Star-Trek Teleportation: A Possibility?
Kiang Wei Kho1
1
School of Physics, National University of Ireland, Galway, Ireland
Abstract
This paper describes a scheme, through which the quantum information as well as the
structural information of a time-reversal invariant system can be teleported over a distance. I
show that my teleportation scheme can be viewed as a form of reversible purification process
by repeated interactions with an auxiliary quantum system.
________________________________________
e-mail: kiang.kho@nuigalway.ie/kho_97@yahoo.com
I. INTRODUCTION
Quantum mechanics or QM has revolutionized our understanding of the universe since its
discovery. An important consequence of QM is the Heisenberg’s uncertainty principle
(HUP), which states that no two non-commuting observable can be simultaneously measured
with infinite accuracy. This flies in the face of classical Newtonian physics, which prescribes
that every particle must have point-like behaviors. Fortunately, what was seemed to be an
unbreakable barrier in the precision of all measuring apparatuses has surprisingly become the
fundamental mechanism behind a new form of communication: quantum teleportation (or
QT) [1]. As described in Bernett’s landmark paper [2], the central element in QT is the
concept of Einstein-Podolsky-Rosen or EPR entanglement, which was born out of an
argument between Einstein and Bohr over the interpretation of HUP [3]. Such a peculiar
quantum phenomenon has since been rigorously tested [4], eventually leading to such
applications as encrypted transactions and high-fidelity quantum logic gates [5, 6].
Concomitantly, the complexity of the quantum system to be teleported has since also
increased, from a simple photonic qubit system [2, 4] to a more complex many-body
fermionic system, such as Bose-Einstein Condensate or BEC [7].
However, QT is not teleportation in the “Star-Trek” sense, since no disembodiment or
reconstruction of any physical object actually occurs. What is actually being teleported in QT
is the information about the quantum state of a physical system [2]. In the current paper, I
propose a plausible scheme for transmitting both the q-information as well as the structural
information of a quantum system over a distance. My theory can be generalized to
accommodate molecular systems, in which case, the original system will undergo dissolution
(dematerialization) at the sending end, and a replicate re-assembling (rematerialization) at
the receiving end. As will be clear later, my teleportation scheme is essentially a form of
reversible purification process, whereby the (quantum) system to be teleported (hereby
referred to as the Alice’s sample, or AS) is reversibly, and gradually, evolved into a specific,
and manageable, eigenstate state through repeated interactions with a train of auxiliary
particle pulses of lower dimension. During the purification process, the state of each of the
auxiliary particles is first registered, after each interrogation, in a quantum memory at Alice’s
side, before being q-teleported or transmitted to Bob’s memory. To reconstruct the original
AS, Bob simply reverses the purification process using the recorded quantum states of the
auxiliary particles, and performs on each of these states a specific symmetry transformation.
Since, in this particular scheme, no Bell operator measurement is needed to be carried out on
the AS, my teleportation scheme is much simplified and is thus, in principle, amenable to any
AS, regardless of its complexity as well as phase (i.e. molecule, gas, liquid etc). Additionally,
my technique generally does not impose a limitation on the maximum allowable thermal
energies of the AS, i.e. the AS needs not be very “cold”. This is in great contrast to the
previous scheme of photon-mediated teleportation of matter-waves, where an efficient
teleportation occurs only within a restricted range of mean particle velocity [8]. The current
proposed scheme therefore put Star-Trek Teleportation (STT) much closer to the reality.
Finally, I will comment on the relevancy of some of the currently existing quantum
technologies to STT and suggest a plausible implementation setup based on the current
model. This paper is divided into two essential parts: Section II and III discuss the general
model of concatenated teleportation via purification process, and its classification, in the
context of STT; in Section IV, I apply my theory in the STT of a physical object.
Alice
t = 0s
Bob
“AS”
Auxiliary light pulses
ϒ1B
AS = ϒ 0A
ϕ 0B 1
ϕ 0A 0
Û Hˆ
ϒA
ˆ
⊗H
Û H−ˆ 1
( ∆t )
ϕA
ϒB
Quantum memory
t = ∆t
ϕ
(∆t )
ϕB
t = 5∆t
ϕ 0B 0
ϕ0A1 → M 0A
ϒA
ˆ
⊗H
tˆ
A1
0
Û Ĥ ( ∆ t )
t = 6∆t
M 0B
Û Ĥ−1 ( ∆ t )
ϒB
t = 4∆t
t = 2∆t
Û
Û Ĥ ( ∆ t )
ϒA
−1
Ĥ B
(∆t )
ϒ
Q-state transfer
