Transferring elements of a density matrix

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 81, Issue 1, 2010, Pages

  Allahverdyan, Armen E ^a   Hovhannisyan, Karen V ^a,b  

^a YEREVAN PHYSICS INSTITUTE   (Armenia)

^b YEREVAN STATE UNIVERSITY   (Armenia)

Author keywords

[No Author keywords available]

Indexed keywords

A-DENSITY; DIAGONAL ELEMENTS; DIAGONAL MATRIX ELEMENTS; DIAGONALIZATIONS; ELEMENT TRANSFER; FINAL DENSITY; FINAL STATE; GENERAL ASPECTS; IMAGINARY PARTS; INITIAL DENSITY; MATRIX; MATRIX ELEMENTS; NONDIAGONAL ELEMENTS; QUANTUM MEASUREMENT; QUANTUM MECHANICS; STATE DISTURBANCE; STATE TRANSFER; UNCERTAINTY RELATION;

COMMERCE; LINEAR ALGEBRA; UNCERTAINTY ANALYSIS;

QUANTUM THEORY;

EID: 74549193574     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.81.012312     Document Type: Article

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51. Consider the differential dλ of the matrix λ over an infinitesimal increments of two independent real variables x Reλac and y Imλac [see Eqs. (7) and (9)]: dλ=Θac(dx+idy) +Θac†(dx-idy).To develop the notion of the matrix gradient (similar to the usual gradient) we first calculate the Euclidean norm ||dλ|| of dλ: ||dλ2||=(dx2+dy2)[tr(A2)nx2+tr(B2)ny2+2tr(AB)nxny],where AΘac+Θac†, Bi(Θac-Θac†) and where (nx,ny)=(dx,dy)/dx2+dy2is the normal vector over the direction of the small increment. The (modulus of) matrix gradient is defined by maximizing over n(nx,ny): 14maxn||dλ2||=12(dx2+dy2)[tr(ΘacΘac†) +||tr(Θac2)||].As follows from tr(AB)≤tr(A2)tr(B2), the quantity maxn||dλ2||4(dx2+dy2)is limited from above by tr(ΘacΘ ac†)=||Θac2||, and from below by 12tr(ΘacΘac†). Thus, ||Θac2|| can be taken as an adequate characteristics of the matrix gradient.