Diagonal unitary and orthogonal symmetries in quantum theory

We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.


Introduction
In Quantum Theory, the set of entangled bipartite states has a very special role: the presence of genuinely quantum correlations between the two parties allow for significant advantage in various information processing and computing tasks. However, this paradigm suffers from a significant theoretical hurdle because of the computational hardness of deciding whether a bipartite (or multipartite) quantum state is separable or entangled [23]. To overcome the NP-hardness of this decision problem in the most general case, a plethora of entanglement (resp. separability) criteria have been discovered and studied: there are computationally efficient methods for certifying that a given quantum state is entangled (resp. separable). The most useful entanglement criterion is that of the positive partial transposition (PPT) [55,32], which turns out to be exact for qubit-qubit and qubit-qutrit systems.
Another way to tackle the separability problem is to restrict the decision problem to special classes of bipartite quantum states. Quantum states satisfying some sort of symmetry are the obvious candidates, since they are physically relevant and mathematically tractable, because of their added structure and reduced number of parameters. One canonical situation is that of bipartite quantum states invariant under the tensor product of the standard representation of the unitary group; two cases need to be considered here: These families of quantum states are respectively known as the Werner [65] and the Isotropic [31] states, and correspond to mixtures of the projections on the symmetric and the anti-symmetric subspaces, respectively to mixtures of the maximally mixed state and the maximally entangled state [64,Example 6.10]. For these one-parameter families (two real parameters if one ignores the unit trace normalization), separability, the PPT property, as well as other relevant properties have been characterized, thanks to their simple structure. Importantly, these states correspond, through the Choi-Jamiołkowski isomorphism, to covariant quantum channels, that is In this work, we analyze the properties of quantum states and channels which are symmetric with respect to the diagonal unitary and orthogonal groups. These classes of states are described by roughly d 2 real parameters, and are of intermediate complexity between the full-unitary invariant states described above and the full set of bipartite quantum states (requiring order d 4 parameters). We shall consider three situations: the first two correspond to the equations above, with the unitary U restricted to the class of diagonal unitary matrices (diagonal matrices with arbitrary complex phases), while the third one corresponds to U being restricted to diagonal orthogonal matrices (diagonal matrices with arbitrary signs). These classes of states, called respectively LDUI, CLDUI, and LDOI, have been introduced in [14,36,52]. We provide a detailed analysis of these matrices, from various points of views: linear algebra, convexity, positivity, separability, etc. Further specializations of these matrices have appeared on numerous occasions in the literature; we thus give a unified treatment of these classes of matrices under one umbrella, in an effort to streamline the different proof techniques used previously.
A focal point of our efforts is the separability problem. It turns out that the classes of states we investigate are still rich enough for the separability problem to be intractable [67,63,36], but the situation is simpler, since the search space for separable decompositions is smaller. We describe the separability properties of these invariant states in terms of two cones of pairs and triples of matrices, called the pairwise completely positive [36] and the triplewise completely positive [52] cone, respectively. Both these notions can be understood as extensions of the classical case of completely positive matrices [4], which is a key notion in combinatorics and optimization [1]. We borrow and extend techniques from these fields to provide several separability and entanglement criteria for our classes of invariant states. Building upon the present work, a novel approach to detect entanglement using critical graph-theoretic techniques has been developed in [59].
In the latter half of this paper, we switch perspectives and discuss everything from the point of view of linear maps between matrix algebras, using the Choi-Jamiołkowski isomorphism. Quantum channels which are covariant with respect to the action of diagonal unitary matrices have appeared in the literature under the name of "mean unitary conjugation channels" (MUCC) [47,49], we dub them here DUC, CDUC, DOC, in parallel with the case of bipartite states. The action of these maps on the space of d × d complex matrices is parameterized by three d × d complex matrices: A, B and C (with diag B = diag C = 0), and has the following form: We provide several equivalent characterizations of these classes of maps, based on their Choi, Kraus and Stinespring representations. We also focus on the composition properties of these maps, thus revealing intimate connections with the corresponding class of LDOI matrices. As is done for bipartite matrices, we present a survey of some important examples of maps which lie in our class, the most notable among these being the Choi map and all its proposed generalizations. Remarkably, the results obtained in this work are used in [60] to show that the PPT square conjecture holds for the diagonal unitary covariant maps, which contain all Choi-type maps as special cases.
Although the presentation is self-contained, we refer the reader for the background material and the proofs of some results to our previous work [52] as well as to the paper of Johnston and MacLean [36], where the notions of CLDUI and PCP matrices were introduced. The paper has two main parts, one focusing on bipartite matrices, the other one on linear maps. The sections are structured as follows. In Section 2 we review the main background material on invariant states from [52], recalling several key results from this paper. Section 3 contains a list of salient examples, already present in the literature, or new. Sections 4 and 5 deal with the linear, resp. convex structure of the sets of invariant states and the different cones associated with them. In Section 6 we change gears, and focus on linear maps between matrix algebras; completely positive maps, and quantum channels in particular are discussed here. Section 7 contains a list of examples of quantum channels which are of particular interest. In the next two Sections (8 and 9) we discuss, respectively, the special form of Kraus and Stinespring representations of covariant maps, and the structure of their invariant subspaces, which we use to present necessary and sufficient conditions for separability of the corresponding invariant states. We conclude our work with an overview of our results and some directions for future work. At several instances in this paper, we employ the diagrammatic language of boxes and strings to represent tensor equations, in order to ease proofs and make the presentation more visually intuitive. For readers who are unfamiliar with this language, Section 3 of our previous work [52] should suffice for a quick introduction.

