EEntanglement–Saving Channels
L. Lami and V. Giovannetti Universitat Aut`onoma de Barcelona, ES-01893 Bellaterra (Barcelona), Spainand Scuola Normale Superiore, I-56126 Pisa, Italy. NEST, Scuola Normale Superiore and Istituto Nanoscienze–CNR, I-56127 Pisa, Italy.
The set of Entanglement Saving (ES) quantum channels is introduced and characterized. Theseare completely positive, trace preserving transformations which when acting locally on a bipartitequantum system initially prepared into a maximally entangled configuration, preserve its entangle-ment even when applied an arbitrary number of times. In other words, a quantum channel ψ is saidto be ES if its powers ψ n are not entanglement-breaking for all integers n . We also characterizethe properties of the Asymptotic Entanglement Saving (AES) maps. These form a proper subsetof the ES channels that is constituted by those maps which, not only preserve entanglement forall finite n , but which also sustain an explicitly not null level of entanglement in the asymptoticlimit n → ∞ . Structure theorems are provided for ES and for AES maps which yield an almostcomplete characterization of the former and a full characterization of the latter. I. INTRODUCTION
Entanglement is a distinctive feature of quantum me-chanical systems and the key resource for quantum dataprocessing [1]. Even though ubiquitous this exotic formof correlations is extremely difficult to create and pre-serve, the problem arising from its monogamous charac-ter and from the tendency of quantum systems to estab-lish spurious connections with environmental degrees offreedom that are not directly under experimental con-trol [2]. A comprehensive study of the processes whichtend to deteriorate entanglement can be conducted inthe context of quantum channels, i.e. completely posi-tive, trace preserving, linear mapping operating on theset of the density matrices which describe the physicalstates of a quantum system. According to the postulateof quantum mechanics, quantum channels represent themost general physical transformations which a quantumsystem can undergo when interacting with an external,initially uncorrelated, environment, e.g. see Ref. [3–5].A very special set of quantum channels is constitutedby the so called entanglement–breaking maps [6]. Theyrepresent the most detrimental form of noise one has toface in any experimental implementation of quantum in-formation processing: when acting locally on an initiallyentangled bipartite system, an entanglement–breakingmap produces an output that is separable (i.e. not entan-gled) no matter how intense the initial entanglement was.Of course not all the completely positive, trace preservingtransformations are entanglement–breaking, the vast ma-jority of maps operating on quantum mechanical systemrepresenting milder forms of noise. In Ref. [7] a classifica-tion of these less disruptive processes was proposed whichis based on the accumulated effect arising under iterativeapplications of a given map. In particular a channel ψ was defined to be entanglement–breaking of order n ifit requires n recursive applications to remove all the en-tanglement initially present in the system. Building upfrom such approach a series of functionals [7, 8] were in-troduced aimed to quantify how noisy a given quantum channel is, under the assumption that correcting filter-ing processes are allowed between two subsequent ap-plications of the latter. Aim of the present work is todevelop further on these ideas by focusing on two veryspecial subsets of quantum channels which we dub Entan-glement Saving (ES) and Asymptotically EntanglementSaving (AES), respectively. An ES channel ψ describesan extremely weak, yet nontrivial, form of noise whichis characterized by the property of preserving the entan-glement of any bipartite maximally entanglement stateeven when applied locally n times recursively on one ofthe two subsystems, with n being a generic integer. Ac-cordingly a map ψ is ES if for all n , its iterative n –foldapplication ψ n is not entanglement–breaking. The AESchannels constitute a special subclass of the red ES set,formed by those maps which drive (as n goes to infinite)a maximally entangled state of the composite system to-ward a final configuration which still contains an explic-itly not null level of entanglement. Structural theoremsand a complete characterization of these maps are de-rived for case of finite dimensional systems.Due to the rather technical character of some of thetheorems we derive, in writing the present manuscript wemade an effort to be as self–contained as possible. In par-ticular, the first two sections are devoted to review someknown facts concerning the theory of quantum channels.Specifically, in Sec. II we set the notation and a providea formal characterization of unitary and entanglement–breaking maps. We also recall the Bloch representationfor qubit channels and the Kadison–Schwarz inequalitywhich will be extensively used in the subsequent sections.While referring to the lecture notes of M. M. Wolf [4]as fundamental reference, in Sec. III instead we give acomprehensive review of the spectral properties of quan-tum channels. Building from this technical introductionwe start hence to present our original contributions. InSec. IV we begin by introducing the notion of universalentanglement–preserving channels, defined as those com-pletely positive, trace–preserving linear maps which pre-serve all forms of entanglement, no matter how weak itmay be at the beginning of the transformation. These can a r X i v : . [ qu a n t - ph ] M a y be seen as the counter–parts of entanglement–breakingchannels and, confirming a result which is intuitively ex-pected, we formally prove that they coincide with the setof unitary transformations (see Theorem 17). The studyof ES channels is then presented in Sec. V. In particu-lar, we present a characterization of these maps basedon assumption that their determinant is not null (seeTheorem 21) and use this result to give a complete clas-sification for qubit systems (see Sec. V D). Section VI ishence devoted to the study of AES maps. Conclusionsand final discussion of the results are then provided inSec. VII. II. BASIC THEORY OF QUANTUMCHANNELSA. Generalities
In this section we shall set the notation and reviewsome basic results on the theory of quantum channels byfocusing on the case of finite–dimensional quantum sys-tems. Recall that a quantum channel provides the propermathematical framework to describe the dynamical evo-lution of an open quantum system. It is well–knownthat such a transformation can be equivalently seen as i) a unitary interaction with an external ancilla which islater discarded (Stinespring representation), as ii) a sumof matrix conjugation operations (Kraus representation),or finally as iii) an abstract linear, completely positive,trace–preserving superoperator (axiomatic approach) [3–5, 9]. B. Notation
In what follows, we will denote by H ( d ; C ) the set of d × d hermitian matrices. In some cases it can be use-ful to consider more generally the set of d × d complex(real) square matrices, denoted by M ( d ; C ) (or M ( d ; R ),respectively). Through the paper, it will be necessaryto distinguish between the operator statements A ≥ A > A ≥ A = 0 (i.e. A has no negativeeigenvalues, but at least one of them is zero). We willrefer to the first case by saying simply that A is positive ,to the second by specifying that A is strictly positive , andto the third by using the expression semipositive . For in-stance, we recall that the states of a d –dimensional quan-tum system are represented by positive matrices, eitherstrictly positive or semipositive.For what concerns the operations between linear sub-spaces H , H , we shall keep the simple notation H + H for the sum, while deserving the notation H ⊕ H for or-thogonal sums, i.e. sums in which we want to specify thatthe addends are indeed orthogonal. Analogously, X ⊕ Y will indicate the direct sum of two operators acting on orthogonal spaces. Now, let us fix the notation concerning the superoper-ators. There are various sets of interesting linear mapsacting on states of a quantum system (or more gener-ally, on square matrices). We will use a set of conve-nient abbreviations to indicate them, all written in boldtypes. In general, block capital letters indicate convexsets and italic capital letters sets defined by nonlinearequations, while small letters impose further linear con-straints. Moreover, a subscript d can be added if neces-sary to specify the dimension of the system (or the size ofthe square matrices) on which the maps are acting. In ourconventions, the letters P , CP , EB , U denote the setsof positive, completely positive, entanglement–breakingor unitary channels, respectively. The small letters t and u are used to specify the trace–preserving or the uni-tal conditions. Thus, with the above rules CPt willbe the set of completely positive, trace–preserving qubitchannels, while U will denote the set of conjugations by3 × T , is a positive but not completely positive map, thatis T ∈ Ptu − CPtu . The partial tranpose of a bipar-tite state ρ only with respect to the second subsystemwill be usually denoted by ρ T B . There is a very natu-ral operation we can define between quantum channels(or more generally between linear maps acting on matri-ces), i.e. their composition . It consists of the consecutiveapplication of two channels (linear maps) ψ and φ . Asusual in linear algebra, the simple juxtaposition φψ of thesymbols denotes the consecutive application of ψ firstly ,and of φ secondly . In the same way, φ n will indicate the n –fold composition of phi with itself.We adopt the standard notation rk L to indicate therank (i.e. the dimension of the image) of a linear op-erator (e.g. a square matrix) and ker L to denote itskernel (i.e. the linear subspace of vectors x such that L x = 0). Another standard notation is σ ( L ) for thespectrum and s ( L ) for the set of singular values of L .Naturally, s ↓ i ( L ) refers to the i th greatest singular valueof L . Actually, they are understood to be multisets ratherthan simple sets. In a multiset each element can be re-peated a number of times equal to its multiplicity. Wedenote by a L ( λ ) and g L ( λ ) the algebraic and geometricmultiplicities of the eigenvalue λ ∈ σ ( L ), respectively.The definition of singular values requires of course thepresence of a positive (semi)definite scalar or hermitianproduct on the vector space. If the vector space we aredealing with is R n or C n we will always choose the stan-dard scalar or hermitian product. Instead, we will re-gard to the set of square matrices in every dimensionas equipped with the Hilbert–Schmidt positive definitehermitian product Tr[ X † Y ]. Observe that the hermitianconjugation φ → φ † naturally induced on the superop-erators is nothing but the Heisenberg representation oftheir action, and that φ ∈ Pt ⇔ φ † ∈ Pu .The Schatten norms of index 1 ≤ p ≤ ∞ are indicatedwith (cid:107) · (cid:107) p and defined by (cid:107)L(cid:107) p ≡ (cid:16) Tr[( L † L ) p/ ] (cid:17) /p . (1)One has (cid:107)L(cid:107) p = (cid:32)(cid:88) i s pi ( L ) (cid:33) /p . (2)Observe that (cid:107) · (cid:107) is precisely the norm induced by theHilbert–Schmidt product. The p = 1 Schatten norm isknown also as trace norm. Furthermore, the natural gen-eralization to the p = ∞ case imposes (cid:107)L(cid:107) ∞ ≡ s ↓ ( L ) . (3)In what follows, | ε (cid:105) = √ d (cid:80) di =1 | i (cid:105) ⊗ | i (cid:105) will alwaysdenote the maximally entangled state of a bipartite sys-tem SS (cid:48) , with dim H S = dim H S (cid:48) = d . The Choi stateassociated with a linear map φ acting on d × d squarematrices is by definition R φ ≡ ( φ ⊗ I )( | ε (cid:105)(cid:104) ε | ) . (4)Finally, we will indicate with S AB the set of separabledensity matrices on a bipartite system AB (the subscriptcan be removed if there is no ambiguity). C. Kadison–Schwarz inequality
An important classical inequality that will turn out tobe very important is called
Kadison’s inequality (afterits discoverer). Its stronger form which exploits thecomplete positivity assumption is known as
Schwarzinequality . For the original proof of the Kadison’sinequality we refer to [10]; otherwise, a more intuitiveargument can be found in [11]. The text [12] provides acomprehensive reference on the subject.
Theorem 1 (Kadison–Schwarz Inequality) . Let ζ ∈ Pu d be a positive unital map. Then ∀ X = X † ∈ H ( d ; C ) , ζ ( X ) ≤ ζ ( X ) . (5) Moreover, if χ ∈ CPu d is completely positive and unital,then ∀ X ∈ M ( d ; C ) , ζ ( X ) † ζ ( X ) ≤ ζ ( X † X ) . (6) D. Characterizing unitary evolutions
A peculiar class of quantum channels is formed by theunitary evolutions, acting as U ( · ) = U ( · ) U † for someunitary matrix U . It is very useful for what follows tohave several less direct characterizations of the unitary channels, that is, several sufficient conditions to claimthat a given channel is indeed unitary. What this kind ofresults have in common is that for their proof a deep factknown as Wigner’s theorem turns out to be fundamental.For the original work we refer the reader to [13], p. 251–254. Otherwise, a direct and mathematically clear proofcan be found in [14]. Theorem 2 (Wigner’s Theorem) . Let T : H → H be a (not necessarily linear) operator ona (not necessarily finite–dimensional) Hilbert space H .Suppose that |(cid:104) T ( x ) | T ( y ) (cid:105)| ≡ |(cid:104) x | y (cid:105)| ∀ x, y ∈ H . (7) Then there exists a real function ϕ : H → R such that T ( x ) ≡ e iϕ ( x ) V x , (8) where V : H → H is an isometry or an anti–isometry. Inparticular, if H is finite–dimensional then V is unitaryor anti–unitary. Now, let us present the main result about the alter-native characterizations of unitary evolutions. Its proofcan be found through Chap. 6 of [4].
Theorem 3 (Alternative Characterizations of UnitaryChannels) . Let φ ∈ CPt be a quantum channel. Then the followingare equivalent:1. φ is a unitary evolution;2. det φ = ± , where the determinant is defined as theproduct of the eigenvalues (see Subsection III A);3. there exists the inverse φ − of φ , and it is again aquantum channel;4. φ maps pure states into pure states, and does nothave the form φ ( X ) = | α (cid:105)(cid:104) α | Tr X for some fixedpure state | α (cid:105) . A few comments are appropriate. In Subsection III Awe will see that | det φ | ≤
1, so that the unitary channelsare the ones and the only ones exhibiting the extremal value of the modulus of the determinant. The third con-dition of Theorem 3 is usually interpreted as the irre-versibility of the evolution of an open quantum system.In Section IV we will see another remarkable characteri-zation of the unitary channels.
E. Entanglement–breaking channels
A class of quantum channels that will play a cen-tral role in what follows is composed of the so–called entanglement–breaking channels [6]. Recall that a quan-tum channel φ ∈ CPt acting on a system A is calledentanglement–breaking (and we will write φ ∈ EBt ) iffor each system B and for each global input state ρ AB of AB , the output ( φ ⊗ I )( ρ AB ) is separable. Remarkably, φ ∈ EB , ψ ∈ CP ⇒ φψ, ψφ ∈ EB . (9)Moreover, it turns out that also EB (just like P or CP )is a closed convex set which is in addition closed underthe operation of taking the hermitian adjoint: φ ∈ EB ⇔ φ † ∈ EB . (10)This equivalence follows also from Theorem 4, which westate in a moment.The problem of the operational characterization of theEB class has been solved in [6]. This solution is thecontent of the following theorem. Theorem 4 (Structure Theorem for EB Channels) . Let φ ∈ EBt be a quantum channel. Then the followingfacts are equivalent:1. φ is entanglement–breaking.2. The associated Choi state R φ (see (4) ) is separable.3. φ can be written in the Holevo form introducedin [15], i.e. there are a (finite) set of density matri-ces { ρ i } and positive operators { E i } satisfying thesum rule (cid:80) i E i = , such that φ ( X ) ≡ (cid:88) i ρ i Tr[ E i X ] ∀ X ∈ M ( d ; C ) . (11)Observe that in the Holevo form (11) we can freelysuppose that the ρ i are pure states and that the E i are(positive) multiples of pure states. This can be seen di-rectly by diagonalizing both operators, and it is also aby–product of the proof. Equation (11) provides exactlythe operative interpretation we were looking for. Indeed,it states that the entanglement–breaking channels are ex-actly those channel which can be implemented by a mea-surement process (POVM) followed by a re–preparationof the system. F. Qubit channels
The qubit case, i.e. the case in which our channels acton a 2–dimensional system, deserves particular attention.Let us recap the main geometrical tools which becomeavailable in this particular framework. It is well–knownthat a qubit state can be written in the
Bloch represen-tation as ρ = + (cid:126)r · (cid:126)σ , (12)where (cid:126)σ = ( X, Y, Z ) is simply the vector of Pauli matri-ces, and | (cid:126)r | ≤
1. Observe that here the pure states areexactly those states ρ whose associated vector (cid:126)r has unit modulus. This can be immediately seen by noting thatthe spectrum of + (cid:126)r · (cid:126)σ is given by σ ( + (cid:126)r · (cid:126)σ ) = { | (cid:126)r | , − | (cid:126)r | } . (13)We can choose to represent a quantum qubit channel φ by means of its action on (12). This means that φ iscompletely specified once we assign the 3 × M and the 3–vector c such that φ (cid:18) + (cid:126)r · (cid:126)σ (cid:19) = + ( M(cid:126)r + (cid:126)c ) · (cid:126)σ . (14)In view of (14), we will sometimes indicate the channel φ with the notation ( M, c ). Remind that in this picturethe unitary evolution U ( · ) = U ( · ) U † , where U = e − i (cid:126)θ · (cid:126)σ/ is a SU(2) matrix, is represented by the counterclockwiserotation R ( (cid:126)θ ) of an angle θ around (cid:126)θ/ | (cid:126)θ | .Since in the (orthogonal) Pauli basis , X, Y, Z the lin-ear map φ is represented by ( c M ), the spectrum of φ asa linear application (see Subsection III A) is simply givenby σ ( φ ) = { } ∪ σ ( M ) , (15)and its determinant bydet φ = det M . (16)The Bloch representation (14) of the qubit quantumchannels allows us to find a useful canonical decomposi-tion for this special case. As pointed out firstly in [16], byapplying unitary evolutions to the left and to the rightof φ = ( M, c ) ∈ CPt the best special singular valuedecomposition we can achieve has the form M = P LQ ,with
P, Q ∈ SO(3) and L = l ( M ) 0 00 l ( M ) 00 0 l ( M ) ≡≡ s ( M ) 0 00 s ( M ) 00 0 sgn det( M ) s ( M ) . (17)Here the symbol sgn denotes the sign function , definedby sgn x ≡ +1 if x >
00 if x = 0 − x < , and s i ( M ) indicates the i th singular value of M . Usu-ally one can suppose | s ( M ) | ≤ s ( M ) , s ( M ), so that l ( M ) , l ( M ) ≥ l ( M ), which has the lowestmodulus, can be negative. These l ( M ) are called spe-cial singular values of the real 3 × M . Oncethe decomposition M = P LQ is obtained, we can define t ≡ P T c and write φ = ( M, c ) = P ( L, t ) Q = U Λ V . (18)Here U , V are the unitary channels corresponding to P, Q ∈ SO(3), and Λ ≡ ( L, t ) is the canonical diagonalform of φ . Remarkably, since the unitary evolutions areone–to–one applications between density matrices, thepositivity, the complete positivity and the entanglement–breaking conditions are not affected if one passes to thecanonical diagonal form. That is, with the notationsof (18) we have φ ∈ Pt ⇔ Λ ∈ Pt , (19) φ ∈ CPt ⇔ Λ ∈ CPt , (20) φ ∈ EBt ⇔ Λ ∈ EBt . (21)What can be said about the entanglement–breakingqubit channels? As a matter of fact, it turns out that theBloch representation allows us to find new useful charac-terizations for the EB qubit channels. Let us recall thefollowing theorem, which is obtained by joining togetherTheorem 1 and 2 of [17]. As usual, ρ T B denotes the par-tial transpose of the bipartite state ρ with respect to thesecond subsystem. Theorem 5 (EB Conditions for Qubit Channels) . Let φ ∈ CPt be a qubit channel. Then the followingfacts are equivalent:1. φ is entanglement–breaking.2. R T B φ ≥ .3. T φ ∈ CPt or φT ∈ CPt ( T is the matrix trans-position channel).4. φ has the “sign–change” property that changing any l i (cid:55)→ − l i of the matrix L defined in (17) and em-ployed in the canonical diagonal decomposition (18) yields another completely positive map.5. (cid:107) R φ (cid:107) ∞ ≤ . III. ADVANCED THEORY OF QUANTUMCHANNELS
Through this section, we will review some more ad-vanced topics in the theory of quantum channels. Theresults we will present have been already studied in theliterature, and a set of references to the previous workswill be provided. However, we find convenient for thesake of clarity to uniform the notation and group to-gether useful facts to be used extensively in the rest ofthe paper.
A. Spectral properties of positive maps
First of all, let us review the main results in the studyof the spectral properties of positive, trace–preservingmaps. Recall that a map φ ∈ Pt is first of all a linearoperator acting on the real space H ( d ; C ) of d × d hermitian matrices. Like all the linear operations ona d –dimensional real space, also φ can be regardedas a d × d real matrix. Therefore, a spectrum σ ( φ ), the related eigenvectors (actually, we should sayeigen matrices !) and the whole Jordan form (see Chap. 3of [18] for an introduction to this standard subject) canbe naturally associated to it. Let us discuss some generalproperties concerning the spectrum of an arbitrary Pt map. The condition of complete positivity (pertaining tothe physical quantum channels) has to be regarded as aparticular case. The knowledge of these basic propertieswill be very useful through the following chapters. Foran excellent overview with all the proofs, we refer thereader to Chap. 6 of [4]. We will approximately followthis text for our exposition. For the sake of simplicity,let us group all together in a theorem. Theorem 6 (Spectral Properties of Pt Maps) . Let φ ∈ Pt be a positive, trace–preserving map with spec-trum σ ( φ ) (counting multiplicities). Then the followingproperties hold.1. The eigenvalues are real or come in complex con-jugate pairs z, z ∗ , with the same multiplicity andJordan structure for z and z ∗ . If λ ∈ σ ( φ ) is realthen the related eigenvector can be chosen hermi-tian. Otherwise, φ ( Z ) = zZ ⇔ φ ( Z † ) = z ∗ Z † . Asa consequence, the linear span of the eigenvectorspertaining to complex conjugated eigenvalues is areal subspace, i.e. it admits a basis composed ofhermitian operators. Finally, the trace–preservingcondition imposes that the eigenvectors associatedwith (cid:54) = λ ∈ σ ( φ ) can be chosen traceless.2. Let X = φ ( X ) be an hermitian fixed point of φ .Denote by X = X + − X − the decomposition of X into its positive and negative spectral parts X ± ≥ .Then X + , X − as well as | X | = X + + X − are (pos-itive definite) fixed points of φ .3. There exists at least a density matrix ρ ≥ whichis fixed by φ (that is, φ ( ρ ) = ρ ).4. All the eigenvalues lie in the complex unit circle(i.e. λ ∈ σ ( φ ) ⇒ | λ | ≤ ). In particular, the deter-minant of the channel satisfies | det φ | ≤ . More-over, the eigenvalues with modulus equal to canonly have trivial Jordan blocks (this is the same asto say that the algebraic and geometric multiplici-ties are always the same for eigenvalues with unitmodulus, or that d k = 1 in (22) , and so no nilpo-tent operator is indeed present on these generalizedeigenspaces).5. Let φ = (cid:80) k ( λ k P k + N k ) ,P h P k = δ hk P k , Tr P k = d k , (cid:80) k P k = ,N d k k = 0 , P k N k = N k P k = N k (22) be the Jordan decomposition for φ . In (22) the λ k are the eigenvalues, the P k the (not necessarilyorthogonal!) projectors onto the generalized sub-spaces, and the N k are nilpotent (super)operators.Then the following combinations of spectral projec-tors are all Pt maps. Moreover, if φ ∈ CPt , thenso are these maps. E φ ≡ (cid:88) k : | λ k | =1 P k , (23) I φ ≡ (cid:88) k : | λ k | =1 λ ∗ k P k , (24) φ ∞ ≡ (cid:88) k : λ k =1 P k . (25) Proof.
1. These are all well–known consequences of thehermiticity–preserving condition, which impliesthat φ is indeed a real endomorphism of the realvector space H ( d ; C ), that is, a real d × d matrix.The trace–preserving condition, moreover, impliesthat φ ( Z ) = λZ, λ (cid:54) = 1 ⇒ Tr[ Z ] = Tr[ φ ( Z )] = λ Tr[ Z ] = 02. Call P + the projector onto the positive part of X , that is P + XP + = X + . Since we know that X = X + − X − = φ ( X ), we can write X + = P + XP + = P + φ ( X + − X − ) P + == P + φ ( X + ) P + − P + φ ( X − ) P + ≤≤ P + φ ( X + ) P + ⇒ P + φ ( X + ) P + − X + ≥ . We would like to prove that indeed P + φ ( X + ) P + − X + = 0. Thanks to the factthat P + φ ( X + ) P + − X + ≥
0, it will suffice to checkthat the trace of this operator is not greater thanzero.Tr [ P + φ ( X + ) P + − X + ] == Tr [ P + φ ( X + )] − Tr[ X + ] ≤≤ Tr [ φ ( X + )] − Tr[ X + ] = 0 .
3. Since φ † ∈ Pu , we know that φ † ( ) = , andso 1 ∈ σ ( φ † ). It is well–known that the spec-trum of the hermitian conjugate matrix is nothingbut the complex conjugate of the original spectrum(with the same multiplicities). As a consequence,1 ∈ σ ( φ ), and moreover the previous point ensuresthat the corresponding eigenvector can be chosenpositive (and of unit trace, of course).4. These nontrivial facts descend from the observa-tion that Pt (or CPt ) are compact sets closed for composition, and an eigenvalue with modulusgrater than 1 produces unbounded powers. More-over, even a nontrivial Jordan block pertaining toan eigenvalue with modulus 1 has unbounded pow-ers, as can be easily verified by direct calculations.5. The proof of this statement relies on exploiting aresult of number theory known as Dirichlet’s theo-rem on simultaneous Diophantine approximationsto show that both E φ and I φ are limit points of thesequence of powers ( φ n ) n ∈ N , and so that they mustbe Pt (or CPt ) if so is φ . Moreover, one can easilysee that φ ∞ is the limit of the means of the powers ,that is φ ∞ = lim n →∞ n n (cid:88) i =1 φ i , and therefore must be again positive (or completelypositive) and trace–preserving, since these proper-ties are preserved under compositions, convex com-binations and limits.Let us fix some nomenclature and notation. In viewalso of Theorem 6, an eigenvalue of unit modulus of apositive, trace–preserving map φ is called a peripheraleigenvalue , belonging to the peripheral spectrum σ P ( φ ).An eigenvector pertaining to a peripheral eigenvalue iscalled a phase point of the map. It is nothing but asquare matrix Z such that φ ( Z ) = e iθ Z for some realnumber θ . The linear span of the phase points, denotedby χ φ , is called phase subspace . Also the fixed subspace ,that is the eigenspace of φ pertaining to the eigenvalue1 (or the set of fixed points ), deserves a special notation,being indicated by η φ .Now, the following central questions naturally arise: what is the most general structure of the phase subspaceof a quantum channel? And what is the most generalaction of φ on this phase subspace? The rest of thissection is devoted to present the answer to this question.The proofs of the central claims are highly nontrivial andquite technical, even if the claims themselves can be writ-ten in a quite simple way. Consequently, we shall breakthe general argument into several smaller constructions,identified by the various subsections.The first step can be immediately understood, thanksto Theorem 6. Suppose that we want identify the struc-ture of the phase subspace χ φ of a completely positivemap φ ∈ CPt . We can then construct the spectral pro-jection E φ as defined in (23), and observe that: • E φ ∈ CPt ; • χ φ = η E φ ; • E φ is idempotent , that is E φ = E φ .Thanks to this simplification, from now on we can restrictourselves to study only the sets of fixed points of idem-potent completely positive maps. This will be enough tounderstand the structure of the phase subspace of everycompletely positive channel. B. Restricting to maps with a strictly positivedefinite fixed point
We know from Theorem 6 that every positive map hasa positive fixed point ρ . Of course, there are maps forwhich ρ > strictly positive definite, and other mapsfor which ρ is still positive but has some zero eigenvalue.Even if it is not a priori obvious, the entire theory of thephase points (or of the fixed points) becomes much sim-pler is we would allowed to make the assumption that ρ >
0. In this subsection we will describe a theoreticalconstruction that allows us to associate with a genericmap ψ ∈ Pt another map ˜ ψ ∈ Pt such that η ψ = η ˜ ψ .Let us begin with a little Lemma. Recall that the support of an hermitian operator is by definition the subspacespanned by its eigenvectors pertaining to nonzero eigen-values (i.e. the orthogonal complement of the kernel). Lemma 7.
Let ψ ∈ Pt , and define the subspace K as the sum overthe positive fixed points of ψ K = (cid:88) ψ ( A ) = A ≥ supp A . (26)
Then the following statements hold.1. The support of every fixed point of ψ is contained in K . Moreover, K is the smallest subspace enjoyingthis property.2. There exists a positive fixed point ρ ≥ of ψ suchthat supp ρ = K . A possible choice is for instance ρ = ψ ∞ ( ) (see (25) ).Proof.
1. By Theorem 6, if X = ψ ( X ) is a fixed point, sois | X | . As a consequence, supp X = supp | X | ⊆ K (because supp | X | is an addend of the sum (26)).That K is the smallest subspace containing all thesupports of the fixed points follows easily if weprove that there exists a (positive) fixed point ρ such that supp ρ = K , which is the next claim.2. If K = supp X + . . . + supp X n , we can choose ρ ≡ | X | + . . . + | X n | . Another legitimate choiceis ρ = ψ ∞ ( ) because on one hand ψψ ∞ = ψ ∞ by the very definition (25) (so that ψ ∞ ( ) is in-deed a fixed point), and on the other hand for all ψ ( A ) = A ≥ ψ ∞ ( A ) = A , and so A = ψ ∞ ( A ) ≤ ψ ∞ ( (cid:107) A (cid:107) ∞ ) = (cid:107) A (cid:107) ∞ ψ ∞ ( ), whichin turn implies supp A ⊆ supp ψ ∞ ( ).Now that we have constructed this subspace K , weshow why it is indeed interesting. Proposition 8.