BS = ϒ 0B
ϒ1A
t = 3∆ t
t = 3∆ t
“BS”
FIG. 1. A general teleportation scheme
II. CONCATENATED-TELEPORTATION OF A COMPLEX QUANTUM SYSTEM
I shall begin with a general teleportation scheme as depicted in Fig. 1, whereby Alice intends
to teleport or map an unknown state AS = ϒ 0A (spatial state, momentum state, etc) of the
AS onto the state of the object that Bob has at hand (referred to as Bob’s sample or BS). Let
ϒ 0A ∈ H
N
ϒA
and ϒ 0B ∈ H
N
ϒB
, where H
N ϒ A and N ϒ B , respectively. Let eiϒ
A
N
ϒA
and H
N
ϒB
are Hilbert spaces with dimensions
ϒ
, i = 1,..., N ϒ A and e j
B
, j = 1,..., N ϒ B , be the
orthogonal basis in H
N
ϒA
and H
N
ϒB
, respectively. Let’s also assume that both the AS and
the BS are complex N-body fermionic systems consisting of the same atomic specie, and that
N ϒ A = N ϒ B . I will also assume that the entire teleportation process is to be carried out in a
piece-wise manner, i.e. in steps of time-interval ∆t (see lower panel of Fig. 1).
Supposed Alice begins the teleportation process by first interrogating the state AS
A0
with a single auxiliary light pulse of state ϕ 0
Û −Hˆ 1 ⊗ Hˆ
ϒ
ϕ
( ∆ t ) , in which
( ϕ 0A 0 ∈ H
N
ϕA
) under the unitary operation
ˆ ⊗H
ˆ is the AS-auxiliary Hamiltonian. Note that I shall assume
H
ϒ
ϕ
the auxiliary pulse to be of a lower dimension, i.e. Nϕ A N ϒ A , and has a pulse-width of ∆t .
In the subsequent ∆t interval following the first interrogation, Alice stores the final state
A1
A1
A
( ϕ 0 ) of the light pulse in her quantum memory M 0A (i.e. ϕ 0 → M 0 ), which has a
coherence time significantly longer than the time needed for the entire teleportation process.
During this ∆t interval, the state AS is allowed to evolve naturally under a unitary
transformation Û Ĥ
ϒ
( ∆ t ) = e − iĤ
ϒ
( ∆t )
, where Ĥ ϒ is the Hamiltonian of the AS. In the
interval [ 2 ∆ t , 3 ∆ t ] (see Fig. 1), the state of Alice’s memory is q-teleported to (or mapped
onto) Bob’s memory; M 0A → M 0B , during which time, AS is again allowed to evolved,
eventually to a final state ϒ1A . Supposed now that Bob wishes to re-create the original state
ϒ 0A , by reversing all the operations carried out by Alice. Bob shall regard the reconstruction
process a success, only if the coefficients for the final state BS = ϒ1B are the same as those
for the original state AS = ϒ 0A , when expressed as linear superposition of their own bases.
To begin the reconstruction process, Bob prepares BS with some initial state BS = ϒ 0B , and
acts on it an inverse unitary transformation Û Ĥ−1 ( ∆ t ) , where Ĥ ϒ is the Hamiltonian of the
ϒ
BS. In the interval [ 4 ∆ t , 5 ∆ t ] , Bob maps the state M 0B of his quantum memory onto the
state ϕ0B 0 (which spans the Hilbert space H
N
ϕB
) of his own auxiliary light pulse, and
performs on this particular light pulse a symmetry transformation tˆ . As in the case for ϒ 0A
A1
and ϒ1B , ϕ0B 0 and ϕ 0 are considered equal if their corresponding coefficients are the
same with respect to their own bases. Bob again let BS evolves naturally during this time
interval under the inverse unitary transformation Û Ĥ−1 ( ∆ t ) . In the final interval [5 ∆ t , 6 ∆ t ] ,
ϒ
both the light pulse and the evolved state Û −Hˆ 1 ( 2 ∆ t ) ϒ 0B
ϒ
−1
unitary Û Hˆ ⊗ Hˆ
ϒ
ϕ
( ∆t ) .
interacts under the action of the
There is no doubt that if Bob could prepare the initial state
BS = ϒ 0B wisely, ϒ 0A shall be recreated in the final state ϒ1B of the BS. The question
here is therefore not so much as to whether Bob can achieve the re-creation, but as to what is
the set of “teleportable” ϒ 0A s - and ultimately how to expand this set to accommodate moregeneral states - if there exists only a classical channel for Alice to convey to Bob information
about the final state ϒ1A of the AS, owing, perhaps, to the difficulties in obtaining a suitable
EPR-pair consisting of members with high enough dimension (at least N ϒ A ), or in carrying
out a Bell-state measurement on the AS-auxiliary system.
To see this, I shall begin by stating the following bijective mapping for the
ϒ
eigenstates, Q̂ ϒ : ei
A
eiϒ
B
ϕ
, Q̂ ϕ : e j
ϒ
which implies equivalency between ei
A
A
eϕj
ϕ
(resp. e j
B
A
, ∀ i = 1,..., N ϒ , ∀ j = 1,..., N ϕ ,
ϒ
), and ei
B
ϕ
(resp. e j
B
), in which
it is assumed that N ϒ A = N ϒ B = N ϒ , N ϕ A = N ϕ B = N ϕ , and N ϕ N ϒ . With this
A
proposition, it thus follows that ϒ =
Nϒ
∑ ai eiϒ
i =1
A
Nϕ
A
ϕ
(resp. ϕ = ∑ b j e j
j =1
A
) and
ϒB =
Nϒ
∑ ai eiϒ
B
Nϕ
∑b
B
(resp. ϕ =
B
) will be considered equivalent. Now, I take
j =1
i =1
ϒ 0A =
eϕj
j
Nϒ
∑ ai eiϒ
A
Nϕ
A0
ϕ
, and ϕ 0 = ∑ b j e j
A
, and express the combined state ϒ 0A ⊗ ϕ 0A 0
j =1
i =1
(which is separable) as,
A
0
ϒ ⊗ϕ
 Nϕ
A
 Nϒ
ϒA 
=  ∑ a m em   ∑ bn enϕ
 m =1
  n =1
A0
0
N ϒ , Nϕ
=
∑
a m bn emϒ
A
enϕ