Local diagonal unitary and orthogonal invariant matrices
In this section, we recall the basic definitions and properties of the families of local diagonal unitary/orthogonal invariant matrices, which were first introduced in [14], and later studied in [36,52]. For more details and proofs of the results stated here, the reader should refer to our previous work [52,Sections 6,7 and Appendix B]. We start by fixing some basic notation.
We use Dirac's bra-ket notation to denote column vectors v ∈ C d as kets |v and their dual row vectors (conjugate transposes) v * ∈ (C d ) * as bras v|; note that we do not require the kets or the bras to have unit norm. In this notation, the standard inner product v * w on C d is denoted by v|w and the rank one matrix vw * is denoted by the outer product |v w|. The standard basis in For a vector |v ∈ C d , diag |v ∈ M d (C) is the diagonal matrix with entries equal to that of |v . For a matrix V ∈ M d (C), we define two kinds of diagonal operations. The first one extracts out the diagonal part of V and puts it back in matrix form: diag V ∈ M d (C). The second one does the same thing but the result is a vector: |diag V ∈ C d . The above definitions are collected below: Vectors |v ⊗ |w in the tensor product space C d ⊗ C d are denoted as |vw . For bipartite matrices X ∈ M d (C) ⊗ M d (C), the partial transposition with respect to the first and second subsystem is denoted by X Γ := ( ⊗ id)(X) and X Γ := (id ⊗ )(X) respectively. A matrix X ∈ M d (C) ⊗ M d (C) is said to be PPT if both X and X Γ are positive semi-definite. By utilizing the isomorphism M d (C) ⊗ M d (C) M dd (C), we denote the sets of all self-adjoint, entrywise non-negative and positive semi-definite matrices in M d (C) ⊗ M d (C) by M sa dd (C), EWP dd and PSD dd respectively.
The groups of diagonal unitary and diagonal orthogonal matrices in M d (C) will play a central role in our paper. These are denoted by DU d and DO d respectively. A random diagonal unitary matrix is a random variable U ∈ DU d , having independent and identically distributed (i.i.d.) complex phases on its diagonal: Similarly, a random diagonal orthogonal matrix is a random variable O ∈ DO d , having independent and identically distributed real signs on its diagonal: With all the notation in place, we begin with the most important definition of this section.
The vector subspaces of LDUI, CLDUI and LDOI matrices in M d (C) ⊗ M d (C) are denoted, respectively, by LDUI d , CLDUI d and LDOI d . We can now begin to set up important bijections between the newly introduced vector spaces and certain families of matrix pairs/triples defined as follows: [52, Propositions 6.4 and 7.2] C d be vector spaces defined in Eqs. (2) and (3), endowed with the usual component-wise addition and scalar multiplication. Then, We take a moment to state the bijections from Proposition 2.2 more explicitly. Notice that given A ∈ M d (C), A denotes the matrix with zero diagonal but with the same off-diagonal entries as A: A := A − diag A, or, in coordinates • X (1) : Coordinate-wise, we have (note that X(i 1 i 2 , j 1 j 2 ) = i 1 i 2 |X|j 1 j 2 ): The general matrix form of a 3 ⊗ 3 LDOI matrix is exhibited below (dots represent zeros): We refer the reader to Proposition 4.1 for the block-structure of the above matrices. The next proposition identifies orthogonal projections on the LDUI/CLDUI/LDOI subspaces with certain local diagonal unitary/orthogonal averaging operations [ where U ∈ DU d (resp. O ∈ DO d ) is a random diagonal unitary (resp. orthogonal) matrix, and E U (resp. E O ) denotes the expectation with respect to the distribution of U (resp. O).
We refer the reader to [52,Theorems 4.8 and 5.5] for a nice graphical method to compute general expectations of the form given in Proposition 2.3.
C d , the above stated bijections imply that Hence, LDUI d and CLDUI d are vector subspaces of LDOI d .
Next, we introduce the notions of completely positive, pairwise completely positive and triplewise completely positive matrices and link them to the separability problem for matrices in LDUI d , CLDUI d and LDOI d [52,Lemmas 6.6 and 7.5]. Recall that a positive semi-definite bipartite matrix X ∈ M d (C) ⊗ M d (C) is said to be separable if there exists a family of vectors {|v k , |w k } k∈I ⊆ C d for a finite index set I, such that Eq. (10) holds (observe that V and W are matrices in M d,|I| (C) with columns given by the vectors {|v k } k∈I and {|w k } k∈I respectively). A positive semi-definite We would like to warn the reader at this point that the notion of a completely positive matrix defined above is unrelated from the one of a completely positive map, discussed in Section 6. Definition 2.6 (Pairwise completely positive matrices). A matrix pair (A, B) ∈ M d (C) ×2 C d is said to be Pairwise Completely Positive (PCP) if there exist matrices V, W ∈ M d,d (C), for arbitrary d ∈ N, such that the following decomposition holds: The next result is due to [36,Theorem 3.4], where the authors prove that the notion of PCP matrices generalizes that of CP matrices. In our discussion of partial transpose invariant LDOI matrices in Example 3.5, we will extend this generalization to TCP matrices as well.
We now arrive at the crucial link between separability of LDOI matrices and the different notions of completely positive matrices introduced.
We prove the equivalence for LDOI matrices, and urge the readers to mimic the same proof for LDUI/CLDUI matrices. Assume first that X (3) (A,B,C) is separable and hence can be decomposed as in Eq. (10), with matrices V, W ∈ M d,|I| (C). Using the coordinate-wise relations in Eq. (7), it is easy to infer the TCP decomposition of (A, B, C): Conversely, assume that (A, B, C) is TCP and let V, W ∈ M d,d (C) form its TCP decomposition. We construct a separable matrix X ∈ M d (C) ⊗ M d (C) using the V, W matrices, as was done in Eq. (10). It is then not too difficult to see that and the proof concludes with the observation that since Proj LDOI is a (classically correlated) local operation, it preserves separability (see Proposition 2.3).
Several elementary results on TCP matrix triples are presented in [52,Appendix B]. Some of these results, which will be of relevance to us in later sections, are presented below. Notice that the entrywise and trace (or nuclear ) norms on M d (C) are denoted by . 1 and . Tr respectively: Before proceeding further, it is essential to recall some definitions.
is defined entrywise as follows: The next result [36,Theorem 4.4] provides easily verifiable sufficient conditions for a pair (A, B) ∈ M d (C) ×2 C d to be PCP and hence yields a non-trivial test for separability of the associated matrices in LDUI d and CLDUI d .
, then all separable matrices X ∈ M d (C) ⊗ M d (C) satisfy: ||R(X)|| Tr ≤ Tr X, see [5,57]. This is known as the realignment criterion of separability. In this paper, we use the notation X R := R(X).

Lemma 2.12. Consider an arbitrary X
3. X In Quantum Mechanics, states ρ of a physical system (called quantum states) are modelled as bounded operators on a separable Hilbert space H, i.e., ρ ∈ B(H), which are positive semi-definite and trace-class with Tr ρ = 1, see [64,Chapter 2], [29,Chapter 2]. For multiparty systems, the Hilbert spaces are naturally required to acquire a tensor product structure, and the relevant states then lie in the space B(H 1 ⊗· · ·⊗H n ). In a finite dimensional bipartite setting: H 1 C d1 , H 2 C d2 , it is obvious that quantum states of a physical system are just positive semi-definite matrices ρ ∈ M d1 (C) ⊗ M d2 (C) with unit trace. The next lemma exploits the results from Lemma 2.12 to define the class of quantum states within the family of local diagonal orthogonal invariant matrices in M d (C) ⊗ M d (C).  In this example, we deal with LDOI matrices which factorize either as a unit rank matrix or as a tensor product, see also Remark 4.2.

Important classes of LDOI matrices
In the former case, B = |y z| and for some non-zero λ ∈ C and i, j ∈ [d].
Proof. Consider a unit rank LDOI matrix is CLDUI. The associated matrices A, B and C can also be shown to have the required form, see Proposition 2.2 and the bijections following it. If |Y is not of the diagonal form, then Y ij = 0 for some i = j. This implies that Z ab = 0 for all (a, b) / ∈ {(i, j), (j, i)}, and we obtain rank one LDUI matrices of the type where α, β, γ and δ are complex numbers. Finally, selecting those matrices from Eqs. (15) and (16) which are both LDUI and CLDUI gives us precisely the matrices in Eq. (14).

Remark 3.2.
The set of LDOI matrices X having a tensor product structure X = Y ⊗ Z can be deduced from the result above by applying the realignment operation, see the Definition used in Lemma 2.12.

Example 3.3 (Werner and Isotropic matrices). [65, 31]
. It is obvious from the definition that the Werner matrices lie in LDUI d , while the isotropic matrices lie in CLDUI d (see Definition 2.1). The structure of these matrices is known to be of the following form, for a, b ∈ C: It is straightforward to check that X wer a,b = X (A,B) and X iso a,b = X (A,B) , for A = b I d + aJ d and B = aI d + bJ d . It is equally easy to see that self-adjointness forces the parameters a, b to be real.
Combining everything together, we deduce that X wer a,b and X iso a,b are PPT if and only if a ≥ 0 and −a/d ≤ b ≤ a. We now prove that this also suffices to guarantee separability of the concerned matrices.
be the Werner and isotropic matrices, parameterized by the pair (a, b) ∈ C 2 . Then, the following equivalences hold: Proof. The idea is to show that the associated pair (A, B) is PCP in the range determined by a and b. To this end, we first fix a ≥ 0 and assume that b = −a/d. It is then evident that B is diagonally dominant and hence Lemma 2.11 tells us that (A, B) is PCP. On the other hand, if b = a, we have It is then clear that A = B is CP =⇒ (A, B) is PCP, see Theorem 2.8. Separability on the entire interval −a/d ≤ b ≤ a then follows from convexity, see also [36,Example 2].
which constitute what is called the Dicke basis for the symmetric subspace. Bipartite matrices which are diagonal in the Dicke basis are known as diagonal symmetric matrices: where Y ∈ M d (R) is an arbitrary symmetric matrix. By defining is separable if and only if it is PPT. For d ≥ 5, every A ∈ DNN d which is not completely positive gives rise to a PPT entangled diagonal symmetric matrix.

