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arXiv:2110.11825 · Communications in Mathematical Physics

Annihilating Entanglement Between Cones

Guillaume Aubrun, Alexander Müller‐Hermes

Published Dec 27, 2022• 8 citations
arXivHTMLPDF

Abstract

Abstract Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform thek-fold maximal tensor product of a cone $${\textsf{C}}_1$$ C1 into thek-fold minimal tensor product of a cone $${\textsf{C}}_2$$ C2 , and the pair $$({\textsf{C}}_1,{\textsf{C}}_2)$$ (C1,C2) is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that $$({\textsf{C}}_1,{\textsf{C}}_2)$$ (C1,C2) is resilient if either $${\textsf{C}}_1$$ C1 or $${\textsf{C}}_2$$ C2 is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.

DOI
10.1007/s00220-022-04621-5

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Abstract Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform thek-fold maximal tensor product of a cone C1{\textsf{C}}_1C1​ C1 into thek-fold minimal tensor product of a cone C2{\textsf{C}}_2C2​ C2 , and the pair (C1,C2)({\textsf{C}}_1,{\textsf{C}}_2)(C1​,C2​) (C1,C2) is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that (C1,C2)({\textsf{C}}_1,{\textsf{C}}_2)(C1​,C2​) (C1,C2) is resilient if either C1{\textsf{C}}_1C1​ C1 or C2{\textsf{C}}_2C2​ C2 is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.
Linear maps between finite-dimensional ordered vector spaces with orders induced by proper cones C A and C B are called entanglement breaking if their partial application sends the maximal tensor product K ⊗ max C A into the minimal tensor product K ⊗ min C B for any proper cone K . We study the larger class of Lorentz-entanglement breaking maps where K is restricted to be a Lorentz cone of any dimension, i.e., any cone over a Euclidean ball. This class of maps appeared recently in the study of asymptotic entanglement annihilation and it is dual to the linear maps factoring through Lorentz cones. Our main results establish connections between these classes of maps and operator ideals studied in the theory of Banach spaces. For operators u : X → Y between finite-dimensional normed spaces X and Y we consider so-called central maps which are positive with respect to the cones C A = C X and C B = C Y . We show how to characterize when such a map factors through a Lorentz cone and when it is Lorentz-entanglement breaking by using the Hilbert-space factorization norm γ 2 and its dual ⁎ γ 2 ⁎ . We also study the class of Lorentz-entanglement annihilating maps whose local application sends the Lorentzian tensor product C A ⊗ L C A into the minimal tensor product C B ⊗ min C B . When C A is a cone over a finite-dimensional normed space and C B is a Lorentz cone itself, the central maps of this kind can be characterized by the 2-summing norm π 2 . Finally, we prove interesting connections between these classes of maps for general cones, and we identify examples with particular properties, e.g., cones with an analogue of the 2-summing property.
Motivated by the importance of the purification principle in quantum theory and generalized probabilistic theories, we study purifications using only the geometry of a finite-dimensional proper convex cone. We prove an existence theorem for indecomposable cones and intermediate tensor cones containing the maximally entangled state; in particular, every interior point of an indecomposable homogeneous cone admits a purification. This applies to Lorentz cones, for example. We also give a criterion for uniqueness up to local automorphisms. On the boundary, we show that if every proper face of CCC is simplicial, then only pure points can admit purifications, and we demonstrate that this conclusion fails in the presence of non-simplicial faces. Examples involving positive semidefinite cones, Lorentz cones, kkk-positive maps, PPT tensors, and polyhedral cones illustrate both existence and non-uniqueness phenomena.
We introduce a 3-parameter class of maps (1) acting on a bipartite system which are a natural generalisation of the depolarizing channel (and include it as a special case). Then, we find the exact regions of the parameter space that alternatively determine a positive, completely positive, entanglement-breaking, or entanglement-annihilating map. This model displays a much richer behaviour than the one shown by a simple depolarizing channel, yet it stays exactly solvable. As an example of this richness, positive partial transposition but not entanglement-breaking maps is found in Theorem 2. A simple example of a positive yet indecomposable map is provided (see the Remark at the end of Section IV). The study of the entanglement-annihilating property is fully addressed by Theorem 7. Finally, we apply our results to solve the problem of the entanglement annihilation caused in a bipartite system by a tensor product of local depolarizing channels. In this context, a conjecture posed in the work of Filippov [J. Russ. Laser Res. 35, 484 (2014)] is affirmatively answered, and the gaps that the imperfect bounds of Filippov and Ziman [Phys. Rev. A 88, 032316 (2013)] left open are closed. To arrive at this result, we furthermore show how the Hadamard product between quantum states can be implemented via local operations.
A pair of proper cones (C1,C2)(\mathsf{C}_1,\mathsf{C}_2)(C1​,C2​) is said to have the Lorentz factorization property (LFP) if every (C1,C2)(\mathsf{C}_1,\mathsf{C}_2)(C1​,C2​)-positive map factors through a direct sum of Lorentzian cones, i.e., cones over Euclidean balls. Clearly, (C1,C2)(\mathsf{C}_1,\mathsf{C}_2)(C1​,C2​) has the LFP if either C1\mathsf{C}_1C1​ or C2\mathsf{C}_2C2​ is a direct sum of Lorentzian cones, and our main goal is to find other examples. We show that such examples cannot be found for pairs (C1,C2)(\mathsf{C}_1,\mathsf{C}_2)(C1​,C2​) where C1=C2\mathsf{C}_1=\mathsf{C}_2C1​=C2​, or in the case where both C1\mathsf{C}_1C1​ and C2\mathsf{C}_2C2​ are polyhedral. We also focus on the case where C1=C□\mathsf{C}_1=\mathsf{C}_\squareC1​=C□​ is the square-based cone in R3\mathbf{R}^3R3. Here, we show that (C□,C)(\mathsf{C}_\square,\mathsf{C})(C□​,C) has the LFP whenever C\mathsf{C}C is a symmetric cone, i.e., a direct sum of Lorentz cones, cones of positive semidefinite matrices over the real numbers, complex numbers or quaternions, and the cone of 3×33\times 33×3 positive semidefinite matrices over the octonions. We leave open the question whether there are more examples, but we show that this list cannot be extended by any strictly convex cone C\mathsf{C}C or for a cone C\mathsf{C}C with dim(C)≤5\text{dim}(\mathsf{C})\leq 5dim(C)≤5. Finally, we discuss an application to a problem in quantum information theory.
We provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors. This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. Using this framework, we construct explicit PPT entangled states in three or more qutrits, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d >= 3 and n >= 3 parties. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions. We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.
We analyze linear maps on matrix algebras that become entanglement breaking after composing a finite or infinite number of times with themselves. This means that the Choi matrix of the iterated linear map becomes separable in the tensor product space. If a linear map becomes entanglement breaking after finitely many iterations, we say that the map has a finite index of separability. In particular, we show that every unital positive partial transpose (PPT) channel has a finite index of separability and that the class of unital channels that have a finite index of separability is a dense subset of the unital channels. We construct concrete examples of maps which are not PPT but have a finite index of separability. We prove that there is a large class of unital channels that are asymptotically entanglement breaking. Our work is motivated by Christandl’s PPT-squared conjecture. This conjecture states that every PPT channel, when composed with itself, becomes entanglement breaking.