Almost all quantum states have non-classical correlations
A. Ferraro, L. Aolita, D. Cavalcanti, F. M. Cucchietti, A. Acin
AAlmost all quantum states have non-classical correlations
A. Ferraro, L. Aolita, D. Cavalcanti,
1, 2
F. M. Cucchietti, and A. Ac´ın
1, 3 ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain Center for Quantum Technologies, University of Singapore, Singapore ICREA-Instituci´o Catalana de Recerca i Estudis Avan¸cats, Lluis Companys 23, 08010 Barcelona, Spain (Dated: October 22, 2018)Quantum discord quantifies non-classical correlations in a quantum system including those notcaptured by entanglement. Thus, only states with zero discord exhibit strictly classical correlations.We prove that these states are negligible in the whole Hilbert space: typically a state picked outat random has positive discord; and, given a state with zero discord, a generic arbitrarily smallperturbation drives it to a positive-discord state. These results hold for any Hilbert-space dimension,and have direct implications on quantum computation and on the foundations of the theory of opensystems. In addition, we provide a simple necessary criterion for zero quantum discord. Finally, weshow that, for almost all positive-discord states, an arbitrary Markovian evolution cannot lead to asudden, permanent vanishing of discord.
PACS numbers: 03.67.Ac, 03.65.Yz, 03.67.Lx
The emergence of quantum information science moti-vated a major effort towards the characterization of en-tangled states, generally believed to be an essential re-source for quantum information tasks that outperformtheir classical counterparts. In particular, the geome-try of the sets of entangled/non-entangled states receivedmuch attention [1] – starting from the fundamental re-sult that the set of separable (non-entangled) states hasnon-zero volume in a finite dimensional Hilbert space [2].In other words, separable states are not at all negligible,which has direct implications on some implementationsof quantum computing [3] and on the definition of entan-glement quantifiers [4].Apart from entanglement, quantum states displayother correlations [5–7] not present in classical systems(meaning, here, systems where all observables commute).Aiming at capturing such correlations, Ollivier and Zurekintroduced the quantum discord [5]. They showed thatonly in the absence of discord there exists a measurementprotocol that enables distant observers to extract all theinformation about a bipartite system without perturbingit. This completeness of local measurements is featuredby any classical state, but not by quantum states, evensome separable ones. Thus, zero discord is a necessarycondition for only-classical correlations.Very recently, quantum discord has received increasingattention [8–15]. A prevailing observation in all resultsobtained so far is that the absence or presence of discordis directly associated to non-trivial properties of states.Thus, it is natural to question how typical are positive-discord states? . Here we prove that a particular sub-set of states that contains the set of zero-discord states,has measure zero and is nowhere dense . That is, it istopologically negligible: typically, every state picked outat random has positive discord; and given a state withzero discord, a generic (arbitrarily small) perturbationwill take it to a state of strictly positive discord. Re-markably, these results hold true for any Hilbert spacedimension and are thus in contrast with expectations based on the structure of entangled states [2]: while theset of separable states has positive volume, the set ofonly-classically correlated states does not. In addition,we provide a novel necessary condition, of very simpleevaluation, for zero quantum discord. With this tool wesuggest a schematic geometrical representation of the setof zero discord, and study the open-system dynamics ofdiscord. We find that for almost all states of positivediscord, the interaction with any (non-necessarily local)Markovian bath can never lead either to a sudden, per-manent vanishing of discord, nor to one lasting a finitetime-interval. In strong contrast