It is shown that the usual one- and two-mode squeezing are based on reducible representations of the SU(1,1) group. Generalized squeezing is introduced with the use of different SU(1,1) rotations on each irreducible sector. Two-mode squeezing entangles the modes and information theory methods are used to study this entanglement. The entanglement of three modes is also studied with the use of the strong subadditivity property of the entropy

Vourdas, A.

English

NASA Technical Reports Server

N94-10580
Generalised equeezin| and informJcton chest7 tpproecb co quintus encenj_emenc
A. VOUR/)AS
Department of Electrical Enlinoerin8 end Electronics
University of Liverpool
P.O. Boz 147
Liverpool L69 3BX
Ic £s shown chac the usual one and ewe-mode squsezin$ are baaed on
reducible representations of the $U(1,1) Stoup. Conoralised squsezin$ is
introduced with the use of different SU(1,1) rotations on each irreducible
sector. Two-mode squeezing onCan$1es the modes and information cheery
methods ere used co study this encan$1omenc. The oncantlomenc of three modes
is also studied wich the use of the scron S subaddicivicy property of the
entropy.
In a recent paper [1] we have explained chat eve-mode squeozin S is bend
on reducible representations of the SU(1,1). The various irreducible sectors
have been identified and different SU(1,1) rotations have been performed on
each of them, Saneralizin$ in this way the concept of squeezins. In this
paper we extend chess ideas. In section 2 we consider one mode squaezins and
prove chac lc Is also baaed on reducible representations of SU(1,1). The ewe
irreducible sectors are identified and different SU(1,1) rotations are
applied on each of them, Sonoralisin$ in this way the concept of one-mode
squsezins. In section 3 the cvo-mode case is considered in connection with
both the SU(1,1) and SU(2) groups. Some of che results presented in [1] are
briefly reviewed here. Each irreducible soccer of the SU(I,1) (or SU(2))
Broup is squeezed independently and the generalised squeezed scats is
characterised by an infintco number of squiezin| parameters. Semllconians
which vtll bead co this type of squsozin|, are presoncsd.
Two-mode squseztnj oncan$1es the eve modes. Especially our senereltsod
squsozin| oncan$1eo them in a very complicated way. One approach co study
this oncanslemonc is by usin S informcion cheery methods. In section _ we
use the subaddiviCy and scron| subaddicivicy properties of the entropy co
define quantities which express the oncanSlemonc of cam and throe quantum
systems. ESlUtcially incerescins are the results for three oncan$1ed sysceaus,
bocauso4bey indicate chac this case to a non-trivia1Seneralisacton of the
eve system encanslemenc. The lector case has of course boon discussed since
che besinninl of quantum mechanics; buc ic is only recently chac some
prelimtrmry discussion of the former case has appeared [2]. Our results
based on information cheery methods susjesc chac che three system
encanslement is a very tncereecin| problem chac requires further study.
2. Generalised one-nmde sauooz£ns
We consider the harmonic oscillator Hilborc space H and express lc as
. + HI (i)H H°
whore H is the subspaco spanned by the even number ol$onstacos and H1 the
subspac_ spanned by the odd number eipnstacos. We also consider the
correspondin| projection operators co chess subspecos:
149
https://ntrs.nasa.gov/search.jsp?R=19940006125 2020-06-16T21:08:33+00:00Z
U, - F I 2.><2NI
o N-0
fl" _ I 2N+I><2N+ 1
N-O
x° + r 1 - 1
The one Bode squeezin| operators Are defined u:
1 LI 1 el #
S (r, e, _) - exp [- - r e" k+ - - r K.] exp (LAKe)
' 2 2
(2)
i
i
l
1 + 1 1 A+2 1 2
- - A A + -- ; K+ - ; K - - AK° 2 4 2 - 2
IXo, X±]-+X± ; IX , x+] -2K °
I 3
K2 - Ko2 - - (K÷ K ÷ K K+) - k(k-l) - - --
2 - - 16
(3)
They Earn A reducible representation of SU(1,1). More specifically,
they form the k - i/4 Irreducible representation when they ArC on H only;
and the k - 3/4 Irreducible representation when they ArC on H1 only°[3].
RelAted to this £l the fA¢¢ Chat:
[S(r, 0, A), .o] = iS(r, e, A), .1] - 0 (4)
The follovLn I unicAZ'y operator squeezes Independently each Irreducible