Example 3.5 (Partial transpose invariant LDOI matrices).
Using Proposition 4.3, it is straightforward to infer that an LDOI matrix is invariant under partial transposition with respect to the first (resp. second) subsystem if and only if the associated triple Moreover, by definition, these matrices are PPT if and only if they are positive semi-definite, which in turn is equivalent to the condition that , see Lemma 2.12. Finally, Proposition 3.4 below (combined with Theorem 2.9) shows that separability of these matrices is equivalent to the separability of the corresponding LDUI/CLDUI matrices with matrix pairs (A, C d , the following sequence of equivalences hold: Proof. We begin with the first row of equivalences. Since the implications pointing from the ends to the center are trivially obtained from part (1) of Lemma 2.10, we start from the center and assume that (A, B) is PCP with V, W ∈ M d,d (C) forming its PCP decomposition, see Definition 2.6. Now, define matrices V , W , V , W ∈ M d,d (C) entrywise as follows: where phase(V ij ) and phase(W ij ) are the complex phases of the entries of V and W : V ij = |V ij | phase(V ij ) and W ij = |W ij | phase(W ij ). Now, observe that since W is entrywise non-negative (and hence W = W ), V , W form a TCP decomposition of (A, B, B) as in Definition 2.7. Similarly, V , W form a TCP decomposition of (A, B, B ). This establishes all the equivalences in the first row. An identical argument does the same for the second row as well. Now, to connect the two rows, we observe that if V, W form a PCP decomposition of (A, B), then V , W (as constructed above) form a PCP decomposition of (A, B ).
Using Proposition 3.4, the conclusion of Theorem 2.8 can be trivially extended to TCP matrices.
, the following equivalences hold: Example 3.6 (LDOI matrices with A = J d ).
In Example 3.1, it was shown that In this example, we investigate the general class of matrices in  We now show that for LDUI/CLDUI matrices with A = J d , the PPT propetry is equivalent to separability.
C d , the following are equivalent for i = 1, 2: Proof. The first equivalence is a straightforward consequence of Lemma 2.12 (combined with Remark 2.4): since A = J d , the condition 1 ≥ |B ij | 2 corresponds to the i, j-minor of B being non-negative. In the final equivalence, the reverse implication is trivial to obtain. To establish the forward implication, we need to show that B ∈ Corr d =⇒ (J d , B) is PCP (see Theorem 2.9). Hence, assume that B ∈ Corr d and consider a decomposition of the form : and the proof is complete.
It seems wise to pause here for a moment to collect some useful facts about the set of correlation matrices Corr d in M d (C): • By definition, Z ∈ Corr d ⇐⇒ Z ∈ PSD d and diag Z = I d .
• Corr d is a compact convex set, with the rank one matrices |z z| being the obvious extreme points, for |z ∈ T d := {|y ∈ C d : |y i | = 1 ∀i}.
• For d ≤ 3, it can be shown that the rank one matrices are the only extreme correlation matrices, and hence Corr d = conv{|z z| : |z ∈ T d }. However, for d ≥ 4, other higher rank extreme points exist, and the preceding conclusion becomes false, see [13,48,22,45].
Now, if we consider the general LDOI matrices with A = J d , we can quickly infer that as before, these matrices are PPT if and only if the associated matrices B, C ∈ Corr d . However, we do not know whether this also suffices to guarantee separability. We do have some partial results in this direction: Let us quickly prove this. Assume B = |b b| and C = |c c| are arbitrary rank one correlations, where |b , |c ∈ T d . Then, we can easily construct vectors |v , |w ∈ T d such that |v w = |b and |v w = |c . Hence, The desired result then follows from convexity.

Example 3.7 (Canonical NPT states).
Extraction of maximally entangled states from several copies of a given bipartite quantum state through the use of local operations and classical communication (LOCC) forms a central task in numerous quantum communication and cryptographic protocols. In the asymptotic limit, if a state ρ ∈ M d1 (C) ⊗ M d2 (C) allows for a non-zero rate of extraction (defined as the ratio of the extracted number of maximally entangled states to the number of input states), it is said to be distillable. Non-distillable entangled bipartite states are called bound entangled. It is well known that distillable states ρ are negative under partial transposition (NPT), i.e., (id ⊗ )ρ is not positive semi-definite. However, it is not known whether every NPT state is distillable or not. Put differently, the existence of an NPT bound entangled state is uncertain (see the excellent review articles [33,42] for a more precise formulation of these concepts). In [20], the authors show that and a, b, c are real parameters. It is not too hard to discern that the states ρ a,b,c ∈ M d (C)⊗M d (C) are LDUI, with the associated matrices A, B ∈ M d (R) defined as follows: Hence, to show that every NPT bipartite state is distillable, it suffices to prove distillability for an arbitrary NPT LDUI state in the above family. The parameter ranges within which the above states are PPT/NPT can easily be calculated using Lemma 2.12, see also [20, Figure 2].

Example 3.8 (PPT entangled edge states).
A PPT entangled matrix X ∈ M d1 (C) ⊗ M d2 (C) is said to be an edge state if there are no product vectors |xy ∈ range(X) such that |xy ∈ range(X Γ ). These states defy the range criterion of separability in a very extreme fashion [11,43,44]. In recent years, there has been a great deal of interest in characterizing edge states X based on their types, which are nothing but pairs of numbers (p, q) such that rank(X) = p and rank(X Γ ) = q. In particular, for the low dimensional 3⊗3 system, all possible types have been identified, and examples for each type have been constructed (see [37] and references therein for a review of all the examples). Noticeably, the authors in [41] show that except for the (4,4) type, all other types of 3 ⊗ 3 edge states can be generated by matrices of the form: where b > 0, −π/3 < θ < π/3 (θ = 0) and |η , |ζ , |ξ ∈ C 3 are vectors such that The zero pattern of these states immediately reveal that they are all 3 ⊗ 3 LDOI matrices. The entries of the associated A, B and C matrices can be read off from the appropriate diagonal/offdiagonal elements of X.
We summarize all the examples discussed in this section in Table 1.

Ex. Name Defining Characteristic Ambient Space
Associated

Tensor product
Tensor product LDOI matrices

Diagonal symmetric
Diagonal in the Dicke basis of the

Linear structure of LDOI matrices
We discuss in this section the linear structure of the sets of bipartite matrices with the invariance properties discussed in Section 2, focusing on different notions of symmetry. We shall make no reference to any positivity or separability notions, this being the topic of the next section. It was shown in [52, Sections 6-7] that the sets CLDUI d , LDUI d , LDOI d are C-vector spaces with dimensions This fact is a consequence of a simple dimension counting for the triples (A, B, C) using the different diagonal restrictions. For example, in the CLDUI d case, the elements are parametrized by pairs (A, B) with A, B being d × d complex matrices (having, in total, 2d 2 complex parameters), with the restriction that diag(A) = diag(B) (fixing d complex parameters), resulting in a total of 2d 2 − d complex parameters, see Eq.
(2) and Proposition 2.2. It is also trivial to see from their definition that CLDUI d and LDUI d are vector subspaces of LDOI d (see Remark 2.4). Moreover, , see also Example 3.1. We represent the general position of these vector spaces in Figure 1. As elements of M d 2 (C), the LDUI/CLDUI/LDOI matrices have interesting block structures, which can be derived by using the explicit form of the isomorphisms stated after Proposition 2.2, see also [52,Eq. (41)].

