Kraus operators and symmetric groups
Alessia Cattabriga, Elisa Ercolessi, Riccardo Gozzi, Erika Meucci
KKraus operators and symmetric groups
Alessia Cattabriga ∗ , Elisa Ercolessi † , Riccardo Gozzi, Erika MeucciFebruary 2, 2021 Abstract
In the contest of open quantum systems, we study a class of Kraus op-erators whose definition relies on the defining representation of the sym-metric groups. We analyze the induced orbits as well as the limit set andthe degenerate cases.
We are interested in studying open quantum systems, that is systems that arefree to interact with the environment or with other systems. The study of opensystems is useful in fields such as quantum optics, quantum measurement theory,quantum statistical mechanics and quantum cosmology. Moreover, the study ofcomposite systems is at the heart of quantum computation and quantum infor-mation, where, for examples, concepts like entanglement can have applicationsin devising algorithms and protocols, such as quantum teleportation, that donot have a classical analogue. ∗ A. Cattabriga has been supported by the ”National Group for Algebraic and GeometricStructures, and their Applications” (GNSAGA-INdAM) and University of Bologna, funds forselected research topics. † E.E. is partially supported by INFN through the project “QUANTUM” and by the projectQuantHEP of the QuantERA ERA-NET Co-fund in Quantum Technologies a r X i v : . [ m a t h - ph ] J a n n elementary quantum mechanics, the state of a closed quantum systemis represented by a ray [1] in a separable Hilbert space v ∈ H , i.e. by anequivalence class of vectors [ v ], v ∈ H , with respect to the relation: v ∼ λ v with λ ∈ C − { } . Such equivalence class can be represented via the densitymatrix ρ v ≡ vv † / (cid:107) v (cid:107) (where v † is in the dual space H ∗ (cid:39) H ), which is arank-one projector or, more precisely, a bounded, self-adjoint, positive definite,unit-trace operator such that: ρ v = ρ v . (1)A system whose density matrix satisfies the latter condition is said to be pure .In general, as we will explain shortly below, it is necessary to consider a moregeneral kind of density matrices that are constructed out of a statistical mixture { ρ j , p j } Nj =1 , where the ρ j ’s are pure density matrices and the p j ’s are probabil-ities, therefore satisfying 0 ≤ p j ≤ (cid:80) Nj =1 p j = 1. Such a density matrix(called mixed ) is obtained by setting ρ = N (cid:88) j =1 p j ρ j (2)and is a bounded, self-adjoint, positive definite, unit-trace operator, with now ρ (cid:54) = ρ .In the following we will be interested in the case in which the Hilbert space isfinite dimensional with dim( H ) = n , a fact we will assume from now on.When considering an open quantum system, i.e. a (sub)system A in inter-action with an environment B , the Hilbert space representing the total sys-tem is given by H A ⊗ H B , where the general element of H A ⊗ H B will be ψ AB = (cid:80) i ∈ I,j ∈ J a ij e Ai ⊗ e Bj where { e Ai | i ∈ I } and { e Bj | j ∈ J } are orthonor-mal basis of H A and H B respectively and (cid:80) i ∈ I,j ∈ J | a ij | = 1. The densitymatrix representing the quantum state of the subsystem A is obtained by tak-ing the partial trace over the environment B : ρ A = Tr B [ ρ ψ AB ]. A very wellknown theorem [2] states that the density matrix ρ A is pure if and only if ψ AB is of the form v A ⊗ v B with v A ∈ H A and v B ∈ H B , i.e. it is a separable state.2n all other cases, i.e. when the state ψ AB is entangled, ρ A will represent amixed state.The space