Quan um Supe posi ion in Ha monic Oscilla o o
Communica ion and C yp og aphy
Abhiji Roy1,*
1IIIT Guwaha i, ECE, Guwaha i, Assam, 781015, India
*abhiji o[email p o ec ed]
ABSTRACT
The speci ics o he quan um pa icle a e gi en by he Sch odinge equa ion. The D B oglie p inciple allows us o measu e he
pa icle’s posi ion o momen um. The pa icle’s ene gy o ime can be measu ed in he same way. As a esul , pa icle ene gy can
also be u ilised o c yp og aphy and communica ion. Main aining he pa icle’s ea u es ac oss a e y long dis ance is challenging.
Howe e , he wa e can go o wa d o be ansmi ed ia he quan um pa icle media. The e o e, communica ion can be done ia
Quan um Wa e. The s a es o he quan um pa icle a e employed o qubi s in quan um compu a ion. Addi ionally, he qubi ’s
ange is cons ained. I a Quan um Wa e is employed, a Quan um Analog Signal o a Quan um Digi al Signal will be p oduced
in place o a quan um pa icle. The Quan um Supe posi ion can play an e icien ole he e. By con e ing he analog signal o
digi al bi s, mo e digi al bi s can be p oduced, depending on he ansmi e and ecei e speci ica ions o Quan um Wa es. In
he s udy, h ee hypo hesis a e discussed. These hypo heses desc ibe he na u e o he Quan um Wa e, Quan um Supe posi ion
and Quan um Rela i i y. P o ocols o ene gy-based quan um signals a e also p oposed he e. Addi ionally, he secu i y o he
ansmi ed Quan um Signal can be imp o ed by using he Quan um Wa e. The He mi e polynomial can be u ilized o ep esen
he Quan um Wa e. Quan um c yp og aphy is o ally dependen on quan um mechanics, in con as o classical enc yp ion’s
g owing compu ing complexi y. The degene acy p ope y o he Quan um Signal also p o ec s he signal’s secu i y. Th ough
a ious echniques, he enc yp ion and dec yp ion o he quan um message signal a e also p oposed o a c yp og aphy applica ion.
Quan um Wa e, Quan um Supe posi ion, Quan um Rela i i y, Quan um En anglemen , Quan um Signal, Quan um
Communica ion, Quan um C yp og aphy
In oduc ion
Fo he wi eless communica ion, classical communica ion is imp o ed om 0
G
o 5
G
and as he Fu u e Gene a ion
communica ion module, 6
G,
7
G
and 8
G
a e also being in es iga ed.Telephone Sys em (AMTS), No wegian o
O en lig Landmobil Tele oni (OLT), Public Land Mobile Telephony and Swedish abb e ia ion o Mobile elephony
sys em Da a (MTD) modules ha e been used ill 0
.
5
G
communica ion. La e No disk Mobile Telephony (NTD),
Ad ance Mobile Phone Se ice (AMPS) and Cellula Digi al Packe Da a ha e been used as he 1
G
communica ion
echnology. Fo 1
G
communica ion FDMA mul iplexing was used. Nex TDMA and CDMA we e in oduced o
Gene al Packe Radio Se ice (GPRS) and Enhanced Da a a es o GSM E olu ion o Enhanced GPRS (EDGE)
wi h 2Gcommunica ion modules. Fu he he oice da a communica ion has been upg aded wi h he Voice, ideo
and da a communica ion in 3
G
communica ion, whe e CDMA based High-Speed Downlink Packe Access (HSDPA)
and High-Speed Uplink Packe Access (HSUPA) mudules we e used. In 4
G
communica ion, MC-CDMA, OFAM
mul iplexing ha e been used o he Long Te m E olu ion (LTE), Ul a Mobile B oadband (UMB) and THE IEEE
802.16 (WiMAX) echnologies o p o ide he in e ne se ice. He e IP-b oadband, LAN, WAN and PAN s anda d
ha e been in oduced. Fo he 5
G
communica ion CDMA based Wo ld Wide Wi eless Web (WWWW) and IP 6 a e
being used. He e he Dynamic In o ma ion access o wea able de ices wi h AI based capabili ies ha e been p o ided
wi h communica ion. La e he in es iga ion is going on o Ai Fibe Technology based 6
G
communica ion, whe e
downloads, uploads, supe - as b oadband In e ne , CCTV moni o ing, mul iple line elephones, ideo con e encing
e e y elecommunica ions equi emen s will be esol ed. In he 7
G
and 8
G
communica ion he s anda ds and
p o ocols o sa elli e o sa elli e communica ion sys em and o cellula o sa elli e sys em will be in es iga ed o
LEO, GEO sa elli es. Bu s ill he classical communica ion is no capable o physical enc yp ion and e y long
ange ex a- e is ial objec communica ion, which can be sol ed in Quan um communica ion1,2.
