Commutativity of operators on the lattice of existence varieties

1997 ◽  
Vol 123 (4) ◽  
pp. 337-364 ◽  
Author(s):  
Norman R. Reilly ◽  
Shuhua Zhang
Keyword(s):  
Commutativity Of Operators

ON APPLICATIONS OF PI-ALGEBRAS IN THE ANALYSIS OF QUANTUM CHANNELS

2012 ◽  
Vol 10 (08) ◽  
pp. 1241007 ◽  
Author(s):  
ANDRZEJ JAMIOŁKOWSKI
Keyword(s):  
Quantum System ◽  
Explicit Form ◽  
Polynomial Identity ◽  
Special Class ◽  
Error Correcting Codes ◽  
Quantum Channels ◽  
Linear Transformations ◽  
Positive Map ◽  
Commutativity Of Operators ◽  
Quantum Error

In this paper, we discuss some constructive procedures which can be used in characterizations of linear transformations which preserve the set of states of a fixed quantum system. Our methods are based on analyzing an explicit form of a linear positive map in its Kraus representation. In particular, we discuss the so-called partial commutativity of operators and its applications to investigation of decoherence-free subspaces. These subspaces can also be considered as a special class of quantum error correcting codes. Using the concept of standard polynomials and Amitsur–Levitzki theorem and other ideas from the so-called polynomial identity algebras (PI-algebras) we discuss some effective algorithms for analyzing properties of quantum channels.

scholarly journals QUANTUM DIAGONALIZATION METHOD IN THE TAVIS–CUMMINGS MODEL

2005 ◽  
Vol 02 (03) ◽  
pp. 425-440 ◽  
Author(s):  
KAZUYUKI FUJII ◽  
KYOKO HIGASHIDA ◽  
RYOSUKE KATO ◽  
TATSUO SUZUKI ◽  
YUKAKO WADA
Keyword(s):  
Differential Geometry ◽  
Explicit Form ◽  
Quantum Physics ◽  
Evolution Operator ◽  
Deep Structure ◽  
Point Of View ◽  
Diagonalization Method ◽  
Commutativity Of Operators ◽  
Quantum Matrix

To obtain the explicit form of evolution operator in the Tavis–Cummings model we must calculate the term e -itg(S+⊗a+S-⊗ a† explicitly which is very hard. In this paper we try to make the quantum matrix A ≡ S+ ⊗ a +S- ⊗ a† diagonal to calculate e -itgA and, moreover, to know a deep structure of the model. For the case of one, two and three atoms we give such a diagonalization which is the first nontrivial examples as far as we know, and reproduce the calculations of e -itgA given in quant-ph/0404034. We also give a hint to an application to non-commutative differential geometry. However, a quantum diagonalization is not unique and is affected by some ambiguity arising from the non-commutativity of operators in quantum physics. Our method may open a new point of view in mathematical or quantum physics.

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