Table Of ContentRandom Solitons Realization of Quantum Me
hani
s
and Sto
hasti
Qubits
1 2
Yu.P. Rybakov , T.F. Kamalov
1
Theoreti
al Physi
s Department People's Friendship University of Russia
Mikluho-Maklay str., 6, Mos
ow, 117198,Russia
e-mail: yrybakovs
i.pfu.edu.ru; yrybakovmtu-net.ru
2
Physi
s Department Mos
ow State Opened University
P. Kor
haginstr., 22, Mos
ow, 107996,Russia
4
e-mail: ykamalovrambler.ru
0
0
2
Abstra
t
n
Sto
hasti
realization of the wave fun
tion in quantum me
hani
s, with the in-
a
J
lusion of soliton representation of extended parti
les, is dis
ussed. The
on
ept of
Sto
hasti
Qubits is used for quantum
omputing modeling.
7
2
1 1 Introdu
tion. Wave(cid:22)Parti
le Dualism and Solitons
v
8
As a first motivation for introdu
ing sto
hasti
representation of the wave fun
tion let us
6
1
onsider the de Broglie plane wave
1 ψ = Ae−ßkx = Ae−ßωt+ß(kr)
0 (1)
4 ω k m
0 for a free parti
le with the energy , momentum , and mass , when the relativisti
/
h relation k2 = ω2 k2 = m2
p − (2)
- ~ = c= 1
t
n holds (in natural units ).
ua Suppose, following L. de Broglie [1℄ and A. Euin(ts,tre)in [2℄, that the stru
ture of the par-
q ti
le is des
ribed by a regular bounded fun
tion , whi
lh =is 1su/mpposed to satisfy some
0
v: nonlinear equation withtheuK(tl,erin) (cid:22) Gordon linear part. Letv = k/ω bethe
hara
teristi
i size of the soliton solution moving with the velo
ity .
X
Now it is worth-while to underline the remarkable fa
t behind this resear
h [3℄, namely,
r
a the possibility to represent the de Broglie wave (1) as the sum of solitons lo
ated at nodes
a l
0
of a
ubi
latti
e with the spa
ing ≫ :
Ae−ßkx = u(t,r+d),
(3)
Xd
d
where marks the positions of latti
e nodes. To show the validity of (3) one
an take into
a
ount the asymptoti
behavior of the soliton in its tail region:
u(x) = d4ke−ßkxg(k)δ(k2 m2)
Z − (4)
1
and then use the well-known formula
2π 3
eß(kd) = δ(k),
(cid:18) a (cid:19) (5)
Xd
implying that 2π 3g(m)
A = .
(cid:18) a (cid:19) 2m
The formula (3) gives a simple illustration of the wave (cid:22) parti
le dualism, showing that
the de Broglie wave
hara
terizes the assemblage of parti
les (cid:22) solitons.
2 Random Hilbert spa
e
The next step to get the sto
hasti
representation of quantum me
hani
s
an be performed
if one admits that the lo
ations of solitons'
enters are not regular nodes of the
ubi
latti
e
φ
but some randomly
hosen points. To realize this pres
ription, suppose that a field
n
des
ribes parti
les (cid:22) solitons and has the form
n
φ(t,r) = φ(k)(t,r),
(6)
kX=1
where φ(k) φ(k′) = 0, k = k′,
supp ∩ supp 6
and the same for the
onjugate momenta
n
π(t,r) = ∂ /∂φ = π(k)(t,r), φ = ∂φ/∂t.
t t
L
kX=1
Let us define the auxiliary fun
tions
1
ϕ(k)(t,r) = (ν φ(k)+iπ(k)/ν )
k k
√2 (7)
ν
k
with the
onstants satisfying the normalization
ondition
~ = d3xϕ(k) 2.
Z | | (8)
R3n x =
Nrow,..w.er define the analog of the wave fun
tion in the
onfigurational spa
e ∋
1 n
{ } as
N n
Ψ (t,r ,...r )= (~nN)−1/2 ϕ(k)(t,r ),
N 1 n j k (9)
jX=1kY=1
2
(k)
N 1 ϕ
where ≫ stands for the number of trials (observations) and j is the one-parti
le
j
fun
tion (7) for the - th trial. It
an be shown [3,4℄ that the quantity
1
ρ = d3nxΨ 2,
N ( )n Z | N|
△∨ (△∨)n⊂R3n
where △∨ is the elementary volume whi
h is supposed to be mu
h greater than the proper
l 3 =
0 0
volume of the parti
le ∨ ≪ △∨, pAlays the role of
oordinMˆate probability density.
A
If we
hoose the
lassi
al observable with the generator , one
an represent it in
the form n
A = d3xπ ßMˆ φ = d3xϕ∗(k)Mˆ(k)ϕ(k),
j Z j A j Z j A j
kX=1
j
for the - th trial. The
orresponding mean value is
N N n
E(A) = 1 A = 1 d3xϕ∗(k)Mˆ(k)ϕ(k) = d3nxΨ∗ AˆΨ +O ∨0 ,
N j N Z j A j Z N N (cid:18) (cid:19) (10)
jX=1 jX=1kX=1 △∨
Aˆ
where the hermitian operator reads
n
Aˆ= ~Mˆ(k).
