Entropic formulation of the uncertainty principle for the number and annihilation operators

0.16 MB·English

by  Alexey E. Rastegin

Entropic formulation of the uncertainty principle for the number and annihilation
operators
Alexey E. Rastegin
Department of Theoretical Physics, Irkutsk State University, Gagarin Bv. 20, Irkutsk 664003, Russia
An entropic approach to formulating uncertainty relations for the number-annihilation pair is
considered. We construct some normal operator that traces the annihilation operator as well as
commuting quadratures with a complete system of common eigenfunctions. Expanding the mea-
sured wavefunction with respect to them,one obtainsa relevant probability distribution. Another
distribution is naturally generated by measuring the number operator. Due to the Riesz-Thorin
theorem, there exists a nontrivial inequality between corresponding functionals of the abovedistri-
butions. We find the bound in this inequality and further derive uncertainty relations in terms of
2 both the R´enyi and Tsallis entropies. Entropic uncertainty relations for continuous distribution as
1 well as relations for discretized oneare presented.
0
2 PACSnumbers: 03.65.Ta,03.67.-a,42.50.Dv
Keywords: number-annihilationuncertainty, R´enyientropy,Tsallisentropy,Riesz-Thorintheorem
n
a
J
I. INTRODUCTION inarymaterialaregiveninSectionII.Theprincipalpoint
9
of our approach is to assign to the annihilation operator

some normal operator in an extended Hilbert space and
] The concept of entropy has found use in many top-
h relatedresolutionoftheidentity. Physically,thiswayim-
ics including a quantification of uncertainty in quantum
p plies the use ofgeneralizedmeasurementwith additional
- measurements. Entropic uncertainty relations still at-
bosonic modes [26]. The simplest construction of such a
nt tract much attention rather due to its connection with kind is described in Section III. In Section IV, we recall
a somerecenttopics[1]. Thefirstrelationfortheposition–
the Riesz-Thorin theorem and derive an inequality be-
u momentum pair in terms of the Shannon entropies was
tweencertainfunctionals of the probabilitydistributions
q derived by Hirschman [2]. Hirschman’s result has later
[ generated by two measurements of interest. In Section
been improved [3]. A general statement of the problem
V, we obtain entropic uncertainty relations for both the
3 and utility of the entropic formulation have been con-
continuous distribution and distribution with respect to
v sidered in the papers [4, 5]. An improvement of lower
finite bins. Section VI concludes the paper.
9 bound from [5] had been conjectured by Kraus [6] and
1
later stated by Maassen and Uffink [7]. The method of
5
the paper [7] is based on Riesz’s theorem. Extensions
1 II. PRELIMINARIES
of this approach to mixed states and generalized mea-
.
7 surements were considered in [8, 9]. The entropic ap-
0
proachalsoallows to formulate the uncertaintyprinciple In this section, the required facts are briefly outlined.
0
1 for trace-preserving quantum operations [10]. Lets={sn}beaprobabilitydistribution. Forrealα>0
and α = 1, the R´enyi entropy of order α of probability
v: There exist numerous approaches to the problem of distribu6 tion s is defined by [27]
finding a proper expression for the number-phase uncer-
i
X tainty [11, 12]. Both the well-defined Hermitian opera-
1
r torofphase andnumber-phaseuncertaintyrelationhave R (s)= ln sα . (1)
a beenfitwithinthePegg-Barnettformalism[13,14]. Var- α 1 α n n
− (cid:16)X (cid:17)
ious measures of quantum phase uncertainty are com-
For real α=1, the Tsallis entropy of degree α is defined
pared in [15]. In the paper [16], the uncertainty relation 6
by [28]
involvingthenumberandannihilationoperatorswaspro-
posedasanalternativetoknownnumber-phaserelations. 1
UsingthePegg-Barnettformalism,thenumber-phaseun- Hα(s)= 1 α nsαn−1 ≡− nsαnlnαsn , (2)
certaintyrelations interms ofthe Shannonentropywere − (cid:16)X (cid:17) X
obtained [17–19]. For canonically conjugate variables, where ln x x1−α 1 /(1 α) is the α-logarithm.