of quantum states can be endowed of interesting geometrical struc-tures. The set of pure states is a complex projective space and indeed a Kahlermanifold [1], that can be embedded in the the Lie algebra of self-adjoint ma-trices as the (co)-adjoint orbit of the unitary group U ( N ) of rank-one densitymatrices. The latter description generalizes also to rank-k mixed states [3, 4],so that the full space of states can be seen as the union of orbits of the unitarygroups, each of them been a complex manifold endowed by a metric, a symplec-tic form and a compatible complex structure.From a more algebraic point of view, let us notice that the space of pure statescan be seen as the extremal points of the positive cone in the algebra of self-adjoint operators, generated by positive definite and unit-trace matrices.The time evolution of a closed quantum system is determined by the Schroedingerequation [2] or, when the Hamiltonian operator H is time-independent, by theunitary operator U ( t ) = exp[ − i H t/ (cid:126) ] via: ρ ( t ) = U ( t ) ρ ( t = 0) U ( t ) † . (3)This is the evolution also of the density matrix of the total system A ∪ B , whichcan be seen as isolated, whereas the evolution of the subsystem A which isobtained by taking the partial trace: ρ A ( t ) = Tr B [ ρ ψ AB ( t )]. Let us notice that,contrary to what happens for a closed system, open quantum dynamics maychange the spectrum as well as the rank of the density matrix.The dynamics of open quantum systems is generated by the so-called Gorini-Kossakowski-Sudarshan- Linbland (GKLS) equation (see ref. [5] for a, alsohistorical, review): L ( ρ ) = − i [ H , ρ ] − N (cid:88) j =1 (cid:110) V † j V j , ρ (cid:111) + (cid:88) j V j ρV † j , (4)where H is the Hermitian Hamiltonian operator and the V j ’s ( N = 1 , , · · · , n −
3) are arbitrary (bounded) operators .Eq. (4) has a clear geometrical interpretation. Indeed, it can be shown [6]that the dynamical evolution described by this equation defines a vector fieldΓ = X + Y + Z, that can be decomposed into three vector fields related withthe three addends of the GKLS equation. More precisely, X is a Hamiltonianvector field whose flow preserves the spectrum of ρ (hence moving on a given co-adjoint orbit), Y is a gradient-like vector field whose flow changes the spectrumbut preserves the rank while Z is a vector field corresponding to a flow thatchanges the rank of ρ .In this paper we take a different, but equivalent perspective [7], according towhich the dynamic of an open quantum system with density matrix ρ A ( t ) ≡ ρ ( t ) is described by means of the so called Universal Dynamical Maps (UDM),that is a trace-preserving linear completely positive definite map defined as E K ([ t ,t ]) : ρ ( t ) → ρ ( t ) = (cid:88) α K α ( t , t ) ρ ( t ) K α ( t , t ) † (5)given an initial configuration of the system at time t = t encoded by thedensity matrix ρ ( t ). The operators K α ( t , t ), for α ∈ A are called Krausoperators : they do not depend on the initial condition ρ ( t ), but, as the indicesshould suggest, just on the time interval [ t , t ]. We will call Kraus map a linearcombination of Kraus operators. Moreover, to ensure Tr[ ρ ( t )] = 1, the Krausoperators must satisfy the following condition (cid:88) α ∈A K α ( t , t ) K α ( t , t ) † = . (6)Whenever the (super)-operators (5) satisfy also: E K ([ s,t ]) ◦ E K [( t, = E K ([ s, , ∀ s ≥ t ≥ , (7) As usual we denote with V † the adjoint of V , with [ · , · ] the commutator and with {· , ·} the anticommutator. Here