Ann. Henri Poincare´ 23 (2022), 3311–3329 c© 2022 The Author(s) 1424-0637/22/093311-19 published online March 6, 2022 https://doi.org/10.1007/s00023-022-01166-0 Annales Henri Poincare´ The PPT2 Conjecture Holds for All Choi-Type Maps Satvik Singh and Ion Nechita Abstract. We prove that the PPT2 conjecture holds for linear maps be- tween matrix algebras which are covariant under the action of the di- agonal unitary group. Many salient examples, like the Choi-type maps, depolarizing maps, dephasing maps, amplitude damping maps, and mix- tures thereof, lie in this class. Our proof relies on a generalization of the matrix-theoretic notion of factor width for pairwise completely positive matrices, and a complete characterization in the case of factor width two. Contents 1. Introduction 3311 2. Review of Diagonal Unitary Covariant Maps 3313 3. Factor Widths 3318 4. The PPT2 Conjecture for Diagonal Unitary Invariant Maps 3323 5. Perspectives and Open Questions 3326 Acknowledgements 3327 References 3327 1. Introduction Quantum channels model the evolution of quantum systems. Mathematically, they correspond to completely positive and trace preserving linear maps be- tween matrix algebras. One important scenario in the rapidly developing field of quantum technologies is the distribution of quantum entanglement: which channels can be used to transmit a quantum particle which is entangled to another system in such a way that some entanglement in the total bipartite system is preserved? Quantum channels Φ which are useless for this task are 3312 S. Singh, I. Nechita Ann. Henri Poincare´ dubbed entanglement breaking [18]: the local application of Φ on any subsys- tem of a bipartite quantum state results in a separable (non-entangled) state. For qubit channels, this property is equivalent to a simpler PPT property, which amounts to saying that both Φ and Φ◦ are completely positive, where  denotes the transposition map. The preceding equivalence ceases to hold for higher-dimensional qudit channels. However, the PPT-squared conjecture posits that the composition of two arbitrary PPT linear maps must be en- tanglement breaking [8,10]. In particular, for a PPT channel, its composition with itself must be entanglement breaking. Conjecture 1.1. The composition of two arbitrary PPT linear maps is entan- glement breaking. This conjecture is relevant for quantum information theory because it imposes constraints on the type of resources that can be distributed using quantum repeaters [3,6]. Since it was first proposed in 2012 by M. Christandl, the conjecture has garnered a lot of attention. It has been shown that the conjecture holds in the asymptotic limit, i.e., the distance between several iterates of a unital (or trace preserving) PPT map and the set of entanglement breaking maps tends to zero in the asymptotic limit [21]. In [26], the authors proved that any unital PPT map becomes entanglement breaking after finitely many iterations of composition with itself; for other algebraic approaches, see [14,17,22]. As noted above, the conjecture trivially holds for qubit maps. For the next dimension d = 3, the conjecture has been proven independently in [10,12]. For higher dimensions however, the validity of the conjecture still remains ambiguous [13,20]. In infinite dimensional systems, the set of Gaussian maps has been shown to satisfy the conjecture [10]. The main result of the current paper is the proof of this conjecture in the case when the two PPT maps are covariant with respect to the action of the diagonal unitary group. More precisely, we consider the class of linear maps