Decomposable Pauli diagonal maps and Tensor Squares of Qubit Maps
- URL: http://arxiv.org/abs/2006.14543v1
- Date: Thu, 25 Jun 2020 16:39:32 GMT
- Title: Decomposable Pauli diagonal maps and Tensor Squares of Qubit Maps
- Authors: Alexander M\"uller-Hermes
- Abstract summary: We show that any positive product of a qubit map with itself is decomposable.
We characterize the cone of decomposable ququart Pauli diagonal maps.
- Score: 91.3755431537592
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: It is a well-known result due to E. St{\o}rmer 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 S. Fillipov and
K. Magadov. 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.
Related papers
- A class of Schwarz qubit maps with diagonal unitary and orthogonal symmetries [0.0]
A class of unital qubit maps displaying diagonal unitary and symmetries is analyzed.
We provide a complete characterization of this class of maps showing intricate relation between positivity, operator Schwarz inequality, and complete positivity.
Our analysis leads to generalization of seminal Fujiwara-Algoet conditions for Pauli quantum channels.
arXiv Detail & Related papers (2024-04-16T20:37:16Z) - Infinite dimensional analogues of Choi matrices [0.0]
Choi matrices are useful to characterize positivity of maps as well as complete positivity.
It turns out that such correspondences are possible for every normal completely bounded map if and only if the factor is of type I.
We also define the notion of $k$-superpositive maps, which turns out to be equivalent to the property of $k$-partially entanglement breaking.
arXiv Detail & Related papers (2023-11-30T04:15:29Z) - Compositions and tensor products of linear maps between matrix algebras [0.0]
We first explain key notions from current quantum information theory and criteria for them in a coherent way.
These include separability/entanglement, Schmidt numbers of bi-partite states and block-positivity.
We show that the description of the dual cone with tensor products is possible only when the involving cones are mapping cones.
arXiv Detail & Related papers (2022-04-05T23:14:58Z) - Near-optimal estimation of smooth transport maps with kernel
sums-of-squares [81.02564078640275]
Under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds.
The object of interest for applications such as generative modeling is the underlying optimal transport map.
We propose the first tractable algorithm for which the statistical $L2$ error on the maps nearly matches the existing minimax lower-bounds for smooth map estimation.
arXiv Detail & Related papers (2021-12-03T13:45:36Z) - Annihilating Entanglement Between Cones [77.34726150561087]
We show that Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.
Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation.
arXiv Detail & Related papers (2021-10-22T15:02:39Z) - Pseudo-Euclidean Attract-Repel Embeddings for Undirected Graphs [73.0261182389643]
Dot product embeddings take a graph and construct vectors for nodes such that dot products between two vectors give the strength of the edge.
We remove the transitivity assumption by embedding nodes into a pseudo-Euclidean space.
Pseudo-Euclidean embeddings can compress networks efficiently, allow for multiple notions of nearest neighbors each with their own interpretation, and can be slotted' into existing models.
arXiv Detail & Related papers (2021-06-17T17:23:56Z) - Guaranteeing Completely Positive Quantum Evolution [2.578242050187029]
We transform an initial NCP map to a CP map through composition with the asymmetric depolarizing map.
We prove that the composition can always be made CP without completely depolarizing in any direction.
We show that asymmetric depolarization has many advantages over SPA in preserving the structure of the original NCP map.
arXiv Detail & Related papers (2021-04-30T20:53:19Z) - The PPT$^2$ conjecture holds for all Choi-type maps [1.5229257192293197]
We prove that the PPT$2$ conjecture holds for linear maps between matrix algebras which are covariant under the action of the diagonal unitary group.
Our proof relies on a generalization of the matrix-theoretic notion of factor width for pairwise completely positive matrices, and a complete characterization in the case of factor width two.
arXiv Detail & Related papers (2020-11-07T17:00:22Z) - Radiative topological biphoton states in modulated qubit arrays [105.54048699217668]
We study topological properties of bound pairs of photons in spatially-modulated qubit arrays coupled to a waveguide.
For open boundary condition, we find exotic topological bound-pair edge states with radiative losses.
By joining two structures with different spatial modulations, we find long-lived interface states which may have applications in storage and quantum information processing.
arXiv Detail & Related papers (2020-02-24T04:44:12Z) - A refinement of Reznick's Positivstellensatz with applications to
quantum information theory [72.8349503901712]
In Hilbert's 17th problem Artin showed that any positive definite in several variables can be written as the quotient of two sums of squares.
Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables.
arXiv Detail & Related papers (2019-09-04T11:46:26Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.