to entanglement – whichtypically vanishes suddenly and permanently at a finitetime [16] –, discord can only permanently vanish in theasymptotic inifnite-time limit, i.e. at the steady state.Our results have wide-range implications. First, from afundamental perspective, they imply that only-classicallycorrelated states are extremely rare in the space of allquantum states. Second, it has been recently discov-ered that an arbitrary unitary evolution for any systemand bath is described (upon tracing the bath out) as acompletely-positive map on the system if, and only if,system and bath are initially in a zero-discord state [11].In view of the rarity of zero-discord states, the funda-mental recipe “unitary evolution + partial trace” is nowin conflict with complete positivity – one of the mostbasic and fundamental requirements that physical evo-lution is demanded to fulfill [17] – for almost all quan-tum states. Another interesting fact is that quantumdiscord is present in typical instances of a mixed-statequantum computation [18], even when entanglement isabsent [9, 10, 14]. This led Datta et al. [9, 10] to sug-gest that discord might be the resource responsible forthe quantum speedup in this computational model. Ifthe mere presence of discord was by itself responsible ofsome speedup, then our results would imply that almostall quantum states are useful resources. Furthermore, Pi-ani et al. [12] introduced a new task – local broadcasting– to operationally distinguish among different varieties a r X i v : . [ qu a n t - ph ] M a r of states with zero quantum discord. They showed thatonly some zero-discord states can be locally broadcasted,which – according to us – now means hardly any quan-tum state. Also, our general results on the Markovian dy-namics of discord complement and generalize the specificresults reported in Refs. 13. There, for particular cases oflocal channels and two-qubit systems, discord was neverobserved to vanish permanently at a finite time. As said,we prove the generality of this behavior. Finally, ourresults also apply to quantifiers of quantum correlationsother than discord. Quantum Discord .– Consider a bipartite system in acomposite Hilbert space H = H A ⊗ H B , of dimension d = d A × d B , with d A = dim( H A ) and d B = dim( H B ),respectively. Given a quantum state ρ ∈ B ( H ) (where B ( H ) denotes the set of bounded, positive-semidefiniteoperators o n H with unit trace), the von Neumann mu-tual information I AB between A and B is defined as I AB ( ρ ) . = S ( ρ A ) + S ( ρ B ) − S ( ρ ) , (1)where S ( ρ ) = − Tr[ ρ log ρ ] is the von Neumann entropyand ρ A,B = Tr
B,A [ ρ ]. Mutual information (1) quantifiesthe total amount of correlations in quantum states [6].A classically equivalent definition of mutual informa-tion is S ( ρ B ) − S ( ρ B | A ), where ρ B | A is the state of B given a measurement in A . Thus, classical mutual in-formation quantifies the decrease in ignorance (gain ofinformation) about subsystem B upon local measure-ment on A . Let us now consider a measurement con-sisting of (non-necessarily orthogonal) one-dimensionalmeasurement elements { M j } on H A . We can write thestate of system B conditioned on the outcome j for A as ρ B | j = Tr A [ M j ρ ] /p (cid:48) j , where the probability of outcome j is given by p (cid:48) j = Tr[ ρM j ]. By optimizing over the mea-surement set { M j } , one can define J AB ( ρ ) . = S ( ρ B ) − min { M j } (cid:88) j p (cid:48) j S ( ρ B | j ) , (2)which quantifies the classical correlations in ρ [6].Despite both definitions for the mutual information be-ing equivalent for classical systems, the quantum gener-alizations I AB and J AB in general do not coincide: Theirdiscrepancy defines the discord: D AB ( ρ ) . = I AB ( ρ ) − J AB ( ρ ) . (3)Notice that quantum discord is always non-negative andit is asymmetric with respect to A and B [5]. Null-discord states .