which are (conjugate-)invariant under the action of diagonal unitaries (see Definition 2.2): ∀X ∈ Md(C) and U ∈ DUd, we have either Φ(UXU∗) = U∗Φ(X)U or Ψ(UXU∗) = UΨ(X)U∗. These maps, dubbed respectively, Diagonal Unitary Covariant (DUC) and Conjugate Diagonal Unitary Covariant (CDUC), were studied at length in [28]. A variety of physically relevant classes of quantum channels are of this kind, like the depolarizing and transpose depolarizing channels, amplitude damp- ing channels, Schur multipliers, etc. Several important properties of (C)DUC maps, such as complete positivity and copositivity, entanglement breaking property, and the like, were analyzed in detail. In this work, we focus on the PPT2 conjecture for these maps, proving a stronger version of the conjecture. The main technical tool in our proof is the characterization of a subclass of en- tanglement breaking covariant maps, which is related to the matrix-theoretic concept of factor width [4] for the cone of pairwise completely positive matrices [19], see Theorems 3.9 and 3.10. The following is an informal statement of our main result, Theorem 4.5: Vol. 23 (2022) The PPT2 Conjecture Holds for All Choi-Type Maps 3313 Theorem 1.2. The composition of two arbitrary (conjugate) diagonal unitary covariant PPT maps corresponds to a pairwise completely positive matrix pair with factor width two; in particular, it is entanglement breaking. The paper is organized as follows. In Sect. 2, we provide several equivalent statements of the PPT2 conjecture and also review some useful facts about diagonal unitary covariant maps; all of them are proven in [28]. In Sect. 3, we introduce the notion of factor width for pairwise and triplewise completely positive matrices. These tools are used in Sect. 4 to prove the PPT2 conjecture for (C)DUC maps. Finally, in Sect. 5 we discuss some open problems and future directions for research. 2. Review of Diagonal Unitary Covariant Maps In this section, we will briefly review several key aspects from the theory of diagonal unitary and orthogonal covariant maps between matrix algebras. For a more comprehensive discussion and proofs of the results stated here, the readers are referred to our previous work [28, Sections 6–9]. Let us first quickly set up the basic notation. We use Dirac’s bra-ket notation for vectors v ∈ Cd and their duals v∗ ∈ (Cd)∗ as kets |v〉 and bras 〈v|, respectively. For |v〉, |w〉 ∈ Cd, the rank one matrix vw∗ is then represented as an outer-product |v〉〈w|. {|i〉}di=1 denotes the standard basis of Cd. We collect all d × d complex matrices into the set Md(C). Within Md(C), the cones of entrywise non-negative and (hermitian) positive semi-definite matrices are denoted by EWPd and PSDd, respectively. We denote the adjoint (conjugate transpose) of A ∈ Md(C) by A∗. Sets of pairs and triples of matrices in Md(C) with equal diagonals are represented as follows Md(C)×2 Cd :={(A, B) ∈ Md(C) × Md(C) ∣ ∣ diag(A) = diag(B)} (1) Md(C)×3 Cd :={(A, B, C) ∈ Md(C) × Md(C) × Md(C) ∣ ∣ diag(A) = diag(B) = diag(C)} (2) The set of all linear maps Φ : Md(C) → Md(C) is denoted by Td(C). A map Φ ∈ Td(C) is called positive if Φ(X) ∈ PSDd for all X ∈ PSDd. If id⊗Φ : Mn(C) ⊗ Md(C) → Mn(C) ⊗ Md(C) is positive for all n ∈ N, where id ∈ Tn(C) is the identity map, then Φ ∈ Td(C) is called completely positive (CP). Every CP map in Td(C) admits a (non-unique) Kraus representation Φ(X) = ∑k j=1 AjXA ∗ j , where {Aj}kj=1 ⊆ Md(C) and k ≤ d2. If Φ ◦  is completely positive, where  acts on Md(C) as the matrix transposition (with respect to the standard basis in Cd), then Φ ∈ Td(C) is called