It is a well-known result due to Størmer [Acta Math. 110, 233–278 (1963)] that every positive qubit map is decomposable into a sum of a completely positive map and a completely copositive map. Here, we generalize this result to tensor squares of qubit maps. Specifically, we show that any positive tensor product of a qubit map with itself is decomposable. This solves a recent conjecture by Filippov and Magadov [J. Phys. A: Math. Theor. 50(5), 055301 (2017)]. We contrast this result with examples of non-decomposable positive maps arising as the tensor product of two distinct qubit maps or as the tensor square of a decomposable map from a qubit to a ququart. To show our main result, we reduce the problem to Pauli diagonal maps. We then characterize the cone of decomposable ququart Pauli diagonal maps by determining all 252 extremal rays of ququart Pauli diagonal maps that are both completely positive and completely copositive. These extremal rays split into three disjoint orbits under a natural symmetry group, and two of these orbits contain only entanglement breaking maps. Finally, we develop a general combinatorial method to determine the extremal rays of Pauli diagonal maps that are both completely positive and completely copositive between multi-qubit systems using the ordered spectra of their Choi matrices. Classifying these extremal rays beyond ququarts is left as an open problem.
It is a well-known result due to Størmer [Acta Math. 110, 233–278 (1963)] that every positive qubit map is decomposable into a sum of a completely positive map and a completely copositive map. Here, we generalize this result to tensor squares of qubit maps. Specifically, we show that any positive tensor product of a qubit map with itself is decomposable. This solves a recent conjecture by Filippov and Magadov [J. Phys. A: Math. Theor. 50(5), 055301 (2017)]. We contrast this result with examples of non-decomposable positive maps arising as the tensor product of two distinct qubit maps or as the tensor square of a decomposable map from a qubit to a ququart. To show our main result, we reduce the problem to Pauli diagonal maps. We then characterize the cone of decomposable ququart Pauli diagonal maps by determining all 252 extremal rays of ququart Pauli diagonal maps that are both completely positive and completely copositive. These extremal rays split into three disjoint orbits under a natural symmetry group, and two of these orbits contain only entanglement breaking maps. Finally, we develop a general combinatorial method to determine the extremal rays of Pauli diagonal maps that are both completely positive and completely copositive between multi-qubit systems using the ordered spectra of their Choi matrices. Classifying these extremal rays beyond ququarts is left as an open problem.
Both completely positive and completely copositive maps stay decomposable under tensor powers, i.e., under tensoring the linear map with itself. But are there other examples of maps with this property? We show that this is not the case: Any decomposable map, that is neither completely positive nor completely copositive, will lose decomposability eventually after taking enough tensor powers. Moreover, we establish explicit bounds to quantify when this happens. To prove these results, we use a symmetrization technique from the theory of entanglement distillation and analyze when certain symmetric maps become non-decomposable after taking tensor powers. Finally, we apply our results to construct new examples of non-decomposable positive maps and establish a connection to the positive partial transpose squared conjecture.
Abstract We analyze positivity of a tensor product of two linear qubit maps, Φ 1 ⊗ Φ 2 . Positivity of maps Φ 1 and Φ 2 is a necessary but not a sufficient condition for positivity of Φ 1 ⊗ Φ 2 . We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both Φ 1 and Φ 2 are completely positive or completely co-positive. We find necessary and (separately) sufficient conditions for n -tensor-stable positive qubit maps, i.e. such qubit maps Φ that Φ ⊗ n is positive. Particular cases of 2- and 3-tensor-stable positive qubit maps are fully characterized, and the decomposability of 2-tensor-stable positive qubit maps is discussed. The case of non-unital maps is reduced to the case of appropriate unital maps. Finally, n -tensor-stable positive maps are used in characterization of multipartite entanglement, namely, in the entanglement depth detection.
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.
An alternative, geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra has been presented. The premise of the proof is the identification of positive maps with operators preserving the Lorentz cone in four dimensions, and it allows to decompose the positive maps with respect to those preserving the boundary of the cone. In addition, useful conditions implying complete positivity of a map of have been given, together with a sufficient condition for complete positivity of an extremal Schwarz map of . Lastly, following the same geometrical approach, a description in topological terms of maps that are simultaneously completely positive and completely copositive has been presented.
Entanglement swapping is a fundamental operation in quantum repeaters for establishing entanglement between distant parties. The positive partial transpose (PPT) squared conjecture asks whether two PPT entangled links can generate terminal entanglement through entanglement swapping, or equivalently, whether the composition of two PPT maps is always entanglement breaking. Motivated by this conjecture, we investigate the map-composition problem beyond the PPT setting. For qutrit completely positive (CP) maps, we prove that the composition of any CP map whose Choi matrix is 1-undistillable with any CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in either order. Moreover, we show that the cone of 1-undistillable CP maps is exactly the largest qutrit cone of CP maps whose composition with every CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in both orders. Finally, although map composition captures only the standard maximally entangled outcome in entanglement swapping, we prove that any 1-undistillable two-qutrit state and any state of Schmidt number at most two cannot generate terminal entanglement under an arbitrary selective measurement on the intermediate systems.
