Entanglement purification in cavity QED using local operations
J. L. Romero,
1
L. Roa,
1
J. C. Retamal,
2
and C. Saavedra
1,
*
1
Departamento de Fı ´sica, Universidad de Concepcio ´n, Casilla 160-C, Concepcio ´n, Chile
2
Departamento de Fı ´sica, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
Received 28 March 2001; revised manuscript received 20 November 2001; published 7 May 2002
We study a physical implementation of an entanglement purification protocol using a cavity quantum
electrodynamics based proposal, in both the microwave and the optical domain. The protocol consists of local
quantum operations of each particle of an entangled system with one auxiliary particle ancilla. After the
above interaction a measurement on ancillas is carried out. In the microwave region the quantum information
is stored in field states inside two distant cavities. We also give a procedure for quantifying the degree of
entanglement between quantum fields, which allows verifying the efficiency of the purification process. In the
optical domain, we study a setup of cold trapped ions inside cavities, where quantum bits are defined by two
electronic levels of ions. This latter proposal is extended to create multiparticle entangled states among distant
quantum systems. Entanglement is achieved through a set of local measurements on pairs of entangled par-
ticles.
DOI: 10.1103/PhysRevA.65.052319 PACS numbers: 03.67.Hk
I. INTRODUCTION
Quantum entanglement plays a fundamental role in the
quantum information theory 1–3. The entanglement con-
cept has been central to the quantum theory since the famous
work of Einstein, Podolsky, and Rosen 4. Most of the quan-
tum information protocols related to transmission, process-
ing, and storing of information make use of the capability of
creating and manipulating quantum entanglement among dis-
tant quantum bits. In recent years, there has been an exten-
sive study of theoretical implications behind quantum en-
tanglement and practical schemes to produce it in a variety
of physical systems. The entangled particles provide a means
for implementing quantum communication channels among
nodes of a quantum network.
The degree of entanglement can be fully or partially de-
stroyed, for instance, due to the presence of interactions with
the environment, or imperfect quantum logic operations.
Thus, the problem of improving the quality of quantum en-
tanglement as a means for protecting or preserving quantum
information has been studied recently. In quantum informa-
tion theory, these processes are usually called quantum puri-
fication protocols 5,6. These protocols consider the exis-
tence of a large number of entangled particles, and that each
one of the components of the entangled system is located in
one node of a quantum network. In a sequential process,
many of these partially entangled systems are disregarded
and the degree of entanglement of the remainder systems is
higher than the initial one. If there exist many partially en-
tangled systems at the same time, the purification can be
implemented by performing bilateral controlled-NOT opera-
tions 5. These can be generalized for many particle opera-
tions, which requires implementing collective measurement
on the particles belonging to a given node of the network
21. This will be addressed at the end of this paper. How-
ever, while establishing entanglement between two nodes of
a quantum network, the most common scenario is to have
access to only one entangled pair at a same time. For in-
stance, this occurs in the experimental implementation of a
quantum cryptography protocol 7 based on Ekert’s pro-
posal 8. Recently, a proposal for a physical implementation
of an entanglement purification protocol has been analyzed
for Gaussian continuous variable entangled states. The pro-
posal makes use of high finesse cavities and cavity enhanced
Kerr nonlinearities 9. In the case of entangled photons, an
alternative method for entanglement purification operating
on two pairs at the same state, without the use of controlled-
NOT gates, has been given recently 10.
Alternatively, enhancement of entanglement in a single
copy of a mixed state can be implemented by filtering opera-
tions, by making use of ancillary systems 11. In this con-
text, Horodecki et al. 12 gave an example of a mixed state
that can be quasidistilled by using local quantum operations
and classical communications LQCC. In this case, the ma-
trix was composed of a maximally entangled and an orthogo-
nal product state. The above-mentioned matrices are of rank
two, i.e., in which a single copy can be quasidistilled by
LQCC as has been recently shown by Verstraete et al. 13.
Here we study the implementation of a purification protocol
based on local operations on ancillary systems interacting
with a partially entangled pair in the cavity QED context.
This process corresponds to a purification protocol based on
filtering operations that can be described by using positive
operator valued measurement POVM14, which consists
of a unitary evolution between one particle of an entangled
pair and an ancillary system, taking place at both nodes of
the network. After the evolution, a measurement is applied
on the ancilla. The initial state of the ancilla is chosen so as
to optimize the purification protocol. A schematic diagram of
the protocol is depicted in Fig. 1. In Sec. II we study a
physical implementation of this purification protocol in the
context of cavity quantum electrodynamics, at the micro-
wave region 15–17. In these cavities quantum bits are de- *Electronic address: carlos.saavedra@udec.cl
PHYSICAL REVIEW A, VOLUME 65, 052319
1050-2947/2002/655/0523199/$20.00 ©2002 The American Physical Society 65 052319-1