α
the uncertainty relations were expressed in terms of the ≡ − −
The Tsallis entropy of degree α = 2 is related to the
Tsallis [20] and R´enyi entropies [21]. Mutually unbiased (cid:0) (cid:1)
so-calleddegree of certainty which is used for expressing
bases [22, 23] and tomographic processes [24, 25] have
complementarityrelationsformutually unbiasedobserv-
been examined within an entropic approachas well.
ables [29]. A physical meaning of the entropy (2) was
In the present work, we aim to formulate the entropic the subject of some discussion (see [30] and references
uncertaintyprincipleforthenumberandannihilationop- therein). In the context of functional form (2), condi-
erators. Asentropicmeasures,wewilluseboththeR´enyi tional and relative entropies were introduced and ana-
andTsallisentropies. Theirdefinitionsandsomeprelim- lyzed [31, 32]. In the limit α 1, both the entropies (1)
→
2
and (2) coincide with the Shannon entropy, namely we have [q,p] = i11 and a = (q+ip)/√2. Considering
twomodes,wewilldenotetheirHilbertspacesas and
x
H
R (s)=H (s)= s lns . (3) . Thetracesover andover willbewritten
1 1 n n y x x y
− n Has Tr ((cid:5)) and Tr ((cid:5))H, respectivelyH. ⊗H
X x xy
Note that the entropies (1) and (2) are both particular
cases of more general class of unified entropies [33, 34].
A quantum state is characterized by the probabilities III. GENERALIZED MEASUREMENT
of the outcomes of every conceivable test [35]. Con-
sider Hermitian operator B = nbnΠn, where the Πn The aim is to build a normal operator A that act as
is orthogonal projector on the subspace corresponding a on a subset of states to be measured. For non-normal
to eigenvalue bn. The set Πn Pof projectors obeys the two-boson operators, such a construction was analyzed
{ }
completeness relation nΠn = 11, where 11 is the iden- in [26]. The simplest way involves one additional mode.
tity. Inthecaseofcontinuousspectrum,thesumsarere- We denote the measured bosonic mode by x and the ad-
P
placedwith the correspondingintegrals. If the state just ditional bosonic mode by y. The y-mode is always pre-
priortothemeasurementisdescribedbydensityoperator pared in the fixed initial state, which is taken as some
ρ, then the outcome bn is obtained with the probability eigenstate n0 of the number operatorn. Consider two-
sbtenym=thweTassre(itnρΠ{oΠnn)ne}o[3fa5tn]h.deIdeteisgnevsniatslyutaeotpeisserocafotmBor,ptρleh.teeWldyihsdterenitbeturhtmeioisnnyesids- bpatrooosrodnuiscotnposeprrm|aactaeoli:rHAxA,⊗=AH†ayx=.⊗I0t1.1iysW+imrit1p1inoxrg⊗taAnat=†y,th(aQactt+itnhigiPsi)no/p√teh2re-,
deterministic and the entropies are zero. Therefore, for
the corresponding quadratures
two and more incompatible tests the sum of relevant en- (cid:2) (cid:3)
tropiesisnontriviallyboundedfrombelow[5]. So,studies A+A†
inthesubjectaremainlydevotedtoobtainingsuchlower Q= √2 =qx⊗11y+11x⊗qy , (8)
boundsforsomemeasurementsofinterest. Aswasshown
A A†
buyniMfyianagssteonolanfodrUthffiesneka[i7m],st.he Riesz theorem provides a P= i−√2 =px⊗11y −11x⊗py , (9)
There may be more than one ways to measure quan-
are commuting operators,i.e. [Q,P]=0.