Let ψ ∈ Pt be a positive, trace–preserving map, anddefine as above the subspace K = (cid:80) ψ ( A ) = A ≥ supp A .Then for all hermitian X = X † such that supp X ⊆ K ,we have also supp ψ ( X ) ⊆ K .Proof. Up to decomposing X into its positive and nega-tive spectral parts, it will suffice to prove the thesis forpositive X . Use Lemma 7 to construct ρ ≥ ρ = K . Then the hypothesis implies there mustbe a number m ∈ R such that 0 ≤ X ≤ mρ , so that0 ≤ ψ ( X ) ≤ mψ ( ρ ) = mρ , and this is possible only ifsupp ψ ( X ) ⊆ K .Then, consider a generic ψ ∈ Pt d , and construct theassociated subspace K such that dim K = r ≤ d and C d = K ⊕ K ⊥ . Thanks to Proposition 8, it makes senseto define the restriction ˜ ψ : M ( r ; C ) → M ( r ; C ) by X = x ⊕ ⇒ ψ ( X ) = ˜ ψ ( x ) ⊕ , (27)where all the block decompositions are understoodto be in accordance with the space decomposition C d = K ⊕ K ⊥ . This new map ˜ ψ enjoys the followingproperties. • ψ ∈ Pt r , and moreover if ψ ∈ CPt d then also˜ ψ ∈ CPt r . • η ψ = η ˜ ψ ⊕
0, once again accordingly with the de-composition C d = K ⊕ K ⊥ . • ˜ ψ admits a strictly positive definite fixed point ρ ∈ H ( r ; C ), which is of course nothing but therestriction of the maximal fixed point provided byLemma 7 to K (and will be again indicated with ρ ). Let us clarify this tricky point. The positivematrix ρ is not invertible on the whole space C d .However, the channel ˜ ψ has been deliberately con-structed as a restriction to the precise subspace onwhich ρ is indeed invertible.Thanks to the above construction and to the fact that η ψ = η ˜ ψ ⊕
0, we can now assume in our study of thefixed subspace that ψ has a strictly positive definite fixedpoint. We shall see that this will guarantee the existenceof a simpler structure. C. Theory for unital (idempotent) maps whosehermitian adjoint has a strictly positive definitefixed point
We already know that the trace–preserving conditionis the hermitian adjoint dual of the unital condition, thatis ψ ∈ ( C ) Pt ⇔ ψ † ∈ ( C ) Pu (remind that the hermitianadjoint is taken with respect to the Hilbert–Schmidt her-mitian product between matrices). Through this sectionwe will develop the theory of the fixed subspace for uni-tal maps, exploiting also the assumptions of idempotenceand of invertibility of maximal fixed point of the adjoint.A good reason for studying the unital maps insteadof their trace–preserving counterpart (hermitian adjoint)comes from the fact that the Kadison’s inequalities con-tained in Theorem 1 are formulated for unital rather thanfor trace–preserving maps. Observe that we can not saya priori that the fixed subspaces η ψ and η ψ † are related ina particular way. Indeed, they will be in general different subspaces. However, they must satisfy some relations.For instance, since we know that the spectrum ψ † is sim-ply the complex conjugated of that of ψ , we can a priorisay that 1 must belong to σ ( ψ † ) with the same multiplic-ity of 1 ∈ σ ( ψ ), i.e. thatdim η ψ = dim η ψ † . (28)The crucial observation, that naturally leads to thefinal classification theorem, has been done by Lindbladin [19]. We will present it through the following theorem,whose central claim is somewhat a priori unexpected andquite surprising. Here is the point in which the assump-tion of complete positivity rather than of simple posi-tivity becomes fundamental. Indeed, the inequality (6),which is stronger than (5), will play a decisive role. Theorem 9 (Lindblad’s Theorem) . Let ζ ∈ CPu be a completely positive, unital map suchthat its hermitian adjoint ζ † ∈ CPt has a strictly positivedefinite fixed point. Then the fixed subspace η ζ is closedunder matrix multiplication.Proof. It suffices to demonstrate that if Z ∈ η ζ then Z † Z ∈ η ζ . Then indeed for all X, Y ∈ η ζ onehas ( X † + Y ) † ( X † + Y ) ∈ η ζ , and so subtracting XX † + Y † Y ∈ η ζ also XY + Y † X † ∈ η ζ . More-over, ( X † + iY ) † ( X † + iY ) ∈ η ζ , and subtracting again XX † + Y † Y ∈ η ζ we obtain XY − Y † X † ∈ η ζ . Finally,summing with the previous identity yields XY ∈ η ζ .Therefore, let us prove that Z ∈ η ζ ⇒ Z † Z ∈ η ζ . Ap-plying (6) gives ζ ( Z † Z ) − Z † Z ≥ . (29)We would like to prove that indeed the operator on theleft–hand side of (29) is zero. Take the strictly positivedefinite fixed point of ζ † , namely ζ † ( ρ ) = ρ >
0, whoseexistence is guaranteed by hypothesis, and note thatTr [ ρ (cid:0) ζ ( Z † Z ) − Z † Z (cid:1) ] == Tr [ ρ ζ ( Z † Z )] − Tr [ ρ Z † Z ] == Tr [ ζ † ( ρ ) Z † Z ] − Tr [ ρ Z † Z ] == Tr [ ρ Z † Z ] − Tr[ ρ Z † Z ] = 0 . (30)Thanks to (29) and to the fact that ρ >
0, this ensuresthat ζ ( Z † Z ) − Z † Z = 0, that is Z † Z ∈ η ζ .Theorem 9 shows that under our assumptions η ζ isa linear complex subspace of matrices which is in addi-tion closed for hermitian adjunction and matrix product.Such a set is a particular instance of what is called in mathematics a von Neumann algebra . Now that we haveproved that η ζ is equipped with such a peculiar structure,we can exploit the powerful characterization theoremsholding for these algebras. Let us recall the main result,that is the classification of all the finite–dimensional vonNeumann algebras up to unitary isomorphisms. In whatfollows we shall adopt the shorthand M n ≡ M ( n ; C ) inorder to develop a more compact notation. Theorem 10 (Classification of Finite–Dimensional vonNeumann Algebras) . Let A be a finite–dimensional von Neumann algebracomposed of bounded operators over a Hilbert space K .Then there exist integers n , . . . , n m ≥ , Hilbert spaces K , . . . , K m and a unitary isomorphism U : K −→ m (cid:77) i =1 C n i ⊗ K i such that U A U † = m (cid:77) i =1 M n i ⊗ K i . (31)What does Theorem 10 mean in practice, when wedeal with a von Neumann algebra formed by square ma-trices? In that case the thesis states that there existsan orthonormal basis ( U ) of the whole Hilbert space ( K )such that when written in that basis ( U ( · ) U † ) all the ma-trices of our von Neumann algebra ( A ) are at the sametime cast into a block–diagonal form ( (cid:76) ), where eachblock corresponds to a subspace to which a structure oftensor product ( C n i ⊗ K i ) can be given in such a wayas to ensure that the matrices of our algebra are exactlythe operators ( M n i ) acting nontrivially only on the firstspace ( C n i ).Theorem 10 characterizes the structure of the fixedsubspace of an appropriate class of completely positivemaps. However, it is not yet clear what is the action ofthe map on a generic matrix (which is not a fixed point).The following theorem, which constitutes the final resultof this section, answers this question. Theorem 11.
Let ζ ∈ CPu d be a completely positive, unital map suchthat its hermitian adjoint ζ † ∈ CPt d has a strictly pos-itive definite fixed point. Then in an appropriate or-thonormal basis the fixed subspace η ζ takes the block form η ζ = (cid:77) i M d (1) i ⊗ d (2) i , (32) where the d (1) i , d (2) i are positive integers. Moreover, if ζ is in addition idempotent (that is, ζ = ζ ), its action ona generic X ∈ M ( d ; C ) can be written as ζ ( X ) = (cid:77) i Tr i, [ P i XP i ( d (1) i ⊗ ρ i, ) ] ⊗ d (2) i , (33) where • P i is the orthogonal projector onto the i th subspace,in accordance with the decomposition (32) (onto U † C n i ⊗ K i in the language of Theorem 10); • ρ i, is a d (2) i × d (2) i density matrix (in the languageof Theorem 10, it acts on K i ); • the symbol Tr i, stands for the partial trace overthe second factor of the restricted i th subspace asindicated in (32) (i.e. K i in the language of Theo-rem 10).Proof. The first claim (32) is a restatement of Theo-rem 10, and so there is nothing new to prove. Sincethe second claim is by far less obvious, let us proceedstep–by–step. • Step 1: ζ preserves the blocks. We are claimingthat if the input matrix X has nonzero elementsonly in the i th block, then the same happens to theoutput matrix ζ ( X ). Let us prove this statement asfollows. If P i is the orthogonal projector onto the i th block, equation (32) claims that it must be afixed point of ζ , i.e. ζ ( P i ) = P i . Denote by { M k } k a collection of Kraus operators for ζ , that is ζ ( · ) = (cid:88) k M k ( · ) M † k . Since M k P i M † k ≥ k , if ζ ( P i ) = P i thenit must be true that each M k maps the i th blockinto itself. This in turn implies that ζ preserves theblock structure. • Step 2: every output of ζ belongs to η ζ . This de-scends directly from the further assumption that ζ = ζ , which implies that for all X ∈ M d we have ζ ( ζ ( X )) = ζ ( X ) = ζ ( X ) . (34)Adopting the shorthand notation X i, ⊗ Y i, = 0 ⊕ . . . ⊕ ⊕ ( X i, ⊗ Y i, ) (cid:124) (cid:123)(cid:122) (cid:125) i th block ⊕ ⊕ . . . ⊕ , (35)what we have proved since now (i.e. step 1and (34)) ensures that ζ ( X i, ⊗ Y i, ) = ( F i ( X i, , Y i, ) ) i, ⊗ i, , (36)where each F i : M d (1) i × M d (2) i −→ M d (1) i is a bilinear function. • Step 3: the F i s defined through (36) act as F i ( X, Y ) = X Tr[ ρ i, Y ] , (37) for some d (2) i × d (2) i density matrices ρ i, s . Let A ∈M d (2) i be such that 0 ≤ A ≤ d (2) i . Then for allpure states | α (cid:105) ∈ C d (1) i one has0 ≤ F i ( | α (cid:105)(cid:104) α | , A ) ⊗ d (2) i = ζ ( | α (cid:105)(cid:104) α | ⊗ A ) ≤≤ ζ (cid:16) | α (cid:105)(cid:104) α | ⊗ d (2) i (cid:17) = | α (cid:105)(cid:104) α | ⊗ d (2) i , (38)where the last passage is a consequence of the struc-ture of the set of fixed points. From (38) we cansee that it must be0 ≤ F i ( | α (cid:105)(cid:104) α | , A ) ≤ | α (cid:105)(cid:104) α | , which in turn implies that F i ( | α (cid:105)(cid:104) α | , A ) is propor-tional to | α (cid:105)(cid:104) α | , that is F i ( | α (cid:105)(cid:104) α | , A ) = | α (cid:105)(cid:104) α | f i ( A ) (39)for some positive, linear, unital functional f i : M d (2) i → R . It is well–known that such a func-tional must have the form f i ( Y ) = Tr[ ρ i, Y ] forsome d (2) i × d (2) i density matrix ρ i, . Using this factand taking linear combinations of (39) yields (37). • Step 4: conclusion . Putting all together, (36)and (37) give ζ ( X i, ⊗ Y i, ) = ( X Tr[ ρ i, Y ] ) i, ⊗ i, == (cid:16) Tr [ ( X ⊗ Y ) ( d (1) i ⊗ ρ i, ) ] (cid:17) i, ⊗ i, . (40)Taking linear combinations, we can see that for all Z ∈ M d (1) i ⊗ M d (2) i we must have ζ ( Z i ) = (cid:16) Tr [ Z ( d (1) i ⊗ ρ i, ) ] (cid:17) i, ⊗ i, , (41)where of course Z i ≡ ⊕ . . . ⊕ ⊕ Z (cid:124)(cid:123)(cid:122)(cid:125) i th block ⊕ ⊕ . . . ⊕ . Note that (41) specify the action of ζ on each block.Taking into account also step 1, we easily see thatthe global action of ζ is given exactly by (33). D. General theory for all quantum channels
At the end of Subsection III A and in Subsection III B,we showed that particular simplifying assumptions canbe made, without loss of generality, in order to studythe structure of the phase subspace of a completely pos-itive, trace–preserving map. Instead, through Subsec-tion III C we developed the theory for the case in whichthese assumptions are added as hypotheses. The task0we must accomplish now is to follow this path backward ,generalizing Theorem 11 to general, completely positive,trace–preserving channels. We summarize the conclusivetheory in the following theorem, which is substantiallyTheorem 6.16 of [4] or Theorem 8 of [20].