A (11)
kX=1
/ 1
0
Thus,uptothetermsoftheorder∨ △∨ ≪ ,weobtainthestandardquantumme
hani
al
rule (10) for the
al
ulation of mean values [4,5℄.
Itisinterestingtounderlinethatthesolitonians
hemeinquestion
ontainsalsothewell-
(k)
ϕ
known spin (cid:22) statisti
s
orrelation [6℄. Namely, if j is transformed under the rotation
D(J) SO(3) J
by irredu
ible representation of , with the weight , then the transposition of
(k)
2π ϕ
two identi
al extended parti
les is equivalent to the relative rotation of j , that gives
( 1)2J Ψ
N
the multipli
ation fa
tor − in .
Ψ /
N 0
It
an be also proved that upto the terms of order ∨ △∨ satisfies the standard
S
hr(cid:4)odinger equation [6℄. To verify the fa
t that solitons
an really possess wave proper-
ties, the gedanken diffra
tion experiment with individual ele
trons (cid:22) solitons was realized.
Solitons with some velo
ity were dropped into a re
tilinear slit,
ut in the impermeable
s
reen, and the transverse momentum was
al
ulated whi
h they gained while passing the
slit,thewidthofwhi
hsignifi
antlyex
eededthesizeofthesoliton. Asaresult,thepi
ture
of distribution of the
enters of s
attered solitons was restored on the registration s
reen,
by
onsidering their initial distribution to be uniform over the transverse
oordinate. It
was
larified that though the
enter of ea
h soliton fell into a definite pla
e of the registra-
tion s
reen (depending on the initial soliton profile and the point of
rossing the plane of
the slit by the
enter), the statisti
al pi
ture in many ways was similar to the well-known
3
diffra
tion distribution in opti
s, i.e. Fresnel's pi
ture at short distan
es from the slit and
Fraunhofer's one at large distan
es [7,8℄.
Fulfillment of the quantum me
hani
s
orresponden
e prin
iple for the Einstein (cid:22) de
Broglie's soliton model was dis
ussed in the works [4(cid:21)6℄.In these papers it was shown that
in the framework of the soliton model all quantum postulates were regained at the limit of
point parti
les so that from the physi
al fields one
an build the amplitude of probability
and the average
an be
al
ulated as a s
alar produ
t in the Hilbert spa
e by introdu
ing
the
orresponding quantum operators (11) for observables. The fundamental role of the
gravitational field in the de Broglie (cid:22) Einstein solitonian s
heme was dis
ussed in [6,9℄.
The soliton model of the hydrogen atom was developped in [10,11℄.
Ψ
N
As a result we obtain the sto
hasti
realization (9) of the wave fun
tion whi
h
an
be
onsidered as an element of the random Hilbert spa
e Hrand with the s
alar produ
t
(ψ ,ψ ) = E(ψ∗ψ ),
1 2 1 2
(12)
E
with standing for the expe
tation value. As a rude simplifi
ation one
an admit that
the averaging in (12) is taken over random
hara
teristi
s of parti
les (cid:22) solitons, su
h as
their positions, velo
ities, phases, and so on. It is important to underline on
e more that
the
orresponden
e with the standard quantum me
hani
s is retained only in the point (cid:21)
N
0
parti
le limit (△∨ ≫ ∨ )Nfor →Ψ∞(.t,Txo) show this [4,5℄ one
an apply the
entral limit
N
theorem stating that for → ∞ behaves as the Gaussian random field with the
varian
e σ2 = ρ(t,x), x R3n,
∈ (13)
ρ(t,x)
Rwh3nere stands for the probability density (partition fun
tion) of solitons'
enters in
.
Random Hilbert spa
es being widely exploited in mathemati
al statisti
s [12℄, for quan-
tum appli
ations they were first used by N. Wiener in [13,14℄. To illustrate the line of
Wiener's argument, we re
all the general s
heme of introdu
ing various representations in
quantumψme
hani
s. Aˆ
Let | i beσa(sAˆta)te ve
tor inathe Hilbert spa
e H and be a self-
onjugate operator with
the spe
trum . Then the (cid:21) representation is given by the wave fun
tion
ψ(a) = aψ ,
i | i
where Aˆa = aa , a σ(Aˆ).
| i | i ∈
q
In parti
ular, the famous S
hr(cid:4)odinger
oordinate (cid:21) representation is given by the wave
fun
tion
ψ(q) = q ψ = q n n ψ ,
i | i i | ii | i (14)
Xn
n
with | i being some
omplete set of state ve
tors in H.
