scholarly journals Systems of Imprimitivity for the Clifford group

2014 ◽  
Vol 14 (3&4) ◽  
pp. 339-360
Author(s):  
D.M. Appleby ◽  
Ingemar Bengtsson ◽  
Stephen Brierley ◽  
Asa Ericsson ◽  
Markus Grassl ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Positive Operator ◽  
Arbitrary Dimension ◽  
Standard Representation ◽  
One Dimensional ◽  
Positive Operator Valued Measures ◽  
Clifford Group ◽  
Machine Calculation ◽  
Hand Calculation ◽  
System Of Imprimitivity

It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of $k$-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).

scholarly journals Galois automorphisms of a symmetric measurement

2013 ◽  
Vol 13 (7&8) ◽  
pp. 672-720
Keyword(s):  
Heisenberg Group ◽  
Galois Group ◽  
Positive Operator ◽  
Finite Dimension ◽  
Great Majority ◽  
Positive Operator Valued Measures ◽  
Clifford Group

Symmetric Informationally Complete Positive Operator Valued Measures (usually referred to as SIC-POVMs or simply as SICs) have been constructed in every dimension $\le 67$. However, a proof that they exist in every finite dimension has yet to be constructed. In this paper we examine the Galois group of SICs covariant with respect to the Weyl-Heisenberg group (or WH SICs as we refer to them). The great majority (though not all) of the known examples are of this type. Scott and Grassl have noted that every known exact WH SIC is expressible in radicals (except for dimension $3$ which is exceptional in this and several other respects), which means that the corresponding Galois group is solvable. They have also calculated the Galois group for most known exact examples. The purpose of this paper is to take the analysis of Scott and Grassl further. We first prove a number of theorems regarding the structure of the Galois group and the relation between it and the extended Clifford group. We then examine the Galois group for the known exact fiducials and on the basis of this we propose a list of nine conjectures concerning its structure. These conjectures represent a considerable strengthening of the theorems we have actually been able to prove. Finally we generalize the concept of an anti-unitary to the concept of a $g$-unitary, and show that every WH SIC fiducial is an eigenvector of a family of $g$-unitaries (apart from dimension 3).

Optimal quantum measurements for spin-1 and spin-3/2 particles

2001 ◽  
Vol 1 (3) ◽  
pp. 52-61
Author(s):  
P Aravind
Keyword(s):  
Positive Operator ◽  
Pure State ◽  
Quantum Measurements ◽  
Positive Operator Valued Measures ◽  
Unknown State

Positive operator valued measures (POVMs) are presented that allow an unknown pure state of a spin-1 particle to be determined with optimal fidelity when 2 to 5 copies of that state are available. Optimal POVMs are also presented for a spin-3/2 particle when 2 or 3 copies of the unknown state are available. Although these POVMs are optimal they are not always minimal, indicating that there is room for improvement.

Transport processes between fluids and clouds of suspended particles: some exact solutions

1957 ◽  
Vol 242 (1231) ◽  
pp. 430-443 ◽  
Keyword(s):  
Exact Solutions ◽  
Turbulent Mixing ◽  
Liquid Fuel ◽  
Transport Processes ◽  
Dimensional System ◽  
Unsteady Heat Transfer ◽  
One Dimensional ◽  
Homogeneous Reactor ◽  
Fourier Equation ◽  
Gas Component

The paper relates to processes such as the burning of pulverized solid fuel and of liquid fuel sprays, absorption of a gas component by a solvent spray, etc., and takes account of convection and turbulent mixing of the cloud. The Eulerian equation of conservation for the steady process is shown to be transformable into the Fourier equation for unsteady heat transfer in a flowing medium. Exact solutions are given for a one-dimensional system, an axially symmetrical system, and the steady-flow homogeneous reactor. Numerical methods are indicated for more complex systems, and the possibility of solution by means of an analogue is pointed out.

Exact Solutions for One-Dimensional Gray Media

2022 ◽  
pp. 467-495
Author(s):  
Michael F. Modest ◽  
Sandip Mazumder
Keyword(s):  
Exact Solutions ◽  
One Dimensional
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