ t∗ . One has for the propagator Vt,s = Λt Λ−1 for s < t∗ , with s F (s) Λs−1 ρ = 1−Fρ (s) − 1−F ωTrρ. The projector Πt∗ is given (s) by Πt∗ ρ = lim Vt∗ ,t∗ −ǫ ρ = lim Λt∗ Λ−1 t∗ −ǫ ρ = ωTrρ. ǫ→0+

ǫ→0+

So we can take Vt,s = Πt∗ for s ≥ t∗ . Again, divisibility means that if Λt ρ relaxes to ω at finite time t∗ it stays at ω forever. CP-divisibility is equivalent to divisibility plus the monotonicity condition F˙ (t) ≥ 0 for t < t∗ . Finally, one finds for the generator Lt ρ = γ(t)(ωtrρ − ρ),

(25)

˙

F (t) with γ(t) = 1−F (t) ≥ 0 for t < t∗ . This formula does not work for t ≥ t∗ due to F (t) = 1. Hence, for t ≥ t∗ the Rt rate γ(t) is arbitrary (provided that 0 γ(τ )dτ = ∞ for t ≥ t∗ ).

Example 3 (Random unitary evolution) Consider the qubit evolution governed by the following time-local generator 3

Lt ρ =

1X γk (t)(σk ρσk − ρ), 2

(26)

k=1

which leads to the following unital dynamical map (timeP3 dependent Pauli channel): Λt ρ = α=0 pα (t)σα ρσα . The map is invertible if its corresponding eigenvalues Z t −Γj (t)−Γk (t) λi (t) = e ; Γj = γj (τ )dτ 0

where {i, j, k} is a permutation of {1, 2, 3}, are different from zero (note, that the remaining eigenvalue λ0 (t) = 1). Now, if for example Γ3 (t1 ) = ∞ at finite time t1 , then λ1 (t1 ) = λ2 (t1 ) = 0, and hence divisibility implies λ1 (t) = λ2 (t) = 0 for t ≥ t∗ . One finds the corresponding CPTP projector Πt1 X =

1 (X + σ3 Xσ3 ). 2

Note that Πt1 σ1 = Πt1 σ2 = 0. Now, if at t2 > t1 one has Γ2 (t2 ) = ∞ (or equivalently Γ1 (t2 ) = ∞), then λ3 (t2 ) vanishes as well and hence divisibility implies λα (t) = 0 for t ≥ t2 (α = 1, 2, 3). One finds the corresponding CPTP projector Πt2 X =

1 I TrX, 2

that is, it fully depolarizes an arbitrary state of the system. To summarize: the evolution is CP-divisible iff all γα (t) ≥ 0 for t < t1 , and γ3 (t) continues to be nonnegative up to t2 . From t2 on the system stays at the maximally mixed state.

5 Conclusions. — In this Letter we provided complete characterization of CP- and P-divisible dynamical maps in terms of monotonicity of trace norm – Theorems 3 and 4. It should be stressed that contrary to the previous attempts this result does not require invertibility of the map and hence it is universal. Interestingly this result sheds new light into the structure of the timelocal generator Lt which gives rise to CP-divisible evolution. For invertible maps, Lt has a structure of timedependent GKLS generator. In particular all dissipation rates γk (t) ≥ 0 for all t ≥ 0. It is no longer true for dynamical maps which are not invertible, that is, they correspond to singular generators [17]. In this case γk (t) ≥ 0 but only for t ∈ [0, t∗ ), where t∗ is the first moment of time where Λt becomes non-invertible. For t ≥ t∗ some γk (t) might be temporally negative and still the evolution might be CP-divisible. The point t∗ at which some γk (t) becomes singular provides an analog of the event horizon: the future behavior of a set of γk (t) does not effect the evolution of the system. A typical example is the evolution reaching equilibrium state at finite time t∗ . Then the system stays at equilibrium forever irrespective of the future (t > t∗ ) time dependence of the generator. Our result fits also the recent result of Buscemi and Datta [13] which states that the evolution is CP-divisible iff the guessing probability decreases for any ensemble of states. Again, it was derived for arbitrary dynamical maps not necessarily invertible. Finally, it should be clear that our result immediately applies for the classical evolution represented by d the stochastic map P Tt : it is P-divisible iff dt ||Tt x||1 ≤ 0, where ||x||1 = k |xk | denotes the Kolmogorov norm (distance), and x is an arbitrary element from Rd [1]. Appendix. — In order to see that Πs is well defined, we employ induced norm on the dual space of B(H):

contraction for ǫ ∈ (0, s) we arrive at k(Vs,s−ǫ Λs−δ − Vs,s−ǫ Λs−ǫ )(X)]k1 kΛs−δ (X)k1 X6=0

kVs,s−ǫ − Vs,s−δ k = sup

k(Λs−δ − Λs−ǫ )(X)]k1 kΛs−δ (X)k1 X6=0

≤ kVs,s−ǫ k sup

k(Λs−δ − Λs−ǫ )(X)]k1 . kΛs−δ (X)k1 X6=0

≤ sup

(28)

Since limǫ→0+ Λs−ǫ = Λs the family Vs,s−ǫ is Cauchy convergent as ǫ → 0+ . Therefore the limit in (15) is convergent as the dual space of B(H) is a Banach space with the induced norm. Similar manipulations show that Πs is idempotent: kΠ2s − Πs k = sup lim + Y 6=0 ǫ,δ→0

= sup lim

X6=0 ǫ,δ→0+

k(Vs,s−ǫ Vs,s−δ − Vs,s−δ )(Y )k1 kY k1 k(Vs,s−ǫ Λs − Λs )(X)k1 kΛs−δ (X)k1

≤ sup lim + kVs,s−ǫ k X6=0 ǫ,δ→0

≤ sup lim X6=0

ǫ,δ→0+

k(Λs − Λs−ǫ )(X)k1 kΛs−δ (X)k1

k(Λs − Λs−ǫ )(X)k1 = 0. (29) kΛs−δ (X)k1

On the other side, from Eq. (14) and again due to the fact that Λs−ǫ is bijective for ǫ ∈ (0, s), we conclude that the range of Vs,s−ǫ is the same as the range of Λs , and so the image of Πs is Im(Λs ). Moreover, since Vs,s−ǫ is positive and TP for ǫ ∈ (0, s), we conclude that Πs is a TP and positive projection onto Im(Λs ). Acknowledgements

DC was supported by the National Science Center project 2015/17/B/ST2/02026. AR acknowledges the Spanish MINECO grants FIS2015-67411, FIS201233152, the CAM research consortium QUITEMAD+ S2013/ICE-2801, and U.S. Army Research Office through grant W911NF-14-1-0103 for partial financial support.

k(Vs,s−ǫ − Vs,s−δ )(Y )k1 kY k1 Y 6=0

kVs,s−ǫ − Vs,s−δ k = sup

k(Vs,s−ǫ − Vs,s−δ )[Λs−δ (X)]k1 kΛs−δ (X)k1 X6=0

= sup

k(Vs,s−ǫ Λs−δ − Λs )(X)]k1 , kΛs−δ (X)k1 X6=0

= sup

(27)

where we have used that Λs−δ is bijective for δ ∈ (0, s). Moreover, since kVs,s−ǫ − Vs,s−δ k = kVs,s−δ − Vs,s−ǫ k we can take δ ≥ ǫ without loss of generality. Then by writing Λs = Vs,s−ǫ Λs−ǫ and using that Vs,s−ǫ is a TP

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