Can Wigner distribution functions with collisions satisfy complete positivity and energy conservation?

To avoid the computational burden of many-body quantum simulation, the interaction of an electron with a photon (phonon) is typically accounted for by disregarding the explicit simulation of the photon (phonon) degree of freedom and just modeling its effect on the electron dynamics. For quantum models developed from the (reduced) density matrix or its Wigner–Weyl transformation, the modeling of collisions may violate complete positivity (precluding the typical probabilistic interpretation). In this paper, we show that such quantum transport models can also strongly violate the energy conservation in the electron–photon (electron–phonon) interactions. After comparing collisions models to exact results for an electron interacting with a photon, we conclude that there is no fundamental restriction that prevents a collision model developed within the (reduced) density matrix or Wigner formalisms to satisfy simultaneously complete positivity and energy conservation. However, at the practical level, the development of such satisfactory collision model seems very complicated. Collision models with an explicit knowledge of the microscopic state ascribed to each electron seems recommendable (Bohmian conditional wavefunction), since they allow to model collisions of each electron individually in a controlled way satisfying both complete positivity and energy conservation.

Complete Positivity of Two Class of Maps Involving Depolarizing and Transpose-Depolarizing Channels

In this note, we mainly study the necessary and sufficient conditions for the complete positivity of generalizations of depolarizing and transpose-depolarizing channels. Specifically, we define [Formula: see text] and [Formula: see text], where [Formula: see text] (the set of all bounded linear operators on the finite-dimensional Hilbert space [Formula: see text] is given and [Formula: see text] is the transpose of [Formula: see text] in a fixed orthonormal basis of [Formula: see text] First, we show that [Formula: see text] is completely positive if and only if [Formula: see text] is a positive map, which is equivalent to [Formula: see text] Moreover, [Formula: see text] is a completely positive map if and only if [Formula: see text] and [Formula: see text] At last, we also get that [Formula: see text] is a completely positive map if and only if [Formula: see text] with [Formula: see text] for all [Formula: see text] where [Formula: see text] are eigenvalues of [Formula: see text].

The Crossed Product of Finite Hopf C*-Algebra and C*-Algebra

Let H be a finite Hopf C*-algebra and A a C*-algebra of finite dimension. In this paper, we focus on the crossed product A⋊H arising from the action of H on A, which is a ∗-algebra. In terms of the faithful positive Haar measure on a finite Hopf C*-algebra, one can construct a linear functional on the ∗-algebra A⋊H, which is further a faithful positive linear functional. Here, the complete positivity of a positive linear functional plays a vital role in the argument. At last, we conclude that the crossed product A⋊H is a C*-algebra of finite dimension according to a faithful ∗- representation.

Utility of p63 and AMACR in Differentiating Benign and Malignant Prostatic Lesions

Background: Prostate cancer is the sixth most common cancer in the world. Since treatment options and prognosis of prostatic adenocarcinomas and benign lesions differ significantly, so, the current study using p63and AMACR was carried out with aim to evaluate the utility of immunohistochemistry in resolving ambiguous lesions of prostate. Methods: The study was conducted on 130 prostatic specimens which included prostate biopsy, TURP and prostatectomy specimens. Routine hematoxylin and eosin staining and immunohistochemical staining using AMACR, p63 monoclonal antibody marker were performed. Results: Total of 130 cases of prostate samples were studied out of which benign lesions were seen in 102 cases (78.5%) and malignant carcinoma in 28 cases (21.5%). This study showed p63 had a sensitivity of 92.86% and specificity of 100% whereas AMACR has a sensitivity of 96.4% and specificity of 95%. BPH with prostatitis was a common finding in majority of benign lesions. All cases of LGPIN 15 cases (11.5%) were histologically associated with BPH, showing complete positivity in 12 cases (80%) and partial positivity in 3 cases (20%) with p63 immunostaining. HGPIN (1.5%) were associated with prostatic adenocarcinoma showing focal positivity whereas adenocarcinoma showed complete negativity of p63 expression (100%) and positive cytoplasmic staining with AMACR. Comparative study done between DRE, PSA, final histopathological diagnosis, expression of p63 and AMACR immunostaining gives highly significant p value of 0.001(<0.05). Conclusion: p63 and AMACR are reliable markers which can be used in morphologically difficult cases.

Complete positivity and distance-avoiding sets

We introduce the cone of completely positive functions, a subset of the cone of positive-type functions, and use it to fully characterize maximum-density distance-avoiding sets as the optimal solutions of a convex optimization problem. As a consequence of this characterization, it is possible to reprove and improve many results concerning distance-avoiding sets on the sphere and in Euclidean space.