scholarly journals Partially Coherent Direct Sum Channels

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 504
Author(s):  
Stefano Chessa ◽  
Vittorio Giovannetti
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Special Structure ◽  
Quantum Channels ◽  
Amplitude Damping ◽  
Direct Sum ◽  
Partially Coherent ◽  
Quantum Capacity ◽  
Necessary And Sufficient

We introduce Partially Coherent Direct Sum (PCDS) quantum channels, as a generalization of the already known Direct Sum quantum channels. We derive necessary and sufficient conditions to identify the subset of those maps which are degradable, and provide a simplified expression for their quantum capacities. Interestingly, the special structure of PCDS allows us to extend the computation of the quantum capacity formula also for quantum channels which are explicitly not degradable (nor antidegradable). We show instances of applications of the results to dephasing channels, amplitude damping channels and combinations of the two.

scholarly journals Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Brendan Goldsmith ◽  
Ketao Gong
Keyword(s):  
Abelian Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Sums ◽  
Direct Sum ◽  
Zero Entropy ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.

scholarly journals On periodic rings

2001 ◽  
Vol 25 (6) ◽  
pp. 417-420
Author(s):  
Xiankun Du ◽  
Qi Yi
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Sum ◽  
Periodic Rings ◽  
Necessary And Sufficient ◽  
Nil Ring ◽  
Positive Integers

It is proved that a ring is periodic if and only if, for any elementsxandy, there exist positive integersk,l,m, andnwith eitherk≠morl≠n, depending onxandy, for whichxkyl=xmyn. Necessary and sufficient conditions are established for a ring to be a direct sum of a nil ring and aJ-ring.

scholarly journals On Left Bipotent Near-Rings

1979 ◽  
Vol 22 (2) ◽  
pp. 99-107 ◽  
Author(s):  
J. L. Jat ◽  
S. C. Choudhary
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Singular Set ◽  
Direct Sum ◽  
Zero Divisors ◽  
Structure Theorems ◽  
Necessary And Sufficient ◽  
Regular Section

A near-ring N is defined to be left bipotent if Na = Na2 for each a in N. Many properties of such near-rings are proved in Section 1, and results of Chandran (4) are generalised. Most of the results are different from, and contrary to, the ring case. Necessary and sufficient conditions have also been obtained under which such near-rings become regular. Section 2 deals with left bipotent near-rings without zero divisors. Some structure theorems for direct sum decompositions and J(N) = (0) are proved and it is shown that for a left bipotent S-near-ring, the singular ‘set’ S(N) = 0. Necessary examples and counter examples are supplied.

Quantum codes from a class of constacyclic codes over finite commutative rings

2019 ◽  
Vol 19 (12) ◽  
pp. 2150003 ◽  
Author(s):  
Hai Q. Dinh ◽  
Tushar Bag ◽  
Ashish K. Upadhyay ◽  
Mohammad Ashraf ◽  
Ghulam Mohammad ◽  
...  
Keyword(s):  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Negacyclic Codes ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Necessary And Sufficient

Let [Formula: see text] be an odd prime, and [Formula: see text] be an integer such that [Formula: see text]. Using pairwise orthogonal idempotents [Formula: see text] of the ring [Formula: see text], with [Formula: see text], [Formula: see text] is decomposed as [Formula: see text], which contains the ring [Formula: see text] as a subring. It is shown that, for [Formula: see text], [Formula: see text], and it is invertible if and only if [Formula: see text] and [Formula: see text] are units of [Formula: see text]. In such cases, we study [Formula: see text]-constacyclic codes over [Formula: see text]. We present a direct sum decomposition of [Formula: see text]-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a [Formula: see text]-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over [Formula: see text], with better parameters than existing ones, are constructed from cyclic and negacyclic codes over [Formula: see text].

Constacyclic codes over the ring Fq[u,v,w]/〈u2 − 1, v2 − 1,w3 − w, uv − vu,vw − wv,wu − uw〉

2021 ◽  
Author(s):  
Shunhua Zhang
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Gray Map ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Direct Sum ◽  
Necessary And Sufficient

Let [Formula: see text] be the ring [Formula: see text], where [Formula: see text] for any odd prime [Formula: see text] and positive integer [Formula: see text]. In this paper, we study constacyclic codes over the ring [Formula: see text]. We define a Gray map by a matrix and decompose a constacyclic code over the ring [Formula: see text] as the direct sum of constacyclic codes over [Formula: see text], we also characterize self-dual constacyclic codes over the ring [Formula: see text] and give necessary and sufficient conditions for constacyclic codes to be dual-containing. As an application, we give a method to construct quantum codes from dual-containing constacyclic codes over the ring [Formula: see text].

Divisibility of Direct Sums in Torsion Theories

1976 ◽  
Vol 28 (1) ◽  
pp. 211-214 ◽  
Author(s):  
B. Sarath ◽  
K. Varadarajan
Keyword(s):  
Sufficient Conditions ◽  
Torsion Theory ◽  
Necessary And Sufficient Conditions ◽  
Countable Subset ◽  
Direct Sums ◽  
Direct Sum ◽  
Hereditary Torsion Theory ◽  
Torsion Theories ◽  
Necessary And Sufficient

Given a hereditary torsion theory on the category Mod R of right R-modules we obtain in this paper necessary and sufficient conditions for the direct sum of a given family of R-modules to be divisible for the torsion theory . Using this criterion we show that if is a family of R-modules having the property that is divisible for every countable subset K ol J then is itself divisible.

scholarly journals Necessary and sufficient conditions on measurements of quantum channels

2020 ◽  
Vol 476 (2236) ◽  
pp. 20190832 ◽  
Author(s):  
John Burniston ◽  
Michael Grabowecky ◽  
Carlo Maria Scandolo ◽  
Giulio Chiribella ◽  
Gilad Gour
Keyword(s):  
Quantum Measurement ◽  
Quantum Channel ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Necessary And Sufficient Conditions ◽  
Quantum Channels ◽  
Completely Positive ◽  
Principle Of Causality ◽  
Necessary And Sufficient

Quantum supermaps are a higher-order genera- lization of quantum maps, taking quantum maps to quantum maps. It is known that any completely positive and trace non-increasing (CPTNI) map can be performed as part of a quantum measurement. By providing an explicit counterexample we show that, instead, not every quantum supermap sending a quantum channel to a CPTNI map can be realized in a measurement on quantum channels. We find that the supermaps that can be implemented in this way are exactly those transforming quantum channels into CPTNI maps even when tensored with the identity supermap. We link this result to the fact that the principle of causality fails in the theory of quantum supermaps.

scholarly journals Different Characterizations of Large Submodules of QTAG-Modules

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Fahad Sikander ◽  
Alveera Mehdi ◽  
Sabah A. R. K. Naji
Keyword(s):  
Homomorphic Image ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules ◽  
Necessary And Sufficient

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. The study of large submodules and its fascinating properties makes the theory of QTAG-modules more interesting. A fully invariant submodule L of M is large in M if L+B=M, for every basic submodule B of M. The impetus of these efforts lies in the fact that the rings are almost restriction-free. This motivates us to find the necessary and sufficient conditions for a submodule of a QTAG-module to be large and characterize them. Also, we investigate some properties of large submodules shared by Σ-modules, summable modules, σ-summable modules, and so on.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Sign in / Sign up

Export Citation Format

Share Document