A
(1)
m , n =1
N ϒ , Nϕ N ϒ , Nϕ
=
∑ ∑
i , j =1 m , n =1
Nϒ
where a m , bn ∈ ,
∑a
2
m
m =1
Nϕ
= 1,
∑b
a b Φ A eϒA
ij
m
 m n
2
n
enϕ
A
 ΦA
 ij
= 1.
n =1
From Eqn. 1 and according to Fig. 1, I obtain for the state ϒ1A ⊗ M 0B at t = 3∆t as,
(
ˆ ˆ
ϒ 1A ⊗ M 0B = U
H
N ϒ , Nϕ
∑
ϒA
( 2 ∆ t ) ⊗ ˆI
) Tˆ
A→ B
ˆ
ˆ
M
ϕ → M U Hˆ
 N ϒ , Nϕ
A
ϒA
 ∑ a m bn Φ ij em
 m , n =1
ϕA
ϒA
ˆ
⊗H
 ϒA
 ei

ϕA
( ∆t )
e
M
j
ˆ ˆ ( 2 ∆ t ) ⊗ ˆI Tˆ
ˆ
ˆ ˆ
Φ ijA =  U
∆ t ) 
(
ˆ
A→ B M ϕ → M U H
H A
⊗H
A
A
ϒ
ϒ
ϕ


−1
=
i , j =1
en
ϒ 0A ⊗ ϕ 0A 0
(2)
B
in which, I have defined
(
where Û Hˆ
ϒ
ˆ
A ⊗H
ϕA
combined system
space
)
( ∆ t ) =e
 ( ∆t )
ˆ
ˆ
-i H
⊗H
 ϒ A
ϕA 

A
e Mj
B
,
ˆ A ⊗H
ˆ A is the total Hamiltonian for the
, and H
ϒ
ϕ
ϒ A ⊗ ϕ A 0 , and M̂ϕ → M represents a bijective mapping of the auxiliary
{ ϕ } onto the memory space { M } ;
A1
0
eiϒ
A
0
M̂ ϕ → M : { ϒ ⊗ ϕ
} → { ϒ ⊗ M } , and
{M }
A
0
T̂A→ B Alice’s memory space
{
T̂ A → B : ϒ ⊗ M
A
{M }
B
0
onto Bob’s memory space
;
to
}→ { ϒ ⊗ M }.
B
A
From Eqn. 2, I can write ϒ 1 as,
ϒ
A
1
 Nϕ N ϒ , Nϕ
A
= ∑  ∑ ∑ a m bn Φ ijA emϒ
i =1  j =1 m , n =1
Nϒ
enϕ
 ϒA
 ei

A
(3)
A
In order for Bob to reconstruct the original state ϒ 0 under an inverse unitary operation, he
A
must bijectively map ϒ 1
ϒ
B
0
B
B
onto ϒ 0 ; ϒ1A ϒ 0B , and prepares ϒ 0
 Nϕ N ϒ , Nϕ
A
A
= ∑  ∑ ∑ a m bn Φ ijA emϒ ⊗ enϕ
i =1  j =1 m , n =1
N ϒ , Nϕ
B
0
 ϒB
 ei ,

Nϒ
B
B
or more precisely, he must prepare ϒ 0 ⊗ M 0
ϒ ⊗M
B
0
=
∑
i , j =1
as,
(4)
as the following,
 N ϒ , Nϕ
A
ϒA
 ∑ a m bn Φ ij em
 m , n =1
enϕ
A
 ϒB
 ei

e Mj
B
(5)
While the satisfaction of Eqn. 5 will always lead to a successful reconstruction of
ϒ 0A in BS, Eqn. 5 is, however, fairly general, and does not take into consideration of the
fact that both Alice and Bob are spatially well separated. In other word, Eqn. 5 may demand
A
B
that the states ϒ 0 , ϒ 0
A
and ϒ 1 be entangled – an impractical requirement for Alice
A
and Bob. To see this, I express, by virtue of Eqn. 1 and 4, the relationship between ϒ 0 and
ϒ 0B as,
ϒ 0A ⊗ ϒ 0B =
 Nϕ
A
ϒA
Φ
a
b
e
∑
∑
m 
n
ij
m
 n , j =1
m , i =1

Nϒ
enϕ
A
 ϒA
 em

eiϒ
B
(6)
A
B
through which, one can easily see that ϒ 0 and ϒ 0
A
B
separablity between ϒ 0 and ϒ 0
are generally entangled. If now the
is assumed, i.e.
 Nϕ
A
a m  ∑ bn Φ ijA emϒ
 n , j =1
enϕ
A

 = f m g i

(7)
and re-writing Eqn. 1 as,
N ϒ , Nϕ
ϒ 0A ⊗ ϕ 0A =
∑
α ij Φ ijA
i , j =1
N ϒ , Nϕ
, where
∑
a m bn Φ ijA emϒ
A
⊗ enϕ
A
= α ij , I can obtain for ϒ1A ⊗ ϒ 0B ,
m , n =1
A
1
ϒ ⊗ϒ
B
0
 Nϕ
 A
= ∑  ∑ α ilα jl  eiϒ
i , j =1  l =1

Nϒ
e ϒj
B
`
(8)
By comparing Eqn. 8 with Eqn. 6, one can further see that although,
Nϕ N ϒ , Nϕ
∑ ∑{
a m bn Φ ijA
j =1 m , n =1
}
emϒ
A
enϕ
A
 Nϒ

=  ∑ fm  gi =
 m =1 
Nϕ
∑α
ij
j =1
A
B
, there is no implication that α ij = α i β j is generally valid; therefore, ϒ 1 and ϒ 0
A
B
generally entangled, even when ϒ 0 and ϒ 0
are
are separable.
A
To this end, and given the fact that Alice and Bob are separated, and that ϒ 1 and
ϒ 0B
are too complex a state to be made entangled, I shall thereafter consider only the
A
B
particular case, where ϒ 1 and ϒ 0
are separable (i.e. α ij = α i β j ). This concomitantly
A
B
B
B
suggests that ϒ 1 ⊗ M 0 , and hence ϒ 0 ⊗ M 0 , be separable as well. Hence, one can
B
B
write, for this particular case, for ϒ 0 ⊗ M 0
N ϒ , Nϕ
ϒ 0B ⊗ M 0B =
∑
i , j =1
α i β j eiϒ
as,
B
e Mj
B
Accordingly, the reconstruction process carried out by Bob (see Fig. 1) is therefore,
B
1
ϒ ⊗ϕ
ˆ ˆ
=U
H
B1
0
ϒB
ˆ
⊗H
( ∆t )
ϕB
−1
(
ˆ -1
ˆ
tˆϕ M
ϕ → M U Hˆ
ϒB
( 2 ∆ t ) ⊗ Iˆ
)
−1
ϒ 0B ⊗ M 0B
N ϒ , Nϕ
∑
=
α i β j Φ ijB
(9)
i , j =1
N ϒ , Nϕ N ϒ , Nϕ
α β e ϒ B
 i j m
∑ ∑
=
m , n =1 i , j =1
For
simplicity,
I
shall
ϒ ⊗ϕ
A0
0
=
∑
tˆϕ = ˆI
set
N ϒ , Nϕ
A
0
ϕB
en
B
Φ ijB  emϒ

