Information-Theoretic Uncertainty Relation and Random-Phase Entropy

0.43 MB·English

by  Kyoung Kon Kim

Information-Theoretic Uncertainty Relation and Random-Phase Entropy
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: January 18, 2010)
Dunkel and Trigger [Phys. Rev. A 71, 052102 (2005)] show that the Leipnik’s joint entropy
monotonouslyincreasesfortheinitiallymaximallyclassicalGaussianwavepacketforafreeparticle.
AfterexpressingthejointentropyofthegeneralGaussian wavepacketsforquadraticHamiltonians
as S(t) = ln(e/2)+ln(2∆x(t)∆p(t)/~), we show that a class of general Gaussian wave packets
0 doesnotwarrant themonotonousincreaseof thejoint entropy. Weproposethat therandom-phase
1 entropy with respect to the squeeze angle always monotonously increases even for non-maximally
0 classical states.
2
PACSnumbers: 03.65.Ta,03.67.-a,05.30.-d
n
a
J
I. INTRODUCTION [5], regardless of an external driving force.
8
1
] The Leipnik’s joint entropy is an important measure II. JOINT ENTROPY FOR GAUSSIAN WAVE
of information loss of quantum states. Recently Dunkel
h PACKETS
p and Trigger [1] have shown that the Leipnik’s joint en-
- tropy monotonously increases for a free Gaussian wave
t We consider the quadratic Hamiltonians of the form
n packet and a wave packet in a monochromatic electro-
a magnetic field when it is initially prepared as a maxi- p2 m(t)ω2(t)
u mally classical state. They have further argued that the H(t)= + x2 f(t)x. (1)
q 2m(t) 2 −
monotonous increase of the entropy may be an intrinsic
[
propertyforothertypesofinitialwavepackets. However, As shown in Ref. [6], the most general quadratic Hamil-

1 it was shown in Ref. [2] that the Leipnik’s joint entropy tonian may be canonically transformed into the form
v foraGaussianwavepacketofaharmonicoscillatoroscil- (1). A harmonic oscillator has constant m and ω
0 0
6
lates between the maximumand the lower bound, which while a free particle is the limit of ω = 0. A charged
6
was also confirmed in Ref. [3]. particle in a monochromatic electric field is described
9
2 In the present paper, we study the Leipnik’s joint en- by f(t) = qEcos(ωt). The Hamiltonian with time-
. tropyforthe generalGaussianwavepacketsofquadratic dependent mass or frequency describes an open system.
1
0 Hamiltonians, whose mass and frequency may depend TofindtheexactGaussianwavepacketfor(1),weadopt
0 on time, and which may be driven by an external force. the invariant method by Lewis and Riesenfeld [7] (see
1 The exact Gaussian wave packets involve the initial po- also Ref. [8]). The Hamiltonian (1) has a pair of lin-
: sition and momentum for the centroid, and the initial ear invariant operators [9–12], which leads to the most
v
i position and momentum variances, which are equivalent general Gaussian wave packet [4]
X
to a squeeze parameter and a squeeze angle [4]. We first
1 1/2
r expresstheLeipnik’sjointentropyintermsoftheuncer- Ψ(x,t) = e−iSc(t)/~eipc(t)x/~
a tainty relationandthen investigatethe conditionfor the (cid:16)√2π~u∗(t)(cid:17)
monotonousincreaseoftheentropy. Thejointentropyof e−i2m~uu˙∗∗(x−xc(t))2. (2)
general Gaussian wave packets for a harmonic oscillator ×
oscillates between the maximum and the lower bound of A direct calculation shows that (2) indeed satisfies the
the joint entropy,while a class of Gaussianwave packets time-dependent Schr¨odinger equation. Here, u is a com-
ofafreeparticlehasthejointentropythatfirstdecreases plex solution to
andthenmonotonouslyincreases. However,therandom-
m˙
phase entropy with respect to the squeeze angle indeed u¨+ u˙ +ω2(t)u=0, (3)
monotonously increases for the general Gaussian wave m
packets for quadratic Hamiltonian systems, including a satisfying the Wronskian condition
free particle, a harmonic oscillator and the Caldirola-
Kanai Hamiltonian with an exponentially varying mass m(uu˙∗ u˙u∗)=i. (4)
−
And, x is the classical position satisfying
c
m˙
∗Electronicaddress: [email protected] x¨c+ mx˙c+ω2(t)uc =f(t), (5)
2
and p (t) = mx˙ (t) the corresponding momentum, and
c c
S (t) = tdt(p2/2m mω2x2/2+fx ) the classical ac-
c 0 c − c c
tion. ThRe two variances are given by
∆(x)(t) = x2 x 2 = ~u∗(t)u(t), 1.5
h i−h i
∆(p)(t) = p p2 p 2 =mp ~u˙∗(t)u˙(t). (6) 1
h i−h i
p p 0.5
The Leipnik’s joint entropy 0 0
0.2 1
0.4 r0.6 4 3the2ta
S(t) = dxψ(x,t)2ln ψ(x,t)2 0.8 1 6 5
−Z | | | |
dpψ˜(p,t)2ln ψ˜(p,t)2 ln(2π~) (7) FIG.1: TheentropychangeS(0)−ln(e/2)forthefreeparticle
−Z | | | | −
as the function of the squeeze parameter r and the squeeze
angle ϑ is drawn in theranges 1≥r≥0 and 2π>ϑ≥0.
