D-divisible quantum evolution families

We propose and explore a notion of decomposably divisible (D-divisible) differentiable quantum evolution families on matrix algebras. This is achieved by replacing the complete positivity requirement, imposed on the propagator, by more general condition of decomposability. It is shown that such D-divisible dynamical maps satisfy a generalized version of Master equation and are totally characterized by their time-local generators. Necessary and sufficient conditions for D-divisibility are found. Additionally, decomposable trace preserving semigroups are examined.


Introduction
The aim of this article is to define, construct and characterize a generalization of CP-divisible (i.e. Markovian) evolution families, or quantum dynamical maps, on matrix algebras onto a certain subclass of much broader, however still mathematically manageable case of decomposable positive maps. We restrict our attention to the case of decomposably divisible families, i.e. such maps Λ t on matrix algebra M d (C), which are divisible and which propagators are trace preserving and decomposable on M d (C). Decomposability is a relatively simple, yet non-trivial generalization of complete positivity, which in turn has been a well-characterized and motivated concept in quantum theory since 1970's (see [1,2,3] and references within), traditionally used to model time evolution of quantum systems. In particular, CP-divisible families [4,5] has been granted a special attention, since CP-divisibility is commonly considered equivalent to Markovianity. We abandon this approach here in favor of D-divisibility, effectively obtaining a new subclass of non-Markovian evolution families (or weakly non-Markovian, using terminology of [6]). We hope that such decomposable dynamical maps might be useful in future for description of physical systems outside a Markovian regime, for example influenced by more sophisticated quantum effects or to mirror the existence of higher-order correlations in the system.
The article is structured as follows. In section 2 we provide some mathematical preliminaries, including notion of decomposable maps over algebra of complex matrices, as well as some basic description of dynamics of open quantum systems. The main part of the article is the section 3, devoted to D-divisible quantum evolution families, where we formulate a necessary and sufficient conditions for D-divisibility expressed in terms of associated time-dependent generators. Construction of such is presented in Theorem 2, which is our main result. In section 4 we remark on a semigroup case and present some results related to their asymptotic behavior (Theorems 4 and 5). Finally, a simple example in M 2 (C) is provided in section 5.

Preliminaries
First, we provide some basic preliminaries including notions of decomposability of positive maps and divisibility (and Markovianity) of quantum dynamics. We will be working a lot with Hilbert-Schmidt bases spanning space M d (C), i.e. bases orthonormal with respect to the Hilbert-Schmidt inner product (also called Frobenius inner product ) on M d (C), given via Amongst all such bases, one consisting of strictly Hermitian matrices will be granted a special attention. Namely, let {F i } d 2 i=1 be a Hilbert-Schmidt basis subject to constraints Such basis may be seen as a generalization of both Pauli and Gell-Mann matrices and may be constructed in similar way (see appendix A.1 for details and for some more properties). By construction, matrices F i can be either non-diagonal and symmetric, antisymmetric or diagonal (where all F i s.t. i < d 2 are traceless). For any d, there is exactly d(d − 1)/2 of both symmetric and antisymmetric matrices and d diagonal ones. We reserve symbol F i for such a basis exclusively throughout the whole article and introduce an accompanying enumeration, such that F i will be: The following composition rule will be of importance: for every F i , F j we have where coefficients ξ ijk may be computed as (2.4) and are expressible in terms of so-called structure constants, which characterize M d (C) as a Lie algebra. It is then a simple exercise to check that the following identities hold: ξ ijk = ξ kij = ξ jki , ξ ijk = ξ jik . Structure of CP maps is characterized by means of the famous Stinespring dilation theorem stating that for every φ ∈ CP(A , B(H)) for A a unital C*-algebra and H a Hilbert space, exists some auxiliary Hilbert space K such that φ admits a (nonunique) representation as a composition for some bounded operator V : H → K and *-homomorphism π : A → B (K). If both A and H in question are finite-dimensional, i.e. φ acts between algebras of matrices, φ : M n (C) → M m (C), one defines the so-called Choi matrix of φ, where E ij are matrix units (i.e. they contain 1 in place (i, j) and 0s everywhere else) spanning M n (C).