for
the
enϕ
B
moment.
By
comparing
N ϒ , Nϕ
α ij Φ
A
ij
∑
=
i , j =1
i , j =1
N ϒ , Nϕ
a m bn =
α i β j Φ ijA , with Eqn. 1, I obtain,
∑
i , j =1
{α β
i
j
emϒ
A
enϕ
A
}Φ
A
ij
(10)
Thus, from Eqn. 9, I can write,
N ϒ , Nϕ N ϒ , Nϕ
ϒ
B
1
=
∑ ∑
m , n =1 i , j =1
{
α i β j emϒ
B
ϕB
en
Φ ijB
}e
ϒB
m
(11)
 Nϕ  ϒ B
= ∑ a m  ∑ bn  em
m =1
 n =1 
Nϒ
ϒ
Based on the above definition on the equality between the eigenstates ei
B
11 therefore shows that ϒ 1
A
ϒ
and ei
B
, Eqn.
A
can still be a perfect reconstruction of ϒ 0 , even when both
A
B
AS and BS (i.e. ϒ 1 and ϒ 0 ) are never entangled.
While Eqn. 11 does not appear much as a surprise, its derivation has implicitly
A
assumed that Bob can precisely “duplicate” the state ϒ 1
in the BS prior to the
reconstruction process, (see Eqn. 4). This assumption is however too strong, and certainly
goes against the no-cloning theorem. In addition, as has assumed earlier on, there is also no
A
quantum channel available to permit Alice to precisely map the state ϒ 1 onto BS . Alice,
A
however, is allowed to communicate with Bob about the state ϒ 1 via a classical channel.
A
To achieve this, Alice will have to perform a classical measurement ( P̂ p ) on ϒ 1 , and
convey to Bob the outcome, so that Bob can begin the reconstruction process with his BS
ϒ
prepared with an appropriate eigenstate. Assuming, the final projected AS state be e p
P̂ p : ϒ
Nϒ
A
1
=
∑k
eiϒ
i
A
→ e pϒ
A
A
, i.e.
(where k p ≠ 0 )
,
i =1
ϒ
and Bob prepares the initial BS to be e p
N ϒ , Nϕ N ϒ , Nϕ
ϒ
B
1
=
. Eqn. 11 then becomes,
{
∑ ∑
δ ip β j emϒ
m , n =1 i , j =1
N ϒ , Nϕ Nϕ
=
B
∑ ∑ {β
j
e
ϒB
m
B
ϕB
en
ϕB
en
Φ
B
pj
m , n =1 j =1
Φ ijB
}
}e
ϒB
m
emϒ
B
(12)
The fidelity of the AS’s original state ( AS = ϒ 0A ) being reconstructed in ϒ1B can be
quantified as,
ϒ
A
0
ϒ
B
1
Nϒ
=∑a
m =1
Nϕ
*
m
∑ {β
j
emϒ
B
n , j =1
enϕ
B
}Φ
B
pj
0
1
From Eqn. 13, it can be seen that the teleportation is successful, i.e. ϒ A ϒ B
(13)
equal to
unity, only if,
Nϕ
∑
n , j =1
β j emϒ
B
ϕB
en
Φ Bpj = a m
(14)
Since there could, at most, be only N ϕ non-trivial bn s, the set of “teleportable” state ϒ 0A
t
that satisfies Eqn. 14, for any given state ϕ0A0 , and any given projection measurement P̂ p
constitutes only a N ϕ -dimensional sub-space
{ϒ }
A t
0
⊂ H N ϒ . This stems simply
P̂ p , ϕ 0A 0
from the fact that the limited dimension of the auxiliary space ( ϕ0A and ϕ0B ) prohibits the
realization of a general teleportation.
If however Alice can repeatedly probe AS for at least N’ times with a sequence of
auxiliary light pulses, such that N ' N ϕ ≥ N ϒ , Alice could still send (a general) ϒ 0A
to
Bob. In this particular case, each AS-light-pulse interaction will partially evolve AS
toward a targeted eigenstate. This is known as concatenated teleportation [9], and is depicted
in Fig. 2 as a cascade series of partial teleportation steps. The entire teleportation process can
be described as,
N ' −1
ϒ
B
N'
⊗∏ϕ
B1
i
i=0
(
)
(
N ' −1
)
ˆ
ˆ
ˆ ˆ
ˆ
ϒ 0A ⊗ ∏ ϕ iA 0
= Oˆ B M
B , M → ϕ Pp T A → B M A ,ϕ → M O A
(15)
i=0
N ' −1
A
0
ϒ ⊗ϕ
in which,
A0
0
⊗ϕ
A0
1
⊗ ... ⊗ ϕ
A0
N '− 2
⊗ϕ
= ϒ ⊗ ∏ ϕ iA 0
A0
N ' −1
A
0
, M̂ A,ϕ → M (resp.
i=0
M̂ B , M →ϕ ) bijectively maps
N ' −1
N ' −1
∏ ϕiA0 onto
∏ M iA0
i =0
N ' −1
(resp.
i =0
N'
N ' −1
∏ M iB 0
onto
i =0
(
∏ϕ
N'
)
), and ,
i =0
−1
ˆ =U
ˆ =  Rˆ U
ˆ −ˆ1 ( 2∆t ) ⊗ ˆI
ˆ ˆ ( 2∆t ) ⊗ ˆI U
ˆ ˆ ˆ ( ∆t )  , O
O
 ˆ Hˆ B ⊗Hˆ B ( ∆t ) U
B
A
N'
H B
H A
H A ⊗H A
ϒ
ϒ
ϕ
ϒ
ϕ
ϒ