of the Gaussian wave packet (2) is determined by the
uncertainty relation as
e 2∆x(t)∆p(t))
S(t)=ln +ln . (8)
2 ~
(cid:16) (cid:17) (cid:16) (cid:17)
4
Note that the centroid (x ,p ) does not give any contri-
c c
3
bution to the joint entropy. So, the question is how to
2
findthevariancesforthepositionandmomentum. There
1
is arbitrariness in choosing a complex solution u to Eq.
(3) since any linear superposition of u and u∗ is also a 0 2 0.2 0
solution. For instance, the Gaussian wave packet for a 4 T6 0.6r0.4
free particle with the minimum uncertainty at t = 0 is 8 10 1 0.8
obtained by the solution
1 t FIG.2: TheentropychangeS(t)−ln(e/2)forthefreeparticle
u0(t)= 1 i , (9) with ϑ = π/2 as the function of time T = t/m0 and the
√2(cid:16) − m0(cid:17) squeezeparameter r.
and for a harmonic oscillator by
1 does not depend on the details of the external force. We
u0(t)= e−iω0t. (10) thus obtain the Leipnik’s joint entropy
√2m ω
0 0
e 1 ϑ ϑ
Then, the most general solution satisfying Eq. (4) may S(t)=ln + ln cos2 +e4rsin2 +
2 2 2 2
involve two parameters r and ϑ as (cid:16) (cid:17) (cid:16) (cid:17)
1 ϑ t ϑ ϑ t ϑ
ln (cos + sin )2+e−4r(sin cos )2 .
u(t)=(coshr)u0(t)+(e−iϑsinhr)u∗0(t), (11) 2 (cid:16) 2 m0 2 2 − m0 2 (cid:17)
(13)
where r 0 and2π >ϑ 0. Here,the r and ϑ havethe
≥ ≥
interpretation as the squeeze parameter and angle [10, The initial entropy at t=0
11], which are equivalent to determining two integration
e 1
constants,∆x(0)and∆p(0). Infact,thegeneralsolution S(0)=ln + ln(1+sin2ϑsinh22r) (14)
2 2
(11)leadstothe Bogoliubovtransformationbetweenthe (cid:16) (cid:17)
Fock bases constructed from u(t) and u0(t): dependsnotonlyonthesqueezeparameterrbutalsothe
squeeze angle ϑ, which is drawn in Fig. 1. One interest-
a(t) = (coshr)a0(t)−(eiϑsinhr)a†0(t), ing point is that the minimum entropy may be provided
a†(t) = (coshr)a†(t) (e−iϑsinhr)a (t). (12) by the zero-squeeze angle (ϑ = 0) even for the non-zero
0 − 0 squeeze parameter r. The entropy (13) monotonously
Though the free particle allows a shift of time, which increases against time in the range of squeeze angle
correspondstochoosingasetofrandϑ,theHamiltonian (π ϑ 0) and has the minimum entropy S(0). This is
≥ ≥
(1)doesnotpreservethetime-translationalsymmetry,so numerically shown in Fig. 2.