The ounda ion o quan um communica ion is he Quan um Key Dis ibu ion (QKD). The lossy ib e is c ucial o
he pho ons’ abili y o a el. E en hough wo communica ing pa ies c ea e a key ha is sa e om ea esd oppe s,
hey s ill need o ely on a hi d pa y o ex end he communica ion’s ange
3
. The li e a u e uses ee space
ansmission o elecom ib e channels o quan um communica ion. The a e age signal s eng h is a he single
pho on le el since he QKD is based on he single pho on. In addi ion o he pola iza ion-based encoding, he
quan um coo dina e sys ems o a e ela ed o he eloci y o he ansmi e and ecei e . Phase a ia ion is also
in oduced by he mi o s and coa ings4.
Quan um en anglemen can be used o sol e he issue. Howe e , he p opaga ion in he quan um channel
deg ades he pa icle-based en anglemen . En anglemen pu i ica ion is necessa y o la ge-scale implemen a ion
due o he ine i able noise. The pho on is abso bed in he quan um communica ion channel o he pho on-based
elepo a ion. I akes a lo o pho ons o communica e in o de o ix he p oblem. The en anglemen s a es likewise
de e io a e apidly wi h he channel leng h. As a esul , many pai s mus be p ope ly in e wined. The p ocedu e is
elian on adi ional communica ion
5
. Quan um communica ion also makes use o con inuous a iable sys ems
ha exploi he elec omagne ic ield’s bosonic modes. Fo quan um c yp og aphy, o he ha dwa e-based secu i y
mechanisms a e also being esea ched6.
The classical channel is used wi h quan um channel in pa allel o he quan um c yp og aphy
7
. The ounda ion
o quan um in o ma ion p ocessing and QKD is Heisenbe g’s unce ain y p inciple and he quan um no-cloning
heo em. QKD uses pho on pola isa ion and adhe es o he B84 p o ocol
8
. Passi e de ices a e used o implemen
he BB84 pa ame ic down-con e sion based applica ion. The BB84 signal s a es we e passi ely gene a ed using
cohe en ligh and a single pho on sou ce
9
. As an al e na i e o single pho on based QKD, quan um con inuous
a iable based quan um c yp og aphy is also being in es iga ed h ough he use o quan um elepo a ion o cohe en
s a es. He e, cohe en s a es a e communica ed in a andom dis ibu ion. Ea esd opping p o ec ion is o e ed by he
no-cloning based p o ocol, which also lacks "non-classical" ea u es
10
.Single pho on sou ces based on quan um do s
a e also in es iga ed. I is possible o de elop quan um and solid s a e in o ma ion. Addi ionally implemen ed is he
change o ea u es h ough he use o se e al quan um pho on sou ces11–1718
Using he wa e cha ac e is ics o he quan um sys em o exp ess he quan um s a es is challenging o quan um
en anglemen . Resea ch in ma hema ics is cu en ly s uggling o de e mine he p ope Sch odinge ’s equa ion
o he en anglemen . Quan um en anglemen esea ch is s ill being conduc ed oday. Va ious p o ocols o
quan um communica ion, quan um compu a ion, and quan um c yp og aphy can be esea ched based on quan um
en anglemen
19–23
. The quan um en anglemen o a quan um ha monic oscilla o is also s udied in p esence o an
ex e nal elec omagne ic ield
24
. The Sch odinge ’s equa ion can be sol ed using he eigen alue and eigens a es.
Because o he ime-dependen elec omagne ic ield, he ha monic oscilla o o he quan um pa icle is always
ime-dependen . As a esul , he Hamil onian also depends on ime. Wi h he excep ion o he equency pa ame e
being ime dependen , he Hamil onian is iden ical o ha o a simple ha monic oscilla o . The quan um pa icle’s
ene gy and angula momen um change o such sys ems. Addi ionally, he elec omagne ic ield is classical. Each
eigens a e unde goes a phase ans o ma ion ha is dependen on ime in o de o sol e Sch odinge ’s equa ion.
The ope a o app oach o he eigens a e ep esen a ions can make use o he ma ix25.
Due o he unce ain y he accu a e posi ion measu emen o a pa icle is challenging ask. Hence he pa ame e s
can be measu ed om indi ec measu emen a he cos o momen um
26
. Ha monic oscilla o is s udied due o
he impo ance o he sys em
27,28
. The gene alized cohe en s a es a e also s udied o he ha monic oscilla o
29
.
The ime e alua ion and he expec a ion alues o he posi ion and momen um ope a o s a e also s udied
30
. Time
independen Sch odinge ’s equa ion o ha monic oscilla o 31.
The cha ac e is ics o wa e pa icle duali y a e whe e quan um mechanics heo y di e ges om classical mechanics.
Each quan um s a e’s wa e and pa icle a ibu es a e ob ained he e. Bo h ea u es can be applied o any expe imen .
