7 research outputs found
Entanglement in mutually unbiased bases
One of the essential features of quantum mechanics is that most pairs of
observables cannot be measured simultaneously. This phenomenon is most strongly
manifested when observables are related to mutually unbiased bases. In this
paper, we shed some light on the connection between mutually unbiased bases and
another essential feature of quantum mechanics, quantum entanglement. It is
shown that a complete set of mutually unbiased bases of a bipartite system
contains a fixed amount of entanglement, independently of the choice of the
set. This has implications for entanglement distribution among the states of a
complete set. In prime-squared dimensions we present an explicit
experiment-friendly construction of a complete set with a particularly simple
entanglement distribution. Finally, we describe basic properties of mutually
unbiased bases composed only of product states. The constructions are
illustrated with explicit examples in low dimensions. We believe that
properties of entanglement in mutually unbiased bases might be one of the
ingredients to be taken into account to settle the question of the existence of
complete sets. We also expect that they will be relevant to applications of
bases in the experimental realization of quantum protocols in
higher-dimensional Hilbert spaces.Comment: 13 pages + appendices. Published versio
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
Galois quantum systems
NoA finite quantum system in which the position and momentum take values in the Galois field GF(p¿l) is constructed from a smaller quantum system in which the position and momentum take values in Zp , using field extension. The Galois trace is used in the definition of the Fourier transform. The Heisenberg¿Weyl group of displacements and the Sp(2, GF(p¿l)) group of symplectic transformations are studied. A class of transformations inspired by the Frobenius maps in Galois fields is introduced. The relationship of this 'Galois quantum system' with its subsystems in which the position and momentum take values in subfields of GF(p¿l) is discussed