scholarly journals Absolute Quantum Theory (after Chang, Lewis, Minic and Takeuchi), and a Road to Quantum Deletion

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 174
Author(s):  
Koen Thas

In a recent paper, Chang et al. have proposed studying “quantum F u n ”: the q ↦ 1 limit of modal quantum theories over finite fields F q , motivated by the fact that such limit theories can be naturally interpreted in classical quantum theory. In this letter, we first make a number of rectifications of statements made in that paper. For instance, we show that quantum theory over F 1 does have a natural analogon of an inner product, and so orthogonality is a well-defined notion, contrary to what was claimed in Chang et al. Starting from that formalism, we introduce time evolution operators and observables in quantum F u n , and we determine the corresponding unitary group. Next, we obtain a typical no-cloning result in the general realm of quantum F u n . Finally, we obtain a no-deletion result as well. Remarkably, we show that we can perform quantum deletion by almost unitary operators, with a probability tending to 1. Although we develop the construction in quantum F u n , it is also valid in any other quantum theory (and thus also in classical quantum theory in complex Hilbert spaces).

Author(s):  
YONINA C. ELDAR ◽  
TOBIAS WERTHER

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinite-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and corresponding positive operators for which the new geometrical interpretation applies.


2008 ◽  
Vol 39 (1) ◽  
pp. 1-7 ◽  
Author(s):  
S. S. Dragomir

In this paper various inequalities between the operator norm and its numerical radius are provided. For this purpose, we employ some classical inequalities for vectors in inner product spaces due to Buzano, Goldstein-Ryff-Clarke, Dragomir-S ´andor and the author.


2021 ◽  
pp. 320-342
Author(s):  
Valia Allori

Quantum mechanics is a groundbreaking theory: it not only is extraordinarily empirically adequate but also is claimed to having shattered the classical paradigm of understanding the observer-observed distinction as well as the part-whole relation. This, together with other quantum features, has been taken to suggest that quantum theory can help one understand the mind-body relation in a unique way, in particular to solve the hard problem of consciousness along the lines of panpsychism. In this chapter, after having briefly presented panpsychism, Valia Allori discusses the main features of quantum theories and the way in which the main quantum theories of consciousness use them to account for conscious experience.


1998 ◽  
Vol 11 (3) ◽  
pp. 477-478
Author(s):  
Peter Marquardt

1962 ◽  
Vol 14 ◽  
pp. 651-659 ◽  
Author(s):  
Bohdan J. Tomiuk

If A is an H*-algebra, then the orthogonal complement of a closed right (left) ideal I is a closed right (left) ideal P. Saworotnow (7) considered Banach algebras which are Hilbert spaces and in which the closed right ideals satisfy the complementation property of an H*-algebra. In our right complemented Banach algebras we drop the requirement of the existence of an inner product and only assume that for every closed right ideal I there is a closed right ideal IP which behaves like the orthogonal complement in a Hilbert space (Definition 1). Thus our algebras may be considered as a generalization of Saworotnow's right complemented algebras.


Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1334
Author(s):  
S. A. Larin

We consider R + R 2 relativistic quantum gravity with the action where all possible terms quadratic in the curvature tensor are added to the Einstein-Hilbert term. This model was shown to be renormalizable in the work by K.S. Stelle. In this paper, we demonstrate that the R + R 2 model is also unitary contrary to the statements made in the literature, in particular in the work by Stelle. New expressions for the R + R 2 Lagrangian within dimensional regularization and the graviton propagator are derived. We demonstrate that the R + R 2 model is a good candidate for the fundamental quantum theory of gravity.


1986 ◽  
Vol 456 (2) ◽  
pp. 279-297 ◽  
Author(s):  
Onishi Naoki

Author(s):  
YONINA C. ELDAR ◽  
TOBIAS WERTHER

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinte-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and the corresponding positive operators for which this geometrical interpretation applies.


2020 ◽  
Vol 18 (01) ◽  
pp. 1941026 ◽  
Author(s):  
Rinie N. M. Nasir ◽  
Jesni Shamsul Shaari ◽  
Stefano Mancini

Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, [Formula: see text], acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of [Formula: see text] when estimating a unitary operator in another. Though, for prime dimension [Formula: see text], the maximal number of MUUBs is known to be [Formula: see text], there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of [Formula: see text]. MUUBs can also exist for some [Formula: see text]-dimensional subspace of [Formula: see text] with the maximal number being [Formula: see text].


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