Proposition 4.1. For all triples
C d , we have the following decompositions: Corollary 4.2. The rank of the invariant matrices X (1,2,3) above are as follows: We now consider various symmetries of the vector spaces LDUI d , CLDUI d and LDOI d , obtained by permuting the tensor legs (i.e. the wires in the graphical representation corresponding to the different tensor factors) of the relevant bipartite matrices.

Proposition 4.3. The vector subspaces CLDUI
where F is the flip operator (F |ab = |ba ), × permutes the legs of a 4-tensor diagonally and R is the realignment operation. Moreover, the space LDOI d has the following additional symmetries: Proof. All the above relations are easily verified using algebra, but expressing them in the graphical language of tensor networks is more insightful. We leave the detailed proofs to the reader, and provide a graphical proof for the flip F X  The considerations above allow us to characterize self-adjoint LDOI matrices as follows.
In particular, the real vector spaces of self-adjoint invariant matrices have the following dimensions: Proof. The main assertion follows from Proposition 4.3: (Ā,B * ,C * ) .

if and only if
Proof. For the first claim, use Proposition 4.3. For the second claim, use P s = (I + F )/2 and Proposition 4.3 to get from which the conclusion immediately follows.
We now show that the CLDUI d , LDUI d , and LDOI d vector spaces are stable under a modified notion of direct sum, that we introduce next.

Definition 4.7. Given two bipartite matrices
With this definition in hand, we have the following result, showing that the vector spaces of invariant bipartite matrices are stable under bipartite direct sum.

Proposition 4.8. Given matrices
, i = 1, 2, the following relations hold: Proof. The result directly follows from the explicit (coordinate-wise) expressions of the bijections X (i) for i = 1, 2, 3, see Equations (5), (6) and (7).  , define the projector P = i∈I |i i| ∈ M sa d (C). Then, the following relation holds: where the locally projected (P ⊗ P )X Proof. Follows trivially from the explicit form of the bijections, stated after Proposition 2.2.
We will see in the next section that the cones of PCP and TCP matrices in M d (C) ×3 C d are stable under the discussed operations of taking direct sums and principal subtriples. We now conclude this section by discussing the partial actions on LDOI d of the two conditional expectations on the diagonal and scalar matrices defined below: with entries equal to the row and column sums of A respectively: Then, the following relations hold: The action of the partial conditional expectations diag and trace are given respectively by We have thus (see also [52,Lemmas 6.9 and 7.7]) Proof. These are simple computations which can be easily obtained using the graphical calculus for tensors, see e.g. Figure 3 for the trace formula.
A B

Convex structure of LDOI matrices
We study in this section the properties of the three sets of local diagonal invariant matrices from the point of view of convex geometry; more precisely, we shall investigate the convex cone of positive semidefinite LDUI/CLDUI/LDOI matrices, as well as other notions of positivity related to quantum entanglement. We start with some basic facts about convex cones.
Definition 5.1. Let V be a real vector space. A convex cone C is a subset of V having the following two properties: • if x ∈ C and λ ∈ R + = [0, ∞), then λx ∈ C.
• if x, y ∈ C, then x + y ∈ C.
In particular, 0 ∈ C. The cone C is said to be pointed if C ∩ (−C) = {0}; in other words, C is pointed if it does not contain any line.
Given a cone C, we define its dual cone by The main examples we shall be concerned with in this work are the cone of entrywise non-negative matrices and the cone of positive semidefinite matrices Their extremal rays are as follows: Let V and W be vector spaces and consider convex cones B ⊆ V and C ⊆ W . We define the following two operations on convex cones: Direct sum: For the corresponding construction for convex sets, see [4] or [3, Section 3.1]. These constructions are dual to each other, see the references above: We have the following result, giving the extremal rays of a Cartesian product of cones in terms of the extremal rays of the factors. By duality, a similar result could be given for the facets of a direct sum of convex cones. Proof. We prove the equality of the two sets by showing separately the two inclusions. For "⊆", consider an extremal ray R + (b, c) of B × C. We first show that one of b or c must be null. Using b ∈ B and c ∈ C, we also have we deduce that the two vectors in (26) must be collinear to (b, c), and thus at least one of b, c must be zero. Assuming, say, c = 0, the extremality of R + b inside B follows easily from the hypothesis. For the reverse inclusion, let R + b be an extremal ray in B, and assume Since C is pointed, c = c = 0. Since b is on an extremal ray in B, it must be collinear to b , b , proving the inclusion and finishing the proof.
Let us now discuss the cones of positive semidefinite local diagonal invariant matrices, and their convex geometry. First, define the three convex cones of interest as sections of the positive semidefinite cone by the corresponding hyperplanes: The elements of these cones were characterized in Lemma 2.12 (see also [52,Lemma 7.6] or [36,Theorem 5.2] for the CLDUI case); this also follows from the following more general result, which gives the spectrum of an arbitrary local diagonal invariant matrix.

Proposition 5.3. For any triple of matrices
C d , the spectra of the matrices X (1,2,3) are given by Proof. The formulas follow immediately from the block structure of the matrices X (1,2,3) , see Proposition 4.1.

Theorem 5.4. The extremal rays of the cones LDUI
are as follows (we abuse notation below, by giving a vector representative v for each extremal ray R + v): The idea of the proof is to decompose the cones as Cartesian products of simpler cones, following the decompositions from Propositions 4.1 and 5.3. Let us start with the case of LDUI + d . In terms of the matrices (A, C), we have: We can thus identify an element of the cone LDUI + d with a vector of 1 + d 2 coordinates, following the decomposition above. Using this block structure, the extremal rays of LDUI + d are the union of the sets of extremal rays of the different factors: Let us now write these elements in the usual form, using the X (1) isomorphism: For CLDUI + d , we start from the decomposition given in Proposition 4.1 where EWP 0 d is the cone of entrywise positive matrices with zero diagonal; we have An extremal ray coming from the i = j factor R + above corresponds to A = |i j| and B = 0 X (A,B) = |i i| ⊗ |j j| .
An extremal ray |x x| for |x = 0 corresponding to the factor PSD d in (27) gives where, for a vector |x ∈ C d , Finally, for LDOI + d , we have the decomposition We leave the details of this case to the reader. We now shift the discussion to the case of PCP and TCP matrices, see Definitions 2.6 and 2.7. Several elementary properties of these matrices can be found in [36,Sections 3,4] and [52,Appendix B] respectively. In particular, it has been shown that PCP (resp. TCP) matrices form convex cones: We now show that these cones are also closed. Proof. The proof is a simple consequence of the fact that the cone of separable bipartite matrices is closed [64, Propositions 6.3 and 6.8]. This is because from Theorem 2.9, we have: where the continuous maps X (·) have been defined in Section 2.
We now examine the stability of the PCP d and TCP d cones under the action of taking direct sums and principal subtriples (refer to Section 4 for the relevant definitions).  For triples (A 1 , B 1 Proof. The forward implication follows from Proposition 5.7. For the reverse implication, we observe that if V 1 , W 1 ∈ M d1,d (C) and V 2 , W 2 ∈ M d2,d (C) form TCP decompositions of (A 1 , B 1 , C 1 ) and (A 2 , B 2 , C 2 ), respectively (see Definition 2.7), then the direct sums Let us now provide two sufficient conditions for membership inside the TCP cone, which come from sufficient separability criteria proven in [24] and [51] respectively.
(A,B,C) ∈ LDOI + d and, moreover, Proof. First, note that X Recall from Proposition 4.11 that, given a matrix A ∈ M d (C), we define the diagonal matrices Proof. The result follows from the separability criterion in [51,Proposition 11], which states that any positive semidefinite matrix X ∈ M d (C) ⊗ M d (C) such that either is separable. Use Proposition 4.11 to write the partial traces of LDOI matrices in terms of A row,col .
Remark 5.12. In order for the condition in the lemma above to hold, the matrix A has to be row-(resp. column-)balanced. For example, in the = row case, one needs that every element must not be much smaller that the average of its row: We now proceed towards characterizing the extreme rays of the PCP d and TCP d cones, which will precisely translate into a characterization of the extreme rays of the cone of separable LDOI matrices through the X (i) isomorphisms (for i = 1, 2, 3) from page 5. It is pertinent to point out here that although the extreme rays of the full separable cone SEP d ⊂ M sa d 2 (C) are already well-known (these are the rank one projectors onto product vectors), the restriction of the domain to the intersection LDOI d ∩ SEP d ⊂ LDOI sa d might create new extremal rays which are not extremal in the full SEP d cone (see also Remark 5.16 in this regard). Theorem 5.13 below resolves these technicalities by providing a complete characterization of the extremal rays in PCP d and TCP d .
A similar result holds for the TCP d cone: the extremal rays are given by ( |v , |w ∈ C d \ {0}) Proof. We prove the case of PCP extremal rays, leaving the similar discussion of TCP matrices to the reader. Let us first show that extremal rays R + (A, B) ∈ ext PCP d are of the form (5.13). Assume A, B = 0 are written as in (11) with V, W ∈ M d,k (C). Let |v (1) , . . . , |v (k) ∈ C d (resp. |w (1) , . . . , |w (k) ) be the columns of V (resp. W ). A simple computation shows that where , proving the claim. Consider now arbitrary |v , |w ∈ C d \ {0}, the corresponding A, B from (5.13), and let us show and the matrix on the left hand side has unit rank (hence it is on an extremal ray in the cone of positive semidefinite matrices in M d (C)), there must exist scalars β t ∈ C such that |v (t) w (t) = β t |v w , which in turn is equivalent to the relation B (t) = |β t | 2 B (hence k t=1 |β t | 2 = 1). In order to conclude, we show next that the same colinearity relation holds for the A matrices:

First, note that the PCP pairs (A (t) , B (t) ) satisfy the PPT condition
ij | 2 with equality, since both the LHS and the RHS above are equal to |v A simple application of the arithmetic-geometric mean inequality then yields: Since the AM-GM inequality above is saturated, we get In other words, we have A ij | 2 = |β t | 4 |B ij | 2 , we obtain (note that we can take square roots since the entries of the matrices A (·) are non-negative) We now have and the proof is complete.

Remark 5.14. The extremal rays of the cone of completely positive matrices (i.e. PCP pairs of the form (A, A)) are well-known [1, Section 2.2]. They are of the form
for some vector |v ∈ C d . The situation is thus similar to the one described above.

Remark 5.15. Similar computations show that the extremal rays of the cone of TCP matrices of the form (A, B, B) are those of the form
for vectors |v , |w ∈ C d .

Remark 5.16.
It is well known that the convex cone of separable matrices in M sa d 2 (C) has extreme rays spanned by tensor products of rank-one projections:

Let (A, B, C) ∈ M d (C) ×3
C d be obtained from the vectors |v , |w = 0 as in Proposition 5.13, i.e., (A, B, C) is on an extremal ray in TCP d . Then, from Theorem 5.13, it is easy to deduce that In other words, the local averaging operation with respect to the random diagonal orthogonal matrices establishes a one-one correspondence between extreme rays of the cone of separable matrices in M sa d 2 (C) and LDOI sa d . Proposition 5.17. Let 0 = |v , |w ∈ C d be two non-zero vectors. The ranks of the extremal invariant separable matrices from Theorem 5.13 are as follows. Writing where σ(z) is the size of the support of the vector |z ∈ C d : In particular, the ranks of extremal separable invariant states can be as high as On the other hand, note that the matrix is singular, so its rank is either zero or one, depending on whether at least one of v i w j or v j w i is non-zero. We have thus Putting everything together, we obtain the result from the statement. For the case of X (2) (A,B) , note that we have, using again Corollary 4.2, rank X (2) (A,B) = rank |v w v w| + |{i = j : |v i | 2 |w j | 2 = 0}| proving the claim. Finally, the case of X (3) is left to the reader, as it can be easily deduced from the first two. Regarding the maximal values of the ranks, the claims can be proven by a careful analysis of the constrained optimization problems. We give below the proof in the CLDUI case, leaving the two others to the reader. Let us write σ(v w) = x, σ(v) = x + a, σ(w) = x + b for non-negative integers x, a, b. Given a triple x, a, b, there exist |v , |w as above if and only if x + a + b ≤ d, which is the only constraint we need to consider. With these new variables, we have which is a non-decreasing function in the integer x ∈ [0, d]. Hence, its maximum is attained at x = d − (a + b). Using this value of x, the claimed upper bound (containing now the indicator function) reads ab − (a + b)(d − 1) ≤ 0, which can easily be checked to be true, equality being attained at a = b = 0 corresponding to fully supported vectors |v , |w .
To illustrate the results above, let us consider the extremal matrices X (1,2,3) corresponding to the choice |v = |w = |diag I d (the all-ones vector). We have in this case (A,C) = I + F − P eq = (I − P eq ) + P s − P a = 2P s − P eq , X (2) (A,B) = I + dP ω − P eq = (I − P eq ) + dP ω , X where F = P s − P a is the flip operator, P s,a are, respectively, the projections on the symmetric and the anti-symmetric subspace of C d ⊗ C d , P ω is the projection on the maximally entangled state, and P eq is the rank-d projection Note that we have P ω ≤ P eq ≤ P s ≤ I in the lattice of projections.
Having described the convex structure of positive semidefinite and separable LDUI/CLDUI/LDOI matrices, we now move on to other convex cones relevant to quantum information theory. In the absence of invariance, there are five proper closed convex cones of M sa d 2 (C) which play crucial roles: These five cones, called respectively separable, positive partial transpose, positive semidefinite, decomposable and block-positive, satisfy the inclusion and duality relations from Figure 4. Observe that duality here is understood in the sense of convex cones. For example, it is evident that X is separable if and only if Tr We are thus led to the following definitions: Figure 5: Inclusion and duality (represented by arrows) relations for LDOI cones.
It is worthwhile to stress here that the above duality relations hold when the respective cones are understood as subsets of the corresponding vector space of self-adjoint LDOI matrices, that is LDOI sa d . Similar inclusions and dualities hold for the LDUI / CLDUI cones, see Figure 6. Note however that the positive semi-definite cones LDUI + d and CLDUI + d are not isomorphic.

Diagonal unitary and orthogonal covariant maps
We denote the set of all linear maps Φ : M d (C) → M d (C) by T d (C). The Choi-Jamiołkowski isomorphism identifies each map Φ ∈ T d (C) with a bipartite matrix J(Φ) ∈ M d (C) ⊗ M d (C) (also called the Choi matrix of Φ). In this section, we will use this isomorphism to study special families of covariant maps in T d (C) by linking them with the families of local diagonal unitary/orthogonal invariant bipartite matrices from Section 2. Before we begin, it is only fair to familiarize the readers with the basic theory of linear maps between matrix algebras. For a more detailed analysis, we refer the reader to [64,Chapter 2], [2,Chapters 2,3].
. Φ is called decomposable if it can be expressed as a sum of a completely positive and a completely copositive map and non-decomposable otherwise. Φ is called PPT if it is both completely positive and completely copositive. Φ is called entanglement breaking if (id ⊗Φ)(X) is separable for all positive semi-definite X ∈ M n (C) ⊗ M d (C). The dual map Φ * is defined as the unique adjoint of Φ with respect to the Hilbert-Schmidt inner product on M d (C). With all the definitions in place, we now state the Choi-Jamiołkowski isomorphism in its full glory. Lemma 6.1 (Choi-Jamiołkowski Isomorphism). [19,35,8] Define the linear bijection J :