Let Φ: Md → M3 and Ψ: M3 → Md be completely positive and completely copositive (PPT) maps, and let W be the Choi matrix of Ψ ∘ Φ. The PPT2 conjecture asserts that W is separable. We study the twirl of W by the group of local diagonal unitaries D ⊗ D̄, whose separability is equivalent to the statement that no diagonal-unitary-covariant witness detects W. The twirl is determined by Aij = ⟨ij|W|ij⟩ and the coherence matrix Bik = ⟨ii|W|kk⟩. We prove that for every d the twirl is separable whenever rank B ≤ 2. The proof factors W through Gram data Ta = Ga*Ha, classifies the two-dimensional spaces span{Ta} into five types, and combines a peeling lemma with known separability criteria for low-rank PPT states and, for two of the five types, with the theorem that composites of PPT maps through a qubit are entanglement breaking. A peel at an index c writes Ψ (or Φ) as the sum of an entanglement-breaking part and a PPT remainder whose coherence vanishes at c. Chains of peels, interleaved with the removal of zero rows of B, close further cases at every coherence rank, and for d = 4 a single peel suffices. For every d ≥ 3 the twirl plus the diagonal operator ((d − 3)/2) ∑i Bii|ii⟩⟨ii| is separable. At d = 4, if the compression of the Choi matrix of Φ or of Ψ* to a set of at least two outer indices is a sum of product states with linearly independent outer factors, then Φ, respectively Ψ, is entanglement breaking. At d = 4 and coherence rank 3 with a linear relation ∑a λaTa = 0 in which some λa vanishes, the twirl is separable, and Ψ ∘ Φ is entanglement breaking when exactly one λa vanishes. A cycle criterion for an arbitrary factorization B = FF* yields explicit product-state certificates. Numerically, this criterion certifies the twirl at each of the 42 generic configurations without peels produced by our random semidefinite searches at d = 4. At some pairs of extreme PPT maps it fails for every factorization of B. In particular, for every choice of Kraus operators, it fails when the columns of F are the diagonals of the products of a Kraus operator of Ψ with one of Φ. The twirl is nevertheless separable at these pairs: the Choi matrices of Φ and of Ψ* are supported on S ⊗ ℂ3 and S′ ⊗ ℂ3 for three-dimensional subspaces S, S′ ⊆ ℂd, and the known case M3 → M3 → M3 of the PPT2 conjecture applies. For d ≤ 4 and every coherence rank, the twirl is separable for every composite Φ* ∘ Ξ ∘ Φ in which Ξ is a sum of conjugations by positive semidefinite matrices. This follows from a criterion of Johnston and MacLean for completely positive coherence matrices. For d ≤ 4 we extend their criterion to coherence matrices that become entrywise nonnegative under conjugation by a diagonal unitary, provided that, around each cycle of the support graph of B, the product of the entries of A is the same in both directions. We also show that the number of product terms in a separable decomposition of the twirl can exceed the coherence rank, even for entanglement-breaking pairs through a qutrit. Finally, given the theorem that composites of PPT maps through a qubit are entanglement breaking, we show that W + (1/5) Φ*(I)⊤ ⊗ Ψ(I) is separable for every d, that for d ≥ 4 no pair-independent average of certificates from this theorem lowers the constant 1/5, even with PPT2 for maps on M3 and the separability of PPT states on ℂ2 ⊗ ℂ3, and that certificates adapted to the pair go beyond it: exact rational certificates show that W − (1/20) Φ*(I)⊤ ⊗ Ψ(I) is separable for an explicit pair at d = 4, and that W + (1/10) Φ*(I)⊤ ⊗ Ψ(I) is separable for an explicit Φ: M4 → M3 and every PPT map Ψ. Given the same theorem, for Φ: M3 → M3 a constant below 1/5 works uniformly: there is c < 1/5 such that W + c Φ*(I)⊤ ⊗ Ψ(I) is separable for every such Φ and every PPT map Ψ: M3 → Mn. Version 2 adds Theorem 10.19: no PPT channel M₃ → M₃ is rigid, so Λ(3) < 1/5. Given the theorem that composites of PPT maps through a qubit are entanglement breaking, it follows that a single constant below 1/5 applies to every pair of PPT maps M₃ → M₃ → Mₙ (Corollary 10.20). Exact certificates, checked in Lean, show that the constant of certificates adapted to one factor is at most 1/10 at two explicit points built from the Tiles unextendible product basis (Remark 10.21). The accompanying Lean code now contains 2688 theorems, 1833 of which are checked by Lean's kernel alone. Mathematics Subject Classification (2020): 81P40; 15A69, 47L07.