Theorem 12 (Structure Theorem for the Phase Sub-space of Quantum Channels) . Let φ ∈ CPt d a quantum channel. Then there exist asubspace K ⊆ C d for which an orthonormal basis canbe found, such that with respect to the decomposition C d = K ⊕ K ⊥ the phase subspace χ φ has the block struc-ture form χ φ = (cid:77) i M d (1) i ⊗ ρ i, (cid:124) (cid:123)(cid:122) (cid:125) acting on K ⊕ (cid:124)(cid:123)(cid:122)(cid:125) acting on K ⊥ , (42) where the ρ i, are density matrices.Moreover, for all the operators X ⊕ (accordingly tothe decomposition K ⊕ K ⊥ ), the action of the spectralprojector E φ (as defined in (23) ) is E φ ( X ⊕
0) = (cid:77) i Tr i, [ P i XP i ] ⊗ ρ i, ⊕ , (43) where P i is the orthogonal projector onto the i th sub-space, in accordance with the decomposition (42) , and Tr i, stands for the partial trace over the second factor ofthe i th subspace.Finally, let us specify the action of φ on its phase sub-space χ φ . Denote by X = (cid:77) i ( X i, ⊗ ρ i, ) ⊕ the generic operator X ∈ χ φ , decomposed accordinglyto (42) . There are d (1) i × d (1) i unitary matrices and apermutation π over the set of indices i , exchanging onlyindices sharing the same dimension d (1) i , such that φ ( X ) = (cid:77) i (cid:16) U i X π ( i ) , U † i ⊗ ρ i, (cid:17) ⊕ for all X ∈ χ φ written in the form (44) .Proof. First of all, define the spectral projection E φ as in (23),and remind (end of Subsection III A) that the phase sub-space of φ coincides with the fixed subspace of E φ , that is χ φ = η E φ . Moreover, E φ is idempotent (i.e. E φ = E φ ).Next, define the subspace K for E φ exactly as in (26),and construct the quantum channel ˜ E φ associated to E φ as in (27), that is X = x ⊕ ⇒ E φ ( X ) = ˜ E φ ( x ) ⊕ , (46)where all the block decompositions are understood to bein accordance to the space decomposition C d = K ⊕ K ⊥ . The discussion in Subsection III B shows that ˜ E φ has astrictly positive fixed point. Moreover, an easy conse-quence of (46) is that also ˜ E φ , just like E φ , is idempo-tent. Another consequence, as already observed, is that η Eφ = η ˜ E φ ⊕ E † φ . Equation (32) gives us η ˜ E † φ = (cid:77) i M d (1) i ⊗ d (2) i . (47)Moreover, from (33) we immediately obtain that for allthe operators X acting on K ˜ E † φ ( X ) = (cid:77) i Tr i, [ P i XP i ( d (1) i ⊗ ρ i, ) ] ⊗ d (2) i . (48)By the very definition of the hermitian adjunctionthrough Tr [ A † ψ ( B )] ≡ Tr [ ψ † ( A ) † B ], it is not difficult tosee that ˜ E φ ( X ) = (cid:77) i Tr i, [ P i XP i ] ⊗ ρ i, . (49)Equation (49) shows immediately that η ˜ E φ = (cid:77) i M d (1) i ⊗ ρ i, , yielding (42). Observe that in general η ˜ E φ (cid:54) = η ˜ E † φ , as an-ticipated. However, these two matrix subspaces havethe same dimension (see (28)). Putting together (46)and (49) we obtain also (43).Naturally, φ maps linearly χ φ into itself. In order toexplicitly write the action of φ on its phase subspace,a crucial observation is that there exists a legitimatequantum channel which inverts this action. This chan-nel is nothing but the I φ of (24), which indeed verifies φ I φ = I φ φ = E φ . Obviously, also I φ maps linearly χ φ into itself. Let us examine the consequences of this ob-servation. Take | α (cid:105) ∈ C d (1) i for some i , and consider theoperator A = | α (cid:105)(cid:104) α | i, ⊗ ρ i, ∈ χ φ , which acts nontriviallyonly on the i th block (see (35)). Since it admits no con-vex decomposition in χ φ , its image under φ must sharethis same property. Otherwise, φ ( A ) = (cid:88) j p j B j , B j ∈ χ φ ⇒⇒ A = E φ ( A ) = I φ φ ( A ) = (cid:88) j p j I φ ( B j )is a nontrivial convex combination of X in χ φ , absurd.This shows that φ ( A ) must be contained inside a sin-gle block and must be of the form φ ( A ) = | β (cid:105)(cid:104) β | j, ⊗ ρ j, for some j . Note that the continuity of φ requires that j depends only on i and not on | α (cid:105) (no “jumps” be-tween different block are allowed). Moreover, since φ hasto be linear and bijective when mapping χ φ into itself,1 d (1) i = d (1) j . This ensures that there exist a permutation π exchanging only blocks with the same d (1) i and quan-tum channels φ i, ∈ CPt d (1) i mapping pure states intopure states in a bijective way such that φ ( X ) = (cid:77) i (cid:0) φ i, ( X π ( i ) , ) ⊗ ρ i, (cid:1) ⊕ X written in the form (44). But it is well–known(Theorem 3) that the only channels φ i, ∈ CPt d (1) i map-ping pure states into pure states in a bijective way are theunitary evolutions. The same conclusion can be drawnif we note that the invertibility of φ in χ φ by means ofa quantum channel ( I φ ) implies that the φ i, s must havecompletely positive inverse., and we apply again Theo-rem 3. Anyway, the unitarity of the φ i, s gives the the-sis (45).From this very general theorem several consequencescan be deduced. One of them, for instance, is the solu-tion of the so–called inverse eigenvalue problem for theperipheral spectrum, as given by Wolf et al. in [20]. Inthat paper, the analogous of Theorem 12 is used to obtaina complete classification of all the peripheral spectra ofcompletely positive, trace–preserving maps, as expressedas follows. Theorem 13 (Peripheral Spectra of Quantum Chan-nels) . Let φ ∈ CPt d be a quantum channel. Then there areintegers n c , d c ∈ N (labeled by an index c ∈ C ) satisfying (cid:80) c n c d c ≤ d , and vectors ω c ∈ C d c whose component arephases (i.e. | ω cα | ≡ ∀ c, α ), such that the peripheralspectrum of φ is σ P ( φ ) = { ω cα ω ∗ cβ e πimcnc : c ∈ C, ≤ m c ≤ n c − , ≤ α, β ≤ d c } . (50) In this way the total number | σ P ( φ ) | of peripheral eigen-values of φ (counting multiplicities) is | σ P ( φ ) | = (cid:88) c n c d c . (51) Conversely, every set of numbers as in (50) is the pe-ripheral spectrum of some φ ∈ CPt (cid:80) c n c d c , which in ad-dition can be chosen unital and with no other nonzeroeigenvalue.Proof. Once we have the explicit form of the action of φ on χ φ , as expressed in (45), it is not too difficult to seehow (50) descends. The main ingredients to be used arethe following. • The spectrum of a unitary channel U ( · ) = U ( · ) U † acting on a d –dimensional system is σ ( U ) = { ω α ω ∗ β : 1 ≤ α, β ≤ d } == σ ( U ⊗ U ∗ ) = σ ( U ) × σ ( U ) ∗ , (52) where σ ( U ) = { ω α : 1 ≤ α ≤ d } is the spectrumof U (composed by phases). This can be explic-itly seen by letting φ act on the operators | ω α (cid:105)(cid:104) ω β | ,where U | ω α (cid:105) = ω α | ω α (cid:105) . • Every permutation can be decomposed into a prod-uct of cycles acting on disjoint input subsets. Forinstance, (1 , , , , → (4 , , , ,
2) is the simulta-neous action of the 3–cycle (1 , , → (4 , ,
3) andof the 2–cycle (2 , → (5 , • The spectrum a n –cyclic block matrix of the form A = . . . A n A A . . . ...... ... . . . ...0 0 . . . . . . , (53)where A i ∈ M ( d ; C ), is composed of allthe n th complex roots of all the eigenval-ues of A n . . . A A . This can be verified(supposing by continuity A n . . . A diagonaliz-able) either by looking for eigenvectors of theform ( λ n − x T , λ n − x T A , . . . , x T A T . . . A Tn − ) T ,where x is an eigenvector of A n . . . A with eigen-value λ n , or by proving by induction the determi-nant formuladet( A − x ) = ( − ( n − d det( A n . . . A − x n ) . Now, φ acts on χ φ as a direct sum of n c –cyclic d c n c × d c n c block matrices as (53), where each A i is a d c × d c matrix representing a unitary evolution(these d c s are the d (1) i s of Theorem 12, possibly re-peated). As a consequence, also A n . . . A is a unitaryevolution, and we know that its spectrum is given by { ν α ν ∗ β : 1 ≤ α, β ≤ d c } . Choosing phases ω α such that ω n c α = ν α gives us exactly equation (50). In order to con-struct examples of channels having a required peripheralspectrum of the form (50), one has only to run this rea-soning backward, using projective measurements to avoidany other nonzero eigenvalue.In what follows, we do not need the whole power ofTheorem 13. Instead, we will find very useful a simple,nice consequence of it. Corollary 14.
Let φ ∈ CPt d be a quantum channel satisfying | σ P ( φ ) | ≥ (where | σ P ( φ ) | is the number of peripheral eigenvalues of φ , counting multiplicities). Then there ex-ists an integer ≤ n ≤ d such that belongs to σ P ( φ n ) with multiplicity strictly greater than .Proof. Let us assume that the multiplicity of 1 ∈ σ P ( φ ) isexactly 1 (otherwise it will be sufficient to choose n = 1).Since 1 is reached in (50) for each α = β, m c = 0, theremust be only one possible c (call it 0), and moreover d = 1. But then there exists 2 ≤ n ≤ d such that σ P ( φ ) = { e πim n : 0 ≤ m ≤ n − } . As a consequence, 1 belongs to σ P ( φ n ) with multiplicity n ≥ φ ∈ CPt ⇒ σ P ( φ ) = { } , { , } , { , − } , { , , e iθ , e − iθ } . (54)The commas in the preceding equation identify the pos-sible alternative spectra. Recalling also (3), we can seethat the last spectrum is the signature of a unitary evo-lution: σ P ( φ ) = { , , e iθ , e − iθ } ⇔ φ ∈ U . (55)This makes sense, because { , e iθ , e − iθ } is exactly thespectrum of a rotation in SO(3), and (15) holds. IV. UNIVERSALENTANGLEMENT–PRESERVING CHANNELS
Entanglement–breaking channels represent the mostdetrimental form of noise a quantum system can undergo.Not surprisingly they have been extensively studied inthe literature and a complete characterization of theirproperties have been obtained – see e.g. Sec. II E. Thenatural counter-part of these maps is constitute by thosetransformations which are always innocuous, in the sensethat they never break the entanglement between Aliceand Bob when acting locally on Alice’s subsystem, nomatter how weak it could be (provided that it existed inthe initial state). In this section we focus on these specialtransformations proving that they coincide with the setof unitary maps, a result which one could have guessedon physical ground but, to be best of our knowledge wasnever formalized before.
Definition 1.
Let φ ∈ CPt be a quantum channel acting on system A . We say that φ is universal entanglement–preserving (UEP) if for each quantum system B and for each globalentangled state ρ AB , ( φ ⊗ I )( ρ AB ) is again entangled: ρ AB / ∈ S AB ⇒ ( φ ⊗ I )( ρ AB ) / ∈ S AB . (56) The requirement that the entanglement preservationmust hold for all the states of the system (even if mixed) is crucial. As noted in [21], we can not restrict this prop-erty to the pure states alone. Indeed, this would modifyDefinition 1 in such a way as to include other channels.For example, in the case of qubit, the channels that pre-serve the entanglement of every pure state are all butthe entanglement–breaking ones (as can be immediatelyseen by observing that every entangled pure state is ob-tained from a maximally entangled one by allowing Bobto use a local, invertible filter). Instead, we shall see thatDefinition 1 is by far more strict.We remark how the concept of universal entanglement–preserving channel is in some sense complementary tothat of entanglement–breaking channel . As the latter al-ways destroys the entanglement, the former always pre-serves it, no matter how much entangled the input stateis. A class of trivial examples of UEP channels is com-posed by the unitary evolutions. The rest of this sectionis devoted to the proof that the unitary evolutions arethe only universal entanglement–preserving channels .The first tool we need is the following technical lemma,concerning the boundary of the convex set S AB of sepa-rable states on a bipartite quantum system AB . Recallthat a point p belonging to a set S (in a normed vectorspace, for instance) is said to be internal to S if thereexists a ball of arbitrary, nonzero radius centered in p and entirely contained in S . The non–internal point of S form the boundary of S , indicated by ∂S . A simple,common way of proving that a point q indeed belongs tothe boundary of S is to find a curve q ε laying outside S for ε > q as limit when ε → Proposition 15.
Let ρ A , ρ B be densities matrices on systems A, B . Denoteby ∂ S AB the boundary of the set of separable states on thebipartite system AB . Then ρ A ⊗ ρ B ∈ ∂ S AB ⇔ det ρ A det ρ B = 0 . (57) Proof.
Suppose that det ρ A det ρ B > ρ A and ρ B are strictly positive definite). Then ρ A ⊗ ρ B can be written as a nontrivial convex combination ofthe maximally mixed state d A d B and a separable (ac-tually, factorized) state. On one hand it is known (seefor example [22]) that the maximally mixed state is al-ways internal to the separable set. On the other hand, asimple reasoning, valid for all the convex sets S , showsthat a nontrivial convex combination of an internal point p ∈ S − ∂S and another point q ∈ S is again internal to S . The following geometric picture can help to visual-ize that reasoning. Since p is internal to the convex set S , we can inscribe a circular cone inside S , whose axiscontains p and q as endpoints, and our nontrivial convexcombination as a non–extremal point. Now, every axialpoint different from the vertex and the base point mustbe internal to the cone, and so to S . This proves that ρ A ⊗ ρ B ∈ ∂ S AB implies det ρ A det ρ B = 0.Let us turn our attention to the converse statement.Suppose for instance that det ρ A = 0; we must prove that3 ρ A ⊗ ρ B ∈ ∂ S AB . Take a vector | (cid:105) such that ρ A | (cid:105) = 0,and | (cid:105) ⊥ | (cid:105) . Consider | Ψ (cid:105) ≡ | (cid:105) + | (cid:105)√ , ρ ε ≡ ε | Ψ (cid:105)(cid:104) Ψ | + (1 − ε ) ρ A ⊗ ρ B , where 0 < ε ≤
1. Then • ρ ε is a density matrix for each ε ≥ • lim ε → ρ ε = ρ A ⊗ ρ B . • Acting with partial transposition T B on ρ ε pro-duces a non–positive operator, for each ε > ρ ε can not be separa-ble for ε >
0. To see that ρ T B ε is not positive,we will prove that its restriction to the subspace W ≡ Span {| (cid:105) , | (cid:105)} has negative determinant.In fact, simple calculations show that ρ T B ε | W = (cid:18) ε/ ε/ − ε ) a (cid:19) , where a is a real number given by a ≡ (cid:104) | ρ A | (cid:105) (cid:104) | ρ B | (cid:105) . As claimed, det( ρ T B ε | W ) < ε >
0, and so ρ T B ε can not be positive definite in that range.Since we can construct entangled matrices arbitraryclose to ρ A ⊗ ρ B , it must be ρ A ⊗ ρ B ∈ ∂ S AB .Before we arrive at the final result of this section, letus formalize a little, nice lemma which will turn out tobe useful many times. Lemma 16.
Let Z be a non–normal complex matrix (that is, [ Z, Z † ] (cid:54) = 0 ) of size d × d . Define Q ( Z ) ≡ (cid:18) ZZ † Z † Z (cid:19) . (58) Then Q ( Z ) is an (unnormalized) entangled state over theHilbert space C d ⊗ C .Proof. Since Q ( Z ) = (cid:0) Z (cid:1) † (cid:0) Z (cid:1) , it is obvious that Q ( Z ) ≥ Q ( Z ) T B is not positive.A straightforward calculation gives (cid:0) Q ( Z ) T B (cid:1) ∗ = (cid:18) Z † Z Z † Z (cid:19) . Now, it is well–known (see for instance [18], p. 472) thata block matrix (cid:0)
A BB † C (cid:1) with A > C ≥ B † A − B . In our case this would give the condition Z † Z ≥ ZZ † , that is [ Z, Z † ] ≤
0. Since Tr [
Z, Z † ] = 0,this would imply that [ Z, Z † ] = 0, which is forbidden byhypothesis. Now we are in position to state and prove the main re-sult about the universal entanglement–preserving chan-nels. This is the content of the following theorem. Theorem 17 (Classification of UEP Channels) . The only universal entanglement–preserving channels arethe unitary evolutions.Proof.
