3 FIG. 4: The entropy change S ( t ) − ln( e/ 2) for the harmonic oscillator is drawn as the function of time T = ω 0 t and the squeeze parameter r for ϑ = 0. FIG. 3: The entropy change S ( t ) − ln( e/ 2) for the free particle with ϑ = 3 π/ 2 as the function of time T = t/m 0 and the squeeze parameter r . The entropy reaches the lower bound ln( e/ 2) at t$_{∗}$ . Now we show that the random-phase entropy always leads to a monotonously increasing entropy for all the Hamiltonians (1) with time-dependent mass or frequency or with the external force. Regarding the squeeze angle ϑ as a random variable for the Gaussian wave packets at t , the random-phase entropy is given by at t$_{∗}$ = − (1 − e − 4 $^{r}$) sin ϑ/ 2 m (sin $^{2}$( ϑ/ 2)+ e − 4 r cos $^{2}$( ϑ/ 2)), which is numerically confirmed in Fig. 3. Thus, the Gaussian wave packets for (1) do not necessarily war- rant the monotonous increase of the joint entropy. This counter-example implies that the argument by Dunkel and Triger is only true for the initially maximally classical states with Δ x (0)Δ p (0) = ℏ / 2 [1]. Indeed, the initially maximally classical state is provided either by ϑ = 0 , π for any r , leading to the monotonously increasing entropy In deriving Eq. (20), we used the integral ∫ 2 π 0 ln( a + b cos x + c sin x ) = 2 π ln(( a $_{+}$√$_{a}$ 2 − b 2 − c $^{2}$) / 2) [13]. The random-phase entropy may be written in terms of the minimal uncertainty as with the upper/lower sign for ϑ = π/ 0 or by r = 0 for any ϑ , leading to another monotonously increasing entropy Second, we consider the joint entropy of the general Gaussian wave packets for the harmonic oscillator with m$_{0}$ and ω$_{0}$ . The Leipnik joint entropy is given by Then, the random-phase entropy is for the free particle and As shown in Fig. 4, the joint entropy oscillates between the maximum and the lower bound ln( e/ 2). This oscillator behavior was first pointed out in Ref. [2]. Thus, we may conclude that the Leipnik’s joint entropy for the free particle and the harmonic oscillator does not always monotonously increase for the whole range of the squeeze parameter and squeeze angle. Then, a ques- tion may be raised whether one may get an information- theoretic entropy in concord with thermodynamics. for the harmonic oscillator. As an open system, we consider the time-dependent mass m ( t ) = m$_{0}$e $^{γt}$, which is well-known as the Caldirola-Kanai Hamiltonian [5]. The solution to Eq. (3) that satisfies (4) and gives the minimum uncertainty is [14] III. RANDOM-PHASE ENTROPY S ( t ) = ln ( e 2 ) + 1 2 ln ( 1 + e ± 4 r t 2 m 2 0 ) , (16) S ( t ) = ln ( e 2 ) + ln ( cosh(2 r ) + 1 2 ) + ln ( 2 m ( t ) | u$_{0}$ ( t ) ˙ u$_{0}$ ( t ) | ) . (20) S ( t ) = ln ( e 2 ) + 1 2 ln ( 1 + t 2 m 2 0 ) . (17) S ( t ) = ln ( e 2 ) + ln ( cosh(2 r ) + 1 2 ) + ln ( 2(Δ x )$_{Ψ}$$_{0}$(Δ p )$_{Ψ}$$_{0}$ ℏ ) . (21) S ( t ) = ln ( e 2 ) + 1 2 ln ( 1 + sinh $^{2}$(2 r ) sin $^{2}$(2 ω$_{0}$t − ϑ ) . (18) S$_{max}$ = ln ( e 2 ) + 1 2 ln ( 1 + sinh $^{2}$(2 r ) (19) S ( t ) = ln ( e 2 ) + ln ( cosh(2 r ) + 1 2 ) + ln ( 1 + t 2 m 2 0 ) (22) S = ln ( e 2 ) + ln ( cosh(2 r ) + 1 2 ) (23) u$_{0}$ ( t ) = e − γt/ 2 √$_{2}$$_{m}$$_{0}$$_{ω}$ e − $^{iωt}$, ω = √ ω 2 0 − ( γ 2 ) $^{2}$. (24) 0 0.2 0.4 0.6 0.8 1 r 0 2 4 6 8 10 T 1 2 3 0 0.2 0.4 0.6 0.8 1 r 0 0.5 1 1.5 2 2.5 3 T 0.5 1 1.5 2 2.5