completely copositive (coCP). We say that Φ ∈ Td(C) is positive partial transpose (PPT) if it is both CP and coCP. If, for all positive semi-definite X ∈ Md(C) ⊗ Md(C), [id⊗Φ](X) lies in the convex hull of product matrices A⊗B with A,B ∈ PSDd, i.e., [id⊗Φ](X) is separable, we say that Φ ∈ Td(C) is entanglement-breaking. By quantum channels, we understand completely positive maps in Td(C) which also preserve trace, i.e., TrΦ(X) = TrX, ∀X ∈ Md(C). Now, in order to reformulate the PPT2 conjecture in the language of bipartite matrices, we need to introduce the notion of locality for CP maps 3314 S. Singh, I. Nechita Ann. Henri Poincare´ between tensor products of matrix algebras. For our purposes, it suffices to look at the tripartite setting. We say that a CP linear map Φ : Md(C)⊗ Md(C)⊗ Md(C) → Md(C) ⊗ Md(C) ⊗ Md(C) is separable if it can be expressed as a finite sum Φ = ∑ i∈I Φ 1 i ⊗Φ2i ⊗Φ3i , where Φji ∈ Td(C) are CP for all i ∈ I and j ∈ {1, 2, 3}. Using the Kraus representation of CP maps in Td(C), it is easy to see that any such separable operation itself admits a Kraus representation of the form Φ(X) = ∑ j∈J(Aj ⊗ Bj ⊗ Cj)X(Aj ⊗ Bj ⊗ Cj)∗, where Aj ,Bj ,Cj ∈ Md(C) for all j ∈ J . Also recall that a bipartite matrix X ∈ Md(C) ⊗ Md(C) is said to be positive under partial transpose (PPT) if both X and XΓ = [id⊗](X) are positive semi-definite. Equipped with the appropriate terminology, we are now prepared to state several equivalent formulations of the PPT2 conjecture. Proposition 2.1. The following statements are equivalent: (1) ∀ PPT linear maps Φ1,Φ2 ∈ Td(C): Φ1 ◦ Φ2 is entanglement breaking. (2) ∀ PPT bipartite matrices ρ, σ ∈ Md(C) ⊗ Md(C): Tr2,3{(ρ ⊗ σ)(I ⊗ |e〉〈e| ⊗ I} is separable. (3) ∀ PPT bipartite matrices ρ, σ ∈ Md(C) ⊗ Md(C), ∀ tripartite separable CP linear maps Λ : Md(C) ⊗ [Md(C) ⊗ Md(C)] ⊗ Md(C) → Md(C) ⊗ [Md(C) ⊗ Md(C)] ⊗ Md(C): Tr2,3{Λ(ρ ⊗ σ)} is separable. Here, |e〉 = ∑di=1 |i〉⊗|i〉 ∈ Cd⊗Cd is a maximally entangled vector, I ∈ Md(C) is the identity matrix, and Tr2,3{·} denotes partial trace over the middle two tensor factors. Proof. The equivalence of (1) and (2) can be readily established by using the Choi-Jamiolkowski isomorphism J : Td(C) → Md(C) ⊗ Md(C) defined as J(Φ) = d∑ i,j=1 Φ(|i〉〈j|) ⊗ |i〉〈j|, which identifies PPT and entanglement breaking linear maps in Td(C) with PPT and separable matrices in Md(C) ⊗ Md(C), respectively. Since J(Φ1 ◦ Φ2) = Tr2,3{(J(Φ1) ⊗ J(Φ2))(I ⊗ |e〉〈e| ⊗ I}, the equivalence of (1) and (2) becomes evident. Let us now assume that (2) holds. Consider an arbitrary separable CP map Λ as given in (3) with Kraus operators {Aj ⊗Cj ⊗Bj}j∈J , where Aj ,Bj ∈ Md(C) and Cj ∈ Md(C) ⊗ Md(C) for all j ∈ J . Then, we can write Λ(ρ ⊗ σ) = ∑ j (I ⊗ Cj ⊗ I)(ρj ⊗ σj)(I ⊗ Cj ⊗ I)∗, where ρj = (Aj ⊗ I)ρ(Aj ⊗ I)∗ and σj = (I ⊗ Bj)σ(I ⊗ Bj)∗ are again PPT. Thus, Vol. 23 (2022) The PPT2 Conjecture Holds for All Choi-Type Maps 3315 ρ σ Alice Bob Charlie Figure 1. The setup for a generalized entanglement swap- ping task Tr2,3{Λ(ρ ⊗ σ)} = ∑ j Tr2,3{(ρj ⊗ σj)(I ⊗ C∗jCj ⊗ I)} = ∑ j∈J d∑ i=1 λij Tr2,3{(ρj ⊗ σj)(I ⊗ |cij〉〈cij | ⊗ I)}, where, for each j ∈ J , ∑di=1 λij |cij〉〈cij | is the spectral decomposition of the positive semi-definite matrix C∗jCj . Now, by writing |cij〉 = (C′ij ⊗ I)|e〉 for C′ij ∈ Md(C) and redefining ij = (I ⊗C′ij)∗ρj(I ⊗C′ij) for all i, j (which are all again PPT), we obtain: Tr2,3{Λ(ρ ⊗ σ)} = ∑ j∈J d∑ i=1 λij Tr2,3{(ij ⊗ σj)(I ⊗ |e〉〈e| ⊗ I)}, whose separability trivially follows from our assumption. Finally, if we assume that (3) holds, then (2) follows by choosing Λ to be defined by a single Kraus operator of the form I ⊗ |e〉〈e| ⊗ I.  