– Let us denote by Ω the set com-posed of all states with zero discord:Ω . = { ρ ∈ B ( H ) s . t . D AB ( ρ ) = 0 } . (4)The members of this set are characterized [5, 10] by be-ing invariant under von Neumann measurements on A insome orthonormal basis { Π j } , that is ρ ∈ Ω ⇐⇒ ∃ { Π j } s . t . ρ = d A (cid:88) j =1 Π j ρ Π j . (5) This implies that the set { Π j } defines a basis of H A withrespect to which ρ is block diagonal [10]: ρ ∈ Ω ⇐⇒ ∃ { Π j } s . t . ρ = d A (cid:88) j =1 p j Π j ⊗ σ j , (6)where σ j are quantum states in B ( H B ) and { p j } definesa probability distribution.The characterization of Ω presented just above is notpractical in the sense that one has to check for the ex-istence of a measurement basis for which conditions (5)and (6) are satisfied. With this motivation, we derive asufficient condition for positive quantum discord that is basis-independent . From condition (6), and denoting by[ , ] the commutator, it follows that Proposition 1 If ρ ∈ Ω then [ ρ, ρ A ⊗ B ] = 0 , (7) where B is the identity operator on H B . Hence, [ ρ, ρ A ⊗ B ] (cid:54) = 0 implies that D AB ( ρ ) > . The converse, however, is not true: there are some stateswith positive discord that commute with their reducedones. States of interest like all pure maximally-entangledstates are an example. Let us introduce the auxiliary set C of all states satisfying Eq. (7): C . = { ρ ∈ B ( H ) s . t . [ ρ, ρ A ⊗ B ] = 0 } . (8)One has that Ω ⊂ C . We prove next that C hasmeasure zero and is nowhere dense, thereby implying thesame properties for Ω [19]. C has measure zero .– The key observation here is thatEq. (7) imposes a non-trivial constraint on ρ that con-fines it to a lower dimensional subspace of B ( H ). Thisalready suggests that the volume of C in B ( H ) is zero,a proof of which we sketch next (a detailed proof is givenin Appendix A). Consider a generic state ρ ∈ B ( H )expressed for example in an orthogonal basis given bythe tensor product between the traceless generators ofthe group SU ( d A ) and those of SU ( d B ). In this basis,the calculation of commutator (7) is straightforward andgives a set of implicit constraints on a state to belong to C . These constraints can be inverted to obtain an ex-plicit differentiable parametrization of the set C whichuses strictly fewer independent real parameters than theones needed to parametrize B ( H ). Since a differentiableparametrization of a set measure zero is also measurezero, C has measure zero in B ( H ). C is nowhere dense .– The set C , apart from beingof zero measure, is also nowhere dense, two concepts apriori independent. A set A is called nowhere dense (in X ) if there is no neighborhood in X on which A is dense.Equivalently, A is said to be nowhere dense if its closurehas an empty interior. In particular, this implies thatwithin an arbitrarily-small vicinity of any state that be-longs to C (Ω ) there are always states out of C (Ω ).Let us next observe that C is closed. This follows fromthe fact that the function f ( ρ ) = [ ρ, ρ A ⊗ B ] is a continu-ous map and the zeros of a continuous map form a closedset. Since any closed set of measure zero is nowheredense, this suffices to conclude that C is nowhere dense(see Appendix B). Being both closed and nowhere denseimplies in particular that a generic perturbation of a stateinside the set will drive it not just to a state outside, butto an entire region (an open set) outside of it. Geometry of the set of zero quantum discord .– First,let us observe that Ω is not a convex set. In fact, anarbitrary convex mixture between two states ρ and ρ that are block diagonal in incompatible local bases is typ-ically not block diagonal. On the other hand, if one mixesstates block diagonal in the same local basis, then the re-sulting state is necessarily block diagonal (in the samebasis), and therefore belongs to Ω . In particular, everystate of zero discord is connected to the maximally mixedstate /d , as the latter is trivially block diagonal in anylocal basis. This already shows us that the set Ω is con-nected. From a geometrical viewpoint, this means thatwhen moving rectilinearly from every state in Ω towards /d , only states in Ω are encountered. Accordingly, thesegment from every state out of Ω to /d is exclusivelycomposed of states out of Ω (being the two-qubit Wernerstate an instructive and simple