It is conjectured that the question of the existence of a set of d + 1 mutually unbiased bases in a ddimensional Hilbert space if d differs from a power of prime is intimatelly linked with the problem whether there exist projective planes whose order d is not a power of prime.Recently, there has been a considerable resurgence of interest in the concept of the so-called mutually unbiased bases [see, e.g., 1-7], especially in the context of quantum state determination, cryptography, quantum information theory and the King's problem. We recall that two different orthonormal bases A and B of a d-dimensional Hilbert space H d are called mutually unbiased if and only if | a|b | = 1/ √ d for all a∈A and all b∈B. An aggregate of mutually unbiased bases is a set of orthonormal bases which are pairwise mutually unbiased. It has been found that the maximum number of such bases cannot be greater than d + 1 [8,9]. It is also known that this limit is reached if d is a power of prime. Yet, a still unanswered question is if there are non-prime-power values of d for which this bound is attained. The purpose of this short note is to draw the reader's attention to the fact that the answer to this question may well be related with the (non-)existence of finite projective planes of certain orders.A finite projective plane is an incidence structure consisting of points and lines such that any two points lie on just one line, any two lines pass through just one point, and there exist four points, no three of them on a line [10]. From these properties it readily follows that for any finite projective plane there exists an integer d with the properties that any line contains exactly d + 1 points, any point is the meet of exactly d + 1 lines, and the number of points is the same as the number of lines, namely d 2 + d + 1. This integer d is called the order of the projective plane. The most striking issue here is that the order of known finite projective planes is a power of prime [10]. The question of which other integers occur as orders of finite projective planes remains one of the most challenging problems of contemporary mathematics. The only "no-go" theorem known so far in this respect is the Bruck-Ryser theorem [11] saying that there is no projective plane of order d if d − 1 or d − 2 is divisible by 4 and d is not the sum of two squares. Out of the first few non-prime-power numbers, this theorem rules out finite projective planes of order 6, 14, 21, 22, 30 and 33. Moreover, using massive computer calculations, it was proved by Lam [12] that there is no projective plane of order ten. It is surmised that the order of any projective plane is a power of a prime.¿From what has already been said it is quite tempting to hypothesize that the above described two problems are nothing but different aspects of one and the same problem. That is, we conjecture that non-existence of a projective plane of the given order d implies that there are less than d + 1 mutually unbiased bases (MUBs) in the corresponding H d , and vice versa. Or, slightly...

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite, and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.

In the present review, we deal with the recently introduced method of spectral parameter power series (SPPS) and show how its application leads to an explicit form of the characteristic equation for different eigenvalue problems involving Sturm–Liouville equations with variable coefficients. We consider Sturm–Liouville problems on finite intervals; problems with periodic potentials involving the construction of Hill's discriminant and Floquet–Bloch solutions; quantum‐mechanical spectral and transmission problems as well as the eigenvalue problems for the Zakharov–Shabat system. In all these cases, we obtain a characteristic equation of the problem, which in fact reduces to finding zeros of an analytic function given by its Taylor series. We illustrate the application of the method with several numerical examples, which show that at present, the SPPS method is the easiest in the implementation, the most accurate, and efficient. We emphasize that the SPPS method is not a purely numerical technique. It gives an analytical representation both for the solution and for the characteristic equation of the problem. This representation can be approximated by different numerical techniques and leads to a powerful numerical method, but most important, it offers a different insight into the spectral and transmission problems. Copyright © 2014 John Wiley & Sons, Ltd.

Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is a constant equal to 1/ 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for general pure quantum electromagnetic states and find them to be related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters.Finally, we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in the study of entanglement and its information aspects.2

We discuss two applications of Riccati equation to Newton's laws of motion. The first one is the motion of a particle under the influence of a power law central potential V (r) = kr ǫ . For zero total energy we show that the equation of motion can be cast in the Riccati form. We briefly show here an analogy to barotropic Friedmann-Robertson-Lemaitre cosmology where the expansion of the universe can be also shown to obey a Riccati equation. A second application in classical mechanics, where again the Riccati equation appears naturally, are problems involving quadratic friction. We use methods reminiscent to nonrelativistic supersymmetry to generalize and solve such problems.