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arXiv:1402.2569 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Squeezing of arbitrary order: the ups and downs

K. Górska, A. Horzela, Franciszek Hugon Szafraniec

Published Oct 8, 2014• 6 citations
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Abstract

We show how using classical von Neumann index theory makes possible a universal treatment of squeezing of arbitrary order. ‘Universal’ means that the same approach applied to displacement (order 1) and squeeze (order 2) operators confirms firmly and unequivocally what is already known as well as provides rigorous arguments that the higher order squeezing cannot be generalized in a ‘naive’ way. We create an environment for answering definitely all the emerging questions in positive ( the ups ) and negative ( downs ). In the latter case, we suggest ways for further development.

DOI
10.1098/rspa.2014.0205

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We show how using classical von Neumann index theory makes possible a universal treatment of squeezing of arbitrary order. ‘Universal’ means that the same approach applied to displacement (order 1) and squeeze (order 2) operators confirms firmly and unequivocally what is already known as well as provides rigorous arguments that the higher order squeezing cannot be generalized in a ‘naive’ way. We create an environment for answering definitely all the emerging questions in positive ( the ups ) and negative ( downs ). In the latter case, we suggest ways for further development.
Recently, Puri revisited the usual uncertainty relation arguments. Utilizing a well-known and sharpened version of the uncertainty relation, he has proposed an alternative squeezing criterion for the case of noncanonical operators. We investigated this in the hope that Puri's proposal might allow one to define a higher-order squeezing criterion that would avoid an anomaly found with the standard Hong-Mandel (HM) scheme, i.e., the possibility of simultaneous-quadrature squeezing. However, it is found that higher-order squeezing based on Puri's squeezing criterion has what one might judge to be an even worse and unphysical defect than the HM scheme, in that even at the squeezing ``trigger level,'' squeezing in one quadrature is necessarily accompanied by considerable antisqueezing in the other quadrature.
Abstract By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a “quantum probability theory”. As such, it is regarded a generalization of the “classical probability theory” due to Kolmogorov. Outside of quantum physics, however, Kolmogorov’s axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann’s axiomatization of quantum mechanics was contingent on the absence of a general theory of probability in the 1920s. This work argues in favor of the latter position. In particular, it shows how to construct a mathematically rigorous theory for non-relativistic N -body quantum systems subject to a time-independent scalar potential, which is based on Kolmogorov’s axioms and physically natural random variables. Though this theory is provably distinct from its quantum-mechanical analog, it nonetheless reproduces central predictions of the latter. Further work may make an empirical comparison possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series discusses the empirical violation of Bell inequalities in the context of this approach. Part III addresses the projection postulate and the question of measurement.
We address the problem of quantifying the information content of a source for an arbitrary information theory, where the information content is defined in terms of the asymptotic achievable compression rate. The functions that solve this problem in classical and quantum theory are Shannon's and von Neumann's entropy, respectively. However, in a general information theory there are three different functions that extend the notion of entropy, and this opens the question as to whether any of them can universally play the role of the quantifier for the information content. Here we answer the question in the negative, by evaluating the information content as well as the various entropic functions in a toy theory called Bilocal Classical Theory.