Let us take a second to interpret the above result. Assume that there are three spatially separated parties: Alice, Bob, and Charlie, such that Charlie shares bipartite states ρ and σ with Alice and Bob, respectively (see Fig. 1). The objective is to transfer any entanglement that Charlie shares with Alice and Bob separately (via ρ and σ) to shared entanglement between Alice and Bob. We can think of this as a generalized entanglement ‘swapping’ task. What the PPT2 conjecture says in this context is that the above task is impossible if ρ and σ are PPT. No matter what local operations the parties might wish to perform on their subsystems, Proposition 2.1 guarantees that the result- ing state shared by Alice and Bob is separable. This has drastic implications in the realm of quantum key distribution using repeater devices, where such entanglement swapping procedures are heavily employed, see [3,6]. We now define the different families of covariant maps in Td(C). 3316 S. Singh, I. Nechita Ann. Henri Poincare´ Definition 2.2. Let DUd and DOd denote the groups of diagonal unitary and diagonal orthogonal matrices in Md(C), respectively. Then, a linear map Φ ∈ Td(C) is said to be • Diagonal Unitary Covariant (DUC) if ∀X ∈ Md(C) and U ∈ DUd : Φ(UXU∗) = U∗Φ(X)U, • Conjugate Diagonal Unitary Covariant (CDUC) if ∀X ∈ Md(C) and U ∈ DUd : Φ(UXU∗) = UΦ(X)U∗, • Diagonal Orthogonal Covariant (DOC) if ∀X ∈ Md(C) and O ∈ DOd : Φ(OXO) = OΦ(X)O. Remark 2.3. [28, Theorem 6.4] Definition 2.2 can be reformulated in terms of bipartite Choi matrices having certain local diagonal unitary/orthogonal invariance properties: • Φ ∈ Td(C) is DUC ⇐⇒ (U ⊗ U)J(Φ)(U∗ ⊗ U∗) = J(Φ) ∀U ∈ DUd. • Φ ∈ Td(C) is CDUC ⇐⇒ (U ⊗U∗)J(Φ)(U∗⊗U) = J(Φ) ∀U ∈ DUd. • Φ ∈ Td(C) is DOC ⇐⇒ (O ⊗ O)J(Φ)(O ⊗ O) = J(Φ) ∀O ∈ DOd. The sets of DUC, CDUC and DOC maps in Td(C) will be denoted by DUCd,CDUCd, and DOCd, respectively. Superscripts i = 1, 2 and 3 will be used to distinguish between maps Φ(i) in DUCd,CDUCd and DOCd, respectively. Using the structure of the invariant bipartite Choi matrices J(Φ(i)) from [28, Proposition 2.3], one can parameterize the action of the corresponding covari- ant maps Φ(i) on Md(C) in terms of matrix triples (A,B,C) ∈ Md(C)×3Cd (Eq. (2)) as follows: Φ(1)(A,B)(X) = diag(A|diagX〉) + B˜  X (3) Φ(2)(A,B)(X) = diag(A|diagX〉) + B˜  X (4) Φ(3)(A,B,C)(X) = diag(A|diagX〉) + B˜  X + C˜  X (5) where B˜ = B − diagB, C˜ = C − diagC, and  denotes the operation of Hadamard (or entrywise) product in Md(C). In a quantum setting, where X = ρ is a quantum state (ρ ∈ PSDd,Tr ρ = 1) and Φ(i) ∈ Td(C) are quan- tum channels, one can interpret the above actions by splitting them into two parts. The first part involves a classical diagonal mixing operation, which is nothing but a transformation on the space of probability distributions in Rd+: |diag ρ〉 → A|diag ρ〉. The second part acts on the off-diagonal part of the in- put state by mixing the well-known actions of Schur Multipliers [24, Chapters 3,8] and transposition maps in Td(C). An important example of the above kind of maps is the Choi map [7], which was introduced in the ‘70s as the first example of a positive non- decomposable map: ΦChoi : M3(C) → M3(C), ΦChoi(X) = ⎛ ⎝ X11 + X33 −X12 −X13 −X21 X11 + X22 −X23 −X31 −X32 X22 + X33 ⎞ ⎠ . Vol. 23 (2022) The PPT2 Conjecture Holds for All Choi-Type Maps 3317 It can be easily seen that the Choi map is a CDUC map ΦChoi = Φ (2) (A,B), with A = ⎛ ⎝ 1 0 1 1 1 0 0 1 1 ⎞ ⎠ and B = ⎛ ⎝ 1 −1 −1 −1 1 −1 −1 −1 1 ⎞ ⎠ = 2I3 − J3, where J3 is the all-ones matrix. In fact, the action of all generalized Choi