(
B0
i
)
N'
ˆ -1  and
R
N'

R̂ defines an operator that left-rotates ϕ 0A 0 ⊗ ϕ1A 0 ⊗ ... ⊗ ϕ NA 0' − 2 ⊗ ϕ NA 0' −1 ;
N'
R̂ ϒ 0A ⊗ ϕ 0A 0 ⊗ ϕ1A 0 ⊗ ... ⊗ ϕ NA 0' −1 = ϒ 0A ⊗ ϕ1A 0 ⊗ ϕ 2A 0 ⊗ ... ⊗ ϕ NA 0' −1 ⊗ ϕ 0A 0
N ' −1
, and the dimension of
∏ϕ
A0
i
is N ' N ϕ .
i =0
As has discussed above, and implied by Eqn. 13 and 14, the pre-requisite for a
ϒ
successful teleportation is that Alice be able to project her AS onto an eigenstate e p
N ' −1
N ' −1
i=0
i=0
A
Oˆ A ϒ 0A ⊗ ∏ ϕ iA 0 = e ϒp ⊗ ∏ ϕ iA1
A
, i.e.
(16)
Eqn. 16 describes essentially a reversible purification of the AS by repeated interactions with
auxiliary light pulses [10, 11]. In other word, if, while coaxing a quantum system to an
eigenstate, Alice can record all the auxiliary states, and then q-teleport or transmit these states
to Bob, he can accordingly (to Eqn. 15) reconstruct the original sate AS = ϒ 0A
by
performing a “reversed purification” on his own quantum system, BS, that has been initially
ϒ
prepared with an appropriate eigenstate state e p
B
. I will discuss the exact nature of such a
purification process in relation to STT in the final section of this paper.
Alice
Bob
“AO”
ϒ BN ' −1
ϒ 0A
t = 0s
ϕ
A0
0
ϕ 0A1
t = ∆t
ϕ
t = 6( N '− 1) ∆ t
B1
N ' −1
ϕ NB 0'−1
t = (6 N '− 7) ∆ t
t = 2∆t
t = (6 N '− 8) ∆ t
t = 3∆ t
ϒ1A
ϕ 1A 0
ϕ NB 1'− 2
ϒ BN '− 2
t = 3(2 N '− 3) ∆ t
ϕ 1A1 ϕ NB 0'− 2
t = 4∆t
t = (6 N '− 10) ∆ t
t = 5∆t
t = (6 N '− 11) ∆ t
t = 6∆t
t = 3( N '− 2) ∆ t
ϒ 2A
ϒ BN '−3
ϕ NA 0'
ϒ NA '− 2
ϕ 0B 1
t = 3( N '+ 1) ∆ t
ϒ1B
t = 3N ' ∆t
ϕ NA1'
ϕ 0B 0
t = (3 N '− 5) ∆ t
t = (3 N '− 1) ∆ t
t = (3 N '− 4) ∆ t
t = (3 N '− 2) ∆ t
t = 3( N '− 1) ∆ t
ϒ NA ' −1
ϒ 0B
t = 3( N '− 1) ∆ t
“BS”
FIG. 2. General scheme for concatenated teleportation
III. DIFFERENT TYPES OF CONCANATED TELEPORTATION
I shall now intend to classify a concatenated teleportation process. As will be clear shortly,
the type of a concatenated teleportation process determines pivotally the minimum size of the
quantum memory required.
Let’s define a permutation operator P̂i , j that swaps any two states ϕ i
and ϕ j
in
N ' −1
ϒ ⊗ ∏ ϕ i , e.g.
i=0
P̂1,2 ϒ ⊗ ϕ 0 ⊗ ϕ1 ⊗ ϕ 2 ⊗ ...ϕ N ' −1 = ϒ ⊗ ϕ 0 ⊗ ϕ 2 ⊗ ϕ1 ⊗ ...ϕ N ' −1
(17)
It can be seen that there exists two types of concatenated teleportation process:
I. Type-1 concatenated teleportation, where
(
ˆ ( ∆ t )-1
U
Rˆ -1 Pˆ i , j
N'
N'
) (
ˆ ( ∆ t )-1
= U
Rˆ -1
N'
)
N'
,i ≠ j , ∀ i , j = 0,..., N '− 1
(18)
,i ≠ j , ∀ i , j = 0,..., N '− 1
(19)
II. Type-2 concatenated teleportation, where
( Uˆ ( ∆t )
-1
Rˆ -1 Pˆ i , j
N'
N'
) ≠ ( Uˆ ( ∆t )
-1
Rˆ -1
N'
)
N'
ˆ ⊗H
ˆ is
Type-1 concatenated teleportation is possible only if the Hamiltonian H
ϒ
ϕ
decomposable as,
ˆ ⊗H
ˆ =
H
ϒ
ϕ
N'
∑ Hˆ
i
ϒ
ˆi
⊗H
ϕ
(20)
i=0
ˆi ⊗H
ˆ i are non-trivial Hamiltonians, and obey the following commutation,
in which, H
ϒ
ϕ
ˆi
ˆi ˆ j
ˆ j
H
 ϒ ⊗ H ϕ , H ϒ ⊗ H ϕ  = 0,
iff i = j
This is the simplest form of concatenated teleportation process that does not require large
quantum memory. In fact, in a Type-1 concatenated teleportation, each of the partial
teleportation steps (see Fig. 2) can be considered as an independent event, and thus the
sequence of which is not entirely important. As such, Type-1 concatenated teleportation
requires Alice and Bob each to possess a quantum memory that is only large enough to store
a single light-pulse state ϕ A (or ϕ B ); the memory can be re-cycled after each partial
teleportation event. Qiang Zhang et al.’s, and M. Al-Amri, et al.’s works in the teleportation
of photonic multi-dimensional qubits composite systems [12, 13] are two examples of Type-1
concatenated teleportation.
However, unlike its photonic counterpart, the total Hamiltonian for the combined state
ϒ ⊗∏ϕ
is generally not decomposable because the constituent fermions interact not
only with the auxiliary photons, but also among themselves. In other word, the expression,
ˆ ⊗H
ˆ =H
ˆ ⊗ Iˆ + ˆI ⊗ H
ˆ +H
ˆ int
H
ϒ
ϕ
ϒ
ϕ
ϒ ⊗ϕ
(21)
cannot generally be decomposed into N’ commutable Hamiltonians. Hence, STT must be a
Type-2 concatenated teleportation, and a huge quantum memory would therefore be required
so as to store the sequence of the states of the light pulses, ϕ0A1 , ϕ1A1 , ϕ2A1 ,..., ϕ NA1'−1 (see
Fig. 2).
IV. STAR-TREK TELEPORTATION
