Effect of noise on quantum teleportation
Deepak Kumar
School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110057, India
P. N. Pandey
ASRD College, Delhi University, Dhaulakuan, New Delhi, India
Received 16 August 2002; published 22 July 2003
The effect of noise on quantum teleportation of a spin-
1
2
state using an entangled pair is studied. We
calculate the time evolution of the density matrix of the three involved particles due to their coupling with the
environmental degrees of freedom. We evaluate the fidelity of transmission as a function of time under a
variety of conditions and compare the fidelities obtained for different entangled states. We find that for a
generic coupling to environment, use of the singlet state for the entangled pair yields the highest fidelity in
noisy conditions.
DOI: 10.1103/PhysRevA.68.012317 PACS numbers: 03.67.Hk, 03.65.Ud, 03.65.Yz
I. INTRODUCTION
A major limitation on quantum devices for information
processing and communication is environmental noise. The
operation of such devices is crucially dependent on the phase
coherence of the evolution of the quantum system. The effect
of noise is a decoherence of this dynamical evolution with
time and hence, successful processing can be done only
within the decoherence time. In the past few years, a consid-
erable amount of work has been done to find practical ways
to counter the effects of decoherence. On one hand, inspired
by classical information theory, there have been develop-
ments such as quantum error correcting codes 1–3 and en-
tanglement purification protocols 4–8. On the other hand,
there are efforts to develop physical realizations of the quan-
tum devices where the decoherence effects are reduced to
practical limits, with photons 9,10, ion traps 11,12, cavity
quantum electrodynamic setups 13, ions in microtrap ar-
rays 14, atoms in optical lattices 15,16, solid-state devices
17,18, and nuclear spins 19.
Clearly, for the operations of quantum devices, it is im-
portant to study quantitatively how the noise-induced deco-
herence affects the various processes. Many studies have
been done in this direction 20–24, such as the effect of
noise i in maintaining memories in qubits, ii operation of
quantum gates for some physical realizations. In this paper,
we follow this work to study the dynamical decoherence of
quantum teleportation for some typical noise processes.
II. QUANTUM TELEPORTATION WITH DECOHERENCE
In quantum teleportation 25, we deal with three spin-1/2
particles. The particle 1 is in an arbitrary, unknown spin
state, |
1
. This particle is with a sender, named ‘‘Alice’’and
its quantum state |
1
is to be transported to a receiver
named ‘‘Bob.’’ Alice and Bob share an entangled pair of
particles 2 and 3, particle 2 with Alice and particle 3 with
Bob. The entangled state is taken to be the spin singlet. The
other entangled states are considered later in the paper. We
assume that the entangled pair and the state |
1
are pre-
pared at t =0. Thus the wave function of the three particles
at t =0 is taken to be
| = cos
/2 e
i /2
| ↑
1
+sin
/2 e
-i /2
| ↓
1
]
1
2
| ↑
2
↓
3
-| ↓
2
↑
3
]. 1
In the usual teleportation scheme 25, Alice performs a Bell-
state measurement on the particles 1 and 2 in her possession,
and sends the result of her measurement to Bob. Alice’s mea-
surement projects the wave function of particle 3 either to
|
1
, or to a state related to it. Bob can recover the state |
1
by performing atmost rotations on particle 3, the axis of
rotation depending upon the result received from Alice. We
now suppose that this arrangement is to be preserved for
time t before the measurement corresponding to the telepor-
tation is made. The time t, for example, could include the
time required for transporting the entangled particles to Alice
and Bob. During this time, the system is subject to environ-
mental noise and is thus undergoing dissipation.
A more general situation is that in which the preparation
times for the entangled pair and the state of the particle 1 are
different. Thus their states decohere for different periods be-
fore the teleportation is effected. Another factor to be ac-
counted for is delay in Bob’s recovery of the projected state
following the Bell-state measurement of Alice. The part of
this delay, which is inevitable, is the time taken by the clas-
sical signal from Alice to Bob. Both these situations are eas-
ily handled by the calculations described below.
The density matrix of the system can be written as a sum
of products of the density matrices of the three particles.
Since the particles are independent, correlations or entagle-
ments are only due to the initial state. Thus the most general
form for the three-particle density matrix has the form
123
t =
1
8 k
l =1
k
I
1
+ P
1
t ; l •
1
I
2
+ P
2
t ; l •
2
I
3
+ P
3
t ; l •
3
, 2
PHYSICAL REVIEW A 68, 012317 2003
1050-2947/2003/681/0123176/$20.00 ©2003 The American Physical Society 68 012317-1