maps in Td(C) [9,11,15] can be similarly parameterized by an arbitrary A ∈ EWPd and B = 2Id − Jd, see [28, Example 7.5]. Besides Choi-type maps, the classes of (C)DUC and DOC maps contain many other important examples, like the depolarizing and transpose depolarizing maps, amplitude damping maps, Schur multipliers, etc. (see [28, Section 7 and Table 2] for a list of examples). Remark 2.4. Note that the maps in DUCd and CDUCd are linked through composition by matrix transposition, i.e., for (A,B) ∈ Md(C)×2Cd , we have Φ(1)(A,B) = Φ (2) (A,B) ◦ . We will shortly observe a reflection of this characteristic in the fact that the PPT and entanglement-breaking properties of these maps entail an equivalent constraint on the corresponding matrix pairs (A,B) ∈ Md(C)×2Cd . Remark 2.5. For a matrix pair (A,B) ∈ Md(C)×2Cd , we have Φ(1)(A,B) = Φ (3) (A,diag A,B) Φ (2) (A,B) = Φ (3) (A,B,diag A) Next, we introduce the cones of pairwise and triplewise completely positive matrices, which were first introduced in [19,23], respectively, as generalizations of the well-studied cone of completely positive matrices [5]. These will be used later in Propositions 2.9 and 2.10 to provide an equivalent description of the entanglement-breaking properties of our covariant families of maps. It is wise to point out that one should not confuse these notions with the earlier defined completely positive maps in Td(C), which are different beasts altogether. Definition 2.6. A matrix pair (A,B) ∈ Md(C)×2Cd is said to be pairwise com- pletely positive (PCP) if there exist vectors {|vn〉, |wn〉}n∈I (for a finite index set I) such that A = ∑ n∈I |vn  vn〉〈wn  wn|, B = ∑ n∈I |vn  wn〉〈vn  wn|. Definition 2.7. A matrix triple (A,B,C) ∈ Md(C)×3Cd is said to be triplewise completely positive (TCP) if there exist vectors {|vn〉, |wn〉}n∈I (for a finite index set I) such that A = ∑ n∈I |vn  vn〉〈wn  wn|, B = ∑ n∈I |vn  wn〉〈vn  wn|, C = ∑ n∈I |vn  wn〉〈vn  wn|. The vectors {|vn〉, |wn〉}n∈I above are said to form the PCP/TCP decom- position of the concerned matrix pair/triple. Notice that (A,B,C) ∈ TCPd 3318 S. Singh, I. Nechita Ann. Henri Poincare´ =⇒ (A,B), (A,C) ∈ PCPd. It is easy to deduce that PCP and TCP matrices form closed convex cones, which we will denote by PCPd and TCPd, respec- tively. For an extensive account of the convex structure of these cones, the readers should refer to [28, Section 5]. Several elementary properties of these cones are discussed in [19, Sections 3, 4] and [23, Appendix B], respectively. We recall some important necessary conditions for membership in the PCPd cone below. Lemma 2.8. Let (A,B) ∈ PCPd. Then, A ∈ EWPd and B ∈ PSDd. Moreover, the entrywise inequalities AijAji ≥ |Bij |2 hold for all i, j. Let us now describe the PPT and entanglement-breaking properties of the covariant maps in terms of constraints on the associated matrix pairs and triples. Proposition 2.9 ([28, Lemmas 6.11, 6.12]). Let (A,B) ∈ Md(C)×2Cd . Then, Φ(1)(A,B) is (1) CP ⇐⇒ A ∈ EWPd, B = B∗ and AijAji ≥ |Bij |2 ∀i, j ⇐⇒ Φ(2)(A,B) is coCP. (2) coCP ⇐⇒ A ∈ EWPd and B ∈ PSDd ⇐⇒ Φ(2)(A,B) is CP. (3) PPT ⇐⇒ A ∈ EWPd, B ∈ PSDd and AijAji ≥ |Bij |2 ∀i, j ⇐⇒ Φ(2)(A,B) is PPT. (4) entanglement breaking ⇐⇒ (A,B) ∈ PCPd ⇐⇒ Φ(2)(A,B) is entangle- ment breaking. Proposition 2.10 ([28, Lemma 6.13]). Let (A,B,C)∈Md(C)×3Cd . Then, Φ (3) (A,B,C) is (1) CP ⇐⇒ A ∈ EWPd, B ∈ PSDd, C = C∗, and AijAji ≥ |Cij |2 ∀i, j. (2) coCP ⇐⇒ A ∈ EWPd, B = B∗, C ∈ PSDd, and AijAji ≥ |Bij |2 ∀i, j. (3) PPT ⇐⇒ A ∈ EWPd, B,C ∈ PSDd and AijAji ≥ max{|Bij |2, |Cij |2} ∀i, j. (4) entanglement breaking ⇐⇒ (A,B,C) ∈ TCPd. 3. Factor Widths The