Measuring the position of a free electron gives it an unknown momentum. A second measurement will give a new position far away. On the other hand, consecutive measurements of spin give the same result. In this talk we attempt to unify these very different behaviors.Position and momentum are complementary observables. An eigenstate of position has no information about momentum and vice versa. For an N‐dimensional Hilbert space we say that two bases |aj〉, and |bk〉 are “mutually unbiased” if all their transition probabilities are equal: |〈aj|bk〉|2 = 1/N. Position and momentum are similar to mutually unbiased bases, but for an infinite dimensional Hilbert space.Svetlichny discovered that Feynman’s (position) path integral can be interpreted as a sum over products of mutually unbiased bases. This raises the question “what happens when we compute spin‐1/2 path integrals over mutually unbiased bases?” There are three pairwise mutually unbiased bases for spin‐1/2. We compute the long time propagators of spin path integrals and show that a natural description of the three generations of elementary particles results. We discuss the lepton masses and other material.
Our previous work on quantum mechanics in Hilbert spaces of finite dimensions N is applied to elucidate the deep meaning of Feynman's path integral pointed out by G. Svetlichny. He speculated that the secret of the Feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases. We confirm the corresponding property of the short-time propagator by using a specially devised N x N -approximation of quantum mechanics in L^2(R) applied to our finite-dimensional analogue of a free quantum particle.
The two observables are incompatible if they cannot be measured simultaneously; however, they become maximally incompatible (complementary) if their eigenstates are mutually unbiased. Only then does the measurement of one observable give no information about the other observable. The spin projection operators onto three mutually orthogonal directions are complementary only for spin 1/2. For higher spin numbers the corresponding eigenstates are no longer unbiased. In this work we examine the properties of spin 1 mutually unbiased bases (MUB) and look for the physical meaning of the corresponding operators. We show that if the computational basis is chosen to be the eigenbasis of the spin projection operator along some direction z , then all the states, which are unbiased to this basis, have to be squeezed. Next, we study the generation and the measurement of MUB states by introducing the Fourier-like transform through spin squeezing. Finally, we try to ascribe some classical interpretation to the operators corresponding to MUB and study what information the observer gains while measuring them. Higher spin numbers are also considered.
It has been stated in the physics literature that the probability amplitude for a quantum system after a continuous position measurement can be expressed by the restricted or the weighted Feynman path integral containing information about the result and the resolution of its measurement. In the present paper it is proved that the restricted Feynman path integrals can be defined in a rigorous sense. More specifically, we consider a quantum system consisting of some particles and simultaneously perform a continuous measurement of the positions of all particles in the time interval
[0,T]. Then, it is shown rigorously that the restricted Feynman path integral after the measurement can be defined in the
L2 space and satisfies the corresponding non-self-adjoint Schroeodinger equation. In addition, our results are generalized to a quantum spin system where a continuous position measurement for all spin components of all particles is simultaneously performed, though the results obtained are inadequate for general use.
The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in
Cd,BandB′ are said mutually unbiased if
∀b∈B,b′∈B′ the scalar product
b⋅b′ has modulus
d−1/2. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state
ψ∈Cd when measured in the mutually unbiased bases
B_jofCd. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when
d is not a power of a prime number. \noindent In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension
d. The method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The Vandermonde matrix based on the
d-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any dimension, to give conditions for existence of more than 3 MUB's for
d even or odd number, and to recover the known result of existence of
d+1 MUB's for
d a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties.