7 research outputs found
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
Multicomplementary operators via finite Fourier transform
A complete set of d+1 mutually unbiased bases exists in a Hilbert spaces of
dimension d, whenever d is a power of a prime. We discuss a simple construction
of d+1 disjoint classes (each one having d-1 commuting operators) such that the
corresponding eigenstates form sets of unbiased bases. Such a construction
works properly for prime dimension. We investigate an alternative construction
in which the real numbers that label the classes are replaced by a finite field
having d elements. One of these classes is diagonal, and can be mapped to
cyclic operators by means of the finite Fourier transform, which allows one to
understand complementarity in a similar way as for the position-momentum pair
in standard quantum mechanics. The relevant examples of two and three qubits
and two qutrits are discussed in detail.Comment: 15 pages, no figure