example [5]). All in all,this leaves us with some sort of star-like hyper-structurefor Ω (with /d at the center), represented in Fig. 1.Some details of the set have been sacrificed in the fig-ure for the sake of clarity. For example, the tips of thestar rays are pure separable states, always at the borderof B ( H ), even though some of them are shown in its in-terior. Also, all the rays are connected not only through /d , but also by (non-convex) continuous trajectories in-duced by local unitaries. This nevertheless, as we alreadyknow, lies fully in a lower-dimensional subspace withoutvolume and is not represented in Fig. 1. The pictureshould thus not be taken as rigorous but just as a picto-rial representation to illustrate the main features of Ω .The geometrical notion of moving rectilinearly toward /d corresponds to the dynamical process of global de-polarization (global white noise). From the above con-siderations, it is clear now that global depolarization cannever induce finite-time vanishing of positive discord. Itonly induces the disappearance of discord in the asymp-totic infinite-time limit, when /d is actually reached. Infact, given the singular geometry of Ω suggested here,it seems highly unlikely that a noisy dynamical evolutioninducing a smooth trajectory in B ( H ) is able to take astate outside of Ω into its interior, and to keep it therepermanently. This is what we discuss next. Open-system dynamics of discord .– We now show forany state ρ / ∈ C – that is, for almost all (positive-discord) states – that the interaction with any (non-necessarily local) Markovian bath can never lead to a sud-den permanent vanishing of discord. Unless the asymp-totic infinite-time limit is reached, a Markovian map cantake ρ through the singular set C (and therefore also FIG. 1: (Color online). Schematic 2D representation of theset Ω of states with zero discord (dark lines). The set of allpossible states B ( H ) (enclosing ellipse) contains the set of sep-arable ones, depicted in grey, with the maximally mixed state1 /d in its center. All block diagonal states, including pureseparable states at the border of B ( H ), compose Ω , and canbe connected to 1 /d through states in Ω . Arbitrary states inΩ , however, cannot in general be combined to form a statein Ω . The whole of Ω lives in a lower dimensional subspaceof B ( H ). The dynamical trajectory of an arbitrary state ρ caused by a Markovian bath is represented in dashed-red. Inthis example the trajectory leads towards 1 /d . During itsevolution, the evolved state can only cross Ω a finite numberof times, and permanent vanishing of discord cannot happenbefore the infinite-time limit, at the stationary state. through Ω ) at most a finite number of times, equal to˜ d λ ( ˜ d λ − / −
1, where ˜ d λ is the number of differenteigenvalues of the map.Consider the system interacts with a generic (non nec-essarily local) bath during an arbitrary time τ . We de-scribe the evolution of the system with a completely-positive, trace-preserving map Λ τ : B ( H ) −→ B ( H ). Inwhat follows we use the notation of Ref. [20]. The mapΛ τ can be written in its (diagonal) spectral decomposi-tion, Λ τ = (cid:80) λ i | µ i ) ( ν i | , where λ i , | ν i ) and | µ i ) are re-spectively the eigenvalues, left and right eigenoperatorsof the map, Λ τ | µ i ) ≡ λ i | µ i ) and ( ν i | Λ τ ≡ λ i ( ν i | . For ageneral map, | ν i ) and | µ i ) span two non-orthogonal com-plete bases of B ( H ) and satisfy the conditions ( ν i | µ j ) ≡ δ ij and ( ν i | ν j ) (cid:54) = δ ij (cid:54) = ( µ i | µ j ), where δ ij is the Kro-necker delta and where ( X | Y ) is nothing but the Hilbert-Schmidt inner product: ( X | Y ) ≡ Tr[ X † .Y ]. In addition,these maps are always contractive, that is, | λ i | ≤ ∀ i and | λ i | = 1 for at least one i . For the specific caseof normal maps (those commuting with their adjoints)the left and right eigenoperators coincide and the basisthey span becomes orthonormal. Also, since we are in-terested in maps that describe some decoherence process,we assume that | λ i | < i , for the case | λ i | = 1 ∀ i corresponds to the case of