In a Hilbert space ℋ, discrete families of vectors {hj} with the property that f=∑j〈hj‖ f〉hj for every f in ℋ are considered. This expansion formula is obviously true if the family is an orthonormal basis of ℋ, but also can hold in situations where the hj are not mutually orthogonal and are ‘‘overcomplete.’’ The two classes of examples studied here are (i) appropriate sets of Weyl–Heisenberg coherent states, based on certain (non-Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ‘‘quasiorthogonal expansions’’ will be a useful tool in many areas of theoretical physics and applied mathematics.
Abstract. Starting with the canonical coherent states, we demonstrate that all the so-called nonlinear coherent states, used in the physical literature, as well as large classes of other generalized coherent states, can be obtained by changes of bases in the underlying Hilbert space. This observation leads to an interesting duality between pairs of generalized coherent states, bringing into play a Gelfand triple of (rigged) Hilbert spaces. Moreover, it is shown that in each dual pair of families of nonlinear coherent states, at least one family is related to a (generally) non-unitary projective representation of the Weyl-Heisenberg group, which can then be thought of as characterizing the dual pair. 1.
Discrete structures in product Hilbert spaces are investigated. For monopartite systems of size
d one relies on the Weyl--Heisenberg group
WH(d), while in the case of composite Hilbert spaces with the local dimensions
di we identify designs covariant with respect to the product group,
⨂iWH(di). In analogy with magic -- a quantity attaining its maximum for states fiducial with respect to
WH(d) -- we introduce a similar quantifier of {\sl product magic}, defined with respect to the product group. The maximum of this quantity over all equimodular vectors yields fiducial states that generate
d=∏idi \textit{a priori} isoentangled mutually unbiased bases (MUBs), which, when supplemented by the identity, form their complete set. Such fiducial states are explicitly constructed in all prime-power dimensions
d=pn with
p≥3. The result for
p≥5 extends the construction of Klappenecker and R{ö}tteler, whereas for
p=3 it is mathematically distinct and is based on Galois rings. The global maximum of the quantifier of product magic for
d=23 yields fiducial states corresponding to the symmetric informationally complete (SIC) generalized measurement of Hoggar. Our approach feeds into a unifying perspective in which highly symmetric quantum designs emerge from fiducial states with extremal properties via structured group-orbit constructions.
Starting with the canonical coherent states, we demonstrate that all the so-called nonlinear coherent states, used in the physical literature, as well as large classes of other generalized coherent states, can be obtained by changes of bases in the underlying Hilbert space. This observation leads to an interesting duality between pairs of generalized coherent states, bringing into play a Gelfand triple of (rigged) Hilbert spaces. Moreover, it is shown that in each dual pair of families of nonlinear coherent states, at least one family is related to a (generally) non-unitary projective representation of the Weyl–Heisenberg group, which can then be thought of as characterizing the dual pair.
We study the properties of regular structures of lines, such as equiangular sets of lines and mutually unbiased bases (MUBs) in a general setting that includes real, complex and quaternionic spaces. We formulate a common generalization of several results in real and complex spaces that also hold in the quaternionic space.
A set of lines is called equiangular if the angle between each pair is the same. A set of MUBs is a collection of orthonormal bases such that the angle between vectors from different bases is constant. Regular structures of lines have been studied in several fields such as digital communication, quantum computing, discrete mathematics and analysis.
Our new concept of a multipartite equiangular set of lines is a common generalization of equiangular lines and MUBs. We prove a bound on the size of such set of lines, which generalizes the well-known absolute upper bounds.
The existence of d + 1 MUBs in Cd is only known for prime power dimensions. We study sets of d + 1 MUBs that are the union of a standard basis and an orbit of the Weyl-Heisenberg group. As an example, we construct such MUBs in prime power dimensions. We also show connections between spherical 2-designs and other structures of lines.
Fiducial vectors have been widely used to construct large sets of equiangular lines. A complex vector is fiducial if its orbit under a Weyl-Heisenberg group is an equiangular set of d2 lines. We give a new characterization of fiducial vectors, one that simplifies and significantly reduces the number of equations that must be solved to find a fiducial vector. We consider some possible classes of fiducial vectors and prove several nonexistence results. For example, using our new characterization we prove that the construction of fiducial vectors in small prime dimensions by Appleby (2005) essentially does not generalize.
Finally, we give some methods for constructing equiangular sets of lines in complex and quaternionic spaces. We also find numerical fiducial vectors with high precision in Cd , d ≤ 21.