Then, Stinespring dilation theorem is equivalent to the famous Choi's theorem [7], which stays that φ is CP iff (if and only if) it is n-positive, which is then true iff C φ ∈ M mn (C) + . Furthermore, as a corollary, it can be shown that for every φ ∈ CP(M n (C), M m (C)) there exists a set of matrices which is the Kraus decomposition of φ (matrices X i are called Kraus operators) associated with φ. The notion of complete positivity proved itself to be very robust concept, both in mathematics and physics. Unfortunately, although the complete characterization of CP maps is known due to results by Stinespring, Choi and Kraus, we lack such in case of merely positive maps and finding it has been a long-standing goal in mathematics for many years.
Throughout this paper, we will be focusing on a special sub-class of positive maps, the so-called decomposable maps, which may be seen as a conceptually simple, however still nontrivial generalization of CP maps. Moreover, from now on we assume all maps under consideration to be exclusively endomorphisms over matrix slgebra M d (C) and we tweak our notation accordingly by writing simply B(M d (C)), P(M d (C)) and CP(M d (C)) for appropriate maps on this algebra.
with respect to some chosen basis in C d . It is easy to see that θ is a positive map, however it is not CP (in fact, it fails to be even 2-positive). Transposition allows to define yet another class of positive maps, the so-called completely copositive maps. One says that a map φ ∈ P(M d (C)) is completely copositive (coCP), if its composition with θ is CP, or that there exists someφ ∈ CP(M d (C)) such that The marriage of notions of both complete positivity and copositivity determines a class of decomposable maps, which will remain at our focus throughout this article: . We say ϕ is decomposable, ϕ ∈ D(M d (C)), if it can be expressed as a convex combination of CP and coCP map, i.e. if there exist φ, ψ ∈ CP(M d (C)) such that Decomposable maps may be also characterized in terms of a following necessary and sufficient condition. Let ϕ ∈ P(M d (C)) and let C ϕ ∈ M d 2 (C) be its corresponding Choi matrix. By identification M d 2 (C) ≃ M d (C) ⊗ M d (C) we introduce a linear map of partial transposition (with respect to second factor) Γ on M d 2 (C), defined by its action on simple tensors as (2.12) Define also two convex cones Then, a following characterization of decomposable maps applies [8,9]: In practice, verifying if a given linear map is decomposable by finding exact decomposition into a combination (2.11) of its CP and coCP part may be a hopeless task, even in low dimensional algebras. Instead, condition stated in theorem 1 can be checked quite sufficiently by means of a semidefinite programming (SDP) routines, as is also the case in this article.
Every decomposable map ϕ is in addition Hermiticity preserving, i.e. it satisfies for some initial ρ 0 , will be called the quantum evolution family, or quantum dynamical map. In order to maintain the probabilistic interpretation of ρ t as density matrix at every t 0, it is required for Λ t to be trace preserving (i.e. tr Λ t (ρ) = tr ρ) and positive. By physical reasoning, one often demands not merely a positivity, but rather complete positivity of Λ t (one can find appropriate explanation e.g. in [1,2,3] and numerous other sources). This restriction, however, will be abandoned in this paper in favor of decomposability.
Definition 2. We say that quantum evolution family (2.18) If in addition V t,s is a positive or completely positive map for every s t, then {Λ t : t ∈ R + } is called P-divisible or CP-divisible in this interval, respectively.
Such two-parameter family of maps {V t,s : s t} is then called the propagator of evolution family (as V t,s propagates Λ s forward in time). If Λ t is invertible then it is immediate that V t,s = Λ t • Λ −1 s . CP-divisibility is commonly identified with Markovianity.
It is most frequently assumed, that the dynamical map in question satisfies the time-local Master Equation in two equivalent forms for some map L t ∈ B(M d (C)), called a generator. All dynamical maps obeying (2.19) are divisible. By celebrated results of Lindblad, Gorini, Kossakowski and Sudarshan [4,5], a necessary and sufficient condition for an invertible map Λ t subject to Master Equation (2.19) to be CP-divisible is that L t must be of a form where H t is Hermitian and [a jk (t)] ∈ M d 2 −1 (C) + for all t ∈ R + ({a, b} = ab + ba is the anticommutator). Equation (2.20) defines so-called standard form (also Lindblad form or LGKS form) of L t . On physics grounds, H t is identified with system's Hamiltonian (which includes Lamb-shift corrections; here one puts = 1 for brevity) and matrix [a jk (t)], being commonly called the Kossakowski matrix, expresses the "non-unitary" part of the evolution due to interactions between system and the environment. If generator L t is time-independent, i.e. L t = L, then a solution of Master Equation (2.19) is a one-parameter contraction semigroup {e tL : t ∈ R + } of trace preserving CP maps, known as the Quantum Dynamical Semigroup.