concept of factor width was first formalized in [4] for real (symmetric) positive semi-definite matrices, although the idea had been in operation before, particularly in the study of completely positive matrices [5, Definition 2.4]. Recall that for |v〉 ∈ Cd, its support is defined as supp |v〉:={i ∈ [d] : vi = 0}. We define σ(v) as the size of supp |v〉, that is the number of non-zero coordinates of |v〉. A real positive semi-definite matrix A ∈ Md(R) is said to have factor width k if there exists vectors {|vn〉}n∈I ⊂ Rd with σ(vn) ≤ k for each n such that A admits the following rank one decomposition: A = ∑ n∈I |vn〉〈vn|. Besides being heavily implemented in the analysis of completely positive matrices [5, Section 4], the concept of factor width has found several applications in the field of conic programming and optimization theory [1]. In Vol. 23 (2022) The PPT2 Conjecture Holds for All Choi-Type Maps 3319 this section, we will extend the notion of factor width to the cones of complex (hermitian) positive semi-definite matrices PSDd, pairwise completely positive matrices PCPd and triplewise completely positive matrices TCPd. In particular, we will obtain a complete characterization of matrices with factor width 2 in PSDd and PCPd, which will later play an instrumental role in proving the validity of the PPT-squared conjecture for covariant maps in DUCd,CDUCd. Without further delay, let us now delve straight into the definition of factor width for matrices in PSDd, PCPd and TCPd. Definition 3.1. A matrix B ∈ PSDd is said to have factor width k ∈ N if it admits a rank one decomposition B = ∑ n∈I |vn〉〈vn|, where {|vn〉}n∈I ⊂ Cd are such that σ(vn) ≤ k for each n ∈ I. The notion of factor width for positive semidefinite matrices has been considered in [25], in relation to measures of coherence for density matrices. This notion has a straightforward generalization for PCP and TCP matrices. Definition 3.2. A matrix pair (A,B) ∈ PCPd (resp. triple (A,B,C) ∈ TCPd) is said to have factor width k ∈ N if it admits a PCP (resp. TCP) decomposition with vectors {|vn〉, |wn〉}n∈I ⊂ Cd such that σ(vn  wn) ≤ k for each n ∈ I. The cones of factor width k matrices in PSDd,PCPd and TCPd will be denoted by PSDkd,PCP k d, and TCP k d, respectively. It should be clear from the definitions that (A,B,C) ∈ TCPkd =⇒ (A,B), (A,C) ∈ PCPkd =⇒ B,C ∈ PSDkd and that the cones PSD k d,PCP k d, and TCP k d are stable by direct sums. The following sequences of inclusions are also trivial consequences of the definitions: PSD1d ⊂ PSD2d ⊂ · · · ⊂ PSDdd = PSDd (6) PCP1d ⊂ PCP2d ⊂ · · · ⊂ PCPdd = PCPd (7) TCP1d ⊂ TCP2d ⊂ · · · ⊂ TCPdd = TCPd (8) Remark 3.3. For all k < d, the inclusions PSDkd ⊂ PSDd, PCPkd ⊂ PCPd, TCPkd ⊂ TCPd are strict. This can be seen by considering extremal rays of the cones (see [28, Theorem 5.13]) generated by vectors of full support. For k = 1, it is evident that PSD1d is precisely the set of diagonal matrices in EWPd. It is equally easy to deduce that PCP1d (resp. TCP 1 d) contain matrix pairs (A,B) ∈ PCPd (resp. triples (A,B,C) ∈ TCPd) where A ∈ EWPd and B = diagA (resp. A ∈ EWPd and B = C = diagA). The k = 2 case is more interesting, and we must familiarize ourselves with some matrix-theoretic terminology before we begin to deal with it. Let us start with the definitions of the so-called scaled diagonally dominant and M-matrices. In what follows, Id denotes the identity matrix in Md(C). For B ∈ Md(C) and k ∈ N, we define a hierarchy of comparison matrices entrywise as: Mk(B)ij = { k|Bij |, if i = j −|Bij |, otherwise (9) 3320 S. Singh, I. Nechita Ann. Henri Poincare´ Notice