A quan um s a e becomes en angled as a esul o supe posi ion on a quan um sys em. Ca S a es a e c ea ed by
supe posing cohe en quan um s a es. Spin, ene gy, spa ial node, and pola isa ion o quan um en anglemen a e
among he a ious cha ac e is ics o a quan um s a e ha can be used o measu emen
3233
. Ca s a es a e c ea ed
ia quan um supe posi ion o wo quan um s a es using quan um op ics. The ampli ude, phase, and quad a u e o
he quan um wa e o pa icle a e used o cha ac e ise he quan um s a es. These s a es hold p omise o quan um
communica ion34.
Pho onic mul iqubi s a es and hei en anglemen o p oduce mul iqubi ca s a es a e di icul o c ea e. In
se e al deg ees o eedom, hype -en anglemen is employed in he solu ion. Howe e , pho onic sub-wa eleng h
phase s abili y is a p oblem o he hype -en anglemen
35
. Fully con ollable mul iqubi quan um compu a ion is
s ill a challenge as he mul iple pa icle en anglemen is a ques ion o quan um p ocesso o quan um in o ma ion
p ocessing3637.
2/11
Resul s
A pa icle is desc ibed by he wa e unc ion Ψ(
x,
), which can be ob ained by sol ing he Sch odinge equa ion
ep esen ed by equa ion 1,
i¯
hdΨ
d =−¯
h2
2m
d2Ψ
dx2+VΨ(1)
By sepa a ing he a iable,
Ψ(x, ) = ψ(x)φ( )(2)
Z∞
−∞ |AΨ(x, )|2dx = 1 (3)
whe e, he pa icle is ep esen ed by he he p obabili y densi y unc ion
|
Ψ(
x,
)
|2
and A is a mul iplica i e
ac o . The equa ion 1and equa ion 3bo h a e consis en . The equa ion 3is he no maliza ion equa ion. The
Sch odinge equa ion conse e he no maliza ion equa ion.
Quan um Wa e Func ion In he Ha onic Oscilla o
Fo he ha monic oscilla o he po en ial ene gy V is ep esen ed by,38
V=1
2mω2x2(4)
The ime dependen Sch odinge equa ion is ep esen ed by equa ion 5,
Eψ =−¯
h2
2m
d2ψ
dx2+1
2mω2x2ψ(5)
I ,
ξ≡ mω
¯
hx(6)
F om equa ion 5and equa ion 6,
d2ψ
dξ2= (ξ2−K)ψ(7)
whe e,
K=2E
¯
hω (8)
By modi ying equa ion 7 o e y la ge ξ,
d2ψ
dξ2≈ξ2ψ(9)
By sol ing equa ion 9,
ψ(ξ)=h(ξ)e−ξ2
2(10)
Whe e,
h(ξ)≈CX1
(j
2)!ξj≈CX1
j!ξ2j≈Ceξ2(11)
K= (2n+1) (12)
F om equa ion 8and equa ion 12,
3/11
E= (n+1
2)¯
hω (13)
Fo n= 0,1,2...
h(ξ)=H(ξ)(14)
The h(ξ)is he polynomial o deg ee n in ξ. This is called He mi e polynomials H(ξ)in equa ion 14.
By no malizing he s a iona y s a es o he ha monic oscilla o is ep esen ed by,
ψn(x) = (mω
π¯
h)1
41
√2nn!Hn(ξ)e−ξ2
2(15)
The equa ion 15 desc ibes he wa e unc ion o quan um oscilla o , which is comple ely di e en om classical
oscilla o wa e. He e he ene gy is quan ized and he p obabili y o ge ing he pa icle ou side classical ange is
no ze o. Unlike classical coun e pa , he e he dis ibu ion is ep esen ed o e an ensemble o iden ically p epa ed
sys ems.
The wa e unc ion can ca y he in o ma ion o a quan um s a e. The ene gy o he s a e can be used o gene a e
analog disc e e signal. The equa ion 13 desc ibes di e en ene gy le els. By gene a ing di e en ene gy le el
E
(
)
wi h espec o ime an analog disc e e signal can be gene a ed. Based on
E
(
)
ψn
(
x
)can be gene a ed wi h espec
o ime. The ene gy da a n o he quan um pa icle is ans e ed by wa e unc ion. Thus ins ead o pa icle, he
quan um wa e unc ion can be used o ansmi he de ails o a quan um s a e by equa ion 15. This da e can be
eco e ed om he ecei ed wa e unc ion. The deg ee n o he He mi e polynomials will be he ansmi ed message
signal. The quan um analog disc e e message signal can be con e ed in o quan um digi al bi s. Fo his me hod he
numbe o he qubi will be inc eased and he numbe o qubi s will be dependen on he quan um analog o digi al
con e e .