3. D-divisible quantum evolution families 3.1. Notion of D-divisibility. In this section we propose and elaborate on the notion of D-divisibility. Let {Λ t : t ∈ R + } again stand for a family of positive and trace preserving maps on M d (C). Then, we define D-divisibility of this family in a manner analogous to CP-divisibility by demanding that the propagator is decomposable: and its associated propagator V t,s is trace preserving and decomposable for all s, t ∈ [t 1 , t 2 ], s t, i.e.
for some maps X t,s , Y t,s ∈ CP(M d (C)) continuously depending on (t, s).
We stress here that although map V t,s is required to be trace preserving as a whole, neither of maps X t,s , Y t,s is a priori expected to be so: ) and is trace preserving, (5) X t,s + Y t,s is trace preserving.
Proof. Property 1 follows immediately from divisibility condition (2.18) after taking s = t. As a consequence V t,t is a decomposable map with its coCP part being zero, so properties 2 and 3 follow. For property 4, see that (2.18) also yields Λ t = V t,0 •Λ 0 = V t,0 and so Λ t is decomposable and trace preserving. Remaining property 5 then follows from linearity of trace and trace preservation of transposition map after simple algebra.

Generators of decomposable dynamics.
In this section we present our main result, i.e. a necessary and sufficient condition for a quantum evolution family to be D-divisible expressed in terms of properties of the associated generator. Before that we briefly discuss some additional notions. Our construction of generator (given in a proof of theorem 2) will be heavily depending on so-called operator sum representation of linear maps on M d (C), including the transposition map. Namely, if T is any linear endomorphism on algebra M d (C), its action on a ∈ M d (C) may be always represented in a form Similarly, the transposition map θ admits an operator-sum representation of a form for coefficients θ i ∈ {−1, 1} given as Proof of this statement is available in the appendix A.3.1. We will use coefficients θ i given above to define a particular 4-index geometric tensor, which will be of crucial importance later on. Recall that basis matrices F i obey composition rule (2.3) for One can easily show (see lemma 1 in section A.4 of the Appendix) thatΩ admits a somewhat more compact and robust representation as which will become useful. Now we are ready to formulate our main result: for coefficients Proof. The proof will follow general guidelines of [3, Theorem 4.2.1]. We are interested in computing dρt dt , where the derivative is to be calculated "from above", i.e.
where θ i are given in (3.4). Therefore, the expression for L t , using composition rule (2.3) and properties (2.5), is is Hermitian for every (t, s), however is not positive semidefinite in general. Next, we subtract from both summations terms with µ, ν = d 2 and obtain, by Now, we define new time-dependent coefficients g µν (t) by setting where existence of all limits is assured by differentiability of Λ t , so our expression for L t (ρ) becomes We demand V t,s to obey the trace preservation condition, which means that L t must nullify the trace, tr L t (ρ) = 0 regardless of ρ. This applied to our expression yields, after some algebra involving cyclic property of trace, By inserting back we therefore end up with a form which despite its visual resemblance is not the standard form, since matrix [g µν (t)] µν is not positive semi-definite in general. However, formula (3.19a) allows to split coefficients g µν (t) into a sum of expressions defined solely via either the CP or the coCP part of the propagator, namely In similar fashion, we have and K t has an identical structure, with η µd 2 (t) in place of γ µd 2 (t). It is then evident that expression (3.25) may be rewritten as a sum of two maps, and M t is of the same structure, with H t replacing K t and γ µν (t) in place of η µν (t). By direct check, matrices H t and K t are Hermitian and complete positivity of map X t,s yields both matrices [x µν (t, s)] and [γ µν (t)] to be positive semidefinite, i.e. map M t is in standard form. It remains to show that coefficients η µν (t) are as claimed.