4
x(s,α) [0,1] s
Wiener
onsidered the real Brownian pro
ess in the interval ∋ , where
α [0,1]
∈ is the generalized number of the Brownian traje
tory and the
orrelator reads
1
′ ′
dαx(s,α)x(s,α) = min(s,s).
Z (15)
0
To obtain the quantum me
hani
al des
ription, Wiener defined the
omplex Brownian
pro
ess 1
z(s α,β) = [x(s,α)+ßy(s,β)];α, β [0,1],
| √2 ∈ (16)
R3 [0,1] R3
andusingthenaturalmapping → , fortheparti
lein ,
onstru
tedthesto
hasti
representation of the wave fun
tion along similar lines as in (14):
α,β ψ = dz(s α,β)ψ(s),
h | i Z | (17)
s∈[0,1]
with the obvious unitarity property
1
ds ψ(s)2 = dαdβ α,β ψ 2
Z | | ZZ |i | i|
0 [0,1]2
stemming from (15).
3 Sto
hasti
qubits and solitons
Now we intend to explain how sto
hasti
qubits (quantum bits)
ould be introdu
ed in the
Φ j
j
solitonian s
hemen. To this end one should dϕe(fki)n(er)the random phase for the - th trial
in our system of solitons (cid:22) parti
les. Let denote the standard (etalon) profile for
k d(k)(t) k j
the - th soliton. The most probable position j of the - th soliton's
enter in - th
trial
an be found from the following variational problem:
d3xϕ∗(k)(t,r)ϕ(k) r d(k) max,
(cid:12)(cid:12)Z j (cid:16) − j (cid:17)(cid:12)(cid:12) → (18)
(cid:12) (cid:12)
(cid:12) (cid:12)
thus giving the random phase stru
ture:
n
Φ = d3xϕ∗(k)(t,r)ϕ(k) r d(k) .
j kX=1 arg Z j (cid:16) − j (cid:17) (19)
The random phase (19)
an be used for simulating quantum
omputing via generating
M
the following random di
hotomi
fun
tions:
f (θ ) = [cos(Φ +θ )], s = 1,M,
s s j s
sign (20)
5
θ
s
with being arbitrary fixed phases. Now re
all that the quantum bit (qubit) is identified
with the state ve
tor
ψ = α0 +β 1 ,
| i | i | i
0 1
orresponding to the superposition of two orthogonal states | i and | i, as for instan
e,
1/2
two polarizations of the photon, or two possible - spin states. Using the well-known
1/2
quantum expression for - spin
orrelation in a singlet state of two parti
les (the latter
being a typi
al entanglement state):
E(σ σ ) = (ab), a = b = 1,
a b
− | | | | (21)
n = 2
one
an
ompare it with the random phases
orrelation for the
ase of parti
les [15℄:
2
E(f f )= 1 θ ,
1 2
− π|△ | (22)
θ = θ θ
1 2
where △ − . The similarity of these two fun
tions (21) and (22) of the angular
M
variable seems to be a good motivation for the qubits simulation by the di
hotomi
random fun
tions (20), popularized in the paper [15℄.
In
on
lusion we express the hope that the random solitons realization of the wave
fun
tion
ould be effe
tively used for quantum
omputing simulation.
Referen
es
[1℄ de Broglie L. Les In
ertitudes d'Heisenberg et l'Interpr(cid:1)etation Probabiliste de la
M(cid:1)e
anique Ondulatoire. Paris: Gauthier-Villars, 1982; Mos
ow: Mir, 1986.- 342 p.
[2℄ Einstein A. Colle
tion of S
ientifi
Works. Mos
ow: Nauka, vol. 4. 1967. - 600 p.
[3℄ Rybakov Yu. P. The Bohm (cid:22) Vigier Subquantum Flu
tuations and Nonlinear Field
Theory // Int. J. Theor. Phys., 1972, vol. 5, No. 2. Pp. 131 - 138.
[4℄ Rybakov Yu. P. On the Causal Interpretation of Quantum Me
hani
s // Found. of
Physi
s, 1974, vol. 4, No. 2. Pp. 149 - 161.
[5℄ Rybakov Yu. P. La Th(cid:1)eorie Statistique des Champs et la M(cid:1)e
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Me
hanism // Bulletin of Peoples' Friendship University (PFU) of Russia. Physi
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Izvestia VUZov. Physi
s, 1982, vol. 25, No. 1. Pp. 36 - 38.
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[8℄ Rybakov Yu. P. Gravitational Me
hanism of Quantization and Solitons // Problems
of Gravitation and Elementary Parti
les Theory. 1986, vol. 17. Pp. 161 - 171. Mos
ow:
Energoatomizdat, 1986.
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hasti
Gravitational Fields and Quantum Corre-
lations // Bulletin of PFU. Physi
s. 2002, No. 10(1). Pp. 5 - 7.
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s. 1995, vol. 25,
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// Phys. Lett., Ser. A, 1996, vol. 222, No. 1. Pp. 5 - 13.
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7