I shall now apply Eqn. 15 to the simplest case of STT as illustrated in Fig. 3. Let’s assumed
that Bob and Alice are placed inside a 3D ellipsoidal reflector, their respective frame of
reference ( L and L' , see Fig. 3) are mirror images of each other, and separated by some
distance d . The origins of both frames are situated at the two foci of the reflector. The AS to
be teleported is magnetically trapped at the focus P at Alice’s end. A shutter S situated on
the z-axis in L will open only momentarily at t = 0,3∆t , 6∆t ,...,3( N '− 1)∆t to allow just one
auxiliary light pulse at a time to enter and interact with the AS at P in the cavity, but close
otherwise. Thus, any light pulse originated from the AS is always reflected, and directed
(green rays in Fig. 3) to a broadband high-fidelity parallel multimode first-in-last-out
quantum memory (FILO-QMM), which divides the cavity of the reflector. It shall be
assumed that the thickness of the FILO-QMM is extremely thin compared to the major
diameter of the reflector, so that the paraxial approximation is applicable here [14]. The
purpose of the FILO-QMM is to allow Bob to receive Alice’s auxiliary pulses in reversed
sequence. Although in this particular scenario, the same memory is shared between Alice and
Bob, it suffices to demonstrate the STT concept, as can be seen later. For simplicity, I shall
assume that the beam width of the incoming auxiliary pulses be comparable to the size of the
AS. Bob possesses a setup similar to Alice’s, except that it is a mirror image of the latter. Bob
also traps his BS magnetically (at the focus P' ). The shutter S' basically operates in a
reverse manner as S , allowing light pulses originated from the BS to exit the cavity.
Incident auxiliary
light pulses
Outgoing auxiliary
light pulses
FILO-QMM
S'
S
z
x
Magnetic
Trap
L
L'
z’
y’
y
x’
BS
AS
3D ellipsoidal
reflector
FIG. 3. Star-Trek Teleportation
Supposed Alice wishes to teleport both the q-information as well as the structural
information about the AS situated at the focus P , to Bob. I shall assume that AS consists
solely of a gas of molecules. Note that the temperature of the AS needs not be very cold or
near-condensation. To commence the teleportation process, Alice must first carry out a
reversible purification process as pointed out in section II (see Eqn. 16). She does so by
allowing a train of N’ light pulses to impinge on the AS, via the shutter S , at a repetition rate
of 1 ( 3∆t ) (see Fig. 2). It will be assumed that she has carefully chosen the q-states of the
light pulses, such that the sequence ϕ0A0 , ϕ1A0 , ϕ 2A0 ,..., ϕ NA0' −1 can effect a purification of the
state AS . The question here is into what eigenstate should AS be purified. Alice must
consider a number of criteria for her choice. Firstly, in order to simplify the teleportation
process, this particular state must be pre-agreed with Bob a priori, so that a classical
measurement on the final purified state of the AS can be omitted. Secondly, it must be a state
that Bob can easily prepare at his side. Thirdly, it must be a state that can be achieved by
optical means (i.e. via the sequence of photon pulses, ϕ0A0 , ϕ1A0 , ϕ 2A0 ,..., ϕ NA0' −1 ). Lastly, it
must be independent of the initial state of the AS. Here, I propose that this particular state
should ideally be an atomic-Bose-Einstein-Condensate (BEC) state.
Hence, assuming that a density matrix ρˆ o corresponding to the initial state
N ' −1
N ' −1
AS ⊗ ∏ ϕ iA 0 = ϒ 0A ⊗ ∏ ϕ iA 0
i
be,
i
N ϒ , Nϕ N ϒ , Nϕ
ρˆ o =
∑ ∑
i , j =1
α i*α m β *j β n emϒ
A
A
eiϒ ⊗ enϕ
A
e ϕj
A
(22)
m ,n
From Eqn. 22, the density matrix for AS is thus,
ρˆ ϒ = Tr Φ
A
A
{ρˆ o }
If the repeated interrogations on AS lead to a reversible cooling process, i.e.
{
ˆ A Tr A
lim Tr H
ϒ
Φ
N '→ ∞
(Oˆ
N'
A
ρˆ o Oˆ -AN '
)} → E
cond.
(23)
, where E cond. is the energy of the atomic-BEC state. Let the BEC state be e ϒp
A
, one can
conclude that, for a sufficiently large N’,
N ' −1
N ' −1
i=0
i=0
A
lim Oˆ A ϒ 0A ⊗ ∏ ϕ iA 0 → e ϒp ⊗ ∏ ϕ iA1
N '→ ∞
(24)
In other word, the condition prescribed by Eqn. 16 for a successful teleportation can thus be
practically realized if AS can be optically cooled by a train of light-pulses. In order to render
the cooling (or purification) process reversible and controllable, Alice can rely on Λ-type