lactose _ +
U(ro 0o Xo ; rl 01 A1) " S<ro' °o' Xo) "o + s(51' el' _t ) "1 (5)
This Is more |ener81 clam the operator of equ.(3). 0nly In the special
(6)
AccIn I vLch the operator (5)
(7)
Call
re " rl ; Po " il ; Ao " A1
the operator (5) reduces co the opAsi+or (3).
on A coherent scare J A >, vo |it A le1"_rAIIled squeezed ;CA_+:
I A ; 5o +o Ao ; 51 I1 A1 >" U(5o +o Ao ; 51 11 A1) i A >
" S(ro' eo' Xo) "o I A • + S(r 1, fl' _1) "1 I A>
In the specLA1 case of equ.(6) this reduces co the usual squeezed
SCACeS.
150
In systems described by the HamiLconlan
+ +2 * 2 + (_l a+2 + "l *a2) fl (8)
ordinal-coherent states v£11 evolve Inca the generalised squeezed states
(7). In the special case n o - _1 the Hamllconian (8) reduces co the
Hamiltonian
+ a2 +2H - _ a a + _ + _ a (9)
vMch is associated to the usual squeezed states.
3. Cenaralised tvo-mod_._3ADl
The appropriate group for the study of two-emde quadratic HasLlconLans
is Sp(_,a) [_]. In chls paper ve shall only consider Los subgroups SU(I,L)
and SU(2) in connection rich cha Haailconians:
H1 " _1 al + al + _2 a2 + a2 + _ al a2 + _ a 1 a 2 (10)
t ÷
H2 " Wl aL+ al + _2 a2+ a2 + " al a2 + + p al a2 (ll)
correspondingly. Bach of chase Ha-ilconians have been used extansLvely in
quan_ optics problems [5].
Scarcir_ rich the SU(I,1) group ve express the eve-mode HLlberc space as
kmol
(12)
vhere Kk is the subspaca spanned by the number eigenscaces
'k" ( J _+k, _> ; N-n.x (o, k) ..... - )
We also Introduce cha correspondinl projection operators
,rk- _.1 re+k, tl><S*k, S I
The eve.mode SU(I,I) squeezing operators are defLned as
(13)
(l_)
S(r, 0, X) - sxp [
l l
- - r • "il K+ + - r • II K.] axp (/. ,_ Ko)
2 2
l
K+ - el+ a2 + ; K. - a 1 a 2, K° -- (el+ a I + a2+ a 2 + 1)
2
I l
K2 . - (el+ al . a2 + a2)2 . _ (15)
I51
They form a reducible representation of SU(1,1).
when =hey act on the space Hk only, they form the
1 + Ikl
' 2
More specifically,
(16)
irreducible representation of $U(1,1) which belongs in the discrete series.
Note also chat
[S(r, e, x). fk] - 0 (17)
The following unitary operator squeezes independently each irreducible
sector:
U({r k, 0k, _k )) - _ S(r k, 0k, _k ) wk
In the special case
(18)
... - r.1 - r o - r I - ...
m 0 m
.... 01"Oo- 1 "'"
.... _-1 " Xo " hi " "'"
the operators (18) reduce co the operators (15).
(19)
Actin$ with the operators (18) on eve-mode coherent states we Set
Seneralised t_o-nodo squeezed states:
U ((r k 0k %)) I A1. A 2 • ° Z S(r k, 0k, _k ) "k [ A1. A2 • (20)
k
In the special c.e o! equ.(19) they reduce co the usual two.mode
squeezed states. In eystaut described by the Hanlltonian
H-w I al + a 1 w2 a2 + a2 + _ (_k al a2 _k*+ + al + a2+)Xk (21)
k
ordinary coherent staten viii evolve inca the scacas of aqu.(20). In the
special case chac all the _ are equal co each ocher, the itmllconian (21)
reducee4_chrltmLtlconian TIO)
In the case of the SU(2) sroup ve express the two-node Hilberc space as
HA x HE - j_ H2J+I
J -0, _, 1 ..... (22)
where H2J+I is the subspece spanned by the number eisenscaces
H2J+I " ( J N, 2J-H> ; N - 0 ..... (2J))
Vo also introduce the correspondin$ projection operators
(23)
152
" _J I ", 2J-br> < N, 2J-,I
w2J+l N-O
_2J+I " 1
The SU(2) squeezlns operators are defined is:
1 el #1 -l# + - r J ] exp (l_J o)T (r, #, _) - sxp [ - - r • J+
2 2 -
1
J+ " el+ a2 ; J_" el a2+ ; Jo "-2 (sl+ °l " "2+ a2)
1 1
j2 . [ - (el + al + a2+ a2)] [ _ (el + al + a2+ a2 + 1)]
2 2
(2;)
(25)
They form s reducible representation of SU(2). When they act on the
space H... only, they form the J irreducible represencacIon of SU(2). Note
also ch|_ t
[T (r, e, ;), w2J+l] - 0 (26)
The followln8 unicary operator performs SU(2) rotaCions independently on each
irreducible sector:
U ({r2j+l, 02J+i, _2J+l}) - _ T(r2j+l, #2J+l' _2J+l ) w2J+l (27)
In the special case:
'2 i iii
A1 A2 ...
The operators (27) reduce to the operators (25).
operators (27) on two-node coherent states we Set the states:
U (4_j+l, e2j+¢, x2j+l}) I x1, A2 • "
z (rZ]+l, e2]+l, _2j+l) w2j+l I _, x2 >
J
They will be formed durInK the cinm evolution of ordinary coherent
states In system described by the Haailtonian:
(28)
Accln$ vlch the