for completely positive we mean that the operator K [ t ,t ] ⊗ B is positive for anypossible extension of H A to H A ⊗ H B . We refer the interested reader to [5] for a detaileddiscussion of why completely positivity and not simply positivity is required. L of eq. (4) is the generator. In this case the dynamics is calledquantum Markovian [7]. Le us remark that the decomposition of such a mapinto Kraus operators is not unique, a question that will be considered in thefollowing.In this paper, we are interested in characterizing the non-unitary part of thedynamic and we set H = 0. Notice that the GKLS equation is invariant underany unitary transformation, since ρ ( t ) → U ρ ( t ) U † , V j → U V j U † , for all j . Also,we can always find a unitary transformation such that, at the initial time, wecan write the density matrix in the diagonal form ρ ( t = 0) = diag( λ , . . . , λ n ),with λ i ≥ λ + · · · + λ n = 1. Thus we can consider a GKLS equation ofthe form L ( ρ ) = − N (cid:88) j =1 (cid:110) V † j V j , ρ (cid:111) + N (cid:88) j =1 V j ρV † j , (8)with ρ diagonal. This situation encompasses a series of interesting cases inphysics, such as (when N=1) the so-called Quantum Poisson and Gaussian Semi-groups [6].In order to describe explicitly this type of dynamics, we take an algebraicapproach and consider Kraus operators associated to elements of the symmetricgroup Σ n via the defining representation. In Section 2, we recall the basicnotions of this representation and associate a Kraus map to each element of C [Σ n ], the group algebra of Σ n . We give conditions on the elements of C [Σ n ]giving rise to Kraus maps with admissible action and reduce the study of theorbits of the dynamic to those associated to cyclic subgroups of Σ n . In Section3, we compute explicitly the orbits in the cyclic case as well as the limit set ofthe dynamics. We will also describe what happens in the degenerate cases, thatis the cases in which the initial density matrix is not generic (i.e., the cardinalityof the spectrum of ρ is less than the order of the matrix).5 Defining representation of Σ n and associatedKraus maps In this section, after recalling some classical notion on the defining representa-tion of Σ n (see [8]), we describe how to associate a Kraus map to elements of C [Σ n ], the group algebra of Σ n .Let Σ n be the symmetric group on n letters. The n -dimensional defining representation χ : Σ n → GL n ( C ) of Σ n is given by χ ( σ ) = R σ with ( R σ ) ij = σ ( j ) = i . (9)The defining representation is unitary (i.e., the image of χ is contained in U ( n ))and reducible. Indeed, the 1-dimensional subspace W spanned by e + e + · · · + e n , with e i the i -th vector of the canonical basis of C n , is invariant under theaction of χ (Σ n ) and χ restricts to the trivial action on GL ( W ). Moreover, χ iscompletely reducible: indeed, if W ⊥ denotes the orthogonal complement withrespect to the standard hermitian product on C n , also W ⊥ is invariant underthe action of χ (Σ n ). It is also possible to prove that the ( n − n induced by χ into GL ( W ⊥ ) is irreducible.If we identify C n with the vector space D n ( C ) of diagonal matrices with complexentries, then Σ n acts on D n ( C ) as σ · diag( λ , . . . , λ n ) = R σ diag( λ , . . . λ n ) R − σ = diag( λ σ (1) , . . . , λ σ ( n ) ) . Since W is an invariant subspace with respect to χ , then the trace of a matrixis invariant under this action.Let C [Σ n ] denotes the group algebra of Σ n . The defining representation ofΣ n