two-photon transitions when evolving the AS to the targeted eigenstate. To obtain the final
atomic-BEC state, Alice can divide the purification process into two different parts. In the
first, the molecular AS is decomposed into its constituent atoms via stimulated photodissociations (dematerialization) with a series of light pulses (entered via the shutter S ) of
appropriate photon-energies [15-17]. While doing so, she registers the q-states of all the light
pulses (both the scattered and the stimulatedly-emitted photons) returned from the AS in the
FILO-QMM. In the second part, she initiates another train of light pulses to cool the resultant
atomic-AS to a condensate state. Again, while doing so, she registers the states of all the
photons (both the scattered and the stimulatedly-emitted photons) returned from the atomicAS in the FILO-QMM. The cooling process must be lossless and made reversible, say, by
employing a combination of Λ-type transitions with circularly-polarized light, linearlypolarized light and a Sisyphus cooling scheme [18-21]. Alternatively, to assist the cooling
process, Alice can first embed the molecules of interest in a BEC gas before loading the
combined system into the magnetic trap at P . Preferably, the BEC gas should be inert to the
spectral energies ( ω ) used for the photo-dissociation (PD) process. The constituent atoms
resulted from the PD can then be cooled sympathetically via cooling of the BEC host using
light pulses with photon-energies differed from those used in the PD. In this case, the BEC
host and the AS may be of different atomic species. Upon completion of the purification
process, Alice releases the light pulse train stored in the FILO-QMM, in reversed sequence,
to Bob.
Before proceeding to discuss how Bob can reconstruct the original molecular AS. The
ϒ
concept of equivalency between the Alice and Bob’s eigenstates (e.g. ei
ϒ
must be re-stated in the context of STT. According to Fig. 3, ei
and
eiϕ
A
ϒ
and ei
A
B
ϒ
and ei
B
ϕ
(resp. ei
)
A
B
)
shall
be
considered
equivalent
if
A
A
B
B
A
B
B
A
x eiϒ = eiϒ ( x ) = x' eiϒ = eiϒ ( x' ) (resp. x eiϕ = eiϕ ( x ) = x' eiϕ = eiϕ ( x' ) )
, where x and x' are vectors defined with respect to L and L' , respectively.
To reconstruct the original state ϒ 0A of the AS, Bob must, in principle, carry out the
inverse unitary operation described in Eqn. 15. This, however, is realistically impossible as it
will require Bob to reverse the arrow of time. Nonetheless, Bob can still “emulate” the timereversal process with the (forward) time-evolution operators, Û Hˆ
ϒ
⊗ Hˆ ϕ
and Û Ĥ , by
ϒ
B0
performing on the state BS and the recalled state ϕ i
s from his quantum memory, a
specific symmetric transformation tˆ (see Fig. 2).
To see how this can be achieved, I note that the AS is a time-reversal invariant
system, and hence the Eugene Wigner’s time-reversal theorem is applicable here [22],
e
((
) )
ˆ ⊗H
ˆ Tˆ −1 ( ∆t ' )
i Tˆ H
ϒ
ϕ
(
ˆ ⊗H
ˆ ) Tˆ −1 )( ∆t ')
− i ( Tˆ ( H
ϒ
ϕ
Tˆ ϒ(−∆t ') ⊗ ϕ (−∆t ') = Tˆ e
ϒ(−∆t ') ⊗ ϕ (−∆t ')
)
ˆ ˆ −1 (∆t ) ϒ(∆t ) ⊗ ϕ (∆t )
= TU
= Tˆ ϒ(t = 0) ⊗ ϕ (t = 0)
(25)
ˆ ∆t ) ϒ(t = 0) ⊗ Φ (t = 0) ,
ϒ(−∆t ') ⊗ Φ (−∆t ') = ϒ(∆t ) ⊗ Φ (∆t ) =U(
where ∆ t ' = −∆ t and
ˆ ˆ ⊗ Tˆ is the combined Wigner’s time-reversal operator acting on ϒ and ϕ .
and T=T
ϒ
ϕ
Now, I define a reversal operator, tˆ ( = tˆϒ ⊗ tˆϕ ), as
ˆˆ ˆˆ ˆ K
ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ
tˆ = TK=KT=T
ϒ ϒ ⊗ Tϕ Kϕ = K ϒ Tϒ ⊗ Kϕ Tϕ
where K̂ is the conjugation. It is clear that tˆ is a symmetry transformation;
ˆˆ ˆ = − pˆ, tlt
ˆˆˆ = −lˆ, etc , and tit
ˆˆ=i.
tpt
From Eqn. 25, we can see that,
e
((
) )
ˆ ⊗H
ˆ Tˆ −1 ( ∆t ' )
i Tˆ H
ϒ
ϕ
( Tˆ ϒ(−∆t ') ⊗ ϕ (−∆t ') ) = Tˆ ϒ(t = 0) ⊗ ϕ (t = 0)
ˆ − i( Hϒ ⊗Hϕ )( ∆t ') ϒ(−∆t ') ⊗ ϕ (−∆t ') = tˆ ϒ(t = 0) ⊗ ϕ (t = 0)
te
ˆ
e
ˆ
(
)
ˆ ⊗H
ˆ ( ∆t )
−i H
ϒ
ϕ
(26)
ˆ tˆ ϒ(t = 0) ⊗ ϕ (t = 0)
tˆ ϒ(∆t ) ⊗ ϕ (∆t ) = K
ˆ ( ∆t ) ( tˆ ϒ(∆t ) ⊗ ϕ (∆t ) ) = K
ˆ ( tˆ ϒ(t = 0) ⊗ ϕ (t = 0)
U
)
Hence, I can replace Oˆ B in Eqn. 15 with Oˆ 'B tˆ , and re-write the equation as,
N ' −1
(
)
(
N ' −1
)
ˆ
ˆ
ˆ ˆ
ˆ
ϒ BN ' ⊗ ∏ ϕ iB 1 = Oˆ B' tˆM
ϒ 0A ⊗ ∏ ϕ iA 0
B , M → ϕ Pp T A → B M A ,ϕ → M O A
i=0
in
which,
ˆ ' = RU
ˆˆˆ
O
B N'
H
ˆ
⊗H B
ϒB
ϕ
( ∆t )
(
ˆˆ
U
H
ϒB
( 2∆t ) ⊗ ˆI
)
 tˆ
ϕ