We already know that a unitary channel is defi-nitely UEP. The problem is to prove the converse state-ment. So, let φ be a universal entanglement–preservingchannel.1. Step 1: φ ( ) >
0. Suppose by contradic-tion that there exists a pure state | α (cid:105) such that (cid:104) α | φ ( ) | α (cid:105) = 0. Then we can rewrite this condi-tion as Tr[ φ † ( | α (cid:105)(cid:104) α | ) ] = 0, that is (using the factthat φ † is a positive map, too) φ † ( | α (cid:105)(cid:104) α | ) = 0. Thisshows that φ † admits a zero eigenvalue (see Subsec-tion III A), which in turn implies that also φ admitsa zero eigenvalue (because σ ( φ ) = σ ( φ † ) ∗ ). Then,consider an input matrix X such that φ ( X ) = 0.Theorem 6 ensures that X can be chosen hermi-tian and traceless, and this in particular shows thatit can not be a multiple of an orthogonal projec-tor. This last property allows us to choose anotherhermitian matrix Y whose support is entirely con-tained in that of X , and such that [ X, Y ] (cid:54) = 0.Defining for Z ε ≡ X + iεY (so that [ Z ε , Z † ε ] (cid:54) = 0 if ε >
0) allows us to exploit Lemma 16 to say that Q ( Z ε ) (as defined in (58)) is an (unnormalized) en-tangled state for all ε >
0. Then( I ⊗ φ ) ( Q ( Z ε )) = (cid:18) φ ( ) iε φ ( Y ) − iε φ ( Y ) φ ( Z † ε Z ε ) (cid:19) == ( I ⊗ φ ) (cid:18) iεY − iεY Z † ε Z ε (cid:19) must be again entangled for all ε >
0, becauseof the very definition of UEP channel. However,we will show in a moment that it is instead sep-arable for sufficiently small ε >
0. This goalwill be reached by arguing that indeed already (cid:16) iεY − iεY Z † ε Z ε (cid:17) is a positive, separable (unnormal-ized) state for sufficiently small ε >
0. Since ev-erything happens inside the support of X , we cansimply project down here and prove the separabil-ity of the resulting matrix. This is the same as tosuppose det X (cid:54) = 0, i.e. X >
0. With this hypoth-esis, we see thatlim ε → (cid:18) iεY − iεY Z † ε Z ε (cid:19) = (cid:18) X (cid:19) is indeed a nontrivial convex combination of thecompletely mixed state (which as already men-tioned is internal to the set of separable state, asproved in [22]) and another separable matrix. Byvirtue of the same reasoning we used in proving4Proposition 15, we can conclude that (cid:0) X (cid:1) isagain internal to the separable set, and so that itsapproximation (cid:16) iεY − iεY Z † ε Z ε (cid:17) is separable for suffi-ciently small ε > Step 2: making φ unital . Thanks to step 1, let usdefine the map ψ by ψ ( X ) ≡ φ ( ) − / φ ( X ) φ ( ) − / . (59)Observe that: • ψ is again completely positive; • ψ is unital (even if no longer trace–preserving). • ψ is again UEP (the definition of UEP makessense also for non–trace–preserving maps), be-cause conjugation by means of an invertiblematrix does not affect separability.3. Step 3: ψ preserves non–invertibility of nonnega-tive matrices . We claim that ∀ ρ ≥ , det ρ = 0 ⇒ det ψ ( ρ ) = 0 . (60)We can suppose that ρ is a normalized density ma-trix. Consider a second system B of dimension d B ≥
2. If ρ ≥ ρ = 0, we know fromProposition 15 that ρ ⊗ d B ∈ ∂ S AB . This entailsthat one can construct a curve R ε (0 < ε ≤ entangled states of AB such thatlim ε → + R ε = ρ ⊗ d B . Since ψ is UEP, it must be ( ψ ⊗ I )( R ε ) / ∈ S AB foreach ε >
0. Moreover, observe thatlim ε → + ( ψ ⊗ I )( R ε ) = ( ψ ⊗ I ) (cid:18) lim ε → + R ε (cid:19) == ( ψ ⊗ I ) (cid:18) ρ ⊗ d B (cid:19) = ψ ( ρ ) ⊗ d B ∈ S AB . Strictly speaking, ψ ( ρ ) is no longer a density ma-trix, because it is not guaranteed to have unit trace.Anyway, it makes sense to say that its normalizedform is indeed separable. We have proved thatthere exists a curve composed of entangled stateswhose limit is the separable state ψ ( ρ ) ⊗ d B . Thisis the same as to say that ψ ( ρ ) ⊗ d B ∈ ∂ S AB , andso Proposition 15 implies thatdet ψ ( ρ ) = 0 . Step 4: ψ preserves non–invertibility of general her-mitian matrices . We have to prove that ∀ X = X † , det X = 0 ⇒ det ψ ( X ) = 0 . (61) Let us exploit step 3 by applying (60) to the posi-tive matrix X :det X = 0 ⇒ det( X ) = 0 ⇒ det ψ ( X ) = 0 ⇒⇒ ∃ | η (cid:105) : (cid:10) η (cid:12)(cid:12) ψ ( X ) (cid:12)(cid:12) η (cid:11) = 0 . (62)Since ψ is positive and unital, we can apply theKadison’s inequality (5) and argue that (cid:10) η (cid:12)(cid:12) ψ ( X ) (cid:12)(cid:12) η (cid:11) ≥ ( ψ ( X ) | η (cid:105) ) † ( ψ ( X ) | η (cid:105) ) . (63)Naturally, (62) and (63) together show exactly thatthere exists | η (cid:105) such that ψ ( X ) | η (cid:105) = 0, i.e. thatdet ψ ( X ) = 0.5. Step 5: ψ preserves the spectrum . From now on,we can proceed on the guideline drawn by [4] (seeChap. 3, p. 66). A crucial fact is that ψ mustpreserve the spectrum of an hermitian matrix as aset , i.e. that ∀ X = X † , λ ∈ σ ( X ) ⇒ λ ∈ σ ( ψ ( X )) . (64)Indeed, by (61) one has λ ∈ σ ( X ) ⇒ det( X − λ ) = 0 ⇒ det ψ ( X − λ ) = 0 ⇒⇒ det ( ψ ( X ) − λ ) = 0 ⇒ λ ∈ σ ( ψ ( X )) . Observe that (64) implies that also the multiplici-ties of the eigenvalues are the same for X and ψ ( X )(i.e. ψ preserves the spectra as multisets ). Indeed,the set of hermitian matrices with non–degeneratespectrum is dense in H ( d ; C ) (as can be easily seenby perturbing the eigenvalues). Take a sequence X ε (with 0 < ε ≤
1) of hermitian matrices en-joying this property, and such that lim ε → + = X .Denote as usual by σ ( · ) the spectrum as a multiset(i.e. counting multiplicities). Then, from (64) wededuce that σ ( ψ ( X ε )) ≡ σ ( X ε ) ∀ ε > . On the other hand, the continuity of the eigenvaluesrequires that σ ( ψ ( X )) = σ (cid:16) lim ε → ψ ( X ε ) (cid:17) = lim ε → σ ( ψ ( X ε )) == lim ε → σ ( X ε ) = σ (lim ε → X ε ) = σ ( X ) . In taking the above limits we have implicitly un-derstood an obvious metric over the spectra.6.
Step 6: preparing the ground for Wigner’s theorem .We now claim that ψ sends pure states into purestates in such a way as to preserve the moduli ofthe scalar products: ψ ( | α (cid:105)(cid:104) α | ) ≡ | α (cid:48) (cid:105)(cid:104) α (cid:48) | ∀ | α (cid:105) , (65) | (cid:104) α | β (cid:105) | ≡ | (cid:104) α (cid:48) | β (cid:48) (cid:105) | ∀ | α (cid:105) , | β (cid:105) . (66)5The proof is as follows. On one hand, for each | α (cid:105) we have σ ( | α (cid:105)(cid:104) α | ) = { , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) d − } ⇒⇒ σ ( ψ ( | α (cid:105)(cid:104) α | ) ) = { , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) d − } ⇒⇒ ψ ( | α (cid:105)(cid:104) α | ) = | α (cid:48) (cid:105)(cid:104) α (cid:48) | . On the other hand, take two states | α (cid:105) , | β (cid:105) , anddenote by | α (cid:48) (cid:105) , | β (cid:48) (cid:105) their images under the actionof ψ . Then { | (cid:104) α | β (cid:105) | , − | (cid:104) α | β (cid:105) | , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) d − } == σ ( | α (cid:105)(cid:104) α | + | β (cid:105)(cid:104) β | ) == σ ( ψ ( | α (cid:105)(cid:104) α | + | β (cid:105)(cid:104) β | ) ) == σ ( | α (cid:48) (cid:105)(cid:104) α (cid:48) | + | β (cid:48) (cid:105)(cid:104) β (cid:48) | ) == { | (cid:104) α (cid:48) | β (cid:48) (cid:105) | , − | (cid:104) α (cid:48) | β (cid:48) (cid:105) | , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) d − } , from which we deduce exactly | (cid:104) α | β (cid:105) | = | (cid:104) α (cid:48) | β (cid:48) (cid:105) | . Step 7: conclusion . Thanks to (66), the hypothesisof Wigner’s Theorem 2 are satisfied. Since an anti–unitary transformation can be represented as thecomplex conjugation in some basis followed by aunitary operation, we must conclude that for allvectors | α (cid:105)| α (cid:48) (cid:105) = e iϕ ( α ) U | α (cid:105) or | α (cid:48) (cid:105) = e iϕ ( α ) U | α ∗ (cid:105) , where U is unitary. Therefore, one has ψ ( | α (cid:105)(cid:104) α | ) ≡ U | α (cid:105)(cid:104) α | U † or ψ ( | α (cid:105)(cid:104) α | ) ≡ U | α ∗ (cid:105)(cid:104) α ∗ | U † ≡ U | α (cid:105)(cid:104) α | T U † . Actually, this implies that for all the input matrices X one has ψ ( X ) ≡ U XU † or ψ ( X ) ≡ U X T U † . The second option has to be discarded, because ψ is completely positive, and the matrix transpositionis only positive. Going back to φ by means of (59),we obtain φ ( X ) ≡ φ ( ) / U XU † φ ( ) / . Since φ has to be trace–preserving, we can easilysee that it must be U † φ ( ) U = , that is φ ( ) = .Hence, we deduce exactly φ ( X ) ≡ U XU † for all theinput matrices X , as we claimed. This proof of Theorem 17, which is only one of the sev-eral different proofs, is technically quite complex. How-ever, this can not distract our attention from its physicalmeaning. Concerning the entanglement between Aliceand Bob, Theorem 17 says a simple, intuitive thing. A truly noisy interaction of Alice’s subsystem with anexternal environment definitely breaks some form ofentanglement between Alice and Bob.
From a conceptual point of view, the context of ourinvestigations is remarkably clarified. This characteriza-tion theorem can be seen as exactly specular to Theo-rem 4. The latter specifies an operational meaning (theHolevo form (11)) for those channels which always breakthe quantum correlations. The former, instead, claimsthat only the unitary evolutions can definitely make theentanglement survive.All that strengthens our belief that the deteriorationto which the entanglement is subjected can be used toquantify the amount of noise introduced by a quantumchannel (as proposed in [7, 8]). In this respect, we haveproved that this kind of measure is faithful : if no entan-glement is wasted, there is no true noise acting on thesystem. So, the main purpose of the following section isto further investigate the classifications induced on theset of quantum channels by the entanglement preserva-tion properties.
V. ENTANGLEMENT–SAVING CHANNELS
In our previous paper [8], we began to investigate theproperties of quantum channels by means of the effectthey have on the entanglement of a bipartite state. Weproposed two (inverse) measures of noise for a genericchannel φ , namely the direct n –index (already definedin [7]) and the filtered N –index . The former is simplythe minimum number of consecutive iterations of φ nec-essary to obtain an entanglement–breaking channel. In-stead, the latter index describes the optimized scenario,in which the application of intermediate quantum chan-nels (called filters ) is allowed in order to save as long aspossible the entanglement. In this paper we will focusonly on the direct n –index. A. Statement of the problem
We begin our study of this functional by posing thefollowing question: which kind of noise is so weak that itnever separates completely a maximally entangled state,even if applied an arbitrary number of times? Within thelanguage developed through [7, 8], these entanglement–saving channels are characterized by an infinite value ofthe direct n –index. So we can give the following defini-tion.6 Definition 2 (Entanglement–Saving Channels) . A quantum channel φ ∈ CPt is called entanglement–saving (ES) if n ( φ ) = ∞ , i.e. if φ n / ∈ EBt ∀ n ∈ N . The main goal of the rest of this section is to find anadequate characterization of these channels. The objec-tive will be completely achieved almost everywhere (i.e.apart from a set of zero measure), and in arbitrary di-mension. As a corollary, the problem in the case of qubitwill be completely solved.The first, elementary property of the ES set is its clo-sure under unitary conjugation, that is φ entanglement–saving ⇔ U φ U † , ∀ U ∈ U . (67)Moreover, observe that if a channel φ ∈ CPt is not entanglement–saving, then the elements of the sequence( φ n ) n ∈ N become eventually entanglement–breaking for sufficiently large n , i.e. for all the n greater than or equalto a certain threshold (which is by definition the directindex n ( φ )).As we shall see, a remarkable simplification in the the-ory of entanglement–saving channels occurs if we restrictour analysis to the set of quantum channels φ such thatdet( φ ) (cid:54) = 0. Here the determinant of a linear map is theproduct of its eigenvalues, as usual (see Subsection III A).Although this assumption could seem rather arbitrary,we will see that it is indeed quite natural at least for asingle qubit. In fact, in that case it causes no loss ofgenerality. In order to take advantage of this restric-tion, we need some preliminary results concerning theentanglement–breaking channels. B. Preliminaries
It is well–known (Theorem 6) that every quantumchannel has a positive fixed point. However, this den-sity matrix can be or not be strictly positive. Recallthat we distinguish between the two alternatives by call-ing strictly positive a matrix
A >
0, and semipositive amatrix A ≥ A = 0). Whenever this distinction is notnecessary, we say simply positive .To appreciate the importance of the question and itslink with the separability problem, recall Proposition 15.We will find this result quite useful also in this context.Indeed, we proved that matrices of the form ρ ⊗ d , with ρ only semipositive, belong to the boundary of the sepa-rable set. Therefore, for an entanglement–breaking chan-nel (whose images are always separable) the presence ofa semipositive fixed point must be a rather delicate sit-uation. Our immediate purpose is to discuss the conse-quences of this possibility. Actually, we will explore amore general circumstance through the following theo-rem. Theorem 18 (Image of Semipositive Matrices ThroughEB Channels) . Let φ ∈ EB d be an entanglement–breaking channel. Sup-pose that there exists a semipositive matrix A ≥ , with rk A = r < d , such that rk φ ( A ) = s verifies r + s < dr .Then dim ker φ ≥ dr − r − s > . (68) Proof.