As he comple e signal is ans e ed in e ms o he He mi e polynomials based quan um analog disc e e signal,
he secu i y o he quan um sys em is inc eased. To inc ease he secu i y se e al quan um enc yp ion me hods can
be applied on he ansmi ing signal. The He mi e polynomials is used o he secu i y pu pose o he quan um
analog disc e e signal.
Rep esen a ion o Quan um S a e in e ms o Ene gy and ela ion wi h Degene acy
The issue o he degene acy o he ene gy le els is c ucial when s udying quan um-mechanical di icul ies. This
degene acy is equen ly linked o basic symme y ea u es o he Sch odinge equa ion, and he symme y condi ions
pe aining o he o a ion- e lec ion g oup and he g oup o pe mu a ions o iden ical pa icles ha e ecei ed a g ea
deal o a en ion.
The ene gy- ime unce ain y p inciple is ep esen ed by equa ion 16.
△E△ ≥¯
h
2(16)
Whe e,
△
is he ime o ake sys em undamen ally. The quan um measu emen by using he ene gy o a
pa icle is di icul . The
△E
should be mode a e o small alue. Fo he apid change, he equa ion 16 is alid o
la ge △Eonly. Which cons ain s he ansmission speed o he quan um signal.
The ene gy o he s a e indica es ha he quan um s a es in he bound sys ems a e disc e e. To c ea e a comple e
se o commu ing obse ables, each ene gy s a e is ep esen ed along wi h he eigen alues o o he obse ables. I a
pa icula ene gy eigen alue is p esen in mo e han one s a e, he sys em is said o be degene a e. Howe e , acco ding
o classical mechanics, a cons ained sys em migh no ha e any dis inc s a es. Consequen ly, i is insu icien o
de ine degene acy in e ms o quan um and classical mechanics. A classical bound sys em is ypically mul i-pe iodic.
Thus, when he e a e deg ees o eedom, i speci ies ha e e y a iable o he sys em can be ex ended wi h
undamen al equencies
ν1,ν2...ν
using Fou ie se ies. I all o hese
℧
equencies a e incommensu able, he sys em
is deemed nondegene a e. The sys em is e e ed o as g- old degene a e i he e a e g ela ions among he equencies
ha ha e he o m o equa ion 17 wi h in ege bj.
The equa ion
d
= (
℧−
1) is said o as comple ely degene a e. I he Hamil onian can be ep esen ed as a unc ion
o ac ion a iables, hen he sepa able mul iply pe iodic sys em is degene a e. In his case, only a linea combina ion
wi h in ege coe icien s allows he Hamil onian o depend on some o hese a iables. Equa ion 17
39
s a es ha
a massi ely pe iodic sys em has some degene acy i i is sepa able and i s Hamil on-Jacobi equa ion is likewise
sepa able in a con inuous amily o coo dina e sys ems.
4/11
℧
X
j=1
bk
jνj= 1; k= 1,...d (1 ≤d≤℧−1) (17)
A sys em is said o be in a ian unde a g oup
G
wi h gene a o s
Hj
i i s Hamil onian
H
is such. The e is a
con inuous amily o coo dina e sys ems in which he Hamil on-Jacobi equa ion can be sepa a ed i he e is only one
coo dina e sys em in which i can be sepa a ed. The o m o he Hamil onian is he same in coo dina es q, p, and q’,
p’ connec ed o each o he by
Xj
, acco ding o he in a iance equi emen exp essed in equa ion 18. Equa ions 19
and 20 gi e i i s name and esul in a sepa able Hamil on-Jacobi equa ion in he a iables q, p. In he a iables q’
and p’, i esul s in an equa ion ha is sepa able in p ecisely he same manne . Consequen ly, he exis ence o he
Xj
es ablishes degene acy i he global ans o ma ions co esponding o equa ions 19 and 20 a e single alued. The
a gumen p o ing degene acy om sepa abili y is i ele an i hese changes ha e in ini ely many alues.In his case,
he p esence o his speci ic g oup does no lead o degene acy. A somewha modi ied a gumen demons a es ha
degene acy again occu s when hese ans o ma ions ha e ini ely nume ous alues, wi h he equencies in ol ed
being a ionally ela ed a he han equal39.