H - _1 al + al + _2 a2+ a2 + _ w2J+l (_2J+1 al a2+ + "2J+l al + a2)
In the special case chat all the _4.1 are equal to each other, the
HaaIlconIan (30) reduces co the HaikilEonian (II),
(29)
(30)
The uncercaIncy properciee of the scares (20), (29) have been studied in
[I]. The results presented chore show that both of chose states exhibit
squeezin&.
153
4. Information _heor V anoroech co auen_um sncanllems_
In ch£s section we use quant_ LnEonution theory methods for the study
o£ two- tnd three-mode corretated systems. Let p be a two-mode density
matrix and <1_1>, <_.> the average number of photons in the two modes. As in
our previous Oork [_] we define the tn£or_ation contained in this density
matrix as
ch th
I " Smax S(p) - S[p 1 (<};1>) x P2 (<32>)] - S(p)
S(p) - - Tr ptn p
th <31>Ni
Pi (<31>) " (1 +<31>)1+_ l I Ni ><_t I ; l- 1, 2
(31)
Pollovlnj the nssentropy idles of Sriilouln ve subtract here the entropy of
the system iron the maxlmum entropy thit the system could have had, wlth the
avera|e number of photons in the t-we modes been kept fixed. The aaxlaua
entropy is equal to the entropy of 8 chermll system vith an oversee number of
photons in the L'VOsodas <311>, <3q2>. Takln I parclal traces, we define:
Pl " Tr2 p ; P2 " Trl p (32)
end express the informatlon (31) as [7. 8]
I - 11 + 12 + I12
1 t - S [pi th (<3L>)] - S(p l)
I12 - S(p 1) + S(p 2) - S(p) (33)
I( Is the inforKatlon in the soda I; and I1_ ts the lnfonaation In the
correlltion becveen the eve sodas. The subaddit[gi_y property ensures that
the I12 is non-nesativo. Numerical evaluation of the quantities I1, I2, I12
for slgeral examples has been presented in [1].
A non-trivial extension of chose ideas occurs in the case of three
correlated nodes. The inforBaclon In this case is &lyon by
I - s (pch) . S(#)
pth_ plt'h (<31>) x p2 th (<32>) x p3 th (<33 >)
Vo define
(34)
PlJ " Trk p ' Pi " Trjk p
th
zi -s[p I (<31>)] . s(p l)
zij - s(p l) + s(p l) - s(pij)
I_ Is the information in the mode 1.
c6rrelation;bscveen the modes (i,j).
in the three-nods system as:
(35)
I+4 is the information in the
W/athos express the total information
I - 11 + 12 + I 3 + 112 + I23 + I(12 ; 23) (36)
154
where
I(12 ; 21) - S(P12) + S(P23) - S(p) - S(p 2) (37)
The moron s subaddlcivicy property [9] ensures chaC the quanclcy I(12 ; 23) is
non-negative. For sysRetry reasons, somebody misht be tempted to split
Z(12 ; 23) as:
I(12 ; 23) - I13 + A (38)
so that he can express the information I o£ equ.(36), as the sum of the three
infornations in the three modes; the three lnformacLons in the correlated
pairs o£ modes; and the quantity X characterisin$ the correlation becween
all modes. Hovever the quanCit 7 k is not necessarily positive and los
interpretation as information would be incorrect. Therefore, the information
! o£ a three-mode system is the sum o£ the three informacions in the three
modes; the eve correlation informacions in eve of the pairs; and the
information 1(12 ; 23) of equ.(37) which describes hey types of correlations
in che three-mode systems. This result can be used as a "suide" of how co
study the encanslemonc of three systems. It is seen that three system
entanslenenc is a non-trivial pneralieacion of eve system entan|lesent.
In man7 cases the concept of squeezin8 Ls based on reducible
representations of the SU(1,1) (or SU(2)) |romp. In creme cases different
SU(1,1) (or SU(2)) rotations on each.irreducible sector lead co leneralLsed
squeezing. These ideas have been applied to both one-mode and eve-mode
squeezin s.
Tvo.mode squsezins correlates the cvo-modes and information cheery
methods have been used for the study of these correlations. The
subaddiciviCy and sirens subadditivicy properties of the entropy have been
used for the study of _wo and three correlated systeis, correspondinsly. Xc
has been shorn that the entanjlemonc of three sTsceu is a non-trivial
seneralisaCion of the entanslenenc of eve syscemo. Further work is required
in this direction.
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156


Generalised squeezing and information theory approach to quantum entanglement

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