tum uncertainty for non-Hermitian operators. One of
possible approach is to extend the standard expression On certain density operators, our extension leads to
for variance. For arbitrary operator C, we can define its thesameaveragevaluesforthequadraturesandthesame
left and right absolute values as measurementstatisticsforthenumberoperator. Forany
density operator ρ on , there hold
x
H
C :=√CC† , C :=√C†C . (4)
L R
| | | | Tr Qρ n n =Tr (q ρ) , (10)
xy ⊗| 0ih 0| x x
WCe,Ch†ave=|C0|.L L=et|Cu|sRpiuftatnhde olenfltyainfdCriigshntovramriaaln,cie.es., Trxy(cid:0)Pρ⊗|n0ihn0|(cid:1)=Trx(pxρ) , (11)
∆ C 0 and ∆ C 0, by the equalities Trxy(cid:0)Aρ n0 n0(cid:1) =Trx(axρ) , (12)
L R ⊗| ih |
(cid:2) ≥(cid:3) ≥
(∆LC)2 =h|C|2Li−hCihC†i , (5) WPe =shoprtly, awnrdi(cid:0)teAthe=abaove. Ir(cid:1)telaistiuosnusalalys shaQidi t=hathqtxhie,
(∆ C)2 = C2 C† C . (6) h i h xi h i h xi
R h| |Ri−h ih i operator A ”traces” the operator a [26]. We also define
ForHermitian operatorB, these expressionsarebothre- the operator N=nx⊗11y such that
duced to the regular expression (∆B)2 = B2 B 2.
The annihilation operator a and the creatihonio−perhatior Trxy (|nihn|⊗11y)(ρ⊗|n0ihn0|) =hn|ρ|ni , (13)
a† enjoy [a,a†] = 11, whence (∆La)2 = (∆Ra)2+1. In Trxy(cid:8)Nνρ⊗|n0ihn0| =Trx(nνxρ(cid:9)) , (14)
the paper [16], the uncertainty relation is given in terms
of ∆ a and ∆n, where the number operator n = a†a. for real (cid:0)power ν 0. Fo(cid:1)r other operators, the moments
R ≥
We could also use ∆ a as a measure of uncertainty of of higher ordersare not the same. Nevertheless, we have
L
the operator a in a given quantum state.
1
Anotherwayistorelatetheannihilationoperatorwith (∆Q)2 =(∆q )2+n + , (15)
x 0 2
a proper resolution of the identity. Here we have to
1
construct some generalized measurement in an extended (∆P)2 =(∆p )2+n + , (16)
Hilbert space. Letus introduce some notation. By n x 0 2
with integer n 0 we denote the number state bas{i|s iin} (∆A)2 =(∆Lax)2+n0 =(∆Rax)2+n0+1. (17)
≥
the Hilbert space . Each ket n obeys nn = nn .
Putting the dimensHionless quadra|tuires | i | i Itis importantherethat the aboveequalities doholdfor
anyinitialstateρofthex-mode. Further,thetwo-boson
a+a† a a† operator A can be measured practically. The measure-
q = , p= − , (7)
√2 i√2 ment of non-normal two-boson operators, which yield A
3
as a particular case, and its application in heterodyne IV. A CONSEQUENCE OF THE
detection are examined in [26]. RIESZ-THORIN THEOREM
We shall now find common eigenfunctions of the com-
muting operators (8) and (9). In the coordinate repre- Recall some mathematical tools. By c = c ∞ we
sentation, the operator Q and P are rewritten as denoteasequenceofcomplexnumbers,andbynlp (np=0 1)
(cid:8) (cid:9) ≥
∂ ∂ ∂ the space of sequences such that
Q=x+y =u, P= i +i = 2i , (18)
− ∂x ∂y − ∂v
∞ 1/p
in terms of new variables u = x + y and v = x y, c p = cn p < . (24)
− k k n−0| | ∞