(H,Xj)PB = 0 (18)
q′=q+ϵ(dXj
dp )(19)
p′=p−ϵ(dXj
dq )(20)
The p esence o in eg als o he equa ions o mo ion o he o m o equa ion 21, whe e he g’s and p’s a e he
coo dina es and conjuga e momen a o he sys em, is he main subjec o he ans o ma ion heo y o classical
dynamics. The in eg als in his case a e no clea unc ions o ime . A se o independen in eg als
F1,F2,F
has
been disco e ed. These ans o ma ions will o m a g oup i he collec ion o in eg als sa is ies condi ions as s a ed
in equa ion 22, acco ding o he Lie heo y o con inuous ans o ma ion g oups. A mos , he coe icien s Cg
xy a e
unc ions o he o al ene gy, bu hey can also be cons an s. Thus, he sea ch o in eg als ha allow one o speci y
he g oup’s cons i uen s educes he challenge o iden i ying he con inuous g oups o symme y ans o ma ions
in ini esimal o a gi en dynamical p oblem. I could be equi ed o symme ically ep esen he in eg als in he p’s
and q’s since he quan um-mechanical heo y equi es ha hey be ep esen ed by He mi ian ope a o s. Howe e ,
in his case, hey only ollow he s anda d commu a ion guidelines. The
F
’s commu a ion wi h he Hamil onian
exp esses hei in eg al ea u e. Fo ou pu poses, his co espondence mus be an algeb aic equi alence in which he
ope a o ela ions o equa ion 23 a e sa is ied by he commu a o s o he
F
ope a o s. Acco ding o he Hamil onian
alone, he Cs in his case ha e o be cons an s o , a mos , ope a o s40.
F(q1,q2...qb;p1,p2...pb)=Cons an (21)
(Fx,Fy) = XCg
xyFg(22)
(Fx,Fy)=i¯
hXCg
xyFg(23)
Discussion
In he s udy, Quan um p o ocols a e p oposed in e ms o algo i hms o secu e Quan um Enc yp ed Communica ion
(QEC). He e, an enc yp ion and a dec yp ion algo i hm a e ep esen ed wi h a new p o ocol o Quan um Communi-
ca ion. He e, he Quan um Iden i ica ion Lock (QIL) based QEC is p oposed, whe e he QIL is dis inc o each
use . As he Quan um Signal (QS), is desc ibed in e ms o Analog QS o Digi al QS, he upg aded communica ion
is compa ible wi h ansmi img an ex ended ange o QS. As he upg aded p oposed model can ansmi complex
QS, he p oposed model equi es mo e secu i y. The enc yp ion and dec yp ion me hods a e also p oposed he e,
which a e pu ely Quan um model-based o he p oposed Qun um Communiac ion p op ocol.
0.1 Hypo hesis o Quan um Wa e
1.
Quan um Wa e is a single- alued unc ion, which can no be de ined by any mul i alued classical unc ion o
classical se ies equa ion.
5/11
2. A Quan um wa e is a disc e e signal, which depends on a dis inc pa icle inside a Quan um Sys em.
3.
Fo a Quan um Sys em, i he e is mo e han one pa icle, ha esides in he Quan um Sys em, each pa icle
has a dis inc Quan um wa e, which can be de ined by dis inc equa ions.
i¯
hdΨQS
d =−¯
h2
2
d2(Ψ1/m1+Ψ2/m2+Ψ3/m3+...Ψn/mj)
dx2+V(Ψ1+Ψ2+Ψ3+...Ψj)(24)
4.
Unlike classical andomess, which is a mix o p obabilis ic and de e minis ic na u e sepa a ely, he Quan um
andomness is no dependen on p e iouly selec ed andom e ms. Ins ead, he Quan um andomness is a
combina ion o bo h p ope ies based on he known pa ame e s, and along wi h hese, he Quan um andomness
is none o hese men ioned p ope ies acco ding o he unp edic abili y o ans o ma ion and degene acy. The
Quan um andomness is uly andom.
5.
Quan um s a e can be de ined specially by he p ope ies, which can be desc ibed as he i ue o he Quan um
wa e and which hold he sus ainabili y o he Quan um wa e. The ances al p ope ies, which is unable o
de ine he in eg i y o ei he he Quan um wa e o he Quan um s a e, can be a successo Quan um p ope y,
bu will no endo se he Quan um ounda ions.
2π¯
hνΨ=−¯
h2
2m
d2Ψ
dx2+VΨ(25)
νΨ=−¯
h
4πm
d2Ψ
dx2+V
2π¯
hΨ(26)
0.2 Hypo hesis o Quan um Supe posi ion
1.
A Quan um Sys em con ains a se o Quan um pa icle, which se o pa icle is in an Ene gy Re e ence F ame
ela ed o he men ioned Quan um Sys em.
2.
Each Ene gy Re e ence F ame has a bounded ene gy band, which is dependen on he se o pa icles in ha
Quan um Sys em.
3.
A e he supe posi ion o Quan um pa icles, a comple ely di e en Quan um S a e is o med wi h a comple ely
di e en Ene gy Re e ence F ame, which ha e a di e en ene gy band wi h espec o he unsupe posed pa icles
and he unsupe posed Quan um S a es be o e supe posi ion.
i¯
hdΨS
d = 2π¯
hνSΨS(27)
4. A e supe posi ion, he supe posed Quan um sys em beha es as a single pa icle Quan um sys em.
i
2π
dΨS
d =νSΨS(28)
5.
A e he supe posi ion o Quan um pa icles, he supe posed pa icles a e in a ac ion, which is de ined by
some unknown equa ion.
6.