naturally induces a representation of C [Σ n ] into M n ( C ), that we still denote6ith χ , given by χ (cid:32) (cid:88) σ ∈ Σ n c σ σ (cid:33) v = (cid:88) σ ∈ Σ n c σ ( R σ v ) , (10)with c σ ∈ C , for all σ , and v ∈ C n .We want now to introduce a time dependence: for each σ ∈ Σ n we choose C -valued smooth functions [0 , + ∞ ) (cid:51) t (cid:55)→ c σ ( t ) ∈ C and consider the map[0 , + ∞ ) (cid:51) t (cid:55)→ (cid:88) σ ∈ Σ n c σ ( t ) σ ∈ C [Σ n ] . (11)If we look at the operator obtained trough χ , we have that: (cid:88) σ ∈ Σ n ( c σ ( t ) R σ ) ( c σ ( t ) R σ ) † = (cid:88) σ ∈ Σ n c σ ( t ) c σ ( t ) R σ R † σ = (cid:32) (cid:88) σ ∈ Σ n c σ ( t ) c σ ( t ) (cid:33) Id n , where c σ ( t ) denotes the complex conjugate function. So if (cid:88) σ ∈ Σ n c σ ( t ) c σ ( t ) = , (12)where denotes the constant function equal to 1, the element (cid:80) σ ∈ Σ n c σ ( t ) σ acts on D n ( C ), via χ , as a Kraus operator.Even if this condition on the coefficients is satisfied, a generic element of (cid:80) σ ∈ Σ n c σ ( t ) σ ∈ C [Σ n ] might not yield a suitable one-parameter semigroup un-less: i) it is completely positive; ii) it satisfies the time condition (7).As for the second condition, let us notice that, if we consider χ (cid:32) (cid:88) σ ∈ Σ n c σ ( t ) σ (cid:33) = (cid:88) σ ∈ Σ n c σ ( t ) R σ , such that N = { σ ∈ Σ n | c σ ( t ) (cid:54) = } is a subgroup of Σ n , it is possible tochoose opportunely the coefficients c σ ( t ) so that the operators satisfy the timeconditions. Indeed, given a subgroup S of Σ n , we associate to it the operator K S = g ( t )Id n + f ( t ) (cid:88) σ ∈ S R σ (13)7iving the evolution function F S ( t, ( ρ (0)) = ρ ( t ) = g ( t ) ρ (0) + f ( t ) (cid:88) σ ∈ S R σ ρ (0) R − σ , (14)where g ( t ) = (cid:115) | S | (1 + ( | S | − e − t ) and f ( t ) = (cid:115) | S | (1 − e − t ) , (15)with | S | the order of S . It is immediate to check that K S satisfies (12).Given S and T subgroups of Σ n , we say that K S and K T are equivalent , andwrite K S ∼ = K T , if they determine the same evolution function that is F S ( t, ( ρ (0)) = F T ( t, ( ρ (0))for each t ∈ [0 , + ∞ ) and each ρ (0) ∈ D n ( C ).Since our aim is to study all the possible evolution functions, we want to look atKraus maps associated to subgroups of Σ n up to equivalence. On this regard,we have the following result. Proposition 1.
Two Kraus maps K S and K T , associated to subgroups S, T ⊆ Σ n , are equivalent if and only if the partition of { , , . . . , n } associated to theorbits of the action of S and T onto { , , . . . , n } is the same.Proof. Notice that K S and K T are equivalent if and only if | S | = | T | and (cid:88) σ ∈ T R σ ρ (0) R − σ = (cid:88) σ (cid:48) ∈ S R σ (cid:48) ρ (0) R − σ (cid:48) Therefore, there is a bijective map T → S that sends σ ∈ T in σ (cid:48) ∈ S . Since the R ’s matrices are permutation matrices, this happens if and only if the orbits ofthe action of S and T onto { , , . . . , n } are the same.The previous proposition allows us to reduce to the case in which S is acyclic subgroup. Indeed, it is enough to select one subgroup S for each par-tition of { , , . . . , n } and we can always choose a cyclic subgroup: given a8artition { p , . . . , p k } of { , , . . . , n } we can take the cyclic subgroup generatedby σ = c · · · c k , where c i is any cycle permuting all the elements of p i , for i = 1 , . . . , k . In other words, given a diagonal matrix ρ , it is always possible tofind a cyclic subgroup that permutes the elements on