BACKGROUND
1. From Quantum Mechanics to Quantum Information Theory
In 1935 Einstein, Podolsky and Rosen [1] have observed that two quantum systems, say two
particles, which interacted at some time in their past and which have become separated later remain
connected in the following way: a measurement performed on one of the two particles instantaneously
influences the quantum state of the other one.
Erwin Schroedinger called this phenomenon
“entanglement” and considered it the essential feature of quantum physics, almost a drawback [2].
Experimentally, quantum mechanics was very well confirmed for statistical ensembles of atoms and
molecules.
In the 1970s, the development of lasers made it possible to confirm the predictions of quantum
physics also for individual systems (particles). Thus, a series of ingenious experiments was performed
[3], mainly on photons, the particles of light, but also on neutrons and atoms, which showed us that
nonlocality and entanglement are the principal features of composite quantum states. Moreover, these
experiments paved the way for a new information technology where individual quantum systems are
the carriers of information. We are thus interested in the new phenomena can occur when we use
quantum systems to represent information and when their processing is determined by the laws of
quantum physics. As an example, let us consider the qubit or quantum bit. As opposed to a classical
bit, a qubit can also be in a superposition of 0 and 1. The great advantage of a qubit is that a
superposition can exist in many different ways, and therefore the qubit has the potential to represent
more information than a classical bit.When dealing with two or more qubits, they can be entangled
with each other. Then observation of one qubit influences the quantum states of the others. While the
value of an individual qubit might be completely random, measurement on one qubit defines the bitvalue of the other one.
The new field of information theory based on quantum principles extends and completes classical
information theory [4]. This theory includes quantum generalizations of classical notions such as
sources, channels, and codes. In quantum mechanics, a quantum state is represented by a vector in a
Hilbert space, or a density operator on that space. Classically, the input system may retain its original
state, while the no-cloning theorem and related results imply that in the quantum case the input system
cannot in general remain in its initial state. Therefore, unlike classical information, quantum
information cannot be read or copied without disturbing it; otherwise said any copy is imperfect [5].
2. Channels
In principle, in quantum information we are dealing with two quantifiable kinds of information—
classical information and quantum entanglement. Both are manipulated and transmitted through
quantum channels. Any physical means, such as an optical fiber, for delivering quantum states more or
less intact from one place to another, may be viewed as a quantum channel. More general, a quantum
channel can be pictured as the transfer of some quantum state from sender to receiver. If the transfer is
intact and undisturbed, the channel is noiseless; if the quantum system interacts en route with some
other system, a noisy quantum channel results [6]. Quantum channels can be used to carry classical
information, and, if they are not too noisy, to transmit intact quantum states and to share entanglement
between remote parties. The efficiency of a communications channel can be measured by its capacity,
namely the highest rate at which information can be reliably transmitted through the channel.
The main goals of quantum information theory are understanding entanglement and calculating the
several capacities of quantum channels. Classical channels are adequately characterized by a single
capacity which cannot be increased by auxiliary resources [7]. By contrast, auxiliary resources have a
major effect on both the classical and quantum capacities of quantum channels. In particular,
entanglement can increase both capacities (classical and quantum) and classical feedback can
increase the quantum capacity of quantum channels capacity [8]. This is the reason for which for
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quantum channels several distinct capacities have been defined: a classical capacity C, for
transmitting classical information, a quantum capacity Q, for transmitting intact quantum states, a
classically assisted quantum capacity ,an entanglement-assisted classical capacity [4].
Work in the area of quantum information processing has increased enormously in the last decade,
with many groups exploring different topics of this newly emerging field. The most interesting areas
explored are quantum communication, quantum cryptography, quantum teleportation and quantum
computation. While copying or cloning is prohibited by quantum mechanics [5], the invention of
quantum teleportation in 1993 [9] suggested another way to achieve disembodied transport. The basic
idea is to transfer the quantum state of one system over to another, distant one in such a way that the
new system becomes identical with the original. The possibility of this process emerges from quantum
entanglement. Already realized in experiments, quantum teleportation is a process perfectly described
as a quantum channel [10].
3. Mathematical description of noisy channels
The fidelity with which states are transmitted through a quantum channel can be increased by using
encoding procedures (at the sender station) and decoding ones (at the receiver station). Together with
the adding of noise during transmission, all such operations can be mathematically described as the
action of a linear, trace-nonincreasing and positivity-preserving map long time ago introduced in open
system theory [11]. Quantum operations have also sometimes been known as completely positive maps
or superscattering operators.
The difficulty in defining a quantum capacity comes from the ambition of considering an optimization
procedure over all possible coding and decoding schemes. This is the reason for which very few
concrete results of capacities are obtained at present. For a recent review see Ref. [12].