for which x = (u + v)/2, y = (u v)/2. A common (cid:16)X (cid:17)
−
eigenfunction Υ (u,v) obeys the equations For the complex-valued function f(x) and p 1, the
ξk
≥
p-norm is defined as
∂Υ
ξk
uΥ (u,v)=ξΥ (u,v) , 2i =kΥ (u,v) ,
ξk ξk ξk
− ∂v 1/p
wheretheeigenvaluesξ ofQandk ofParearbitraryreal f p = dx f(x)p , (25)
k k | |
numbers. The solutions are written in a form (cid:18)Z (cid:19)
Υ (u,v)= 1 δ(u ξ)eikv/2 (19) with f ∞ = sup f(x) : x R for the special case
ξk 2√π − p = k .kBy Lp w{e| deno|te th∈e sp}ace of functions such
∞
that f < . We say that the linear mapping T from
and satisfy the normalization condition p
k k ∞
Lp to Lq is bounded when its (p,q)-norm,
du dvΥξk(u,v)∗Υξ′k′(u,v)=δ(ξ−ξ′)δ(k−k′) . Tf
Z Z T =sup k kq , (26)
p,q
In terms of the variables x and y, these eigenfunctions k k f6=0 f p
k k
are reexpressedas
is finite. In general, it is hard to find exactly values of
Ψ (x,y)=√2Υ (x+y,x y)
ξk ξk (p,q)-norms. Say, the answer is known for the Fourier
−
1 transform in both the discrete and continuous cases. In
= δ(x+y ξ)eik(x−y)/2 , (20)
√2π − the former, it is given by the Young-Hausdorff inequali-
ties; in the latter, it has been found by Beckner [3]. For
sincetheJacobiandeterminant∂(u,v)/∂(x,y)= 2. So,
− arbitrary mapping, we can employ the Riesz-Thorin in-
the functions (20) also enjoy the desired property
terpolation theorem (see, e.g., theorem 1.1.1 in [36]). It
should be noted, however, that this theorem gives only
dx dyΨξk(x,y)∗Ψξ′k′(x,y)=δ(ξ−ξ′)δ(k−k′) . an upper estimate on (p,q)-norms. Here the conjugate
Z Z
indices p,q [1; ] obey 1/p+1/q=1.
If the state of the xy-system right before the joint mea- The Rie∈sz-T∞horin theorem. Assume that p = p ,
surement of Q and P was Φ(x,y), then the probability 0 6 1
T =M and T =M ; thenfor allθ (0;1),
density is calculated by k kp0,q0 0 k kp1,q1 1 ∈
2 T M1−θMθ , (27)
w(ξ,k)= dx dy Ψξk(x,y)∗Φ(x,y) . (21) k kpθ,qθ ≤ 0 1
(cid:12)(cid:12)Z Z (cid:12)(cid:12) where the indices pθ and qθ are put as
The average va(cid:12)lues are calculated in the sta(cid:12)ndard way,
for instance, A(cid:12) = Q +i P √2 and (cid:12) 1 1 θ θ 1 1 θ θ
h i h i h i = − + , = − + . (28)
Q (cid:0) (cid:1)(cid:14) ξ pθ p0 p1 qθ q0 q1
h i = dξ dk w(ξ,k) . (22)
P k
∞
(cid:26) h i (cid:27) Z Z (cid:26) (cid:27) Supposethat ϕ (x) isanorthonormalsetinthe
n n=0
For α > 0 and continuous distributions, we will use a space L2 and n is a fixed number. To each wave func-
norm-like α-parametric functional defined as tion f(x)ϕ (y)0(cid:8)of the(cid:9)two modes with f(x) L2, we
n0 ∈
1/α assign its decomposition with respect to the eigenfunc-
w = dξ dk w(ξ,k)α . (23) tions Ψ (x,y), i.e.
k kα ξk
(cid:18)Z Z (cid:19)
fTuhnicstiiosnaoctfutahlelynuamnboerrmopfoerraαto≥rn1.byDϕeno,ttihnegpnrtohbaebigileitny- f(x)ϕn0(y)= dξ dk f˜(ξ,k)Ψξk(x,y) . (29)
n
Z Z
of obtaining outcome in the x-mode is written as
Here the integral kernel f˜(ξ,k) is represented as
s = dxϕ (x)∗ dx′ϕ (x′) dyΦ(x,y)Φ(x′,y)∗ .
n n n
1
Z Z Z
Ψ ,fϕ = dx dy δ(x+y ξ)
We aim to relate the functionals w α and s β = h ξk n0i √2π −
k k k k Z Z
1/β
∞n=0sβn , whenever Φ(x,y)=f(x)ϕn0(y). ·e−ik(x−y)/2f(x)ϕn0(y) . (30)
(cid:16) (cid:17)
P
4
It is clear that the transforms ϕ˜ (ξ,k) of the basis func- For the pre-measurement state Φ(x,y) = f(x)ϕ (y),
n n0
tions ϕ (x) form an orthonormal set, i.e. ϕ˜ ,ϕ˜ = the probability density (21) is represented as
n m n
h i
δ . Consider the linear map T from lp to Lq such that
mn
w(ξ,k)= f˜(ξ,k)2 . (41)
∞ | |
c c ϕ˜ (ξ,k) (31)
n n
7→ n=0 We have w = f˜ 2 and s = c 2, where α = q/2
X k kα k kq k kβ k kp
and β =p/2. Squaring (35) and (36), we obtain
for any sequence c . The right-hand side of (31) can
n
{ }
be viewed as the transform f˜(ξ,k) of the function
w η2(1−β)/β s , (42)
α β
k k ≤ k k
f(x)= ∞n=0cnϕn(x) , (32) kskα ≤η2(1−β)/βkwkβ , (43)
X
povidedthat1/α+1/β =2andα>1>β. Theinequali-
and we write T(c)=f˜. It follows from the orthonormal-
ties(42)and(43)directlyleadtotheentropicuncertainty
ity of the ϕ˜ (ξ,k)’s that f˜ = c , whence the norm
n 2 2 relation for the number and annihilation operator.
k k k k
T =1. Putting the quantity
2,2
k k
η =sup ϕ˜ (ξ,k) : ξ,k R, n N , (33)
| n | ∈ ∈ V. ENTROPIC UNCERTAINTY RELATIONS
(cid:8) (cid:9)
we also have
Using the probabilities (40), we take the sums in the
f˜ = sup ∞ c ϕ˜ (ξ,k) η ∞ c , formulae (1) and (2) from n = 0 up to n = . For the
∞ n n n ∞
k k ξ,k∈R n=0 ≤ n=0| | probability density (41), we write down
(cid:12)X (cid:12) X
(cid:12) (cid:12)
i.e. f˜ ∞ η c(cid:12) 1 and T 1,∞ (cid:12)η. Using the Riesz- R (w)= 1 ln w α , (44)
Thorkinktheo≤remkwkiththepkaikrs(p ≤=2,q =2)and(p = α 1 α k kα
0 0 1 −
1,q = ), we obtain the inequality 1
1 ∞ Hα(w)= 1 α kwkαα−1 . (45)
T p,q ηθ . (34) − (cid:0) (cid:1)
k k ≤ The nth normalized eigenfunction of the operator n is
where the conjugate indices p and q are such that p = written as
2/(1+θ) < 2 and q = 2/(1 θ) > 2. By θ = (2 p)/p
and (26), the inequality (34)−results in − ϕn(x) xn = √π2nn! −1/2e−x2/2hn(x) , (46)
≡h | i
f˜ η(2−p)/p c . (35) where hn(x) is the (cid:0)nth Herm(cid:1)ite polynomial [14]. The
q p
k k ≤ k k factor (33) is the supremum of modulus of the quantity
Assuming 1<p<2 andq >2,we alsohave the relation
1
Ψ ,ϕ ϕ = dxϕ (x)e−ikx/2 dyϕ (y)
c η(2−p)/p f˜ , (36) ξk n n0 √2π n n0
k kq ≤ k kp (cid:10) (cid:11) Z 1 Z
δ y (ξ x) eiky/2 = eikξ/2 dx ϕ (x)
which is derived from (35) as follows. Let us write γn = · − − √2π n
c q/c∗ if c =0, and γ =0 otherwise. Then we have Z
| n| n n 6 n ϕ(cid:0) (ξ x)e−(cid:1)ikx = 1 ϕ ,φ . (47)
∞ ∞ · n0 − √2π h n i
c q = γ∗c = g˜,f˜ , (37)
n=0| n| n=0 n n h i
wHho¨ledreerg˜i(nξe,Xqku)a=lity ng˜γ,nf˜ϕ˜nX(ξ,kg˜) byf˜defi,ntihtieorne.hDouldesto the Hnfoterhrwechowieecffihpcukietφnktth2ie=nfutkhnϕecnte0ixoknp2a=φn(sx1io.)n=Sinecikeξ/t2heϕnv0a(lξue−hxϕ)ne,−φiikxis,
P|h i|≤k kqk kp
∞
c q g˜ f˜ η(2−p)/p γ f˜ , (38) φ(x)= ϕ ,φ ϕ (x) , (48)
k kq ≤k kqk kp ≤ k kpk kp n=0h n i n
where we have applied (35) with g˜ and γ instead of f˜ ∞n=0X|hϕn,φi|2 =kφk22 =1 , (49)
and c respectively. Transposing the factor X
the modulus of each coefficient does not exceed one.