We have As we show in lemma 2 in the Appendix, the above limiting procedure under the summation defines a positive semidefinite matrix for all t ∈ [t 1 , t 2 ], i.e. we have and η µν (t) admits a form (3.10). This proves sufficiency. To show necessity, we start with re-expressing N t , basing on expression (3.29), as which is achieved by: (1) expressing η µν (t) via (3.10), (2) expanding the geometric tensor Ω jk µν according to (3.5), (3) applying the operator-sum representation (3.3) of transposition map θ, (4) expressing [ω jk (t)] ∈ M d 2 (C) + as ω jk (t) = α c jα (t)c kα (t) for some new matrix [c ij (t)] and finally (5) substituting for A k = l θ l ξ lkd 2 F l (see the derivation in section A.5 in the Appendix). The matrix D t − D * t is clearly skew-Hermitian, so it is of a form D t − D * t = −iE t for some Hermitian E t . Now, recall M t was in standard form, so matrix [γ µν (t)] is positive semidefinite, i.e. it may be cast into a form which is sometimes referred to as the second standard form of a generator. All of this allows to rewrite expression for L t as Now, select an increasing sequence (τ j ) n j=0 ⊂ [s, t] of instants such that τ 0 = s and τ n = t. Then, we can express the propagator V t,s in a form i.e. we approximate the exact propagator by a composition of semigroups; this is known as the time-splitting formula [3]. Denote τ j+1 − τ j = ∆ j . Applying decomposition (3.8) we have, by Lie-Trotter product formula, (3.39) We now have to specify properties of three maps exp (1) Case k = 0. Let us define for fixed t, j and a mapping ξ → f ξ ∈ CP(M d (C)) by setting Then, by direct calculation one can easily check that we have i.e. the identity holds for all ξ ∈ R, i.e. {f ξ : ξ ∈ R} is a group of completely positive maps. In particular, exp In the result, the map appearing under the limit in expression (3.39) is decomposable for every n (as a composition); this shows e ∆jLτ j is also decomposable, since it is a limit of a sequence of decomposable maps in closed cone D(M d (C)). This very same fact then shows that V t,s given in (3.38) is also decomposable. Finally, one checks by direct calculation that L t = M t + N t nullifies the trace, i.e. tr L t (ρ) = 0. This yields that a family {e τ Lt : τ ∈ R + } must be trace preserving for every choice of t ∈ R + ; in consequence, every map e ∆jLτ j in decomposition (3.38) is also trace preserving and so is the whole propagator V t,s . This concludes the proof.

Decomposable semigroups and asymptotic complete positivity
Here we remark on the semigroup case. It is immediate that by suppressing all time dependence in decomposition (3.8) we obtain a general characterization of Ddivisible trace preserving semigroups over M d (C), for any d. Clearly, a semigroup is D-divisible if and only if it is decomposable, so we have a following corollary of theorem 2: Theorem 3. A semigroup {e tL : t ∈ R + } is trace preserving and decomposable iff L is of a form stated in Theorem 2, with all matrices time independent.
In order to confirm validity of our results, we verified if semigroups given by L in proposed form were indeed decomposable. We checked for condition stated in theorem 1 by minimizing the functional ρ → tr C e tL ρ over a convex set V d ∩ V Γ d . This was achieved via an application of SDP optimization in Wolfram Mathematica software for a very wide range of different forms of L in different dimensions and values of t.
In general, decomposability properties of D-divisible dynamical maps -including semigroups -turn out to be quite surprising, as we were able to check numerically. For instance, it may happen that Λ t suddenly becomes completely positive, despite the fact that the propagator V t,s remains truly decomposable, i.e. has a non-zero coCP part. Behavior of Λ t in this manner may be quite complex and ranges from being simply CP to even fluctuating between complete positivity and decomposability. Under particular circumstances, i.e. under specific choice of the generator, an interesting phenomenon of Λ t is observed: namely, it is possible that initially Λ t is decomposable and then it switches to being only CP and remains such as time progresses. This observation justifies a following definition of asymptotic complete positivity of decomposable maps: Definition 5. We will call a family {Λ t : t ∈ R + } asymptotically CP if there exists t 0 > 0 such that Λ t is CP and trace preserving for all t t 0 .
