278 research outputs found
Representations of the Weyl group and Wigner functions for SU(3)
Bases for SU(3) irreps are constructed on a space of three-particle tensor
products of two-dimensional harmonic oscillator wave functions. The Weyl group
is represented as the symmetric group of permutations of the particle
coordinates of these space. Wigner functions for SU(3) are expressed as
products of SU(2) Wigner functions and matrix elements of Weyl transformations.
The constructions make explicit use of dual reductive pairs which are shown to
be particularly relevant to problems in optics and quantum interferometry.Comment: : RevTex file, 11 pages with 2 figure
Unitary transformations for testing Bell inequalities
It is shown that optical experimental tests of Bell inequality violations can
be described by SU(1,1) transformations of the vacuum state, followed by photon
coincidence detections. The set of all possible tests are described by various
SU(1,1) subgroups of Sp(8,). In addition to establishing a common
formalism for physically distinct Bell inequality tests, the similarities and
differences of post--selected tests of Bell inequality violations are also made
clear. A consequence of this analysis is that Bell inequality tests are
performed on a very general version of SU(1,1) coherent states, and the
theoretical violation of the Bell inequality by coincidence detection is
calculated and discussed. This group theoretical approach to Bell states is
relevant to Bell state measurements, which are performed, for example, in
quantum teleportation.Comment: 3 figure
Lost and found: the radial quantum number of Laguerre-Gauss modes
We introduce an operator linked with the radial index in the Laguerre-Gauss
modes of a two-dimensional harmonic oscillator in cylindrical coordinates. We
discuss ladder operators for this variable, and confirm that they obey the
commutation relations of the su(1,1) algebra. Using this fact, we examine how
basic quantum optical concepts can be recast in terms of radial modes.Comment: Some minor typos fixed
Wigner function for SU(1,1)
In spite of their potential usefulness, Wigner functions for systems with
SU(1,1) symmetry have not been explored thus far. We address this problem from
a physically-motivated perspective, with an eye towards applications in modern
metrology. Starting from two independent modes, and after getting rid of the
irrelevant degrees of freedom, we derive in a consistent way a Wigner
distribution for SU(1,1). This distribution appears as the expectation value of
the displaced parity operator, which suggests a direct way to experimentally
sample it. We show how this formalism works in some relevant examples.Comment: Version accepted in Quantu
SU(N)-symmetric quasi-probability distribution functions
We present a set of N-dimensional functions, based on generalized
SU(N)-symmetric coherent states, that represent finite-dimensional Wigner
functions, Q-functions, and P-functions. We then show the fundamental
properties of these functions and discuss their usefulness for analyzing
N-dimensional pure and mixed quantum states.Comment: 16 pages, 2 figures. Updated text to reflect referee comment
A complementarity-based approach to phase in finite-dimensional quantum systems
We develop a comprehensive theory of phase for finite-dimensional quantum
systems. The only physical requirement we impose is that phase is complementary
to amplitude. To implement this complementarity we use the notion of mutually
unbiased bases, which exist for dimensions that are powers of a prime. For a
d-dimensional system (qudit) we explicitly construct d+1 classes of maximally
commuting operators, each one consisting of d-1 operators. One of this class
consists of diagonal operators that represent amplitudes (or inversions). By
the finite Fourier transform, it is mapped onto ladder operators that can be
appropriately interpreted as phase variables. We discuss the examples of qubits
and qutrits, and show how these results generalize previous approaches.Comment: 6 pages, no figure
Exchange Gate on the Qudit Space and Fock Space
We construct the exchange gate with small elementary gates on the space of
qudits, which consist of three controlled shift gates and three "reverse"
gates. This is a natural extension of the qubit case.
We also consider a similar subject on the Fock space, but in this case we
meet with some different situation. However we can construct the exchange gate
by making use of generalized coherent operator based on the Lie algebra su(2)
which is a well--known method in Quantum Optics. We moreover make a brief
comment on "imperfect clone".Comment: Latex File, 12 pages. I could solve the problems in Sec. 3 in the
preceding manuscript, so many corrections including the title were mad
- âŠ