entanglement breaking if and only if J(Φ) is separable.
In Lemma 6.1, part (5) appears in [61], while part (6) is due to [30]; see also [21] for a unified approach to the above presented results.
The action of a map Φ and its adjoint Φ * on M d (C) can be retrieved from the Choi matrix J(Φ), as is depicted through the following equations: This enables us to transform the unital and trace preserving property of Φ into partial trace conditions on its Choi matrix J(Φ), which forms the subject of the next Lemma. In Quantum Mechanics, physically allowed operations (called quantum channels) on quantum states are completely positive and trace preserving linear maps between the spaces of bounded operators on separable Hilbert spaces: Λ : B(H) → B(H ), see [28,29]. While positivity and trace preserving property is expected to ensure that Λ maps quantum states in B(H) to quantum states in B(H ), complete positivity stems from the physical restriction that a local quantum operation on an arbitrary multiparty system must also be positive. In a finite dimensional setting: H C d , H C d , quantum channels are precisely those linear maps Φ : M d (C) → M d (C) which are completely positive and trace preserving. Entanglement breaking maps represent noisy physical operations, so much so that a local partial action of such a map on a bipartite physical system destroys all entanglement -no matter how strong -present in the input state.
Positive but not completely positive maps, while not physically realizable, are important nevertheless, due to their crucial role in detecting entanglement of bipartite matrices. Using the duality relations in Figure 4 and the Choi-Jamiołkowski isomorphism, it can be shown that a positive semi-definite matrix for all positive maps Φ ∈ T d (C). In other words, for every entangled which is said to "detect" the entanglement in X. Moreover, if X is PPT entangled, then such a Φ must be non-decomposable. Obviously, Eq. This marks the end of our brief digression on the theory of linear maps between matrix algebras. We are now fully prepared to study different families of covariant maps in T d (C). The DUC and CDUC maps were introduced in [47,49], where they were dubbed mean unitary conjugation channels; we use here a different name to mirror the case of invariant bipartite matrices. We will denote the sets of DUC, CDUC and DOC maps in T d (C) by DUC d , CDUC d and DOC d respectively. Using the Choi-Jamiołkowski isomorphism, we can immediately start to construct links between the diagonal unitary/orthogonal covariant maps in T d (C) and the local diagonal unitary/orthogonal invariant matrices in M d (C) ⊗ M d (C). The following result is a pivotal step in this direction. With the help of Theorem 6.4, the task of extending the isomorphisms from Proposition 2.2 to the vector spaces of diagonal unitary/orthogonal covariant maps in T d (C) becomes effortless: We collect the explicit actions of Φ Recall that B = B − diag B and C = C − diag C. Let us take a moment to discuss the action of DOC maps on M d (C) in some detail. In the quantum setting, where Φ (A,B,C) is a quantum channel and Z = ρ is a quantum state, the first term in Eq. (38) can be interpreted as a "classical" quantum operation. It takes in the classical probability distribution |diag ρ ∈ R d + defined by the diagonal entries of the state ρ and returns A |diag ρ , which is again guaranteed to be a probability distribution since Φ

Remark 6.5. From the above discussed isomorphisms, it should be clear that
We now begin to study the properties of positivity and decomposability for maps in DOC d .
Proof. We obtain the characterization of positive maps in DOC d , leaving a near-identical discussion on decomposability to the reader. To begin with, we use the form of LDOI matrices from the discussion following Proposition 2.2 to compute the expression Tr[X and Remark 5.16 tells us that all the extremal TCP rays are of the form Proj LDOI (|ζ ⊗ η ζ ⊗ η|), for some |ζ , |η ∈ C d \ {0}. This completes the proof.
Although elegant, Theorem 6.6 will seldom be of practical use, as the stated conditions are too hard to check in practice. Drawing motivation from [15], we try to remedy this situation in the next couple of results. We first derive some easily verifiable constraints on matrix triples (A, B, C) which are necessary to ensure that the corresponding maps in DOC d are positive, see Proposition 6.7. Then, we present a set of sufficient conditions on triples (A, B, C) which guarantee that the associated maps in DOC d are both positive and decomposable, see Proposition 6.8.
(A,B,C) ∈ DOC d is positive. Then, • A is entrywise non-negative and B, C are self-adjoint, Proof. Since Φ (A,B,C) is positive, it is hermiticity preserving as well. Lemma 6.12 then tells us that A is real and B, C are self-adjoint. Now, consider an (arbitrary) extremal TCP ray (D, E, F ) ∈ M d (C) ×3 C d of the form D = |ζ ζ η η|, E = |ζ η ζ η| and F = |ζ η ζ η|, for some non-zero vectors |ζ , |η ∈ C d , see Theorem 5.13. By invoking Theorem 6.6, we can write: If we fix k, l ∈ [d] and choose |ζ , |η ∈ C d such that ζ k = η l = 1 and ζ i = η j = 0 if (i, j) = (k, l), then f (ζ, η) ≥ 0 ⇐⇒ A kl ≥ 0 for all k, l ∈ [d], i.e., A is entrywise non-negative. Now, let us try to further expand the above expression: If we impose the constraint that k < l and choose |ζ , |η ∈ C d such that ζ i = η i = 0 for all i / ∈ {k, l}, it is clear that It is now possible to choose {ζ i , η i } i=k,l in such a way that f (ζ, η) ≥ 0 implies the following inequality, which holds for all x k , y k , x l , y l ≥ 0 (recall that f (ζ, η) ≥ 0 holds for all |ζ , |η ∈ C d ): Assuming that A kk , A ll , A kl and A lk are non-zero, we can set and x k = y l = 1 to obtain: which is equivalent to the desired inequality as stated in the Lemma: If one or more entries of A are zero, we can set them equal to an arbitrarily small non-zero value to ensure that the above inequalities hold, and the final result will then follow by taking limits. C d be such that • A is entrywise non-negative and B, C are self-adjoint, Then Φ (A,B,C) ∈ DOC d is decomposable and thus positive.
Proof. Since positivity trivially follows from decomposability, it is sufficient to show that Φ (D,E,F ) ∈ LDOI d . Recall from Lemma 2.12 that . We now prove that the desired trace expression is non-negative: where the ante-penultimate inequality follows from the arithmetic-geometric mean inequality, the penultimate inequality follows from the fact that X (D,E,F ) ∈ LDOI d is PPT, and the final inequality follows from the hypothesis of the Proposition. This completes the proof.
It is clear that the conditions in Propositions 6.7 and 6.8 are equivalent for d = 2. This leads us to the following complete characterization of positivity for maps in DOC 2 , which generalizes similar results in [46,40] -these were obtained for the restricted class of positive maps which preserve diagonals (see Example 7.7). Recall that a matrix A in M d (C) is called row (resp. column) stochastic if it is entrywise non-negative and the sum of entries in each row (resp. column) equals one. By combining the results from Lemmas 6.12 and 6.13, it is straightforward to present a characterization of the class of quantum diagonal orthogonal covariant channels in T d (C).
. We now discuss the symmetries of the Choi matrices described in Proposition 4.3 in terms of the corresponding maps.

Important classes of DOC maps
This section contains a number of examples of (classes of) DOC maps. The examples of LDOI bipartite matrices from Section 3 can be seen, through the Choi-Jamiołkowski isomorphism, as examples of DOC maps. We list some of them, together with some important classes of maps discussed in the literature, below. One of the main achievements of the current work is realizing that all these linear maps fall under the same umbrella, and hence can be studied within a unified framework. A neatly summarized list of all the examples is presented in Table 2 towards the end of the section.

Example 7.1 (Identity and transposition).
The identity map id ∈ T d (C) corresponds to a CDUC map with A = I d and B = J d , whereas the transposition map ∈ T d (C) corresponds to a DUC map with A = I d and C = J d . While the identity map is clearly completely positive, transposition, on the other hand, is the most common example of a positive but not completely positive map. These maps are special examples of the more general class of diagonal-preserving maps, which is discussed in Example 7.7.