In fact, asymptotic complete positivity is observed even in simplest semigroup case, as an example (see below) demonstrates, and is analyzed by examining the spectrum of Choi matrix C Λt . Since Λ t is Hermiticity preserving, C Λt is Hermitian and therefore it suffices that spec C Λt ⊂ R + for Λ t to be CP, which in turn is guaranteed if the smallest eigenvalue λ min (C ϕ ) is non-negative. Therefore one should be interested at least in finding some well-behaved and computable lower bounds for smallest eigenvalues. One such bound was specified by Wolkowicz and Styan in [11, Theorem 2.1]. Let A ∈ M n (C) be a matrix of real spectrum, spec A = {λ i (A) : 1 i n}, λ i (A) ∈ R. Then, the smallest eigenvalue λ min (A) satisfies inequality for µ A = 1 n tr A and ν 2 A = 1 n tr (A 2 ) − µ 2 A . This allows to formulate a following sufficient condition for complete positivity:

2)
where a 2 = √ tr a * a stands for the Hilbert-Schmidt norm of a ∈ M d (C).
Proof. Clearly ϕ ∈ CP(M d (C)) if λ min (C ϕ ) is non-negative. By a simple algebra involving trace preservation of ϕ one checks that since E ij = E * ji and ϕ is Hermiticity preserving. This allows to check that λ min (C ϕ ) satisfies which comes from (4.1) after putting A = C ϕ , n = d 2 . Finally, demanding the above lower bound to be non-negative yields the claim.
A systematic asymptotic behavior of D-divisible evolution governed by a general time-dependent L t appears to be highly nontrivial and lays way beyond the scope of this article. Therefore, we focus here on a much more tractable case of Λ t being a decomposable semigroup, Λ t = e tL , hoping that this restriction might be lifted in future considerations. A following criterion of asymptotic complete positivity of such semigroups then arises: then the semigroup is asymptotically CP.
Proof. Let g(t) = d i,j=1 e tL (E ij ) 2 2 . If indeed lim t→∞ g(t) < d 2 d 2 −1 then by definition of a limit there exists t 0 0 such that g(t) < d 2 d 2 −1 for all t > t 0 and we have λ min (C e tL ) 0, e tL ∈ CP(M d (C)) by proposition 2, i.e. a semigroup is asymptotically CP. In some cases, the limit appearing in theorem 4 may be computed exactly. For example, if L is diagonalizable, its value turns out to be determined by the biorthogonal system of eigenbasis and associated dual basis of L: Theorem 5. Let L be diagonalizable and let 0 ∈ spec L be of multiplicity 1. Let ε ∈ ker L be an associated eigenmatrix. Then, where β ∈ M d (C) is an element of dual basis of L such that β, ε 2 = 1.
Proof. Let again g(t) = d i,j=1 e tL (E ij ) 2 2 and assume that L is diagonalizable, i.e. that there exists a linearly independent set {e i } spanning C d 2 of (not necessarily orthogonal) normalized eigenvectors ofL ∈ M d 2 (C), the matrixized version of L, as elaborated in section A.2. Then, one can show that there always exists so-called dual basis (or reciprocal basis) {b i }, also spanning C d 2 , which is subject to relation b i , e j = δ ij , or that ({e i }, {b i }) constitutes for a biorthogonal system. Then, every operatorÂ acting on C d 2 may be cast into a form for coefficients a ij = b i ,Âe j . In particular, when basis {e i } is chosen as an eigenbasis ofÂ, we haveÂ where β i = vec −1 b i , ε i = vec −1 e i are eigenmatrices of L and β i , ε j 2 = δ ij . From general theory of positive unital maps, we know that spec e tL lays inside unit circle (being a trace norm contraction), contains 1 (as a result of trace preservation) and is closed with respect to complex conjugation, i.e. e µit , e µit ∈ spec e tL (by Hermiticity preservation property) [12]. This implies that 0 ∈ spec L and spec L \ {0} consists of pairs {µ i , µ i : Re µ i < 0} and possibly some negative reals. Let us then set µ 1 = 0. We have e µ1t = 1 and e µit = e −| Re µi|t e i Im µit , 1 i d 2 − 1, (4.10) where we write −| Re µ i | to emphasize negativity of real parts. Decomposition (4.9) allows to re-write expression for g(t). First, one easily confirms that which comes from properties of inner product and property tr E ij [a ij ] = a ji . Substituting this into formula for g(t) we have e (µ k +µ l )t z kl (4.12) for shorthand notation z kl = β l , β k 2 ε k , ε l 2 . Applying properties of eigenvalues µ i we recast this into Clearly, both sums vanish exponentially as t → ∞, so (4.14) By Schwartz inequality, 1 = β 1 , ε 1 2 β 1 2 ε 1 2 , so indeed lim t→∞ g(t) 1, as claimed.

