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Signal Processing

A quantum mechanical derivation of the general uncertainty relation for real signals in communication theory

Emilio Borchi, G. Pelosi

Published Jul 1, 1980• 8 citations
Publisher
DOI
10.1016/0165-1684(80)90026-2

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In this paper, the generic uncertainty relation (UR) for kernel-based transformations (KT) of signals is derived. Instead of using the statistics approach as shown in the literature before, here we employ quantum mechanical operator approach for directly deriving the UR for KT's. We are able to do this because we have found the quantum operator realization of KT. Our new method is concise and applicable to any kinds of KT's with continuous and discrete parameters and variables. An explicit result of UR for a family of KT's including FrFT, generalized fractional transformation (GFrT) and linear canonical transformation (LCT) is provided as an application of our new method.

arXiv:1410.5507•Oct 21, 2014•0 citations
We consider a range of "theories'' that violate the uncertainty relation for anti-commuting observables derived. We first show that Tsirelson's bound for the CHSH inequality can be derived from this uncertainty relation, and that relaxing this relation allows for non-local correlations that are stronger than what can be obtained in quantum mechanics. We continue to construct a hierarchy of related non-signaling theories, and show that on one hand they admit superstrong random access encodings and exponential savings for a particular communication problem, while on the other hand it becomes much harder in these theories to learn a state. We show that the existence of these effects stems from the absence of certain constraints on the expectation values of commuting measurements from our non-signaling theories that are present in quantum theory.
We consider a range of "theories" that violate the uncertainty relation for anti-commuting observables derived in [JMP, 49, 062105 (2008)]. We first show that Tsirelson's bound for the CHSH inequality can be derived from this uncertainty relation, and that relaxing this relation allows for non-local correlations that are stronger than what can be obtained in quantum mechanics. We continue to construct a hierarchy of related non-signaling theories, and show that on one hand they admit superstrong random access encodings and exponential savings for a particular communication problem, while on the other hand it becomes much harder in these theories to learn a state. We show that the existence of these effects stems from the absence of certain constraints on the expectation values of commuting measurements from our non-signaling theories that are present in quantum theory.
In this paper, the generic uncertainty relation (UR) for kernel-based transformations (KT) of signals is derived. Instead of using the statistics approach as shown in the literature before, here we employ quantum mechanical operator approach for directly deriving the UR for KT's. We are able to do this because we have found the quantum operator realization of KT. Our new method is concise and applicable to any kinds of KT's with continuous and discrete parameters and variables. An explicit result of UR for a family of KT's including FrFT, generalized fractional transformation (GFrT) and linear canonical transformation (LCT) is provided as an application of our new method.