weselectthe wavepacketsatt=0by adjustingr andϑ. However,in the other range (2π >ϑ>π) the entropy
First, we consider the joint entropy for the free parti- has the lower bound
cle with ω = 0. As the external force only governs the e
centroid of the Gaussian wave packet, the joint entropy Smin =ln , (15)
2
(cid:16) (cid:17)
3
2.5
3 2
1.5
2
1
1 0.5
0
0 0 0 0
2 0.2 0.5 0.2
4 T6 0.6r0.4 1T1.5 2 0.6r0.4
8 0.8 2.5 0.8
10 1 3 1
FIG.3: TheentropychangeS(t)−ln(e/2)forthefreeparticle FIG. 4: The entropy change S(t)−ln(e/2) for the harmonic
with ϑ = 3π/2 as the function of time T = t/m0 and the oscillator is drawn as the function of time T = ω0t and the
squeeze parameter r. The entropy reaches the lower bound squeezeparameter r for ϑ=0.
ln(e/2) at t∗.
III. RANDOM-PHASE ENTROPY
att = (1 e−4r)sinϑ/2m(sin2(ϑ/2)+e−4rcos2(ϑ/2)),
∗
which is−nu−merically confirmed in Fig. 3. Thus, the Now we show that the random-phase entropy always
Gaussian wave packets for (1) do not necessarily war- leads to a monotonously increasing entropy for all the
rant the monotonous increase of the joint entropy. This Hamiltonians(1)withtime-dependentmassorfrequency
counter-example implies that the argument by Dunkel or with the external force. Regarding the squeeze angle
andTrigerisonlytruefortheinitiallymaximallyclassical ϑ as a randomvariable for the Gaussianwavepacketsat
states with ∆x(0)∆p(0) = ~/2 [1]. Indeed, the initially t, the random-phase entropy is given by
maximally classical state is provided either by ϑ = 0,π
e cosh(2r)+1
foranyr,leadingtothemonotonouslyincreasingentropy S(t) = ln +ln
2 2
(cid:16) (cid:17) (cid:16) (cid:17)
S(t)=ln e + 1ln 1+e±4r t2 , (16) +ln(cid:16)2m(t)|u0(t)u˙0(t)|(cid:17). (20)
2 2 m2
(cid:16) (cid:17) (cid:16) 0(cid:17) 2π
In deriving Eq. (20), we used the integral ln(a +
0
withtheupper/lowersignforϑ=π/0orbyr=0forany bcosx+csinx)=2πln((a+√a2 b2 c2)/2)R[13]. The
− −
ϑ, leading to another monotonously increasing entropy random-phase entropy may be written in terms of the
minimal uncertainty as
e 1 t2
S(t)=ln + ln 1+ . (17) e cosh(2r)+1
2 2 m2 S(t) = ln +ln
(cid:16) (cid:17) (cid:16) 0(cid:17) 2 2
(cid:16) (cid:17) (cid:16) (cid:17)
2(∆x) (∆p)
+ln Ψ0 Ψ0 . (21)
Second, we consider the joint entropy of the general ~
(cid:16) (cid:17)
Gaussian wave packets for the harmonic oscillator with
Then, the random-phase entropy is
m and ω . The Leipnik joint entropy is given by
0 0
e cosh(2r)+1
S(t) = ln +ln
S(t)=ln e + 1ln 1+sinh2(2r)sin2(2ω t ϑ) .(18) (cid:16)2(cid:17) (cid:16) 2 (cid:17)
(cid:16)2(cid:17) 2 (cid:16) 0 − (cid:17) t2
+ln 1+ (22)
m2
As shown in Fig. 4, the joint entropy oscillates between (cid:16) 0(cid:17)
the maximum for the free particle and
e 1 e cosh(2r)+1
S =ln + ln 1+sinh2(2r) (19) S =ln +ln (23)
max (cid:16)2(cid:17) 2 (cid:16) (cid:17) (cid:16)2(cid:17) (cid:16) 2 (cid:17)
for the harmonic oscillator.
and the lower bound ln(e/2). This oscillator behavior
As an open system, we consider the time-dependent
was first pointed out in Ref. [2]. mass m(t) = m eγt, which is well-known as the
0
Thus,wemayconcludethattheLeipnik’sjointentropy Caldirola-Kanai Hamiltonian [5]. The solution to Eq.