Example: decomposable semigroup
As an example, we examine a decomposable semigroup on M 2 (C). We give it in a simplest possible way for readability of obtained formulas. We set where σ i is the usual basis of Pauli matrices, i.e. we explicitly neglect the M generator from decomposition (3.8) and the commutator part of (3.9). The geometric tensorΩ may be then computed by applying (3.6); its only non-zero coefficients Ω jk µν read Ω 11 11 = Ω 11 22 = Ω 12 12 = Ω 13   for s ij = w i + w j . Next, performing the vectorization of L (which we omit here for brevity) we obtain its spectrum, as well as corresponding eigenmatrices ε i such that L(ε i ) = µ i ε i , which in this particular case happen to be equivalent to Pauli matrices, In such case, a dual basis is identical, The Choi matrix C e tL is Hermitian as expected and reads     (5.10) and its spectrum is found to be Depending on actual values of w i , the smallest eigenvalue of C e tL may change sign and monotonicity. It is then possible for the semigroup to exhibit a mixed behavior: (1) it may be CP for all t 0, when λ min (C e tL ) is everywhere non-negative; exemplary plot regarding such situation is shown in fig. 1, (2) it may be decomposable (with both CP and coCP parts non-zero) for all t 0, when λ min (C e tL ) < 0 everywhere; see fig. 2, (3) and finally, it can be decomposable in some interval (0, t 0 ] and then become CP for t t 0 , i.e. it may be asymptotically CP, as presented in fig. 3. Matrices {F i } can be then constructed explicitly in a following way [13,14]. Let again E jk denote matrix units, i.e. they contain 1 in position (j, k) and 0s elsewhere. Let us define matrices W d kj , K d k ∈ M d (C) such that such that j, k ∈ {1, ... , d 2 − 1}, j = k, as well as where k ∈ {1, ... , d − 1}. Then, the set {W jk , K k , 1 √ d I d } contains d 2 matrices and is orthonormal (with respect to Hilbert-Schmidt inner product) and complete, being a basis of M d (C). Its elements are then labeled F i for 1 i d 2 . Matrices W jk are either symmetric off-diagonal or antisymmetric and matrices K k are diagonal and of zero trace. By simple counting, there is then exactly 1 2 d(d− 1) of both symmetric off-diagonal and antisymmetric matrices and d diagonal matrices (including F d 2 = 1 √ d I).
One then introduces the so-called structure constants f ijk and g ijk , which respectively define the commutation and anticommutation relations amongst matrices F i , being defined as It is worth noting that structure constants characterize M d (C) as a Lie algebra. These allow us to derive a following composition rule . It is then very common and convenient to utilize these identifications in order to represent matrices as (column) vectors and linear maps on M d (C) as matrices of size d 2 .
Every bijection M d (C) → C d 2 defines so-called vectorization scheme [15,16]. i.e. by putting rows of [m ij ] one behind another, or in a lexicographic order. We remark here, that the convention of vectorization we use in this article is by no means universal. For example, some authors prefer the matrix flattening not in a row-by-row manner, but rather in column-by-column manner, which is sometimes called a reshaping. For details, see [15] and references within. The inverse operation vec for some family of matrices {A i } and real coefficients λ i [17]. If in addition all λ i 0, then T is completely positive. Assume T is an endomorphism over M d (C). Expanding matrices A i , B i in basis {F i } one quickly checks that (A.8) can be equivalently expressed as for some coefficients t ij ∈ C.
which is the claim.
Define also a set Then, the transposition map θ admits an operator-sum representation of a form for coefficients θ i ∈ {−1, 1} given explicitly as which therefore yields Proof. From proposition 3 we know that the transposition map may be put in its operator-sum representation for matrix [θ ij ] ∈ M d 2 (C) calculated from formula (A.11), whereT is chosen as a matricial representation of θ under the vectorization scheme. It is not difficult to show that general structure ofT iŝ where E ij are matrix units.T then consists of d 2 square blocks containing only single 1 at some location and 0s elsewhere and in fact is a permutation matrix (in literature, those are sometimes called SWAP matrices). As an example, below we demonstrate appropriate matrices for d = 2 and 3: Now, by Hermiticity of F i we have since canonical basis {E ij } is yet another (nonhermitian) Hilbert-Schmidt orthonormal basis. Notice that F T j = ±F j depending on symmetry of F j and so In the remaining case d 2 − d + 1 i d 2 the resulting diagonal matrices F i are naturally also symmetric, so we still have θ i = 1, as claimed.
after easy algebra. Ad (2). Analogously, let again [m ij ] = U * θ U for some arbitrarily chosen unitary U = [u ij ]. Then, if one defines G i = j u ji F j then immediately we have ij m ij G i aG * j = i θ i F i aF i = a T and there exists such a basis.
Proof. Recall that, since φ may be represented as a complex square matrix of size d 2 , one can always express e φ as a limit where all maps of a form id + 1 n φ n , n ∈ N, are also CP. Then, the limit also defines a CP map since the cone CP(M d (C)) is closed.
Proof. Let ϕ = θ • φ for φ = 0 completely positive (case φ = 0 gives e θ•φ = id which is trivially decomposable). Then, it suffices to express e ϕ by putting θ •φ in place of φ in formula (A.28) and to notice that all maps under the limit are decomposable, for all n ∈ N, as is the limit itself by the fact, that D(M d (C)) is closed.
Lemma 1. Geometric tensorΩ may be re-expressed in a form . This fact implies that Ω jk µν may be, after using cyclicity of trace, put in a form where id M d (C) is the identity map on M d (C) and P as. is the orthogonal projection onto M d (C) as. given as Let {e i } be a canonical basis in C d . By dimension count, it is easy to see that space M d (C) may be identified with a Hilbert space tensor product C d ⊗ C d , with a mapping ζ : C d ⊗ C d → M d (C) defined by its action on basis elements as ζ(e i ⊗ e j ) = E ij = |e i e j | (A.33) and then extended by linearity, being a natural bijection. Under action of ζ, every vector x = ij x ij e i ⊗ e j ∈ C d ⊗ C d can be isomorphically represented as a matrix [x ij ] ∈ M d (C) and vice versa. This implies, that M d (C) as. is identified with C d ∧C d , the antisymmetric subspace of C d ⊗ C d . In result, operator P as. = ζ −1 • P as. • ζ is the corresponding projection onto C d ∧ C d . We know however, that such projection may be expressed in a form with V being the swap operator on C d ⊗ C d defined via V (x ⊗ y) = y ⊗ x, x, y ∈ C d . Since matrix [y jk (t, s)] was uniquely identified with a CP map Y t,s appearing in the propagator, it is positive semidefinite for all t s, so clearly f v (t, s) 0 for every v ∈ C d 2 and t s. Moreover, from proposition 1 we have Y t,t = 0 and so f v (t, t) = 0. Let then t ∈ [t 1 , t 2 ] be arbitrary and assume indirectly, that f v (· , t) is decreasing in some interval [t 0 , ξ 0 ] for some t 0 t. Then there exists ξ t 0 such that f (ξ, t 0 ) < f (t 0 , t 0 ) = 0, which is a contradiction. This yields that ξ → f v (ξ, t), where ξ t, must be non-decreasing for every t 0. We will use this reasoning in a following computation. The formula for matrix [ω jk (t)] can be rewritten as ω jk (t) = lim ǫց0 1 ǫ y jk (t + ǫ, t) (A.40) = lim ǫց0 y jk (t + ǫ, t) − y jk (t, t) ǫ = ∂y jk (ξ, t) ∂ξ t , since y jk (t, t) = 0, i.e. as a derivative wrt. first variable of a matrix [y jk (ξ, t)], computed at ξ = t. This however yields, for every v ∈ C d 2 , F ν so they can be transformed by applying nearly exactly the same steps. After some easy algebra, we obtain d 2 j,k=1 for the second term, as well as after some effort.