unitary evolutionof the composite system.We consider now all maps Λ t that can be expressedas the successive composition of n times Λ τ : Λ t = (cid:80) λ ni | µ i ) ( ν i | , with t = nτ . All Markovian maps fall intothis category. From a strictly mathematical viewpoint,it is possible that some of the eigenvalues of Λ τ are null.Nevertheless, since the initial condition Λ τ =0 ≡ τ for which all eigenvalues are non-null.With this physically motivated observation in mind, werestrict our discussion to all maps such that λ i (cid:54) = 0 ∀ i .The initial state ρ is expanded in the basis {| µ i ) } as ρ = (cid:80) ρ i | µ i ), with ρ i ≡ ( ν i | ρ ), and after time t it evolvesto ρ t ≡ Λ t ( ρ ) = (cid:80) ρ i λ ni | µ i ). Now we can show thatfor a generic (positive-discord) initial state ρ such that[ ρ, ρ A ⊗ B ] (cid:54) = 0, there exists no t s ∈ [0 , ∞ ) such that ρ t ∈ C (and in particular such that ρ t ∈ Ω ) for all t > t s . We do it by reductio ad absurdum . Assume thenthe opposite is true. This means that there exists a state ρ t that satisfies [ ρ t , ρ At ⊗ B ] = 0, with ρ At ≡ Tr B [ ρ t ],for all t > t s . This, however, defines an infinite set of lin-early independent equations (as many as n > n s ≡ t s /τ ),which can never be satisfied. An analogous contradic-tion is obtained also if it is assumed that ρ t satisfies[ ρ t , ρ At ⊗ B ] = 0 only during the finite-time interval( t s , t s + ∆ t ], with any ∆ t >
0. Furthermore, we provethat ρ t can enter C (and in consequence also Ω ) a max-imum of ˜ d λ ( ˜ d λ − / − d λ is the numberof different eigenvalues λ i (see Appendix C). Discussion .– We have shown here that a random quan-tum state have strictly positive discord and a genericsmall perturbation of a state with zero discord will gen-erate discord. These results imply that only-classicallycorrelated quantum states are extremely rare. An inter- esting analogy can now be established: almost all statespossess discord just as almost all pure states possess en-tanglement. This means that the mere presence of pos-itive quantum discord lacks per se informative content(for example as a computational resource), being it acommon feature of almost all quantum states. Of course,this by no means excludes the possibility that a morequantitative characterization of the discord gives valu-able assessment of a state’s usefulness for some task. Ina future perspective, our results call for a better under-standing of the conflict between the standard approach toopen quantum systems and complete positivity of maps.A final comment about experimental implications. Wehave shown that states with zero discord are (densely)surrounded by states with positive discord. As a conse-quence, ruling out the presence of quantum discord is,strictly speaking, experimentally impossible (unless fur-ther assumptions are taken). The reason is that any mea-surement with a non-null error range is compatible witha positive amount of discord. This is in striking contrastto what happens for entanglement, whose presence caninstead be strictly ruled out in experiments.We thank A. Datta, R. Drumond, I. Garc´ıa-Mata, M.T. Cunha, J. Wehr, A. Winter and W. Zurek for dis-cussions; and the European QAP, COMPAS and PER-CENT projects, the Spanish MEC FIS2007-60182 andConsolider-Ingenio QOIT projects, the Generalitat deCatalunya, and Caixa Manresa for financial support. [1] I. Bengtsson, K. ˙Zyczkowski
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CoherentMatter Waves, Proceedings of the Les Houches SessionLXXII , edited by R. Kaiser, C. Westbrook, and F. David(Springer Verlag, Berlisn, 2001).[18] E. Knill and R. Laflamme, Phys. Rev. Lett. , 5672(1998).[19] Notice that these properties automatically extend also to“two-side measures” of quantumness [8] based on the dis-turbance caused by local measurements on both subsys-tems [7, 12]. This is because discord yields a general lowerbound to such measures [8], for the disturbance whenmeasuring both particles is never less than that causedwhen only one particle is subject to measurements.[20] C. M. Caves, J. of Supercond. , 707 (1999). Appendix A: The set C has measure zero We express ρ ∈ B ( H ) in the basis given by the trace-less, orthogonal generators γ Ai ⊗ γ Bj of the product group SU ( d A ) ⊗ SU ( d B ) (we use the same notation as Eq. (5.2)of Ref. [K. ˙Zyczkowski and I. Bengtsson, arXiv: quant-ph/0606228]): ρ = 1 d A d B AB + d A − (cid:88) i =1 τ Ai γ Ai ⊗ B + d B − (cid:88) j =1 τ Bi A ⊗ γ Bi + d A − (cid:88) h =1 d B − (cid:88) k =1 β hk γ Ah ⊗ γ Bk (cid:35) . (A1)The expression above maps the Hilbert space H to R d ( d = d A d B −