∞ 1−1/q Combining this with (47) gives
γ = c q = c q−1 (39)
k kp n=0| n| k kq
(cid:16)X (cid:17) sup Ψ ,ϕ ϕ : ξ,k R, n N 1 . (50)
from (38), we finally obtain (36). ξk n n0 ∈ ∈ ≤ √2π
For the number operator, the probabilities are (cid:8)(cid:12)(cid:10) (cid:11)(cid:12) (cid:9)
(cid:12) (cid:12)
The case n = 0, when no photons are initially being
0
s = c 2 = ϕ ,f 2 . (40) in the y-mode, is easier to realize experimentally. For
n n n
| | |h i|
5
n=0, direct calculations give be maximal sizes of bins on these axes. By theorem 192
of the book [37] for integral means, we have
1
eikξ/2 dxϕ (x)ϕ (ξ x)e−ikx = (51)
√2π Z 0 0 − 1 ξl+1 km+1 α , α>1
√12π √1π e−ξ2/4 dze−z2−ikz = √12π e−(ξ2+k2)/4 , ∆ξl∆km Zξl dξZkm dk w(ξ,k)! (cid:26)≥≤, α<1(cid:27)
Z 1 ξl+1 km+1
dξ dk w(ξ,k)α , (59)
wherez =x ξ/2. Thesupremumof(51)withrespectto ∆ξ ∆k
kandξiseq−ualto(2π)−1/2,thatisexactlyη =(2π)−1/2. l m Zξl Zkm
By (50), we rewrite the inequalitie (42) and (43) as whence ∆ξ∆k (1−α)/α r w for α > 1, and
α α
k k ≤ k k
kwkα ≤(2π)−(1−β)/βkskβ , (52) kwkβ ≤(cid:0) ∆ξ∆k(cid:1) (1−β)/βkrkβ for β < 1. The last in-
equalities are obtained by summing the inequality (59)
kskα ≤(2π)−(1−β)/βkwkβ , (53) and raisin(cid:0)g these(cid:1)sums to the powers 1/α and 1/β re-
spectively. Since (α 1)/α=(1 β)/β, we then deduce
where 1/α+1/β = 2 and α > 1 > β. The uncertainty
− −
from (52) and (53) that
relation in terms of the R´enyi entropies is given by
R (w)+R (s) ln2π , (54) ∆ξ∆k (1−β)/β
α β ≥ r s . (60)
α β
k k ≤ 2π k k
underthecondition1/α+1/β =2. Takingthelogarithm (cid:18) (cid:19)
(1−β)/β
of (52) and doing some algebra, we just obtain ∆ξ∆k
s r . (61)
α β
k k ≤ 2π k k
1 α β (cid:18) (cid:19)
− R (w) ln2π+R (s) . (55)
α 1 β α ≤− β Onlyfor∆ξ∆k <2πtheseinequalitiescontainanontriv-
− ial constraint ( (cid:5) 1 by α>1, (cid:5) 1 by β <1).