A e supe posi ion he equa ion o he supe posed Quan um S a e is de ined by a combina ion o he equa ions
o he unsupe posed pa icles by some unknown ma hema ical ope a ion.
i¯
hdΨS
d =−¯
h2
2
d2{Ψ1/m1;Ψ2/m2;Ψ3/m3;...Ψn/mj}
dx2+V{Ψ1;Ψ2;Ψ3;...Ψj}(29)
7.
The pa icle which ies o ge ou o he supe posed Ene gy Re e ence F ame, equi es mo e ene gy o be
ou o he supe posed Ene gy Re e ence F ame. He e he equi ed ene gy will no be added la e wi h he
unsupe posed pa icle, a e b eaking ou he supe posi ion.
i¯
hd{ΨS−(Ψ1;Ψ2;Ψ3;...Ψj)}
d =2π¯
h{νSΨS−(ν1Ψ1;ν2Ψ2;ν3Ψ3;...νnΨj)}=Cons an Unused Rdia ed Ene gy =ER
(30)
2π¯
h{νSΨS−(ν1Ψ1;ν2Ψ2;ν3Ψ3;...νnΨj)}=h{ΨS
λS−(Ψ1
λ1
;Ψ2
λ2
;Ψ3
λ3
;...Ψn
λj
)}=ER=hνR=h1
λR
(31)
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0.3 Hypo hesis o Quan um Rela i i y
1. A Quan um sys em is de ined by a disc e e Ene gy S a e, named as Ene gy Re e ence F ame.
2. Inside an Ene gy Re e ence F ame all o he Quan um ules a e equally ollowed.
3.
A Quan um Sys em wi h a di e en Ene gy Re e ence F ame has a Rela i e Ene gey and is called a ela i e
ene gy e e ence ame.
4.
I a se o Quan um Sys ems in Rela i e Ene gy, con ains Quan um pa icles in e y high speed, hen all o
he Quan um Sys ems will be equally a ec ed by a pa icula Po en ial Ene gy.
The equa ion 24,25 and 26 ep esen s he na u e o he quan um wa e. He e, he equa ion 27,28,29,30
and 31 ep esen s he na u e o quan um supe posi ion. The equa ion 30 and 31 a e ep esen ing he Quan um
Supe posed-S a e Radia ion E ec . The equa ion 31 ep esen s he cause o abso p ion and adia ion o ene gy,
whe e di e en wa es can be abso bed o adia ed in quan um mechanical e ec s. Due o his, he e a e ligh and
da kness in he c ea ion and des uc ion in he cosmological e ec .
E e y quan um signal con ains secu e quan um in o ma ion abou he QIL o bo h o he ansmi e and he
ecei e (i.e.
QILT
and
QILRe
). Each anscei e con ains a unique QIL. E e y quan um communica ion se ice
use has he de ails o all o he a ailable anscei e s wi h he co esponding QILs and he co esponding use
de ails. The QIL o he ansmi e will be en angled wi h he QIL o he ecei e . Fo each communica ion o a
anscei e one QIL is always pe manen . E e y anscei e will gene a e signals o all o he egis e ed QILs o
he sea ching pu pose. The quan um communica ion p o ide mus use a ce ain alue o
n
o he con i ma ion o
he secu e communica ion channel be ween he ansmi e and he ecei e . I
n
is di e en he communica ion
de ice can no be con i med. The de ice con i ma ion is achie ed based on he degene acy o he quan um signal.
A e he de ice con i ma ion bo h anscei e will se he alue o equency and QILs o u he communica ion.
Once he communica ion is con i med he ansmi e and ecei e can send he message signals. As he quan um
signal is used o communica ion, due o he bene i o he degene acy, se e al communica ion use s can use a single
equency o communica ion.
Besides he communica ed signal also con ains in o ma ion abou he message signal. The message signal will be
in he o m o he ene gy
E
(i.e.
M
(
τ
) =
E
(
τ
)). The
M
(
τ
)can be con e ed o
m
(
τ
)by using equa ion 8. He e
acco ding o he equa ion,
E
(
τ
)=(
m
(
τ
)+
1
2
)
¯
hω
he message signal can be ansmi ed. He e he message signals
m
(
τ
) o he ime
τ
is gene a ed. Thus he message signal is a disc e e analog o digi a signal. The ecei e will
gene a e signals o se e al
m
(
τ
). Acco ding o he degene acy p inciple, only same signals will be ma ched and
o he signals will be anished. Thus he ecei e will cap u e he message signal. The disc e e analog message signal
can be con e ed o digi al signal, whe e he bi s o he digi al signal is dependen on he quan um analog o digi al
con e e .
Me hods
P o ocols o QIL based Secu e Quan um Communica ion and C yp og aphy
Algo i hm 1 P o ocol o gene a ion and usage o Quan um Iden i ica ion Lock
1:
Each quan um de ice ha e a unique iden i ica ion module. The module is named Quan um Iden i ica ion Lock
(QIL). The QIL o a de ice can no be changed o modi ied.
