A class of optimal positive maps in $M_n$

• URL: http://arxiv.org/abs/2207.03821v2
• Date: Tue, 11 Apr 2023 09:28:01 GMT
• Title: A class of optimal positive maps in $M_n$
• Authors: Anindita Bera, Gniewomir Sarbicki and Dariusz Chru\'sci\'nski
• Abstract summary: It is proven that a certain class of positive maps in the matrix algebra $M_n$ consists of optimal maps. This class provides a generalization of a seminal Choi positive map in $M_3$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: It is proven that a certain class of positive maps in the matrix algebra $M_n$ consists of optimal maps, i.e. maps from which one cannot subtract any completely positive map without loosing positivity. This class provides a generalization of a seminal Choi positive map in $M_3$.

Related papers

• Optimality of generalized Choi maps in $M_3$ [0.0]
  We investigate when generalized Choi maps are optimal, i.e. cannot be represented as a sum of positive and completely positive maps. This property is weaker than extremality, however, it turns out that it plays a key role in detecting quantum entanglement.
  arXiv  Detail & Related papers  (2023-12-05T14:57:11Z)
• 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)
• Optimizing positive maps in the matrix algebra $M_n$ [0.0]
  We conjecture how to optimize a map $tau_n,k$ when $GCD(n,k)=2$ or 3. For $GCD(n,k)=2$, a series of analytical results are derived and for $GCD(n,k)=3$, we provide a suitable numerical analysis.
  arXiv  Detail & Related papers  (2023-09-18T09:47:55Z)
• A minimal completion theorem and almost everywhere equivalence for Completely Positive maps [0.0]
  It is shown that whenever such a completion is feasible there exists a unique minimal completion. This theorem is used to show that under some very general conditions a completely positive map almost everywhere equivalent to a quasi-pure map is actually equal to that map.
  arXiv  Detail & Related papers  (2023-06-28T06:25:40Z)
• Supervised Training of Conditional Monge Maps [107.78770597815242]
  Optimal transport (OT) theory describes general principles to define and select, among many possible choices, the most efficient way to map a probability measure onto another. We introduce CondOT, a multi-task approach to estimate a family of OT maps conditioned on a context variable. We demonstrate the ability of CondOT to infer the effect of an arbitrary combination of genetic or therapeutic perturbations on single cells.
  arXiv  Detail & Related papers  (2022-06-28T19:34:44Z)
• Learning Sound Localization Better From Semantically Similar Samples [79.47083330766002]
  Existing audio-visual works employ contrastive learning by assigning corresponding audio-visual pairs from the same source as positives while randomly mismatched pairs as negatives. Our key contribution is showing that hard positives can give similar response maps to the corresponding pairs. We demonstrate the effectiveness of our approach on VGG-SS and SoundNet-Flickr test sets.
  arXiv  Detail & Related papers  (2022-02-07T08:53:55Z)
• Nearly Optimal Regret for Learning Adversarial MDPs with Linear Function Approximation [92.3161051419884]
  We study the reinforcement learning for finite-horizon episodic Markov decision processes with adversarial reward and full information feedback. We show that it can achieve $tildeO(dHsqrtT)$ regret, where $H$ is the length of the episode. We also prove a matching lower bound of $tildeOmega(dHsqrtT)$ up to logarithmic factors.
  arXiv  Detail & Related papers  (2021-02-17T18:54:08Z)
• Positively Factorizable Maps [6.09170287691728]
  We study linear maps on $M_n(mathbbC)$ that factor through a tracial von Neumann algebra. The Choi matrix of a map of this kind which factors through an abelian von-Neumann algebra turns out to be a completely positive (CP) matrix.
  arXiv  Detail & Related papers  (2020-12-04T06:27:59Z)
• Decomposable Pauli diagonal maps and Tensor Squares of Qubit Maps [91.3755431537592]
  We show that any positive product of a qubit map with itself is decomposable. We characterize the cone of decomposable ququart Pauli diagonal maps.
  arXiv  Detail & Related papers  (2020-06-25T16:39:32Z)
• Signed Graph Metric Learning via Gershgorin Disc Perfect Alignment [46.145969174332485]
  We propose a fast general metric learning framework that is entirely projection-free. We replace the PD cone constraint in the metric learning problem with possible linear constraints per distances. Experiments show that our graph metric optimization is significantly faster than cone-projection schemes.
  arXiv  Detail & Related papers  (2020-06-15T23:15:12Z)
• Graph Metric Learning via Gershgorin Disc Alignment [46.145969174332485]
  We propose a fast general projection-free metric learning framework, where the objective $min_textbfM in mathcalS$ is a convex differentiable function of the metric matrix $textbfM$. We prove that the Gershgorin discs can be aligned perfectly using the first eigenvector $textbfv$ of $textbfM$. Experiments show that our efficiently computed graph metric matrices outperform metrics learned using competing methods in terms of classification tasks.
  arXiv  Detail & Related papers  (2020-01-28T17:44:01Z)

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.