the matrix according tothe evolution.Let us consider now the following trivial but important: Remark 2. If S and T are conjugated subgroups, then the evolution function F T ( t, can be deduced from the evolution function F S ( t, because • | S | = | T | and hence g T ( t ) = g S ( t ) and f T ( t ) = f S ( t ) ; • (cid:88) σ ∈ T R σ ρ (0) R − σ = (cid:88) σ (cid:48) = τστ − ∈ S R σ (cid:48) ρ (0) R − σ (cid:48) , where τ ∈ T and R σ (cid:48) = R τ R σ R − τ . Therefore, by Proposition 1 and Remark 2, in order to understand the dy-namical evolution, it is enough to consider actions of cyclic subgroups of Σ n onto C n up to conjugation. It is important to recall that the number of conjugacyclasses of elements in Σ n , that corresponds to the number of cyclic subgroupsof Σ n up to conjugacy, depends on the number of partitions of n as follows.First we recall that a partition µ of n is a vector ( µ , . . . , µ r ), whose entriesare positive integers and satisfy µ + · · · µ r = n and µ i ≥ µ i +1 . Given an el-ement σ ∈ Σ n , let σ = c · · · c r be its decomposition into disjoint cycles and,up to renumbering the cycles, suppose that | c i | ≥ | c i +1 | , where | c i | denotes thelength of the i -th cycle. We can associate to σ a partition µ σ of n given by( | c | , . . . , | c r | ). The following facts hold:a) given two element σ , σ ∈ Σ n , they are conjugated if and only if µ σ = µ σ ; 9) the map [ σ ] → µ σ is a one to one correspondence on the level of conjugacyclasses of elements in Σ n . n Given S = (cid:104) σ (cid:105) be a cyclic subgroup of Σ n , we have K S = K σ = g ( t )Id n + f ( t ) | σ |− (cid:88) i =1 R iσ (16)and ρ ( t ) = g ( t ) ρ (0) + f ( t ) | σ |− (cid:88) i =1 R iσ ρ (0) R − iσ . (17)We want to compute an explicit analytic expression for ρ ( t ).Suppose that σ = c · · · c r is the decomposition into disjoint cycles, includingcycles of length one, and with | c i | ≥ | c i +1 | . Let | c i | = µ i , for i = 1 , . . . , r andset µ = 1. Clearly µ + µ + · · · µ r = n . Since we work up to conjugacy, wecan assume that the permutation has the following form σ = (1 2 · · · µ ) ( µ + 1 µ + 2 · · · µ + µ ) · · · r − (cid:88) j =1 µ i r − (cid:88) j =1 µ i · · · n . So the i -th cycle is c i = i − (cid:88) j =0 µ j i − (cid:88) j =0 µ j · · · i (cid:88) j =1 µ j . Note that | σ | = LCM { µ , . . . , µ r } , where LCM stands for the least commonmultiple.A straightforward computation shows that if ρ (0) = diag( λ , . . . , λ n ) and weset B i = (cid:32) µ i (cid:88) h ∈ c i λ h (cid:33) Id µ i (18)10or i = 1 , . . . , r , then ρ σ ( t ) = ρ (0) e − t + (1 − e − t ) B, (19)where B is the block diagonal matrix B ⊕ B ⊕ · · · ⊕ B r .We observe that, by equations (18) and (19), the eigenvalues of the matrix ρ σ ( t )are linear combinations of the eigenvalues of ρ (0) with non negative coefficientsand at least one non-zero coefficient.We can then formulate the following: Theorem 3.
The action associated to each cyclic subgroup < σ > of Σ n satisfiestwo properties: 1) it is completely positive and 2) it satisfies the time condition(7).Proof. We consider ρ ( t ) = g ( t ) ρ (0) + f ( t ) | σ |− (cid:88) i =1 R iσ ρ (0) R − iσ . (20)Since a map A → BAB ∗ is completely positive [9], and the sum of completelypositive operators is completely positive, we can deduce that the action associ-ated to each cyclic subgroup < σ > is completely positive.Now we prove the time condition, F ( s,t ) ◦ F ( t, ( ρ (0)) = F ( s, ( ρ (0)). Noticingthat F ( s, ( ρ (0)) = ρ σ ( s ) = g ( s ) ρ (0) + f ( s ) | σ |− (cid:88) i =1 R iσ ρ (0) R − iσ ,