2: E e y de ice an send signal o a ecei e de ice by using he QIL o he ecei e de ice.
3: The QIL o he ansmi e de ice will gene a e a unc ion wi h he QIL o he ecei e de ice.
4:
Based on he gene a ed unc ion o he ansmi e de ice, he ansmi ing signal will be gene a ed a he
ansmi e end.
5: The ansmi ing signal con ains he in o ma ion o he QIL o bo h o he ansmi e and he ecei e .
6: The ecei e can cap u e he ecei ed signal i he signal has in o ma ion abou he QIL o he ecei e .
The Algo i hm 1desc ibes he p o ocol o he gene a ion and usage o QIL. The QIL is a quan um solid s a e
challenge. The QIL will be di e en o each quan um de ice (QD). A quan um de ice will be iden i ied by a unique
QIL o o he de ice. Each de ice has wo se o QIL. Inside he i s se i con ains i s own QIL, while on he o he
se i con ains QIL o o he de ices wi h which de ices i is wishing o communica e. Each QD is a communica ion
de ice, so i con ains a ansmi e and a ecei e module. Be o e ansmi ing and ecei ing a QD will gene a e a
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quan um unc ion (QF) wi h i sel and he communica ing QD wi h which i is ying o communica e. Now he QF
will be used o gene a e he ansmi ing signal. On he o he side he ecei e con inuously gene a es QF wi h all
o he exis ing QILs. Due o quan um degene acy he ecei e QD can only cap u e he signal which is gene a ed o
he ecei e QD.
Algo i hm 2 P o ocol o public discussion be ween ansmi e and ecei e
1: Alice will send he signal which con ains he QIL o bo h o Alice and Bob.
2:
As he signal con ains he QIL o Bob, he will be able o cap u e he signal. F om he signal he will ge
in o ma ion abou he QIL o Alice.
3: Bob will send a signal, which also con ains he QIL o bo h o Alice and Bob o con i ma ion.
4: As he signal con ains he in o ma ion abou he QIL o Alice.
5:
The QILs o Alice and Bob a e inalized by Bob and Alice. Now hey can communica e by using he inal keys.
The Algo i hm 2 ep esen s he p o ocol o secu ing he communica ion be ween he ansmi e and he ecei e
QD. He e Alice’s QD gene a es and ansmi s a signal o Bob’s QD. A e cap u ing he signal by Bob’s QD ecei e ,
Bob’s ansmi e QD sends a signal o Alice’s ecei e QD o con i ma ion. As he ansmi ing signal con ains QF
which includes Alice’s QIL, he signal will be cap u ed by Alice’s ecei e QD. In he p ocedu e di e en pa ame e s
will be se be ween bo h o Alices’s and Bob’s QDs. Those pa ame e s a e he inal keys o he communica ion
be ween Alice and Bob. Those pa ame e s can be changes each imes hey communica e. A e con i ming he inal
key he communica ion is es ablished be ween Alice and Bob.
Algo i hm 3 P o ocol o Quan um communica ion o signal based on message
1:
The message is an analog disc e e signal. Based on he message a signal will be gene a ed o he communica ion.
2: The communica ing signal will be ansmi ed by he ansmi e .
3: The communica ing signal will be ecei ed by he ecei e .
4: F om he ecei ed signal he message will be gene a ed a he ecei e end.
The Algo i hm 3 ep esen s he p o ocol o he quan um communica ion o he signal based message. He e he
message is a quan um analog disc e e signal, which ha e a pa icula alue o each ime. Based on he message
signal he ansmi ing signal will be gene a ed by a QD. A e ansmi ing he signal he ecei e QD o o he
de ice will ecei e and decode he signal, i o he ecei e QD he signal ha e been gene a ed. F om he decoded
signal, he message signal will be econs uc ed a he ecei e QD.
Algo i hm 4 P o ocol o Quan um c yp og aphy o e he quan um communica ion
1: The QIL o Alice and Bob is dis ibu ed o Bob and Alice.
2:
Then ansmi ed signal does no con ains he QIL o he E e. Thus he signal can no be cap u ed by E e’s
ecei e .
3:
I he E e ansmi s a signal along wi h Alice, hen he ansmi ed signal o Alice will no be cap u ed by Bob’s
ecei e i he communica ion is al eady es ablished be ween Bob and E e. Bob ha e he eedom o selec he
ansmi e , wi h whom he wan o es ablish he communica ion (i.e. Alice o E e).
4:
Only i bo h o Alice and Bob include he QIL o E e in hei communica ing signal, hen he e will be only
one way ha E e can join he communica ion. A ha condi ion E e acn cap u e he signals o Alice and Bob.
Besides E e can send signal o alice and Bob.