for the free particle andthe harmonic oscillatordoes not (3) that satisfies (4) and gives the minimum uncertainty
alwaysmonotonously increase for the whole range of the is [14]
squeeze parameter and squeeze angle. Then, a ques-
tion may be raisedwhether one may get aninformation- e−γt/2 γ 2
u (t)= e−iωt, ω = ω2 . (24)
theoretic entropy in concord with thermodynamics. 0 √2m0ω r 0 −(cid:16)2(cid:17)
4
Then, the random-phase entropy is squeeze angle as shown in Eqs. (13) for the free particle
and (18) for the harmonic oscillator.
e cosh(2r)+1 1 γ2
S =ln +ln + ln 1+ . (25) ThegeneralGaussianwavepacketshavetheinitialen-
2 2 2 4ω2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) tropy (14) that depends on the squeeze parameter and
Even for r = 0, the damping effect contributes to the the squeeze angle. It is shown that the Leipnik’s joint
lower bound of entropy by the amount of the last term, entropy satisfies the lower bound S(t) ln(e/2) and
≥
which is related with the generalized minimum uncer- monotonously increases for the range of squeeze angle
tainty [14]. π ϑ 0, while for the remaining range 2π > ϑ > π it
≥ ≥
For time-dependent Hamiltonians (1), we may find first decreases until the lower bound, ln(e/2), is reached
a useful lower and upper bound. Using the identity, and then it monotonously increases. This counter-
ln(2m(t)u (t)u˙ (t))=ln(1+(m(t)du (t)2/dt)2)/2,we example implies that the joint entropymonotonouslyin-
0 0 0
find the l|ower boun|d for the random-|phase| entropy creases only for the initially maximally classical states
that have the minimum uncertainty. Similarly, the joint
e
entropy of the squeezed vacuum states of the initially
S(t) ln +ln cosh(2r)+1 . (26)
≥ (cid:16)2(cid:17) (cid:16) (cid:17) maximally classical state for the harmonic oscillator os-
cillates between the maximum and the lower bound,
Similarly, from H(t) ~ω(t)m(t)u (t)u˙ , with re-
h iΨ0 ≥ | 0 0| ln(e/2).
spect to the Gaussian wave packet Ψ obtained using
0 In this paper, we have proposed that the random-
u (t), we find the upper bound
0 phase entropy with respect to the squeeze angle leads to
monotonously increasing entropy for the general Gaus-
e 1 H(t)
ln +ln cosh(2r)+1 + ln h iΨ0 S(t)(.27) sian wave packets, which is shown explicitly for a free
(cid:16)2(cid:17) (cid:16) (cid:17) 2 (cid:16) ~ω(t)/2 (cid:17)≥ particle, a harmonic oscillator and the Caldirola-Kanai
Hamiltonian. For a quadratic Hamiltonian with time-
dependent mass or frequency, the joint entropy satis-
IV. CONCLUSION
fies the lower bound (26) and the upper bound (27).
It remains an open question whether the random-phase
WehaveinvestigatedtheLeipnik’sjointentropyofthe
entropy monotonously increases for all time-dependent
general Gaussian wave packets for quadratic Hamiltoni-
quadratic Hamiltonians.
ans. TheGaussianwavepackets(2)havefourintegration
constants: two constants for the classical position and
momentum, and two variances for the position and mo-
mentum. Thefirsttwoconstantsareequivalenttoacom-
plexparameterforthecoherentstateandthesecondtwo
Acknowledgments
constants to the squeeze parameter and the squeeze an-
gle. Thejointentropy(8)isexpressedentirelyintermsof
theuncertaintyrelation,whichisindependentofthecen- SPK appreciates APCTP for the warm hospitality,
troid but depends on the position and momentum vari- where this paper was finished. This work has been sup-
ances. We computedthe jointentropyforafreeparticle, ported by KORDI researchprogram(PG47100)through
a harmonic oscillator and the Caldirola-Kanai Hamilto- KRCF(Korea Research Council of Fundamental Science
nian with an exponentially varying mass. The joint en- and Technology) of Ministry of Education, Science and
tropy depends only on the squeeze parameter and the Technology.
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