On this page, I describe some of the more interesting results which I have obtained over the years.
Quantum Information
In a Hilbert space of dimension d, one can find (d+1) mutually unbiased bases if d is the power of a prime number. It has been conjectured that no more than three such bases exist for other dimensions.
- Strong numerical evidence has been provided that the conjecture holds in dimension d=6.
- Using algebraic calculations based on Buchberger's
algorithm, no
more than triples of mutually unbiased bases have been
shown to exist in dimension 6, in a large number of
cases.
- It is instructive to construct all MU bases in small dimensions.
A complete classification has been found, leading to a pair of
inequivalent triples for d=5 and a three-parameter family of triples
for d=4.
The eigenstates of the position and momentum operators provide a well-known pair of mutually unbiased (or complementary) bases for continuous variables.
- For a quantum particle with a single degree of freedom, it is possible to find three observables which are pairwise complementary - not just two.
Highly efficient quantum algorithms to search a database or to factor a large integer number sparked considerable interest in the idea to implement and hence process information quantum mechanically.
- I have shown that bare projective measurements possess computational power: the eigenvalues of a given Hermitean matrix of size NxN can be found from suitable measurements performed on a spin with s=(N-1)/2.
- In a similar way, the real zeros of a polynomial can be obtained as the outcomes of quantum mechanical measurements.
PT Symmetry
Some non-Hermitean Hamiltonians exhibit purely real spectra if they are invariant under simultaneous reversal of parity and time.
- Wigner's normal form of anti-unitary operators turns out to be a natural framework to describe the spectra of PT-symmetric operators.
Quantum State Reconstruction
The Pauli problem for a quantum particle is an early instance of the question which quantum state reconstruction later answers: do the probability distributions of position and momentum determine a unique pure state?
- I have obtained all pure states compatible with the two probability distributions and a third one, corresponding to a measurement along a slightly rotated postion axis.
Quantum Chaos and Integrability
It is often claimed that Schrödinger's equation as a linear equation cannot generate truly chaotic motion.
- The dynamics of the Configurational Quantum Cat Map is coincides with the dynamics of Arnold's classical Cat Map.
Other Results
- A theory which predicts the frequency shifts of an AFM cantilever immersed in a dense medium has been developed and tested experimentally.
- The Casimir
force increases when suitable squeezed
states of the electromagnetic field are excited.
- A useful method...