α β
Here the multiplier of R (w) is equal to ( 1). Hence we k k ≤ k k ≥
α The uncertainty relations are now written as
−
get (54) in the case when the entropy of the distribution
w(ξ,k) has larger order. Otherwise, we repeat the same 2π
R (r)+R (s) ln , (62)
with the inequality (53). The derivation is slightly more α β ≥ ∆ξ∆k
complicated with the Tsallis entropies. We claim that (cid:18) (cid:19)
2π
H (r)+H (s) ln , (63)
α β µ
H (w)+H (s) ln 2π , (56) ≥ ∆ξ∆k
α β ≥ µ (cid:18) (cid:19)
where 1/α+1/β = 2 and µ = max α,β . Indeed, we where 1/α+1/β =2 and µ=max α,β .
{ } { }
rewrite the left-hand side of (56) as the function Theaboveentropicuncertaintyrelationscanallbeex-
tended to mixed initial states of the x-mode. For the
t 1 τ 1
density operator ρ = λ λ λ, we put the functions
g(t,τ)= − + − (57) λ | ih |
1 α 1 β f (x)= xλ and the total probability distributions
− − λ h | i P
ainndterτm=s ofsthβe=variabsleβs.tA=sskuwmkiααng=α >dξ1 d>kβw,(ξt,hke)rαe w(ξ,k)= λλwλ(ξ,k) , sn = λλs(nλ) . (64)
k kβ n n R R X X
holds t ≤ 1 and τP≥ 1 due to w(ξ,k) ≤ η2 < 1 and Here sn(λ) = ϕn,fλ 2, wλ(ξ,k) = f˜λ(ξ,k)2, and the
the normalization relation. Adding (52) in the form |h i| | |
f˜(ξ,k) is defined by (30) with f (x) instead of f(x).
(2π)1−βtβ/α τ, we want to minimize g(t,τ) under the λ λ
≤ Foreachλ,thereholdtheinequalities(52)and(53)with
aboveconditions. In view ofAppendix of the paper [10],
the minimum is reached for t0 = (2π)1−α, τ = 1 and wλ(ξ,k)andsn(λ). BytheMinkowskiinequality,thesame
equal to g(t ,1) = ln 2π. In the same manner, we use inequalities are valid for the total distributions (64). We
0 α
(53) andresolvethe casewhen the Tsallisentropy ofthe refrain from presenting the details here (cf. equation
distribution s has largerdegree. It is properthat the en- (3.9)of[9]). Sotheuncertaintyrelations(54),(56),(62),
tropic uncertaintyrelations(54)and(56)coincide in the and (63) can all be derived.
limit µ 1.
→
Therelations(54)and(56)involvethe continuousdis-
VI. CONCLUSIONS
tributionw(ξ,k). We canalso obtainsimilarrelationsin
terms of probabilities
Wehaveconsideredanapproachtoformulatenumber-
ξl+1 km+1 annihilation uncertainty by means of entropic measures.
r = dξ dk w(ξ,k) , (58)
lm To the annihilation operator we have assigned the nor-
Zξl Zkm
maltwo-bosonoperatorenjoyingthesameaveragevalues
where ξ is a partitionof the ξ-axis and k is a par- for measured density matrices. Its measurement can be
l m
{ } { }
tition of the k-axis. Let ∆ξ = max∆ξ , ∆k = max∆k implemented in practice. The corresponding Hermitian
l m
6
quadraturesare commuting with a common setof eigen- tum degree of freedom occupies a (dimensionless) phase
functions. The decomposition of any wave function of cellbysizeatleast2π. Thesameentropicboundsremain
interest with respect to the common eigenfunctions al- valid for an impure measured state.
lows to obtain a proper probability distribution. The
non-trivial entropic bounds for two generated probabil-
ity distributions have been derived in terms of both the
R´enyi and Tsallis entropies. Entropic uncertainty rela-
Acknowledgments
tions of a state-independent form are written down for
the continuousdistribution(see (54) and(56))aswellas
for discretized one (see (62) and (63)). The discretized I am grateful to S. Abe for bringing [17] to my atten-
distribution is taken with respect to some partitions on tion and to anonimous referee for constructive criticism.
the corresponding axes. Nontrivial entropic bounds are This work was supported in a part by the Ministry of
given for the case, when any product of two bins is less Education and Science of the Russian Federation under
than2π. Thelatterconcurswiththefactthatonequan- grants no. 2.2.1.1/1483,2.1.1/1539.
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