Sum rule of quantum uncertainties: coupled harmonic oscillator system with time-dependent parameters
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Sum rule of quantum uncertainties: coupled harmonic oscillator system with time-dependent parameters DaeKil Park1,2
· Eylee Jung1
Received: 6 February 2020 / Accepted: 10 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Uncertainties (x)2 and ( p)2 are analytically derived in an N -coupled harmonic oscillator system when spring and coupling constants are arbitrarily time-dependent and each oscillator is in an arbitrary excited state. When N = 2, those uncertainties are shown as just arithmetic average of uncertainties of two single harmonic oscillators. We call this property as “sum rule of quantum uncertainty”. However, this arithmetic average property is not generally maintained when N ≥ 3, but it is recovered in N coupled oscillator systems if and only if (N − 1) quantum numbers are equal. The generalization of our results to a more general quantum system is briefly discussed. Keywords Quantum uncertainty · Coupled harmonic oscillators
1 Introduction Uncertainty [1–4] and entanglement [5–7] are two major cornerstones of quantum mechanics. These characteristics cause quantum mechanics to differ from classical mechanics. Quantum uncertainty provides a limit on the precision of measurement for incompatible observables. The most typical expression of uncertainty relation is x p ≥ /2, where is the standard deviation. Recently, researchers have analyzed different expressions of uncertainty relations, such as entropic uncertainty relations [8,9] from the context of quantum information and generalized uncertainty principle [10] from the context of Planck scale physics. Even though entanglement has been studied since the discovery of quantum mechanics [5], it has been extensively explored for the last few decades with the development of quantum technology. Entanglement is used as a physical resource in various quantum information processing, such as quantum teleportation [11,12], superdense coding [13], quantum cloning [14], quan-
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DaeKil Park [email protected]
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Department of Electronic Engineering, Kyungnam University, Changwon 631-701, Korea
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Department of Physics, Kyungnam University, Changwon 631-701, Korea 0123456789().: V,-vol
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tum cryptography [15,16], quantum metrology [17], and quantum computer [18,19]. Furthermore, with many researchers trying to realize such quantum information processing in the laboratory for the last few decades, quantum cryptography and quantum computer seems to approaching the commercial level [20,21]. Although these two phenomena seem to be distinct properties of quantum mechanics, there is some connection, albeit unclear, between them because of the fact that both are strongly dependent on the interaction between subsystems. For example, the uncertainty of a given system was computed in Refs. [22,23] to discuss on the effect of the Feynman’s rest of universe [24]. The ignoring of the effect of the rest of the universe was shown to increase uncertainty and entropy in the target system
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