Let us write the action of the entanglement–breaking channel φ in Holevo form 11: φ ( X ) = (cid:88) i ∈ I ρ i Tr XE i . Here the ρ i are density matrices, and the operators E i are positive definite. Calling A (cid:48) ≡ φ ( A ), one has A (cid:48) = (cid:88) i ∈ I ρ i Tr AE i . Recall that supp X is the support of a hermitian matrix X , i.e. the orthogonal complement of the kernel. Clearly,thanks to the positivity of the operators ρ i and E i , ∀ i ∈ I , supp ρ i ⊆ supp A (cid:48) or supp E i ⊆ ker A . (69)Let us define a bipartition of I distinguishing betweenthese two possibilities: I ≡ { i ∈ I : supp ρ i ⊆ supp A (cid:48) } , I ≡ I \ I . Thanks to (69), we can write ∀ i ∈ I , supp E i ⊆ ker A . (70)Denote by | (cid:105) , . . . , | r (cid:105) an orthonormal basis of supp A ,and by | (cid:105) , . . . , | d (cid:105) one of its completions to a global or-thonormal basis. Consider the vector subspace of hermi-tian matrices V spanned on R by the operators • | α (cid:105)(cid:104) β | + | β (cid:105)(cid:104) α | , with 1 ≤ min { α, β } ≤ r and α (cid:54) = β ; • i | α (cid:105)(cid:104) β |− i | β (cid:105)(cid:104) α | , again with 1 ≤ min { α, β } ≤ r and α (cid:54) = β ; • | α (cid:105)(cid:104) α | , with 1 ≤ α ≤ r .Moreover, define also W ≡ { X = X † : supp X ⊆ supp A (cid:48) } . The dimensions of V and W are easy to calculate:dim V = 2 d (cid:88) k =1 ( d − α ) + r = 2 dr − r , dim W = (rk A (cid:48) ) = s . Observe thatdim V − dim W = 2 dr − r − s > ∀ i ∈ I , ∀ X ∈ V, Tr XE i = 0 . Therefore, for all X ∈ V we have φ ( X ) = (cid:88) i ∈ I ρ i Tr XE i == (cid:88) i ∈ I ρ i Tr XE i + (cid:88) i ∈ I ρ i Tr XE i == (cid:88) i ∈ I ρ i Tr XE i ∈ W .
As a consequence, it makes sense to consider the restric-tion φ | V : V → W . Thanks to (71), one hasdim ker φ ≥ dim ker φ | V = dim V − rk φ | V ≥≥ dim V − dim W = 2 dr − r − s > . With Theorem 18, we have explored a nontrivial fea-ture of the entanglement–breaking channels. However,our result does not look conceptually transparent at all.In order to clarify its meaning, let us examine the follow-ing simpler corollary (which is what we really need in therest of the paper).
Corollary 19.
Let φ ∈ EBt be an entanglement–breaking channel. Sup-pose that φ has a semipositive fixed point. Then dim ker φ ≥ d − , and in particular det φ = 0 .Proof. It suffices to apply Theorem 18 with r = s ≤ d − φ ≥ dr − r = 2 r ( d − r ) ≥ d − > , which implies det φ = 0.Before we can state and prove our main result, anothertechnical lemma is necessary. Lemma 20.
Let φ ∈ Pt be a positive, trace–preserving map whosespectrum contains with multiplicity strictly greater than (that is, whose fixed subspace verifies dim η φ > ).Then φ admits a semipositive fixed point.Proof. Let us call ρ the positive fixed point of φ whoseexistence is guaranteed by Theorem 6. If ρ is semipos-itive we can immediately conclude. Otherwise, suppose ρ > X = X † (independent from ρ )such that φ ( X ) = X . Denoting by λ min ( Y ) the minimumeigenvalue of the hermitian Y , define A ≡ X − λ min ( ρ − / Xρ − / ) ρ == ρ / (cid:16) ρ − / Xρ − / − λ min ( ρ − / Xρ − / ) (cid:17) ρ / . Then φ ( A ) = A , and moreover A must be semipositive,since λ min (cid:16) ρ − / Xρ − / − λ min ( ρ − / Xρ − / ) (cid:17) = 0 . C. A characterization theorem
Now, we are in position to easily prove the followingtheorem, which is the main achievement of this section.We postpone our comments on the meaning of this resultafter its statement and proof. In what follows, | σ P ( φ ) | will denote the number of peripheral eigenvalues of thechannel φ , counting multiplicities. Moreover, recall thatwe denote with a φ ( λ ) the algebraic multiplicity of theeigenvalue λ ∈ σ ( φ ). Theorem 21 (ES Channels with Nonzero Determinant) . Let φ ∈ CPt d be a quantum channel satisfying a φ (0) < d − (in particular, det φ (cid:54) = 0 is a sufficientcondition). Then the following are equivalent.1. φ is entanglement–saving.2. φ has a semipositive fixed point, or | σ P ( φ ) | ≥ .3. There exists ≤ n ≤ d such that φ n has a semi-positive fixed point.Proof. ⇒ φ (cid:54) = 0. Suppose by contradictionthat φ has a fixed point ρ > σ P ( φ ) = { } (recall that 1 always belongs to σ ( φ ), by Theo-rem 6). Then it is not difficult to prove thatlim n →∞ φ n = D ρ , where the depolarizing chan-nel is defined by D ρ ( X ) ≡ ρ Tr X . This can beseen by noting that lim n →∞ φ n ≡ D is a (well–defined) channel whose output is always propor-tional to the positive eigenvector associated withthe eigenvalue 1 (because the others eigenvaluestend to zero when raised to arbitrary large pow-ers). This can be written as D ( X ) = ρ f ( X ) forall X , where f : M ( d ; C ) → C is a linear func-tional, and we can normalize Tr ρ = 1. Applyingthe trace–preserving condition yields immediately f ( X ) ≡ Tr X .Next, let us observe that the Choi–Jamiolkowskiisomorphism is linear (in particular, continuous),and solim n →∞ R φ n = R lim n →∞ φ n = R D ρ = ρ ⊗ d . Since ρ >
0, by Proposition 15 the limit of thesequence is internal to the set of separable states,and this implies n ( φ ) < ∞ , which is absurd.82 ⇒ φ (cid:54) = 0. If φ admits a semipositive fixed pointwe can immediately conclude. Otherwise, thanks toCorollary 14, there exists 1 ≤ n ≤ d such that thespectrum of φ n contains 1 with multiplicity strictlygreater than 1. In that case, Lemma 20 again guar-antees the existence of a semipositive fixed point.3 ⇒ a φ (0) ≤ d −
1) comes into play. Firstly, iffor some n = n the map φ n has a semipos-itive fixed point, it is immediate to see thatthe same happens for each multiple of n , i.e.frequently in n ∈ N . Assume by contradiction that n ( φ ) < ∞ . Then there exists N ∈ N such that φ N is entanglement–breaking and has a semipositivefixed point. By Corollary 19, this would implydim ker( φ N ) ≥ d − φ N ) ≤ a φ (0) (as can be easily seenusing the Jordan decomposition (22)), we woulddeduce a φ (0) ≥ d − φ (cid:54) = 0, then Corollary 19 immediately gives theabsurd equality det φ N = 0.Theorem 21 completely solves the problem of findingan explicit characterization of the entanglement–savingproperty for a wide class of channels, i.e. those verifying a φ (0) < d −
1) (or, simplifying, det φ (cid:54) = 0). Among itsconsequences, observe that we can immediately concludethat the set of ES channels has measure zero (since boththe initial restricting condition and the other conditionsgiven in the thesis define sets of measure zero).From a geometrical point of view, we could say thatTheorem 21 characterizes the ES set almost every-where , that is, apart from a set of measure zero. Ac-tually, this does not ensures a priori that the statementof Theorem 21 is useful (i.e. identifies a nonempty setof ES channels), because also the ES set has measurezero. As a matter of fact, there are many entanglement–saving channels whose determinant is equal to zero (forinstance). We will see that a large class of examplesemerges in a natural way in the context of Section VI.However, we will see with the explicit example d = 2(see Subsection V D) that what we have just proved in-deed gives a useful characterization. Namely, the restric-tion det φ (cid:54) = 0 causes no loss of generality for the simplestnontrivial case, i.e. those of qubit channels. In this sense,it is less severe than what we could imagine. D. Entanglement–saving qubit channels
Through this subsection, we explore some conse-quences of Theorem 21. In particular, we show that thisresult gives a complete characterization of the ES classin the case of channels acting on a single qubit. To pro-ceed further, we need some simple lemmas. The first one discusses the consequences of the equation det φ = 0 fora quantum qubit channel. Lemma 22.
Let φ ∈ CPt be a qubit channel such that det φ = 0 .Then φ is entanglement–breaking.Proof. Let φ = ( M, c ) denote the Bloch representa-tion (14) for the channel φ . By (16) and (17), we havedet φ = det M = l ( M ) l ( M ) l ( M ) = 0 . Then, at least one special singular value l i ( M ) of M mustbe zero. Consequently, φ must necessarily have the sign–change property expressed in the fourth condition of The-orem 5, and so it must be entanglement–breaking.What is shown in (14) is that every quantum channelacting on a two–dimensional system can be seen as anaffine transformation sending the Bloch sphere into itself.Therefore, the image of the set of density matrices isrepresented by an ellipsoid contained in the Bloch sphere(we called it image ellipsoid ). Its principal axes’ lengthsare nothing but the singular values of M . The followingresult states some geometrically intuitive facts. Lemma 23.
Let ( M, c ) ∈ Pt be a positive, trace–preserving, qubitmap. Then we must have (cid:107) M (cid:107) ∞ ≤ . Moreover, if (cid:107) M (cid:107) ∞ = 1 then c = 0 , i.e. the map is unital, and theimage ellipsoid contains a pure state.Proof. Consider a generic unit vector n ∈ R . Then thematrices ± (cid:126)n · (cid:126)σ are positive, because of (13). Since( M, c ) is a positive map, ± ( (cid:126)c + M(cid:126)n ) · (cid:126)σ must be againpositive operators, that is (see again (13)) | M ( ± n ) + c | ≤ . Taking one half the sum of these equations, one obtains | M n | + | c | ≤ . Since we can certainly choose | M n | = (cid:107) M (cid:107) ∞ , we musthave (cid:107) M (cid:107) ∞ ≤
1, where the equality sign can hold if andonly if c = 0. Moreover, we have already observed thatthe singular values of M are the lengths of the principalaxes of the image ellipsoid. Therefore, the image ellipsoidof a unital qubit channel with (cid:107) M (cid:107) ∞ = 1 is necessarilytangent to the surface of the Bloch sphere, that is, itcontains a pure state.Thanks to Lemma 22, we can see that the (simpli-fied) restriction det φ (cid:54) = 0 we considered in Theorem 21causes no loss of generality in the d = 2 case. In fact,quantum channels with zero determinant are easily clas-sified as entanglement–breaking (and so they are notentanglement–saving, of course). We are ready to useTheorem 21 to obtain a classification of the ES qubitchannels.9 Theorem 24 (ES Qubit Channels) . Let φ ∈ CPt be a qubit channel. Then φ isentanglement–saving if and only if det φ (cid:54) = 0 and φ fixesor inverts a pure state. Here the “ inversion ” is intendedas the geometrical inversion − in the Bloch sphere. Ob-serve that a map which inverts a pure state is necessarilyunital.Proof. If det φ (cid:54) = 0 and φ fixes or inverts a pure state,then surely φ fixes one of them. In that case, The-orem 21 guarantees the entanglement–saving property,because of the fact that a pure state is (as a density ma-trix) semipositive.Let us turn our attention to the converse statement.If φ is entanglement–saving, then certainly det φ (cid:54) = 0 byLemma 22. Moreover, either φ has a semipositive fixedpoint (i.e. fixes a pure state), or | σ P ( φ ) | ≥ M has an eigenvalue with unit modulus, then (cid:107) M (cid:107) ∞ ≥ φ is unital. This fact willbe useful in a moment, but first observe that (54) restrictsthe possible peripheral spectra to σ P ( φ ) = { , } , { , − } , { , , e iθ , e − iθ } . Since Lemma 20 implies that φ must necessarily fix a purestate if { , } ⊆ σ P ( φ ), let us examine the case σ P ( φ ) = { , − } . Recall that by Theorem 6 the − n · (cid:126)σ . Moreover, up to a simple rescaling, we can freelysuppose | n | = 1. In that case, using also the unitality, weobtain φ (cid:18) + n · (cid:126)σ (cid:19) = − n · (cid:126)σ . This is the same as saying that φ inverts the pure state + n · (cid:126)σ in the Bloch sphere.Theorem 24 gives us a geometrical characterization ofthe ES set for a single qubit. With this tool at hand, wecan find an explicit parametrization of the ES set in the d = 2 case. This is the content of the following theorem. Theorem 25 (Explicit Form for ES Qubit Channels) . Let φ ∈ CPt be a qubit channel represented in the Paulibasis (as in (14) ) by a matrix M ∈ M (3; R ) and a vector c ∈ R . Then φ is entanglement–saving if and only ifone of the following two possibilities holds.1. There exist O ∈ SO (3) , θ ∈ R , < λ ≤ , λ ≤ µ ≤ , α ≥ such that the complete positivitycondition α ≤ (1 − µ )( µ − λ ) holds, and M = O M + ( λ, θ, α, µ ) O T ≡≡ O λ cos θ λ sin θ α − λ sin θ λ cos θ
00 0 µ O T , (72) c = O c + ( α, µ ) ≡ O − α − µ . (73)
2. The channel is unital (that is, c = 0 ), and thereexist O ∈ SO (3) , θ ∈ R , < λ ≤ , such that M = O M − ( λ, θ ) O T ≡≡ O λ cos θ λ sin θ λ sin θ − λ cos θ
00 0 − O T . (74) Proof.