The Algo i hm 4 ep esen s he p o ocol o quan um c yp og apgy o e he quan um communica ion. As he
ansmi ing end o he ansmi e QD he signal is gene a ed o he pa icula ecei e QD, he signal can no be
decoded by any hi d pa y. He e, i Alice’s ansmi e QD gene a es signal o he Bob’s ecei e QD, hen i can
no be decoded by ea esd oppe E e. I E e’s ansmi e QD also sends signal along wi h he Alice’s ansmi e
QD, hen Bob has he eedom o es ablish he communica ion wi h ei he E e o Alice. Only i , bo h o he Alice
and Bob’s ansmi e QD includes he E e’s QIL while hey a e gene a ing and ecei ing signals, hen only E e’s
QD in e e e wi h he communica ion be ween Alice and Bob.
P o ocols o Quan um C yp og aphy o Enc yp ion and Dec yp ion o he Message signals
The Algo i hm 5 ep esen s he p o ocol o Alice’s key c ea ion. The Algo i hm 6p ep esen s Bob’s enc yp ion
and a las Algo i hm 7 ep esen s he Alice’s dec yp ion algo i hm. He e, Alice is asking o Bob o an enc yp ed
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Algo i hm 5 Alice key c ea ion
Requi e: QILs,QIL ,ξ,n∈N
Ensu e: Ψn,(s, )(ξ)∈R
1: Se a na u al numbe n
2: Gene a e ξand calcula e Ψn,(s, )(ξ)∈R
3: Alice’s p i a e Key is QILs; while he public key is (ξ,Ψn,s, (ξ))
message signal. Alice made he p i e and public key o he communica ion. La e , Bob uses Alice’s gene a ed key
o enc yp he message, which is eco e ed by Alice a e ecei ing he message signal.
Algo i hm 6 Bob enc yp ion algo i hm
Requi e: ξ,Ψn,s, (ξ),M(τ)∈R
Ensu e: Ψn,s, (ξ)∈R,QILs, n
1: Choose Alice’s public key (ξ,Ψn,s, (ξ))
2: Message M(τ)p esen as a nume ical alue
3: Gene a e m(τ) om M(τ)
4: Calcula e Ψn, ,s,Ψm(τ), (τ)
5: Send o Alice Ψn, ,s,Ψm(τ), ,s(τ)
The Algo i hm 5 ep esen s a unc ion Ψ
n,(s, )
, which is gene a ed by a speci ic He mi e polynomial-based wa e
ea u e o a speci ic alue o
n
in he ha monic oscilla o . He e,
ξ
is he pa ame e o he unc ion Ψ.
ξ
can be
measu ed by using equa ion 6. The
s
and
is dependen on he QIL o he sende and he ecei e . A e ensu ing
n,s
and
he He mi e polynomial based wa e o m is gene a ed in he Quan um domain. He e, he majo wo k is o
in o m he alue o n o he ecei e . Once comple ed, he ecei e Bob can send he message o Alice.
Algo i hm 7 Alice’s dec yp ion algo i hm
Requi e: QIL ,Ψn, ,s,Ψm(τ), ,s(τ)
Ensu e: M(τ)∈R
1: Using QILscalcula e m(τ) om Ψm(τ), ,s(τ)
2: Reco e message M(τ) om m(τ)
The Algo i hm 6 ep esen s he enc yp ion o message signal Bob. He e,
n
is eplaced by
m
(
τ
) o di e en
ene gy le els, o gene a e a quan um analog signal. The m(τ)is p oduced om ac ual message signal M(τ). Now,
he Ψn, ,s and Ψm(τ), ,s a e he ansmi ing signal, which a e gene a ed by Bob, who is now a sende .
The Algo i hm 7 ep esen s he dec yp ion o message signal by Alice. He e, Alice knows he
QIL
, which is
ep esen ed by Bob. Now, Alice will ge he message signal om Bob, om which Alice can eco e he
m
(
τ
)and
la e he ac ual message
M
(
τ
). The degene acy p ope y p o ec s he message signal in he quan um domain, om
he ea esd oppe E e.
Righ s and Access
This wo k is licensed unde CC BY-NC-ND 4.0. Download and ci a ion a e allowed; euse o modi ica ion equi es
au ho ’s pe mission.
Re e ences
1.
Mondal, S., Sinha, A. & Rou h, J. A su ey on e olu ion o wi eless gene a ions 0g o 7g. In . J. Ad . Res. Sci.
Eng. (IJARSE) 1, 5–10 (2015).
2. Mih e , E. & Haile, G. 4g, 5g, 6g, 7g and u u e mobile echnologies. J Comp Sci In o Technol 9, 75 (2021).
3.
Bhaska , M. K. e al. Expe imen al demons a ion o memo y-enhanced quan um communica ion. Na u e 580,
60–64 (2020).
4. Naue h, S. e al. Ai - o-g ound quan um communica ion. Na . Pho onics 7, 382–386 (2013).
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