scholarly journals Symmetric informationally complete–positive operator valued measures and the extended Clifford group

2005 ◽  
Vol 46 (5) ◽  
pp. 052107 ◽  
Author(s):  
D. M. Appleby
Keyword(s):  
Positive Operator ◽  
Positive Operator Valued Measures ◽  
Clifford Group

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.

scholarly journals Operational link between mutually unbiased bases and symmetric informationally complete positive operator-valued measures

2013 ◽  
Vol 88 (3) ◽  
Author(s):  
Roberto Beneduci ◽  
Thomas J. Bullock ◽  
Paul Busch ◽  
Claudio Carmeli ◽  
Teiko Heinosaari ◽  
...  
Keyword(s):  
Positive Operator ◽  
Mutually Unbiased Bases ◽  
Positive Operator Valued Measures

scholarly journals Probing qubit by qubit: Properties of the POVM and the information/disturbance tradeoff

2014 ◽  
Vol 12 (02) ◽  
pp. 1461012 ◽  
Author(s):  
Carlo Sparaciari ◽  
Matteo G. A. Paris
Keyword(s):  
Positive Operator ◽  
Projective Measurement ◽  
Matrix Elements ◽  
Positive Operator Valued Measures ◽  
Unitary Coupling ◽  
Probe System ◽  
Initial Preparation ◽  
Free Parameters

We address the class of positive operator-valued measures (POVMs) for qubit systems that are obtained by coupling the signal qubit with a probe qubit and then performing a projective measurement on the sole probe system. These POVMs, which represent the simplest class of qubit POVMs, depends on 3 + 3 + 2 = 8 free parameters describing the initial preparation of the probe qubit, the Cartan representative of the unitary coupling, and the projective measurement at the output, respectively. We analyze in some detail the properties of the POVM matrix elements, and investigate their values for given ranges of the free parameters. We also analyze in detail the tradeoff between information and disturbance for different ranges of the free parameters, showing, among other things, that (i) typical values of the tradeoff are close to optimality and (ii) even using a maximally mixed probe one may achieve optimal tradeoff.

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