that M1(B) coincides with the usual comparison matrix M(B) for B ∈ Md(C) Definition 3.4. A matrix B ∈ Md(C) is called diagonally dominant (DD) if |Bii| ≥ ∑ j =i |Bij | and |Bii| ≥ ∑ j =i |Bji| ∀i. For B ∈ Md(C), if there exists a positive diagonal matrix D such that DBD is diagonally dominant, then B is called scaled diagonally dominant (SDD). Note that, for a hermitian matrix B, the two conditions |Bii| ≥ ∑ j =i |Bij | and |Bii| ≥ ∑ j =i |Bji| are equivalent. Also note that hermitian DD and SDD matrices are clearly positive semi-definite. Definition 3.5. A matrix B = s Id − P with s ≥ 0 and P ∈ EWPd is called an M -matrix if s ≥ ρ(P ), where ρ(P ) denotes the spectral radius of P . From the above definition, it is easy to see that if P ∈ EWPd is symmetric, then B = s Id − P is an M-matrix if and only if it is positive semi-definite. Before proceeding further, let us equip ourselves with an important result from the Perron–Frobenius theory of non-negative matrices [16, Chapter 8]. Recall that a matrix B ∈ Md(C) is reducible if it is permutationally similar to a block matrix in Eq. (10) (where B1, B3 are square matrices) and irreducible otherwise. ( B1 B2 0 B3 ) (10) Lemma 3.6. For an irreducible non-negative matrix P ∈ EWPd, the spectral radius ρ(P ) is an eigenvalue (called Perron eigenvalue) of unit multiplicity with a positive eigenvector (called Perron eigenvector) |p〉 ∈ Rd+: P |p〉 = ρ(P )|p〉. With all the required tools now present in our arsenal, we begin to analyze the structure of the PSD2d and PCP 2 d cones. We start by showing that diagonal dominance is a sufficient condition to guarantee membership in these cones. It would be insightful to compare the following results with [2, Theorem2] and [19, Theorem4.4]. Lemma 3.7. If B ∈ Md(C) is a (hermitian) diagonally dominant matrix, then B ∈ PSD2d. Proof. Let us use the symbol in Eq. (11) to define a d × d matrix which has zeros everywhere except in the i, j-principal submatrix, where the entries are defined by the matrix present in the notation. ( a b c d ) i,j∈[d] :=a|i〉〈i| + b|i〉〈j| + c|j〉〈i| + d|j〉〈j| ∈ Md(C) (11) Then, the following decomposition of a hermitian DD matrix shows that it has factor width 2 B = ∑ 1≤i 2. More precisely, we have found PPT matrix pairs (A,B) ∈ Md(C)×2Cd such that (A,B) /∈ PCP2d, where (A,B,B) is obtained by composing the triple (A,B,B) with itself in the above fashion. Hence, in order to prove the PPT2 conjecture for the more general class of DOC maps, one likely requires stronger criterion for separability, in terms of sufficient conditions for membership in both the PCPd and TCPd cones. Vol. 23 (2022) The PPT2 Conjecture Holds for All Choi-Type Maps 3327 Acknowledgements We thank Alexander Mu¨ller-Hermes for valuable correspondence on the PPT2 conjecture, and Salman Beigi and Ludovico Lami for directing our attention to the reference [25], where the notion of factor width has been employed to study coherence of quantum states. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and re- production in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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Quantum 5, 519 (2021) Satvik Singh Department of Applied Mathematics and Theoretical Physics (DAMTP) University of Cambridge Cambridge United Kingdom e-mail: satviksingh2@gmail.com Vol. 23 (2022) The PPT2 Conjecture Holds for All Choi-Type Maps 3329 Ion Nechita Laboratoire de Physique The´orique, CNRS, UPS Universite´ de Toulouse Toulouse France e-mail: nechita@irsamc.ups-tlse.fr Communicated by Matthias Christandl. Received: March 10, 2021. Accepted: January 31, 2022.