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arXiv:0708.3079

Feynman's Integral is About Mutually Unbiased Bases

George Svetlichny

Published Aug 22, 2007• 7 citations
arXivHTMLPDF

Abstract

The Feynman integral can be seen as an attempt to relate, under certain circumstances, the quantum-information-theoretic separateness of mutually unbiased bases to causal proximity of the measuring processes.
DOI
10.48550/arxiv.0708.3079

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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.
Mutually Unbiased bases has various application in quantum information procession and coding theory. There can be maximum d + 1 MUBs in C^d and d/2 +1 MUBs in R^d. But , over R^d MUBs are known to be non existent when d is odd and for most of the other even d there are mostly 3 Real MUBs. In case of C^d the construction for complete set of MUBs are known for only Prime Power dimension. Thus in general large set of MUBs are not known, particularly for composite dimensions which are not of the form of prime powers. Because of this, there are many constructions of Approximate version of MUBs. In this paper we make an attempt to define certain measures to characterise the AMUBs. Our construction of measures derives its inspiration from the applications of MUBs, and based on them, we define certain quantifiable measures, which are can be computed and gives estimates of how close the Approximate MUBs are to the MUBs. We use geometric interpretation, projective design features of MUBs and applications like Optimal State determination and Entropic Uncertainty of MUBs. We show generic relationship between these measures and show that it can be evaluated for APMUBs without known the exact construction details, thereby showing that definition of APMUB is sufficient completely characterise it. We also evaluate these measure for an interesting class of AMUBs called Weak MUBs and certain AMUBs constructed using RBDs.
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.