1) via the parameters τ Ai , τ Bi , and β hk .The partial trace of ρ over H B gives: ρ A = 1 d A (cid:34) A + d A − (cid:88) i =1 τ Ai γ Ai (cid:35) . (A2)The generators form a closed set with respect to commu-tation: [ γ Ai , γ Aj ] = 2 i (cid:80) k f ijk γ Ak , where f ijk is a rank-3antisymmetric tensor called the structure constant of thegroup SU ( d A ) (see, for example, [G. Mahler and V. A.Weberrus, Dynamics of Open Nanostructures (SpringerVerlag, Berlin, Germany, 1998]). The calculation of com-mutator (7) is straightforward in this representation,[ ρ, ρ A ⊗ B ] = 2 i d A − (cid:88) h,l,m =1 d B − (cid:88) k =1 β hk τ Al f hlm γ Am ⊗ γ Bk . (A3)Since matrices γ Am , γ Bk are orthogonal, imposing [ ρ, ρ A ⊗ B ] = 0 accounts for constraining parameters β hk and τ Al in the following way, d A − (cid:88) h,l =1 β hk τ Al f hlm = 0 (A4)for all k and m . These equations can be inverted. Inparticular, even the inversion of only one of them is suffi-cient for our purposes. Doing this, one obtains an explicitdifferentiable parametrization of the set C with strictlyfewer real independent parameters than d , i.e. , the onesrequired to parametrize B ( H ). Thus, C has Lebesguemeasure zero in B ( H ). (cid:3) Appendix B: The set C is nowhere dense Let us first show that C is closed. Since the partialtrace is a contractive map – meaning that the (trace) dis-tance between any two operators is larger than, or equalto, that between the operators resulting from the applica-tion of the map –, the map f : B ( H ) −→ f ( B ( H )) is con-tinuous. The operator zero (the operator whose matrixrepresentation is composed only of zero elements) in turnforms a closed subset of the set image of f , f ( B ( H )). Bythe topological definition of a continuous map, the preim-age of a closed set is also closed. Thus, C is closed, beingthe preimage of the closed set “operator zero”. To complete the proof, recall that the closure of aclosed set is – by definition – the set itself. Then a closedset of measure zero is nowhere dense because its beingmeasure zero implies that it has no interior point. Weshow the latter with our example of interest C : Supposethat there exists an interior point in the closed, zero-measure set C . By the definition of interior point, thiswould mean that there exists a state ρ ∈ C surroundedby an open ball of positive radius entirely contained in C (the metric used to define the ball is not relevant,since we are considering finite dimensions). Neverthe-less, since open balls have positive Lebesgue measure in R n for any n this would contradict the fact that C hasmeasure zero. Then, there exists no interior point of C ,implying that the set is nowhere dense. (cid:3) Appendix C: No finite-time according
For any initial state ρ such that [ ρ, ρ A ⊗ B ] (cid:54) = 0, weprove here that there exists no finite time t s after whichthe evolved state Λ t ( ρ ) belongs to C neither for all t ≤ t s nor for t ∈ ( t s , t s + ∆ t ], with any ∆ t >
0. Followingthe notation from the text above, we write the condition[ ρ t , ρ At ⊗ B ] = 0 ∀ t > t s explicitly as a system ofequations to see for their linear independence: d (cid:88) i,j =1 ρ i ρ j ( λ i λ j ) n (cid:2) | µ i ) , (cid:12)(cid:12) µ Aj (cid:1) ⊗ B (cid:3) = 0 ∀ n > n s ∈ N , (C1)where (cid:12)(cid:12) µ Aj (cid:1) = Tr B [ | µ j )] and the expansion ρ = (cid:80) ρ i | µ i )has been used. Let us relabel the pair of indexes ( i, j ) us-ing a single index k = 1 , ..., d × d and define R k = ρ i ρ j , L k = λ i λ j , and D k = (cid:2) | µ i ) , (cid:12)(cid:12) µ Aj (cid:1) ⊗ B (cid:3) . Then Eqs.(C1) above can be recast in the form of linear equationsin R k ’s: (cid:88) k R k L n s +1 k D k = 0 , ... (cid:88) k R k L n s + mk D k = 0 , (C2)for any m ∈ N , which have to be satisfied conditioned onthe initial condition [ ρ, ρ A ⊗ B ] (cid:54) = 0, (cid:88) k R k D k (cid:54) = 0 . (C3)We can already intuit that operator equations (C2) com-pose a set of m linearly independent equations from thefact that coefficients L k (with 0 < | L k | ≤
1) appear all ina geometric progression. We demonstrate this formallyby writing Eqs. (C2) and (C3) in a matrix representation,and thus recasting them as a set of linearly independentequations for complex numbers.In Eq. (C2) and (C3) we keep only the ¯ d terms suchthat D k and R k are both different from zero. Thus, ¯ d (cid:88) k =1 R k L nk D k = 0 , ¯ d (cid:88) k =1 R k D k (cid:54) = 0 . (C4)for n s < n ≤ n s + m . Let us now express the operators D k in an arbitrary matrix representation and focus on theirmatrix elements [ D k ] p,q . The initial condition Eq. (C4)implies that there exists at least a couple ( p ¯0 , q ¯0 ) suchthat (cid:80) d (cid:48) k =1 R k [ D k ] p ¯0 ,q ¯0 (cid:54) = 0, for some d (cid:48) ≤ ¯ d . Focusing onsuch a couple ( p ¯0 , q ¯0 ), and denoting d k ≡ [ D k ] p ¯0 ,q ¯0 (cid:54) = 0,we have that Eqs. (C4)) above reduce to ordinary equa-tions with complex coefficients d k and L k : d (cid:48) (cid:88) k =1 R k L nk d k = 0 , (C5) d (cid:48) (cid:88) k =1 R k d k (cid:54) = 0 . (C6)We now change variables to the non-null coefficients r k . = R k d k : d (cid:48) (cid:88) k =1 r k L nk = 0 , (C7) d (cid:48) (cid:88) k =1 r k (cid:54) = 0 , (C8)for all n s < n ≤ n s + m . If the coefficients L k are degen-erate, one can define another set of variables by groupingtogether all the r k ’s that correspond to the same degen-erate L k . Namely, we introduce s h = (cid:80) r k , where thesum extends to the r k ’s corresponding to the same L h .Denoting by ˜ d the number of different L h ’s we have thatEqs. (C7) and (C8) are equivalent to: ˜ d (cid:88) h =1 s h L n s +1 h = 0... ˜ d (cid:88) h =1 s h L n s + mh = 0 , (C9) ˜ d (cid:88) h =1 s h (cid:54) = 0 , (C10) with L h (cid:54) = L h (cid:48) if h (cid:54) = h (cid:48) and n s < n ≤ n s + m . Equa-tions (C9) are linear in s h with complex, non-null, non-degenerate coefficients in geometric progression. Fromthe properties of eigenvalues λ i mentioned in the text, wesee that coefficients L h necessarily satisfy | L h | ≤ ∀ h ,with | L h | = 1 for some h and | L h | < h ’s.Thus, Eqs. (C9) yield an homogenous system of m inde-pendent linear equations for ˜ d unknowns s h . For m < ˜ d there are m nontrivial solutions that are also compatiblewith (C10). For m ≥ ˜ d though, Eqs. (C9) become auniquely-determined homogenous system, whose uniquesolution is the trivial one s h = 0 for all h = 1 , ..., ˜ d . Thissolution, however, is not acceptable, since it contradictsthe initial condition Eq. (C10). (cid:3) As said, trajectories that cross C at most ˜ d − d λ different λ i -eigenvalues, it is straightfor-ward to count that there are ˜ d = ˜ d λ ( ˜ d λ − / L h ’s. As an example, we can now easily calculate anupper bound to the number of times C can be crossedby usual maps, such as a local depolarizing or dephas-ing channels (three different eigenvalues, two times), orthe global depolarizing channel (two different eigenval-ues, never).On the other hand, also from Eqs. (C9) one can seethat ρ t ∈ C , for t → ∞ if, and only if, the steady stateof the map is itself a state inside C . This is clear whenone considers the limit n s → ∞ in (C9), where all powersof L h , from L n s +1 h to L n s + mh , are exactly equal to zero for | L h | <
1, and equal to 1 for the single L h equal to one.Eqs. (C9) simply converge to the single condition s H = 0,where H is the one h for which L H = 1. This condition isin turn not in conflict with (C10) and therefore providesan acceptable solution. The coefficient s H is associated tothe projection of the initial state onto the map’s steadystate. So it simply gives the trivial fact that the finalstate will end up in C if and only if the steady state ofthe map is itself in C0