ˆˆˆ
N ' RUH
ˆ
⊗H B
ϒB
ϕ
(27)
i=0
( ∆t )
(
ˆˆ
U
H
ϒB
( 2∆t ) ⊗ ˆI
)



N ' −1
.
Note that the expression for Ô' does not contain any inverse unitary term Û -1 . From Eqn.
27, it thus follows that if Alice can perform a unitary described in Eqn. 24 on AS to obtain a
BEC state e ϒp
A
, Bob should be able to reconstruct the original ϒ 0A state by first preparing
his own N-body fermionic system (of same atomic species as the AS) with a BEC state
BS = e ϒp
B
, and then reversing the cooling (and the photo-dissociation) process using the
sequence of light-pulse states ( ϕ0A1 , ϕ1A1 , ϕ2A1 ,..., ϕ NA1'−1 ) he has received from Alice. To
elicit a “time reversal”, Bob simply performs on each of these states a reversal transformation
tˆϕ . Fortunately, the use of a 3D ellipsoidal reflector automatically effects such an operation at
Bob’s side. Additionally, he should act upon his own condensate ( e ϒp
B
) a reversal
transformation tˆϒ . Since a trapped condensate generally possesses no momentum, tˆϒ can be
approximated as an identity for a BEC state, i.e. e ϒp
B
≈ tˆϒ e ϒp
B
We can therefore see that Eqn. 27 should lead to,
N ' −1
(
N ' −1
)
ˆ ⊗Q
ˆ K
ˆ ϒA ⊗
ϒ BN ' ⊗ ∏ ϕ iB 1 = Q
∏ ϕ iA 0
ϒ
ϕ
0
i=0
i=0
or rather,
x 'ϒ ϒ BN ' = x ϒ Kˆ ϒ ϒ 0A = ϒ BN ' (x 'ϒ ) = ϒ 0A (x ϒ ) *
The final equation implies that Bob can re-create (rematerialisation) the physical structure of
the AS ( ϒ0A ( x ϒ ) ) without the need to reverse the time-arrow, i.e. without having had to
−1
perform the inverse operations Û Hˆ
ϒB
ˆ
⊗H
ϕB
( ∆t )
ˆ −1
and U
Ĥ
ϒB
( 2∆t ) ⊗ ˆI . Note that however the
reconstructed object will be an mirror image of the original, since L' is a mirror image of
L . If Bob is very stringent about the symmetricity of the final object, a double STT process
via an intermediate party, say Victor, would be necessary, so that the sample can be inverted
back to its original symmetry at Bob’s side.
V. CONCLUSION
I have shown that a time-reversal invariant system can be teleported in a “Star-Trek” manner.
Owing to the complexity of the AS involved, STT is a type-2 concatenated teleportation
process and a broadband large-capacity FILO-QMM is required. Such a memory can likely
be constructed based on the technique recently reported by Mahdi Hosseini, et al. [23]. My
subsequent papers will look into various issues affecting the fidelity of the STT. These
include quantum chaos and de-coherence due to vacuum fluctuations.
REFERENCES
[1]
A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
[2]
C. H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993).
[3]
A. Galindo, and M. A. Martin-Delgado, Rev. of Mod. Phys. 74, 347 (2002).
[4]
D. Bouwmeester et al., Nat. 390, 575 (1997).
[5]
L. Lydersen et al., Nat. Photon. 4, 686 (2010).
[6]
J. L. O'Brien, A. Furusawa, and J. Vuckovic, Nat. Photon. 3, 687 (2009).
[7]
A. Dantan et al., Phys. Rev. Lett. 94, 050502 (2005).
[8]
A. S. Bradley et al., in arXiv:0706.0062v1.
[9]
S. Bandyopadhyay, and B. C. Sanders, Phys. Rev. A 74, 032310 (2006).
[10]
H. Nakazatom, T. Takazawa, and K. Yuasa, Phys. Rev. Lett. 90, 060401 (2003).
[11]
K. Jacobs, Phys. Rev. A 67, 030301R (2003).
[12]
Q. Zhang et al., Nat. Phys. 2, 678 (2006).
[13]
M. Al-Amri, J. Evers, and M. S. Zubairy, Phys. Rev. A 82, 022329 (2010).
[14]
T. Golubev et al., Phys. Rev. A 83, 053810 (2011).
[15]
H. R. Thorsheim, J. Weiner, and P. S. Julienne, Phys. Rev. Lett. 58, 2420 (1987).
[16]
A. Vardi et al., J. Chem. Phys. 107, 6166 (1997).
[17]
Y. B. Band, and P. S. Julienne, Phys. Rev. A 51, R3417 (1995).
[18]
B. Hoeling, and R. J. Knize, Opt. Comm. 104, 202 (1994).
[19]
D. E. Pritchard, Phys. Rev. Lett. 51, 1336 (1983).
[20]
N. R. Newbury et al., Phys. Rev. Lett. 74, 2196 (1995).
[21]
M. D. Barrett, J. A. Sauer, and M. S. Chapman, Phys. Rev. Lett. 87, 010404 (2001).
[22]
J. Martin, B. Georgeot, and D. L. Shepelyansky, Phys. Rev. Lett. 100, 044106 (2008).
[23]
M. Hosseini et al., Nat. 461, 241 (2009).