Example 7.2 (Classical maps).
Equations (36) and (37) entail that a map Φ ∈ DOC d is both DUC and CDUC if and only if both the associated matrices B and C are diagonal. These maps are then parameterized by a single matrix A ∈ M d (C), and have the following action: From the above equation, one understands that these maps completely discard the off-diagonal entries of its input, and act only on the diagonal part through the matrix A. In the quantum setting, these maps only change the classical probability distribution |diag ρ associated with the input state ρ, which earns them their classical nature. Notable examples from this class include the completely depolarizing and the completely dephasing maps, for which A = J d (see Example 7.6) and A = I d (see Example 7.7). Positivity, complete positivity and entanglement breaking property for maps in this class are all equivalent to the condition that A ∈ EWP d , as is clear from the discussion of the corresponding diagonal Choi matrices in Example 3.1.
Clearly, Φ S = Φ (A,B) ∈ CDUC d ⊂ DOC d for B = S and A = diag(S). Using Lemma 6.11, we can easily see that Φ S is completely positive if and only if S ∈ PSD d . The same lemma tells us that complete copositivity forces S to be diagonal, implying that the map Φ S is PPT if and only if S is diagonal and entrywise non-negative, in which case, it is entanglement breaking as well.
Quantum channels within the class of Schur multipliers are also known as generalized dephasing or Hadamard channels. From Lemma 6.14, it is clear that Φ S is a quantum channel if and only if S is a correlation matrix (Definition 3.6), see also [64,Section 4.1.3]. These channels are used to describe a decoherence type noise in quantum systems, since their action on a quantum state preserves the diagonal entries (see Example 7.7) and reduces the magnitude of the off-diagonal entries. The extreme case occurs when S = I d , in which case the associated channel entirely discards the off-diagonal part of the input, thus resulting in a completely dephasing action. It is perhaps worthwhile to mention that quantum channels (Lemma 6.14) within the CUC and UC classes of linear maps are important models of quantum noise and are known as depolarizing [38] and transpose depolarizing [18] channels, respectively.

Example 7.5 (Choi-type maps).
Consider the Choi map Φ ch ∈ T 3 (C) defined as: This was the first example of a positive non-decomposable map between matrix algebras, presented by Choi in the '70s [7,9,11]. Since then, many generalizations of this map have been proposed.
In [6], the authors introduced the family {Φ I (a,b,c) ∈ T 3 (C) : a, b, c ≥ 0} and studied constraints on the triple (a, b, c) ∈ R 3 which guarantee that the corresponding map is positive/decomposable: A slightly different variant Φ II (a,c1,c2,c3) ∈ T 3 (C) of the above maps was introduced in [39]: where a, c 1 , c 2 , c 3 ≥ 0. In higher dimensions, the family {τ d,k ∈ T d (C) : d ∈ N, k = 1, 2, . . . , d − 1} has received considerable attention [62,53,66,25], where the maps are defined in terms of a cyclic permutation matrix S ∈ M d (C): Finally, the most general d−dimensional Choi maps (parameterized by an entrywise non-negative matrix A ∈ M d (C)) have been analyzed in [26,17,15]: for matrices B, C ∈ M d (C) with diag B = diag C = I d , and correspond precisely to the LDOI Choi matrices from Example 3.6. The same example informs us that these maps are PPT if and only if B and C are correlation matrices, see Definition 3.6. If we restrict ourselves to the DUC/CDUC maps in this class, which are of the form then Proposition 3.7 can be immediately applied on the corresponding Choi matrices to deduce the following sequence of equivalences for i = 1, 2: Example 7.7 (Diagonal-preserving maps). [40,46] In [40,46], the authors studied the class of linear maps in T d (C) which fix diagonals. The positive maps in this class were shown to be of the form whereX,Ỹ ∈ M sa d (C) have zero diagonals. A distinguished element of this class is the completely dephasing map Z → diag(Z), which corresponds to the choiceX =Ỹ = 0 and is an element of DUC d ∩ CDUC d , see Example 7.3 as well. More generally, for A = I d , B =X + I d and C =Ỹ + I d , it is clear that ΦX ,Ỹ = Φ (3) (A,B,C) ∈ DOC d . We utilize Lemma 6.12 to infer that ΦX ,Ỹ is completely positive if and only ifỸ = 0 and B =X + I d is a correlation matrix, i.e. ΦX ,Ỹ is a Schur multiplier, see Example 7.3. Positivity of the maps ΦX ,Ỹ ∈ T d (C) was shown to be equivalent to decomposability if and only if d ≤ 3. This is clearly not true for positive maps in DOC d , as the celebrated Choi map in CDUC 3 ⊂ DOC 3 is positive and non-decomposable.
Example 7.8. In [50], the map Λ 3 ∈ T 3 (C) (defined in Eq. (53)) was shown to be positive and non-decomposable. This was later generalized in [58] to the positive non-decomposable map Λ d ∈ T d (C) for arbitrary d ∈ N, see Eq. (54). These maps were introduced in an effort to understand the structure of stable subspaces of extremal bistochastic (positive maps which are unital and trace-preserving) maps between matrix algebras, see [50].
By defining the matrices A, B and C ∈ M d (C) entrywise as follows we observe that Λ d = Φ (A,B,C) ∈ DOC d for all d ∈ N. We summarize the ensemble of cases discussed in this section in Table 2.

Kraus and Stinespring characterizations of DOC maps
This section aims to study maps in DOC d in terms of their so-called Kraus and Stinespring representations. We begin with a brief review of these representations for arbitrary linear maps in T d (C), and then proceed to give a general uniqueness result which links different minimal Kraus and Stinespring representations of a given map. This result will then be used to provide necessary

DOC d
Eqs. (55), (56) [50], [58]  and sufficient conditions on the minimal Kraus/Stinespring representation of a given map, in order for it to be DOC. Given a linear map Φ ∈ T d (C), it admits a representation of the form known as a Kraus representation, where are known as Kraus operators associated with the stated representation of Φ. For a given map Φ, the minimal number r of operators needed for such a representation to exist is known as the Choi rank of the map, which can be easily shown to be equal to the rank of its Choi matrix J(Φ); in the case of invariant matrices, see Corollary 4.2. A representation of Φ which uses the minimal number of Kraus operators n = r = rank(J(Φ)) is said to be minimal.
Given such a representation, we can define tensors P, Q ∈ C n ⊗ M d (C) (these can be thought of as mappings P, Q : C d → C n ⊗ C d ) as follows: where C n acts as an auxiliary space, such that the action of the map can be expressed as follows: This is known as a Stinespring representation of the given map. It is not too difficult to retrace one's steps in order to interchange between the Stinespring and Kraus representations. Graphically, the strings popping out vertically from the boxes correspond to the auxiliary space C n . If the map Φ ∈ T d (C) is completely positive, then the Kraus operators are forced to be equal, i.e., P i = Q i ∀i ∈ [n], and we get a representation of the form With the relevant background in place, we now link different minimal Kraus representations of an arbitrary map Φ ∈ T d (C) in the following Lemma.
Then, there exists an invertible matrix Z ∈ M r (C) such that the following equivalent relations hold: where I d ∈ M d (C) is the identity matrix. In case Φ ∈ T d (C) is completely positive so that P i = Q i and R i = S i ∀i ∈ [r], the invertible matrix Z ∈ M r (C) above is also unitary.
Proof. From the given representations of Φ, it is easy to see that the Choi matrix J(Φ) has the following rank one decompositions: Now, since r = rank(J(Φ)), Eq. (61) represents two full rank factorizations of J(Φ). Hence, from the uniqueness of full rank factorizations (see [56,Theorem 2]), there exists an invertible matrix Z ∈ M r (C) such that the required relations hold: Graphically, the above uniqueness result can be visualized by imagining that the wire connecting the R and S * matrices in Eq. (61) is replaced by the identity matrix I r = ZZ −1 . Finally, if Φ is completely positive, it is trivial to see that Z ∈ M r (C) must be unitary.
Equipped with Lemma 8.1, we now present the two main results of this section.
Theorem 8.2. Consider a linear map Φ ∈ T d (C) with r = rank(J(Φ)), such that Proof. Let us first assume that Φ ∈ T d (C) is DOC. Then, it is clear from Definition 6.3 that for A swift application of Lemma 8.1 then yields us the desired invertible matrix Z o ∈ M r (C). Conversely assume that such a Z o ∈ M r (C) exists for every diagonal O ∈ DO d . Then, a straightforward computation shows that the map Φ is DOC: Theorem 8.3. Let Φ ∈ T d (C) be completely positive with r = rank(J(Φ)), such that