Thanks to Theorem 24, we know that φ is entanglement–saving if and only if det φ (cid:54) = 0, and it fixes or inverts apure state. Let us begin with the first possibility. Inthe following, recall the elementary property (67), whichcorresponds to the degree of freedom represented by O in(72), (73) and (74). Therefore, by applying if necessaryan orthogonal matrix before the channel and its inverseafter, we can suppose without loss of generality that thefixed point is | (cid:105)(cid:104) | = + e · (cid:126)σ (with e = (0 , , T ), i.e. M e + c = e . (75)The positivity condition which has to be imposed on( M, c ) can be written (exactly as in Lemma 23) : | M n + c | ≤ ∀ n ∈ R : | n | = 1 . (76)Since the left–hand side of (76) reaches its maximum at n = e , here its first–order variation must be zero. Then2 δn T M T M e + 2 δn T M T c ≡ ∀ δn ⊥ e ⇒⇒ M T ( M e + c ) ∝ e . Together with (75), this gives M T e = µe for some real − ≤ µ ≤ (cid:107) M (cid:107) ∞ ≤ m ∈ M (2; R ) and − ≤ α, β ≤ M = m m αm m β µ , c = − α − β − µ . It will be more simple to adopt the parametrization m = (cid:18) s + d a + ba − b s − d (cid:19) . Until now we have used only the positivity of φ . In orderto exploit the complete positivity, we have to impose thatthe Choi matrix R φ = ( φ ⊗ I )( | ε (cid:105)(cid:104) ε | ) must be positive.In what follows, we will use for the bipartite system thecomputational basis sorted in lexicographical order, i.e. | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) . With this convention, one has R φ = 12 s + ib − µ d − ia − α + iβ d + ia s − ib − α − iβ µ . × R φ ≥ ⇒ ≤ det (cid:18) − µ d − iad + ia (cid:19) == − d − a ⇒ d = a = 0 . Let us call s = λ cos θ and b = λ sin θ , with θ ∈ R .Observe that λ = 0 is prohibited by det φ (cid:54) = 0, and λ > (cid:107) M (cid:107) ∞ >
1. Since this would contradictLemma 23, we can require 0 < λ ≤
1. Then R φ = 12 λe iθ − µ − α + iβ λe − iθ − α − iβ µ . Exploiting Silvester’s criterion on principal minors (see p.404 of [18]), it is not difficult to prove that the positivityof this matrix is equivalent to0 ≤ det λe iθ − µ − α + iβλe − iθ − α − iβ µ == (1 − µ )( µ − λ ) − α − β . Until now, we have proved that, if det φ (cid:54) = 0and φ = ( M, c ) fixes a pure state, then there exists O ∈ SO(3), θ ∈ R , 0 < λ ≤ λ ≤ µ ≤ α, β ∈ R satisfying the condition α + β ≤ (1 − µ )( µ − λ ) suchthat M = O ˜ M + ( λ, θ, α, β, µ ) O T ≡≡ O λ cos θ λ sin θ α − λ sin θ λ cos θ β µ O T ,c = O ˜ c + ( α, β, µ ) ≡ O − α − β − µ . To show that every such a pair ( ˜ M + , ˜ c + ) is entanglement–saving, observe that (cid:16) ˜ M + ( λ, θ, α, β, µ ) , ˜ c + ( α, β, µ ) (cid:17) n == (cid:16) ˜ M + ( λ n , nθ, α n , β n , µ n ) , ˜ c + ( α n , β n , µ n ) (cid:17) , with (cid:18) α n β n (cid:19) ≡ (cid:18) (cid:18) λ cos θ λ sin θ − λ sin θ λ cos θ (cid:19) + µ (cid:19) n (cid:18) αβ (cid:19) . Therefore, by taking the partial transpose of R φ n , oneobtains R T B φ n = 12 − µ n λ n e inθ − α n + iβ n λ n e − inθ − α n − iβ n µ n . The 2 × λ >
0, and this shows that R T B φ n can not be positivedefinite. Then the PPT criterion implies that φ n can notbe entanglement–breaking. Observe that it is possible tosuppose β = 0 and α ≥ e before the channel and of its inverse after. Inthis way, one obtains (72) and (73). This concludes thefirst part of the proof.Now, let us concern ourselves with the second possi-bility. Suppose that det φ (cid:54) = 0 and that φ inverts a purestate. Proposition 23 shows that such a channel mustbe unital ( c = 0), since (cid:107) M (cid:107) ∞ = 1. As in (75), we cansuppose M e = − e . Moreover, to avoid (cid:107) M (cid:107) ∞ > M can not contain any other nonzeroelement, i.e. there must exists (cid:18) s + d a + ba − b s − d (cid:19) ∈ M (2; R )such that M = s + d a + b a − b s − d
00 0 − . Now, the corresponding Choi matrix becomes R φ = 12 s + ib d − ia d + ia s − ib . The positivity condition for such an object im-plies s = b = 0, and d = λ cos θ, a = λ sin θ , again with0 < λ ≤ θ ∈ R . In order to show that every sucha pair ( M − ( λ, θ ) , < λ ≤
1, is entanglement–saving, it suffices to use the previous reasoning by ob-serving that M − ( λ, θ ) n = ˜ M + ( λ n , , , , . Observe that the tho cases ( M + , c + ) and ( M − ,
0) aretruly different only if λ < λ = 1 it is alwayspossible to bring back the second channel into the firstform.Thanks to this result, the set of ES qubit channels isessentially characterized (up to a unitary conjugation)1by four parameters, that we called λ, θ, α, µ . It could beuseful to write once for all the action of the two maps φ + λ,θ,α,µ ≡ ( M + ( λ, θ, α, µ ) , c + ( α, µ ) ) , (77) φ − λ,θ ≡ ( M − ( λ, θ ) ,
0) (78)on a generic 2 × φ + λ,θ,α,µ (cid:18) a bb ∗ c (cid:19) ≡ (cid:18) a + (1 − µ ) c λe iθ b − α cλe − iθ b ∗ − α c µ c (cid:19) , (79) φ − λ,θ (cid:18) a bb ∗ c (cid:19) ≡ (cid:18) c λe − iθ b ∗ λe iθ b a (cid:19) . (80)Observe that the two real parameters λ, θ can be joinedtogether in order to form an unique complex parameter z ≡ λe iθ which satisfies 0 < | z | ≤ Example 1 (Amplitude Damping Channels as ES) . The Amplitude Damping channels are a well–knownset of qubit channels which reproduce the action of aspontaneous emission process, the system being coupledto a zero–temperature environment. They form a setparametrized by ≤ p ≤ and defined by AD p (cid:18) a bb ∗ c (cid:19) ≡ (cid:18) a + (1 − p ) c √ p b √ p b ∗ p c (cid:19) . An easy inspection reveals that AD p / ∈ EBt as soon as p > . Moreover, since the composition rule is simply AD np ≡ AD p n , (81) we deduce that for p > their direct n –index takes thevalue + ∞ , that is, that they are entanglement–saving.As expected, a comparison with (79) shows that that AD p = φ + λ,θ,α,µ with λ = √ p, θ = 0 , α = 0 , µ = p . VI. ASYMPTOTICALLYENTANGLEMENT–SAVING CHANNELS
Through this section, we define and study an inter-esting subset of the entanglement–saving set, which wecall asymptotically entanglement–saving class. This fur-ther classification is based on the behaviour of the limitpoints of the sequence ( φ n ) n ∈ N . In order to state clearlya well–posed definition, we need some technical prelimi-naries. A. Definition
Recall that a limit point of a sequence is by definitionthe limit of one of its subsequences. Naturally, if a se-quence admits more than one limit point, then it does not converge (e.g. the sequence (( − n ) n ∈ N has the twolimit points +1 and − φ n ) n ∈ N . Lemma 26.
Let A ∈ M ( m ; C ) be a complex square matrix. Then thesequence ( A n ) n ∈ N has some (finite) limit points if andonly if every eigenvalue z ∈ C of A verifies | z | ≤ , andfor each eigenvalue z of modulus the corresponding Jor-dan blocks are trivial (that is, there are no off–diagonalelements). Every quantum channel φ ∈ CPt d has theseproperties.Proof. It is enough to put A in Jordan block form tosee that the condition expressed in the thesis are nec-essary and sufficient in order to guarantee that the se-quence ( A n ) n ∈ N is not unbounded. Then, the Bolzano–Weierstrass theorem ensures that every bounded se-quence in a finite–dimensional euclidean space (such as M ( d ; C )) admits a limit point. Theorem 6 states thatthe quantum channels enjoy these properties. Indeed,the boundness requirement for the sequence of powers(which are again CPt maps, and
CPt d is a compact set)is exactly the way we proved that spectral properties forquantum channels.Using this discussion as well as the proof of Theorem 6as guidelines, the following Lemma should be quite obvi-ous. Lemma 27.
Let A ∈ M ( m ; C ) be a complex square matrix. Asin (22) , write a Jordan decomposition for A , that is A = (cid:88) k ( λ k P k + N k ) , (82) where the λ k are eigenvalues, the P k are projectors ontothe generalized subspaces, and the N k are nilpotent appli-cations. If the sequence ( A n ) n ∈ N has some (finite) limitpoints, then E A ≡ (cid:88) k : | λ k | =1 P k , (83) I A ≡ (cid:88) k : | λ k | =1 λ ∗ k P k (84) are two of them. Moreover, every limit point is diagonal-izable in a Jordan basis for A , and has the form (cid:88) k : | λ k | =1 z k P k , | z k | ≡ ∀ k . (85)Now we are in position to prove a useful observation.The following Proposition explores the algebraic struc-ture of the set of the limit points of a sequence ( A n ) n ∈ N .2 Proposition 28 (Limit Points of the Powers of a Matrixas a Group) . Let A ∈ M ( m ; C ) be a complex square matrix. Consider G A ≡ { limit points of ( A n ) n ∈ N } . (86) Then G A , if not empty, is an abelian compact group withthe standard operation of matrix multiplication.Proof. We will prove that G A is closed under multiplica-tion, possesses an identity element, is closed and limitedas a set, and moreover that each element has an inverse. • If S, T ∈ G A , then it must be ST ∈ G A . Infact, there exist subsequences ( k n ) n ∈ N , ( h n ) n ∈ N such that S = lim n →∞ A k n , T = lim n →∞ A h n . But then ST = lim n →∞ A k n + h n ∈ G A . • Let us explicitly construct an identity element. If S ∈ G A , then Lemma 27 ensures that S is diagonal-izable in a Jordan basis for A . Therefore, with thesame notation of (83), equation (85) implies that S E A = S . And so, E A is an identity element for G A . • Observe that G A is closed as a set because of itsdefinition. To see this, we consider a limit point of G A and show that it actually belongs to G A itself.Let ( S k ) k ∈ N be a sequence of elements belongingto G A . Then for every k there exists a sequence ofpowers of A which converges to S k . In other words,we can write˜ S = lim k →∞ S k , S k = lim n →∞ A h ( k ) n . For each k ≥
1, define an integer n k such that (cid:13)(cid:13) S k − A h ( k ) nk (cid:13)(cid:13) ∞ ≤ k . Then˜ S = lim k →∞ S k = lim k →∞ A h ( k ) nk ∈ G A . • A consequence of Lemma 27 is that G A must be lim-ited as a set. In fact, all matrices S ∈ G A can be si-multaneously diagonalized using a Jordan basis for A . In this basis our claim is obvious, because (85)guarantees that all the eigenvalues belong to thecomplex circumference | z | = 1. • Let us prove that each generic S ∈ G A has an in-verse internal to G A (and so, such an object mustbe unique). Observe that S k ∈ G A for each k ∈ N ,and that Lemma 27 claims that I S (defined as in (84)) is indeed a limit point of this sequence (whichis limited because contained inside G A , and there-fore has some limit points). Thanks to the propertyof closure of G A , we can deduce that I S ∈ G A . Be-cause of its definition, it must be S I S = E A , so I S is an inverse of S .Now, consider a quantum channel φ ∈ CPt d . Thanksto Lemma 26, Proposition 28 applies, and we can definethe corresponding (non–empty) set of limit points G φ .Since CPt d is a closed set, it can immediately seenthat G φ ⊆ CPt d . Our immediate goal is to classifythe entanglement–breaking properties of the elementsbelonging to G φ . Fortunately, this task is quite easy; theanswer is the content of the following proposition. Proposition 29.
Let G φ be the non–empty set of limit points of the powersof a quantum channel φ ∈ CPt . There are only twopossibilities: G φ ⊆ EBt or G φ ∩ EBt = ∅ . Proof.
The only thing we have to prove is that G φ ⊆ EBt if there exists S ∈ G φ ∩ EBt . Thanks to the groupproperties of G φ , taken a generic S ∈ G φ we can certainlywrite S = S ( S − S ) , S − S ∈ G φ ⊆ CPt . Recall the property (9) of the entanglement–breakingchannels. Since S ∈ EBt , it must be also S ∈ EBt . Alternative proof (more direct).
