Information-Theoretic Uncertainty Relation and Random-Phase Entropy arXiv:1001.2966v1 [quant-ph] 18 Jan 2010 Kyoung Kon Kim and Sang Pyo Kim ∗ Department of Physics, Kunsan National University, Kunsan 573-701, Korea Sok Kuh Kang Korea Ocean Research & Development Institute, Ansan 426-744, Korea (Dated: November 6, 2018) Dunkel and Trigger [Phys. Rev. A 71 , 052102 (2005)] show that the Leipnik’s joint entropy monotonously increases for the initially maximally classical Gaussian wave packet for a free particle. After expressing the joint entropy of the general Gaussian wave packets for quadratic Hamiltonians as S ( t ) = ln( e/ 2) + ln(2Δ x ( t )Δ p ( t ) / ℏ ), we show that a class of general Gaussian wave packets does not warrant the monotonous increase of the joint entropy. We propose that the random-phase entropy with respect to the squeeze angle always monotonously increases even for non-maximally classical states. PACS numbers: 03.65.Ta, 03.67.-a, 05.30.-d The Leipnik’s joint entropy is an important measure of information loss of quantum states. Recently Dunkel and Trigger [1] have shown that the Leipnik’s joint en- tropy monotonously increases for a free Gaussian wave packet and a wave packet in a monochromatic electro- magnetic field when it is initially prepared as a maxi- mally classical state. They have further argued that the monotonous increase of the entropy may be an intrinsic property for other types of initial wave packets. However, it was shown in Ref. [2] that the Leipnik’s joint entropy for a Gaussian wave packet of a harmonic oscillator oscil- lates between the maximum and the lower bound, which was also confirmed in Ref. [3]. In the present paper, we study the Leipnik’s joint en- tropy for the general Gaussian wave packets of quadratic Hamiltonians, whose mass and frequency may depend on time, and which may be driven by an external force. The exact Gaussian wave packets involve the initial po- sition and momentum for the centroid, and the initial position and momentum variances, which are equivalent to a squeeze parameter and a squeeze angle [4]. We first express the Leipnik’s joint entropy in terms of the uncer- tainty relation and then investigate the condition for the monotonous increase of the entropy. The joint entropy of general Gaussian wave packets for a harmonic oscillator oscillates between the maximum and the lower bound of the joint entropy, while a class of Gaussian wave packets of a free particle has the joint entropy that first decreases and then monotonously increases. However, the random- phase entropy with respect to the squeeze angle indeed monotonously increases for the general Gaussian wave packets for quadratic Hamiltonian systems, including a free particle, a harmonic oscillator and the Caldirola- Kanai Hamiltonian with an exponentially varying mass As shown in Ref. [6], the most general quadratic Hamil- tonian may be canonically transformed into the form (1). A harmonic oscillator has constant m$_{0}$ and ω$_{0}$ while a free particle is the limit of ω = 0. A charged particle in a monochromatic electric field is described by f ( t ) = qE cos( ωt ). The Hamiltonian with time- dependent mass or frequency describes an open system. To find the exact Gaussian wave packet for (1), we adopt the invariant method by Lewis and Riesenfeld [7] (see also Ref. [8]). The Hamiltonian (1) has a pair of lin- ear invariant operators [9–12], which leads to the most general Gaussian wave packet [4] We consider the quadratic Hamiltonians of the form [5], regardless of an external driving force. A direct calculation shows that (2) indeed satisfies the time-dependent Schr¨ odinger equation. Here, u is a com- plex solution to satisfying the Wronskian condition And, x$_{c}$ is the classical position satisfying Ψ( x, t ) = ( 1 √$_{2}$$_{π}$ ℏ u $^{∗}$( t ) ) 1 / $^{2}$e − iS c ( t ) / $^{ℏ}$e ip c ( t ) x/ ℏ × e − i m ˙ $_{u}$∗ 2 ℏ $_{$_{u}$∗}$( x − x c ( t )) $^{2}$. (2) ¨ u + ˙ m m ˙ u + ω $^{2}$( t ) u = 0 , (3) m ( u ˙ u ∗ − ˙ uu $^{∗}$) = i. (4) ¨ x$_{c}$ + ˙ m m ˙ x$_{c}$ + ω $^{2}$( t ) u$_{c}$ = f ( t ) , (5) H ( t ) = p 2 2 m ( t ) + m ( t ) ω $^{2}$( t ) 2 x 2 − f ( t ) x. (1) II. JOINT ENTROPY FOR GAUSSIAN WAVE PACKETS I. INTRODUCTION ∗ Electronic address: sangkim@kunsan.ac.kr