DOC maps and triplewise complete positivity
In this section, we provide an alternate characterization of the family of diagonal orthogonal covariant maps in terms of invariant subspaces, which will be used to derive necessary and sufficient conditions for triplewise complete positivity of matrix triples in M d (C) ×3 C d (or equivalently, for the separability of matrices in LDOI d ). We will employ this characterization to provide an example of a non-TCP triple (A, B, C) ∈ M d (C) ×3 C d such that both (A, B) and (A, C) are PCP. Recall that if B = C, this is not possible, since a triple of the form (A, B, B) is TCP if and only if the pair (A, B) is PCP, as was shown in Proposition 3.4. In this process, we explicitly compute the partial action of a map Φ ∈ DOC d on a matrix X ∈ LDOI d , which is then connected to the operation of map composition in DOC d . Without further delay, we delve straight into the promised alternate characterization of the set DOC d .
Proof. For an arbitrary O ∈ DO d , Φ ∈ T d (C) and X ∈ LDOI d , it is evident that (Φ ⊗ id)(X) ∈ LDOI d if and only if the equality in the diagram given below holds, which is clearly equivalent to the condition that J(Φ) ∈ LDOI d or Φ ∈ DOC d . The case with the map id ⊗Φ can be proven similarly.
Then, for (A 1 , B 1 , C 1 ), (A 2 , B 2 , C 2 ) ∈ M d (C) ×3 C d , the following holds true: (A2,B2,C2) ) = X Proof. We wish to explicitly compute the following action: (A2,B2,C2) → X Proceeding diagrammatically, it is clear that where the equalities follow from Eq. (30) and Remark 6.5. By exploiting the isomorphism from Proposition 2.2, we can express the above diagram as in Figure 12. Figure  Notice that, as before, A i , B i and C i are matrices with the same off-diagonal entries as A i , B i and C i respectively, but with diag( A i ) = diag( B i ) = diag( C i ) = 0, for i = 1, 2. This leads us to the final expression in Figure 13 With the composition rule from Lemma 9.3 in hand, let us now consider two particular instances of it, for matrix pairs (A 1 , B 1 The following definition formulates these new rules in a more succinct fashion.

Remark 9.5. It is obvious from the above definition that
Next, we state and prove an important proposition, which connects all the composition rules on matrix pairs/triples introduced so far to the operations of map composition in DUC d , CDUC d and DOC d . But first, we need familiarity with the notion of stability under composition.
Proof. The stability results follow directly from Definition 6.3. It is also trivial to check that if Φ 1 ∈ DUC d and Φ 2 ∈ CDUC d (or vice-versa), then Φ 1 • Φ 2 ∈ DUC d , since ∀ U ∈ DU d and Z ∈ M d (C), the following equation holds: where (A, B, C) = (A 1 , B 1 , C 1 ) • (A 2 , B 2 , C 2 ). Notice that the Choi-Jamiołkowski isomorphism for DOC maps was implemented in obtaining the first and last implications above: Then, an amalgamation of the recently proved result and Eqs. (64), (65) immediately yields the desired composition rules.
The composition rule from Lemma 9.3 allows us to construct necessary and sufficient conditions on a triple (A, B, C) ∈ M d (C) ×3 C d which guarantee that it is triplewise completely positive -these are presented in Theorem 9.8 below. The reader is advised to keep the discussion from Section 5 in mind before proceeding further.
(D,E,F ) ∈ DOC d is positive. Proof. For the fist part, we observe that the following isomorphisms can be established from the discussion in Section 5 and the Choi-Jamiołkowski isomorphism (Lemma 6.1(2)): which shows that X is separable. The other direction of the proof is trivial. The second part follows directly from the first, since we know that X The simple necessary conditions that follow from triplewise completele positivity of (A, B, C) ∈ M d (C) ×3 C d (see Lemma 2.10) can be easily derived with the help of the previous Theorem.
Proof. Choose the (obviously positive) identity and transposition maps in DOC d from Example 7.1 and apply Theorem 9.8 to obtain the desired result.
Although the conditions in Corollary 9.9 are necessary for triplewise complete positivity, the following example elucidates that they are not sufficient, which is equivalent to the fact that PPT entangled matrices exist in LDOI d .

Conclusions and future directions
We have presented an elaborate study of the family of local diagonal unitary and orthogonal invariant bipartite matrices (LDUI d , CLDUI d , and LDOI d ), along with the accompanying class of diagonal unitary and orthogonal covariant maps between matrix algebras (DUC d , CDUC d , and DOC d ). By easing the analysis of several important properties of objects in these classes, the isomorphisms with the family of matrix pairs and triples with equal diagonals (M d (C) ×2 C d and M d (C) ×3 C d ) play an instrumental role in our endeavors. In particular, we show that the cone of separable LDOI matrices admits an equivalent description in terms of the cone of triplewise completely positive matrices, which generalizes the well-studied cone of completely positive matrices. We entirely determine the extreme rays of these cones, along with the cone of positive semi-definite LDOI matrices. We also spend considerable time on describing the linear structure of the vector space Ever since Choi discovered the first example of a positive non-decomposable map in the '70s, there has been immense interest in studying its generalizations, especially after the relatively recent Entanglement Theory associations were unraveled. We have seen that all Choi-type maps are a particular example of a much broader class defined by a unique covariance property. One of the merits of our work is to provide a unifying framework for the study of these maps, leveraging tools from linear and multi-linear algebra, and convex geometry, to obtain powerful characterizations of the relevant properties of these maps.
Several research prospects stem from our work. The membership problem in the cone of triplewise completely positive matrices (or equivalently, in the cone of separable LDOI matrices) is the most glaring one. Simple and easily verifiable sufficient conditions to guarantee that a matrix triple is or is not TCP are desirable. A significant attempt in this direction is made in [59], where crucial graph-theoretic techniques are implemented to explore a new variety of entanglement in both LDOI and arbitrary bipartite matrices. Other entanglement-theoretic properties of positive semi-definite LDOI matrices (like entanglement of formation, distillation, cost, and concurrence, to name a few) deserve further scrutiny. The cones of positive/decomposable linear maps between matrix algebras have evaded simple characterizations for quite some time now, which translates into similar difficulties while dealing with the intersection of these cones with the DOC d subspace. Characterization of these cones' convex structure has the potential to provide new insights into the theory of entanglement.
PPT square conjecture [12] posits that the composition of a PPT map in T d (C) with itself is entanglement breaking. In [60], we prove that this conjecture holds for DUC and CDUC maps, thus establishing its validity for a very large class and generalizing many known results from the literature. The analysis in [60] is based on the tools developed in the current paper, such as the composition formulas from Lemmas 9.3 and 9.7, as well as the separability results for LDOI matrices from Theorem 2.9 and Lemma 2.11.