These statements can beproved without invoking nor the particular group struc-ture of G φ , neither the other technical lemmas. Thefact that G φ is not empty descends from the com-pactness of CPt d , as previously observed. Now, sup-pose that S ∈ G φ ∩ EBt . Then there exists a di-vergent sequence ( n k ) k ∈ N such that lim k →∞ φ n k = S .Take another limit point S = lim h →∞ φ m h ∈ G φ ,and construct the divergent sequence ( h k ) k ∈ N , where h k ≡ min { h ∈ N : m h ≥ n k } . Then, for all k ∈ N wecan write φ m hk = φ m hk − n k φ n k , (87)with m h k − n k ≥ k →∞ φ m hk = S and lim k →∞ φ n k = S . Onthe other hand, the sequence ( φ m hk − n k ) k ∈ N is bounded(as it is composed by quantum channels), and so it musthave at least one limit point S ∈ CPt . Taking the limitof (95) only on the subsequence that produces that limitpoint, we obtain S = S S , and then from (9) we inferagain S ∈ EBt .Proposition 29 makes a clear distinction between thetwo behaviours of the limit points. The following defini-tion makes sense now; actually, it seems quite natural.3
Definition 3 (Asymptotically Entanglement–SavingChannels) . Let φ ∈ CPt be a quantum channel, and denote by G φ the non–empty set of limit points of the sequence ( φ n ) n ∈ N . If G φ ∩ EBt = ∅ then φ is called asymptot-ically entanglement–saving (AES). Remind that the set
EBt is closed, and so its comple-ment in
CPt is open. Consequently, every AES chan-nel is also ES, but the converse is not necessarily true.Moreover, consider a limit point S ∈ G φ of the sequenceof powers of an AES channel φ . We know (by definition)that S is not entanglement–breaking. But not only: since G φ is a closed set (see Proposition 28), it turns out that S itself must be an AES channel! This means that φ is AES ⇔ G φ is entirely composed of AES channels.(88)This observation leads to an elementary way to constructexamples of ES channels with determinant equal to zero,that is, ES channels that might elude the classificationprovided by Theorem 21. Indeed, consider S ∈ G φ , φ being a non–unitary AES channel. On one hand, (88)ensures that S is AES and so ES. On the other hand, The-orem 3 and Theorem 6 impose the constraint | det φ | < S = lim n →∞ det( φ n ) = lim n →∞ (det φ ) n = 0 . An explicit example of this construction is given in Ex-ample 2.Another useful characterization of the AES set de-scends directly from its definition. Since Proposi-tion 29 guarantees that either all the limit points areentanglement–breaking or the channel is AES, we can re-strict ourselves to check only a particularly simple limitpoint, for instance the E φ defined through (23). φ is AES ⇔ E φ / ∈ EBt . (89)What is the physical meaning of Definition 3? TheseAES channels represent a particularly innocuous kind ofentanglement–saving noise, in the following sense. It canhappen that a quantum channel never breaks completelythe entanglement, even if it is applied many times; thisis the entanglement–saving property. However, these re-peated application can reduce the quantum correlationsto an arbitrary low value, and destroy them only in thelimit . An asymptotic entanglement–saving channels doesnot exhibit such a behaviour. Instead, a finite amount ofentanglement is present also in the limit . In other words,suppose that Alice makes sure that only an AES noiseis acting on her half of the global system. Then she isguaranteed that the bipartite system will always containa significant and concretely usable quantity of entangle-ment. B. Simple results
This subsection presents some partial results concern-ing the AES channels. These results can be easily seenas corollaries of the general theory to be discussed later,but here we will deduce them independently. Then, itwill be very instructive to see as they fit into the generalscheme drawn by Theorem 32.Firstly, let us present a simple link between the AESproperty and the size of the peripheral spectrum. Indeed,a large peripheral spectrum is enough to guarantee theasymptotic entanglement saving, as we will see in a mo-ment. We start by recalling that the trace norm (cid:107) A (cid:107) of a generic matrix A can be bounded by the sum of themoduli of its eigenvalues { λ i ( A ) } , that is (cid:107) A (cid:107) ≥ (cid:88) i | λ i ( A ) | . (90)We refer the reader interested in the proof to [25], p. 172.Moreover, let us recap the content of the so–called reshuf-fling separability criterion (see the earlier works [26], [27]and [28], or [29] p. 355 for a good review with proofs).Let ρ AB be a separable state on a bipartite system AB ,with dim H A = dim H B = d . Denote by φ ∈ CP d theunique linear map on states of A associated to ρ AB viathe Choi–Jamiolkowski isomorphism (4)), i.e. verifying R φ = ρ AB . Considering φ as a d × d complex matrix,its trace norm can not exceed d . In short, R φ ∈ S ⇒ (cid:107) φ (cid:107) ≤ d . (91) Proposition 30 (AES Channels and Peripheral Spec-trum) . Let φ ∈ CPt d be a quantum channel. If φ has more than d peripheral eigenvalues, i.e. | σ P ( φ ) | > d , then φ is AES.Proof. Suppose | σ P ( φ ) | > d . Then (85) guarantees thatevery limit point S ∈ G φ also verifies | σ P ( S ) | > d .Thanks to (90), this implies that (cid:107) S (cid:107) ≥ (cid:88) i | λ i ( S ) | > d . By invoking (91), we can see that this forbids S ∈ EBt d .The last partial result we are going to present concernsthe AES qubit class. Indeed, if d = 2 then Lemma 22and Theorem 3 give us powerful tools to characterize thewhole set of AES channels. Proposition 31.
A qubit channel is AES if and only if it is unitary.Proof.
Obviously a unitary channel U is AES, becausethe limit points of ( U n ) n ∈ N are again unitary channels.Let us concern ourselves with the converse. Thanks toTheorem 3, we have only to prove that every AES qubitchannel φ satisfies the property | det φ | = 1. Assume4by contradiction that | det φ | <
1. Then the elementaryequality det( φ n ) = (det φ ) n shows that the limit points of( φ n ) n ∈ N have zero determinant. This is absurd, becauseLemma 22 shows that this mere property implies thatthey are entanglement–breaking. C. General characterization of AES channels
Through this subsection, we will prove the theoremthat completely solves the characterization problem forthe AES channels. This section is where the very gen-eral theory presented in Section III (and in particularTheorem 12) comes heavily into play. We postpone thediscussion of our results, as well as some clarifying exam-ples, after the statement and proof of the central result.
Theorem 32 (Complete Characterization of AES Chan-nels) . Let φ ∈ CPt d be a quantum channel. Then the followingfacts are equivalent:1. φ is asymptotically entanglement–saving.2. With the notation of Theorem 12, at least one ofthe d (1) i associated to φ is strictly greater than .3. The von Neumann algebra η ˜ E † φ (see (47) ) is non-commutative.4. φ admits at least two noncommuting phase points.Proof. Through this proof, the notation will follow theone developed for Theorem 12.1 . ⇔ . The equivalence (89) states that φ is AES if andonly if E φ (as defined in (23)) is not entanglement–breaking. Observe that (43) gives us an explicitexpression for the action of E φ . So we could thinkthat this is enough to decide whether E φ is or notentanglement–breaking. The problem here is thatthis expression works only when the input matrix isof the form X ⊕
0, with respect to the decomposition C d = K ⊕ K ⊥ . With the nomenclature of (27), wecan say that we only know the action of ˜ E φ , notthe one of E φ .But the amazing fact is that the difference between E φ and ˜ E φ does not matter when we only careabout the entanglement–breaking behaviour. In-deed, E φ ∈ EBt ⇔ ˜ E φ ∈ EBt . (92)The implication ⇒ is trivial, because ˜ E φ is a restric-tion of E φ , and a restriction of an entanglement–breaking channels must be again entanglement–breaking. In order to prove the converse, take ageneric input matrix X . Since E φ = E φ becauseof the very definition (23), E φ ( X ) must be a fixedpoint for E φ , that is its support must be contained in K . Denoting by P : M d → M r (with r ≡ dim K )the superoperator that restricts every input to K ,we can write E φ ( X ) = P E φ ( X ) ⊕
0, or more gen-erally E φ = P E φ ⊕
0. But then E φ = E φ = E φ ( P E φ ⊕ E φ P E φ ⊕ . (93)This equality and a slight generalization of (9) showthat E φ must be entanglement–breaking if so is ˜ E φ .Thanks to (92), we know that φ is AES if and onlyif ˜ E φ / ∈ EBt . Moreover, (43) shows that˜ E φ ( · ) = (cid:77) i Tr i, [ P i ( · ) P i ] ⊗ ρ i, . (94)A straightforward inspection of (94) immediatelyreveals that ˜ E φ is entanglement–breaking if andonly if d (1) i = 1 for all i . Indeed, if d (1) i = 1 forall i then Tr i, [ P i ( · ) P i ] becomes a scalar function,and (94) takes the form (11), meaning that ˜ E φ isentanglement–breaking. Conversely, if at least oneof the d (1) i is strictly greater than 1, then thereexists a nontrivial, preserved qudit “inside” theHilbert space. To be more precise, consider an en-tangled state of the form A = | ε (cid:105)(cid:104) ε | i, i, (cid:48) ⊗ ρ i, ,where | ε (cid:105) i, i, (cid:48) = 1 (cid:113) d (1) i d (1) i (cid:88) j =1 | j (cid:105) i, ⊗ | j (cid:105) ( i, (cid:48) is a partial maximally entangled state over twocopies of the ( i, E φ fixes that en-tangled state A , ˜ E φ can not be entanglement–breaking.2 . ⇔ . Obvious consequence of (47).2 . ⇔ . Observe that φ admits two noncommuting phasepoints if and only if at least two noncommutingelements belong to χ φ . Equation (42) allows us toconclude that χ φ is noncommutative as a set if andonly d (1) i > i . Alternative proof of . ⇒ . . Let us present now an al-ternative, direct proof of the implication 3 . ⇒ . , whichdoes not use the heavily technical theory of the phasesubspace summarized in Theorem 12. Consider two her-mitian, noncommuting matrices X, Y ∈ η ˜ E † φ , and define Z ≡ X + iY ∈ η ˜ E † φ . Since [ X, Y ] (cid:54) = 0, one can eas-ily verify that [ Z, Z † ] (cid:54) = 0. Then, Lemma 16 allows us toconstruct the entangled state Q ( Z ). Moreover, thanks tothe *–algebra property of η ˜ E † φ , we know that Z † Z ∈ η ˜ E † φ ,so that indeed (cid:16) I ⊗ ˜ E † φ (cid:17) ( Q ( Z )) = Q ( Z ). Since Q ( Z ) is5entangled, this imply that ˜ E † φ / ∈ EBu , that is (see (10))˜ E φ / ∈ EBt . Equations (89) and (92) allows us to con-clude that φ is AES. D. Examples and discussion
What is the intuitive meaning of Theorem 32? In ouropinion, the most easily understandable requirement con-tained in Theorem 32 is condition 4. Remind that theonly input states which survive in the limit of an infinitenumber of iterations are the phase points (by virtue ofTheorem 6). Then, condition 4 says simply that the en-tanglement (the most genuine quantum mechanical prop-erty) can survive in that limit only if a noncommutationrelation (the most basic feature of quantum mechanics)exists among the suriving states, i.e. the phase points.Before we give some general class of examples of AESchannels, let us discuss how the simple results containedin Proposition 30 and Proposition 31 fit into the generalscheme drawn by Theorem 32.
Proof of Proposition 31 (revisited).
The fact that theonly AES qubit channels are the unitary evolutions canbe easily proved now. In fact, in the qubit case the secondcondition of Theorem 32 imposes that K = C is indeedthe whole space, and that there is only one addend inthe direct sum (42). Moreover, also the tensor productdecomposition in that addend must have a trivial secondfactor, that is ρ i, = 1. Thanks to (45), this implies that φ is an unitary evolution. Proof of Proposition 30 (revisited).
We have to provethat a peripheral spectrum strictly larger than d (count-ing multiplicities) invariably denotes an AES behaviour.This can be easily regarded as a consequence of condition4 (or 3) of Theorem 32. Indeed, suppose by contradic-tion that | σ P ( φ ) | > d and that the associated (at least) d + 1 phase points commute with each other. Then thelinear span χ φ of the phase points has dimension at least d + 1. Since φ is hermiticity–preserving, χ φ is a real sub-space (see Theorem 6), and so it contains d + 1 linearlyindependent (and commuting) hermitian matrices. It isa well–known fact that commuting hermitian operatorscan be simultaneously diagonalized. We would obtain d + 1 linearly independent, diagonal matrices. Since thediagonal is composed of only d entries, this is clearly ab-surd.Finally, we present the simplest possible class of exam-ples of AES channels. Example 2 (Simple AES Channels) . Once again, we refer the reader to Theorem 12 for thebasic notation. Consider a Hilbert space C d = K decom-posed in the form C d = (cid:77) i K i = (cid:77) i K (1) i ⊗ K (2) i , (95) with at least one of the d (1) i ≡ dim K (1) i strictly greaterthan . Call P i the orthogonal projector onto K i . Takeunitary matrices U i acting on K (1) i , density matrices ρ i, defined on K (2) i and a permutation π over the set of in-dices i which exchanges only indices sharing the samedimension d (1) i . Then construct the channel S given bythe formula S ( · ) ≡ (cid:77) i U i (cid:0) Tr π ( i ) , [ P π ( i ) ( · ) P π ( i ) ] (cid:1) i, U † i ⊗ ρ i, . (96) These channels enjoy the following properties. • If at least one of the d (1) i is strictly greater than (as we supposed), then S is asymptoticallyentanglement–saving. This can be easily seen byobserving that its phase subspace is indeed χ S = (cid:77) i M d (1) i ⊗ ρ i, , and so it is not commutative (by Theorem 32 thisis enough to conclude). • These channels S are the simplest AES channelsin the sense that they have only eigenvalues ei-ther of unit modulus or equal to . Indeed, it canbe easily verified that the S defined through (96) has (cid:80) i d (1) i eigenvalues of unit modulus (see theconstruction reported in the proof of Theorem 12for more details). Moreover, it has two differentkinds of zero eigenvalues: the first group is com-posed of (cid:80) i d (1) i (cid:16) d (2) i − (cid:17) zeros correspondingto eigenvectors of the form (cid:76) i a i, ⊗ x i, , wherethe x i, satisfies Tr x i, ≡ for all i . The sec-ond group corresponds to the (cid:80) i (cid:54) = j d (1) i d (2) i d (1) j d (2) j independent hermitian input matrices having onlyoff–diagonal elements with respect to the block de-composition (95) . Observe that the total numberof eigenvalues gives correctly (cid:16) (cid:80) i d (1) i d (2) i (cid:17) . Thepresence in the spectrum of only zero or unit com-plex eigenvalues by the way implies that S itself isa limit point of the sequence ( S n ) n ∈ N of its powers.This in turn shows that S is an ES channel thatmight be impossible to classify according to Theo-rem 21, because it has a large number of zero eigen-values. VII. CONCLUSIONS
In the recent years enormous progresses have been putforward in the development of a brand new form of tech-nology based on a clever use of quantum mechanical sys-tems. In particular, a lot has been learned on how toattenuate the detrimental effects of noise arising from6the interaction of a system of interest with its surround-ing environment, either by building better devices (hard-ware approach), or by exploiting complex coding pro-cedures (software approach), yielding as an outcome aneffectively noise whose intensity is milder than the orig-inal one. In view of all this, it makes sense to studymore closely the properties of those processes which in-duce only small (yet nontrivial) perturbations on the sys-tem of interest. In order to characterize such a class oftransformations, in the present work we focus on theirability of preserving entanglement of a bipartite quan-tum system when acting locally on one of the subparts.This yields to the introduction of three special set of su-peroperators, namely the universal preserving channels(i.e. those mapping which preserve any form of entangle-ment initially present in the system, no matter how weakit may be), the set of entanglement saving channels (i.e.maps which preserve entanglement of a maximally entan-gled state even after n iterations, n being an arbitrary integer), and the set of asymptotically entanglement sav-ing channels (that make the entanglement survive evenin the limit n → ∞ ). First of all, we proved that the onlyuniversal entanglement preservers are the unitary evolu-tions. Then we found a partial characterization theoremfor the entanglement saving channels (which in turn al-lows to fully understand the qubit case) and a completecharacterization result holding for the asymptotically en-tanglement saving ones.Most of the major open problems here concern the lackof a full understanding of the ES class. It is apparentthat an ES channel has to produce very little noise inthe system, but in what sense? Are there some easilyverified criteria that can certify that a given channel is(or is not) entanglement saving? We are confident that amore complete knowledge of this kind of questions couldreveal us some interesting phenomena appearing whenthe concatenation of quantum noises takes place. [1] R. Horodecki, P. Horodecki, M. Horodecki, and K.Horodecki, Rev. Mod. Phys. , 865 (2009).[2] W. Zurek, Rev. Mod. Phys. , 715 (2003).[3] A. S. Holevo and V. Giovannetti, Rep. Prog. Phys. ,046001 (2012).[4] M. M. Wolf, Quantum Channels & Operations , lecturenotes (2012).[5] A. S. Holevo