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arXiv:1505.06054

Each normalized state is a member of an orthonormal basis: A simple proof

Iman Sargolzahi, Ehsan Anjidani

Published May 22, 2015• 0 citations
arXivHTMLPDF

Abstract

In a finite dimensional Hilbert space, each normalized vector (state) can be chosen as a member of an orthonormal basis of the space. We give a proof of this statement in a manner that seems to be more comprehensible for physics students than the formal abstract one.
DOI
10.48550/arxiv.1505.06054

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This article presents a simple characterization for entangled vectors in a finite dimensional Hilbert space HHH. The characterization is in terms of the coefficients of an expansion of the vector relative to an orthonormal basis for HHH. This simple necessary and sufficient condition contains a restriction and we also present a more complicated condition that is completely general. Although these characterizations apply to bipartite systems, we show that generalizations to multipartite systems are also valid.
From here, we take up the task of reformulating quantum mechanics in the language of inner product space. We start by introducing vector space through the familiar example of a set of finite dimensional, complex, column matrices under addition and multiplication by scalars. The inner product is then introduced as a dot product generalized to complex vectors leading to the idea of an inner product space . This is adequate to show how the concept of a quantum state can be described by a normalized vector in an inner product space so that the state space can be identified with the inner product space. Amplitudes are associated with inner products , and observables with orthonormal bases in this space. Hence the laws of quantum mechanics are reformulated. We then point out that the now reformulated laws are independent of the space of description and the reader is urged to contemplate an universality which would reflect the independence with respect to the representation space . This creates the stage for the introduction of the abstract inner product space and the formulation of quantum mechanics in this setting. The chapter concludes with this description. The transformed laws are now called postulatesof quantum mechanics .
I discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. Stated informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of measurements and states that is already very suggestive of quantum mechanics. In particular, in any theory satisfying these axioms, measurements can be represented by orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space -- in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short.
An N-tangle can be defined for the finite dimensional Hilbert space H = C2N , with N = 3 or N even.We give an orthonormal basis which is fully entangled with respect to this measure.We provide a spin Hamilton operator which has this entangled basis as normalized eigenvectors if N is even. From these normalized entangled states a Bell matrix is constructed and the cosine-sine decomposition is calculated. If N is odd the normalized eigenvectors can be entangled or unentangled depending on the parameters.