This study delves into the efficacy of the Petz recovery map within the context of two paradigmatic quantum channels: dephasing and amplitude-damping. While prior investigations have predominantly ...focused on qubits, our research extends this inquiry to higher-dimensional systems. We introduce a novel, state-independent framework based on the Choi-Jamiołkowski isomorphism to evaluate the performance of the Petz map. By analyzing different channels and the (non-)unital nature of these processes, we emphasize the pivotal role of the reference state selection in determining the map's effectiveness. Furthermore, our analysis underscores the considerable impact of suboptimal choices on performance, prompting a broader consideration of factors such as system dimensionality.
•Performance of the Petz map for qudit channels.•Introduction of a state-independent framework that facilitates the extension to higher dimensions.•Robustness of the Petz map for higher dimensions.
Three affine SL(2,8)-unitals Möhler, Verena
Beiträge zur Algebra und Geometrie,
06/2023, Letnik:
64, Številka:
2
Journal Article
Recenzirano
SL
(
2
,
q
)
-unitals are unitals of order
q
admitting a regular action of
SL
(
2
,
q
)
on the complement of some block. We introduce three non-classical affine
SL
(
2
,
8
)
-unitals and their full ...automorphism groups. Each of those three affine unitals can be completed to at least two non-isomorphic unitals, leading to six pairwise non-isomorphic unitals of order 8.
We introduce the concept of approximately invertible elements in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at ...hand, and, on the other side, it is a direct extension of topological invertibility to non-unital algebras. Basic observations relate approximate invertibility with concepts of topological divisors of zero and density of (modular) ideals. We exemplify approximate invertibility in the group algebra, Wiener algebras, and operator ideals. For Wiener algebras with approximate identities (in particular, for the Fourier image of the convolution algebra), the approximate invertibility of an algebra element is equivalent to the property that it does not vanish. We also study approximate invertibility and its deeper connection with the Gelfand and representation theory in non-unital abelian Banach algebras as well as abelian and non-abelian C*-algebras.
In this article we look at the geometric structure of the feet of an orthogonal Buekenhout–Metz unital 𝓤 in PG(2,
). We show that the feet of each point form a set of type (0, 1, 2, 4). Further, we ...discuss the structure of any 4-secants, and determine exactly when the feet form an arc.
Graphs cospectral with NU(n + 1,q2), n ≠ 3 Ihringer, Ferdinand; Pavese, Francesco; Smaldore, Valentino
Discrete mathematics,
November 2021, 2021-11-00, Letnik:
344, Številka:
11
Journal Article
Recenzirano
Let H(n,q2) be a non–degenerate Hermitian variety of PG(n,q2), n≥2. Let NU(n+1,q2) be the graph whose vertices are the points of PG(n,q2)∖H(n,q2) and two vertices P1,P2 are adjacent if the line ...joining P1 and P2 is tangent to H(n,q2). Then NU(n+1,q2) is a strongly regular graph. In this paper we show that NU(n+1,q2), n≠3, is not determined by its spectrum.
Abstract
This paper addresses a number of problems concerning Buekenhout-Tits unitals in
$${{\,\textrm{PG}\,}}(2, q^2)$$
PG
(
2
,
q
2
)
, where
$$q = 2^{2e + 1}$$
q
=
2
2
e
+
1
and
$$e \ge 1$$
e
≥
1
.... We show that all Buekenhout-Tits unitals are equivalent under
$${{\,\textrm{PGL}\,}}(3, q^2)$$
PGL
(
3
,
q
2
)
addressing an open problem in Barwick and Ebert (Unitals in Projective Planes. Springer Monographs in Mathematics. Springer, New York, 2008), explicitly describe their stabiliser in
$$\textrm{P}\Gamma \textrm{L}(3, q^2)$$
P
Γ
L
(
3
,
q
2
)
expanding Ebert’s work in Ebert (J Algebraic Comb 6(2):133–140, 1997), and show that lines meet the feet of points not on
$$\ell _{\infty }$$
ℓ
∞
in at most four points. Finally, we show that feet of points not on
$$\ell _{\infty }$$
ℓ
∞
are not always a
$$\{0, 1, 2, 4\}$$
{
0
,
1
,
2
,
4
}
-set, in contrast to what happens for Buekenhout-Metz unitals Abarzúa et al (Adv Geom 18(2):229–236, 2018).
We show that a unital
in PG(2,
) containing a point
, such that at least
−
of the secant lines through
intersect
in a Baer subline, is an ovoidal Buekenhout–Metz unital (where
≈ 2
for
even and
≈
/2 ...for
odd).
We give a general construction for unitals of order q admitting an action of SU(2,q). The construction covers the classical Hermitian unitals, Grüning’s unitals in Hall planes and at least one ...unital of order four where the translation centers fill precisely one block. For the latter unital, we determine the full group of automorphisms and show that there are no group-preserving embeddings into (dual) translation planes of order 16.
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