11e have F ( s,t ) ◦ F ( t, ( ρ (0)) = g ( s − t ) g ( t ) ρ (0) + g ( s − t ) f ( t ) (cid:80) | σ |− i =1 R iσ ρ (0) R − iσ ++ f ( s − t ) g ( t ) (cid:80) | σ |− i =1 R iσ ρ (0) R − iσ + f ( s − t ) f ( t ) (cid:80) | σ |− i,j =1 R i + jσ ρ (0) R − ( i + j ) σ == { g ( s − t ) g ( t ) + ( | σ | − f ( s − t ) f ( t ) } ρ (0)++ (cid:80) | σ |− i =1 { g ( s − t ) f ( t ) + f ( s − t ) g ( t ) + ( | σ | − f ( s − t ) f ( t ) } R iσ ρ (0) R − iσ == | σ | { [1 + ( | σ | − e − s + t ][1 + ( | σ | − e − t ] + ( | σ | − − e − s + t ][1 − e − t ] } ρ (0)++ | σ | (cid:80) | σ |− i =1 { [1 + ( | σ | − e − s + t ][1 − e − t ] + [1 − e − s + t ][1 + ( | σ | − e − t ]++( | σ | − − e − s + t ][1 − e − t ] } R iσ ρ (0) R − iσ == | σ | { | σ | − e − s + t + ( | σ | − e − t + ( | σ | − e − s + | σ | − − ( | σ | − e − s + t + − ( | σ | − e − t + ( | σ | − e − s } ρ (0) + | σ | (cid:80) | σ |− i =1 { − e − t + ( | σ | − e − s + t − ( | σ | − e − s ++1 − e − s + t + ( | σ | − e − t − ( | σ | − e − s + | σ | − − ( | σ | − e − s + t − ( | σ | − e − t ++( | σ | − e − s } R iσ ρ (0) R − iσ == | σ | [1 + ( | σ | − e − s ] ρ (0) + | σ | (1 − e − s ) (cid:80) | σ |− i =1 R iσ ρ (0) R − iσ = ρ σ ( s ) . Using the explicit description of Formula (19), we can easily deduce a de-scription of the associated orbit. First of all, notice thatlim t → + ∞ ρ σ ( t ) = B. (21)Moreover we have ρ (0) − ρ σ ( t ) = (1 − e − t )( ρ (0) − B ) , so it is easy to checkthat ρ (0) − ρ ( t ) is a diagonal matrix whose diagonal entries satisfy the systemof equations x + x + · · · + x µ = 0 x µ +1 + x µ +2 + · · · + x µ + µ = 0... x n − µ r + x n − µ r +1 + · · · + x n = 0 . (22)Notice that if we start with a matrix ρ (0) having n different eigenvalues, thatis a generic initial condition, the limit point of the orbit of K σ with σ = c · · · c r ,lies in a closed subspace containing the matrices having at most r distinct eigen-values ν , . . . , ν r . 12f we start with a matrix ρ (0) having at least one eigenvalue with multiplicitygreater then one, all the previous results hold. Nevertheless, we have less free-dom in movements: indeed, there exists non-trivial elements of Σ n acting triv-ially on the submanifold containing it. More precisely, the elements of Σ n thatpermutes the eigenvalues that are equal act trivially on ρ (0). In order to havea non-trivial action, the different eigenvalues must be shuffled by the permuta-tion. Hence, if, given a partition λ = ( λ , . . . , λ r ) of n , we denote with M λ thesubmanifold containing matrices having r -different eigenvalues with multiplici-ties λ , . . . , λ r , the trivial action is carried by the stabilizer of M λ in Σ n . Thissubgroup can be characterized as that containing σ such that µ σ = ( µ , . . . , µ k )is a subpartition of λ , that is there exist indices 1 ≤ j ≤ j ≤ . . . ≤ j r ≤ k suchthat λ i = µ j i + µ j i +1 + · · · + µ j i +1 − + µ j i +1 , for i = 1 , . . . , r . Let’s try to have a more geometric picture. Given n points P , . . . , P n ∈ C n ingeneral position, we denote the ( n − P , . . . , P n as verticeswith ∆ n − = ∆( P , . . . , P n ). To each matrix ρ (0) = diag( λ , λ , . . . , λ n ) wecan associate a point λ P + λ P + · · · + λ n P n in ∆( P , . . . , P n ). Each element σ ∈ Σ n clearly acts on the vertices of ∆ n − and, by linearity, on the points of thesimplex. Given a cycle c = ( i i · · · i µ ), we denote by L ( c ) ⊂ C n the subspacespanned by the n − P i − P i j , with j = 2 , . . . , µ . Moreover, with thenotation Bar( c ) we indicate the barycenter of the ( µ − P i , . . . , P i µ . Notice that c fixes Bar( c ).Since K σ satisfies the range condition, the orbit ρ σ ( t ) gives a path inside∆ n − . In this setting, (22) tells us that the orbit is contained in the affine sub-space passing trough the point associated to ρ (0) and parallel to the subspace L ( c ) ⊕ L ( c ) ⊕ · · · ⊕ L ( c r ), with σ = c · · · c r . Moreover, the limit of the orbitis the intersection point between this affine subspace and the affine closure of13he points Bar( c ) , . . . , Bar( c r ).Figure 1: The simplex of diagonal density matrices for (a) n = 1, i.e. a qubit,and (b) n = 3, i.e. a qutrit. The arrows show the direction of the time evolutionof the operators defined in the text.As a first example, we can take the case of a qubit, i.e. a Hilbert spaceof dimension n = 2, so that a generic diagonal density matrix is of the form ρ = diag( λ , λ = 1 − λ ). By setting X = λ − λ ∈ [ − , as the segment (convex cone) generated by the two points P , P with coordinate X = +1 , − σ : ( λ , λ ) (cid:55)→ ( λ , λ ). A simple calculation shows that it generates the timeevolution: ρ σ ( t ) = diag( e − t λ + (1 − e − t )( λ + λ ) / , e − t λ + (1 − e − t )( λ + λ ) / e − t λ + (1 − e − t ) / , e − t λ + (1 − e − t ) /
2) (23)which tends to the limit point ρ ∞ = diag(( λ + λ ) / , ( λ + λ ) / n = 3, i.e. the case of a qutrit, whose diagonaldensity matrices are of the form: ρ = diag( λ , λ , λ = 1 − λ − λ ). Setting X = ( λ − λ ) / X = ( λ + λ ) / − /
3, we can represent the simplex ∆ inthe X − X -plane as the (equilateral) triangle with vertices: P = (1 , √ , P =( − , √ , P = (0 , − / √ S : ( λ , λ , λ ) (cid:55)→ ( λ , λ , λ ), which correspondsto a cycle of length 1 and one of length 2. Then, the density matrix evolves intime through a UDM F , as follows: ρ F ( t ) = λ e − t λ + (1 − e − t )( λ + λ ) / e − t λ + (1 − e − t )( λ + λ ) / (24)which tends to the limit point ρ ∞ = diag( λ , ( λ + λ ) / , ( λ + λ ) / P P of the triangle.Similar orbits, but now parallel to the other sides P P and P P are obtainedby considering the cyclic subgroups: S : ( λ , λ , λ ) (cid:55)→ ( λ , λ , λ ) and S :( λ , λ , λ ) (cid:55)→ ( λ , λ , λ ) respectively.We can also consider the maximal cyclic subgroup S : ( λ , λ , λ ) (cid:55)→ ( λ , λ , λ ),which yields: ρ F ( t ) = e − t λ + (1 − e − t ) / e − t λ + (1 − e − t ) / e − t λ + (1 − e − t ) / (25)whose limit point is the barycenter of the triangle, i.e. the maximally mixedmatrix ρ ∞ = diag(1 / , / , / eferences [1] E. Ercolessi, G. Marmo, and G. Morandi, From the equations of motionto the canonical commutation relations , La Rivista del Nuovo Cimento 33(2010) 401.[2] J. Preskill,
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Completely Positive Linear Maps on Complex Matrices , LinearAlgebra and its Applications 10 (1975).16lessia CATTABRIGADepartment of Mathematics, University of BolognaPiazza di Porta San Donato 5, 40126 Bologna, ITALYe-mail: [email protected]
Elisa ERCOLESSIDepartment of Physics and Astronomy, University of Bolognaand INFN, Sezione di Bologna, via Irnerio 46, 40126 Bologna, ITALYe-mail: [email protected]
Riccardo GOZZIInstituto Superior Tecnico, Universidade de LisboaAv. Rovisco Pais 1, 1049-001Instituto de Telecomunica¸coe, Lisboa, PORTUGALe-mail: [email protected]
Erika MEUCCISchool of Advanced International Studies, Johns Hopkins Universityvia Beniamino Andreatta 3, 40126 Bologna, ITALYe-mail: [email protected]@jhu.edu