4. Our previous work
The team addressing this proposal has a good expertise and results in the theory of quantum
information processing. Note that an important issue in quantum information theory is the ability to
distinguish between different quantum states, either pure or mixed. A practical tool in exploring
distinguishability is a carefully defined transition probability between any pair of quantum states
(pure or mixed). Therefore, In recent work, we had evaluated and used the Uhlmann fidelity [13],
which is largely accepted as a good measure of distinguishability . Evaluation of the fidelity in the
infinite dimensional Hilbert space of the density operators is now of greatest importance due to the
experimental interest in quantum information processing of field states [14]. We focused on the
Gaussian states’ fidelity. Especially useful in experiments, the Gaussian states are defined by the
exponential form of their density operators. In recent years the Uhlmann fidelity became an important
tool in quantifying nonclassicality (in the one-mode case [15]) and inseparability (for two-mode states
[16]). Measuring the entanglement, which is an avoidable ingredient in all processes, for two-mode
Gaussian states was one of our main tasks in the last years [16]. The explicit formula of the one-mode
fidelity was then used to quantify the accuracy of teleportation of mixed one-mode Gaussian states
through a Gaussian channel in Refs.[17]. In a very recent paper [18], we gave a novel description of
the continuous-variable teleportation protocol in terms of the characteristic functions of the quantum
states involved. The Braunstein-Kimble protocol was written for an unbalanced homodyne
measurement and arbitrary input and resource states. We have shown that the output of the protocol is
a superposition between the input one-mode field and a classical one induced by measurement and
classical communication. We described the input state distortion through teleportation by the average
photon number of the measurement-induced field. This was a novel idea in describing the
teleportation channel valid for non-Gaussian resource state as well. We thus intend to continue our
work, by exploiting this important result. Also, very recently it was found that the capacity of a
continuous variable channel and the entanglement of formation are related quantities [19]. We intend
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to deepen this idea in the general case based on some recent results on EoF for two-mode Gaussian
states [20].
In finite-dimensional settings, the analyzed processes are quantum teleportation and further
generalizations. The principal aspects treated by Iulia Ghiu in Refs.[21-23] and in her thesis
APPLICATIONS OF THE QUANTUM INFORMATION PROCESSING (June, 2007, University of
Bucharest, under the supervision of Professor Tudor A. Marian) are local manipulation of
entanglement for tripartite states [21], asymmetric broadcasting of entanglement by using local and
nonlocal optimal universal asymmetric cloning machines[22]. She also proposed a new protocol of
many to many teleportation [23] , where the information of a quantum system is distributed from N
senders to M receivers.
REFERENCES
[1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
[2] E. Schroedinger, Naturwiss. 23, 807 (1935).
[3] A. Aspect, J. Dalibard, P. Grangier, and G. Roger, Phys. Rev. Lett. 49, 1804 (1982).
[4] Charles H. Bennett and Peter W. Shor, IEEE Transactions on Information Theory 44, 2724
(1998).
[5] W. K. Wootters and W. H. Zurek, Nature 299, 802-803 (1982).
V. Buzek and M. Hillery, Phys. Rev. A 54, 1844 (1996).
[6] Howard Barnum, M. A. Nielsen, and Benjamin Schumacher, Phys. Rev. A. 57, 4153 (1997).
Howard Barnum, John A. Smolin, and Barbara M. Terhal, Phys. Rev. A. 58, 3496 (1997).
[7] C.E. Shannon, Bell. Syst. Tech. J. 27, 379 (1948).
[8] C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, Phys. Rev. Lett. 83, 3081 (1999);
ibidum IEEE Transactions on Information Theory 48, 2637 (2002).
A. S. Holevo, IEEE Transactions on Information Theory 44, 269 (1998).
[9] C. H. Bennett, et al., Phys. Rev. Lett. 70, 1895 (1993).
[10] M. M. Wolf, D. Perez-Garcia, and G. Giedke, Phys. Rev. Lett. 98, 130501 (2007)
[11] K. Hellwig and K. Kraus, Commun. Math. Phys. 16, 142(1970);
K. Kraus, States, Effects, and Operations ~Springer-Verlag, Berlin, 1983.
[12] I. Devetak and P. Shor, quant-ph/0311131/2004.
[13] A. Uhlmann, Rep. Math. Phys. 9, 273 (1976); Rep. Math. Phys. 24, 229 (1986).
[14] A. Furusawa s. a, Science 282, 706 (1998).
[15] Paulina Marian, Tudor A. Marian and Horia Scutaru, Phys. Rev. Lett. 88, 153601 (2002);
Paulina Marian, T.A. Marian, and H. Scutaru, Phys. Rev. A. 69, 022104 (2004).
[16] Paulina Marian, Tudor A. Marian and Horia Scutaru, Phys. Rev. A 68, 062309 (2003).
Paulina Marian and Tudor A. Marian, the e-print arXiv:0705.1138v1
[17] Paulina Marian, T. A. Marian, and H. Scutaru, Rom. J. Phys. 48, 727 (2003). See also the e-print
quant- ph/0601045.
M. Ban, Phys. Rev. A 69, 054304 (2004).
[18] Paulina Marian and T. A. Marian, Phys. Rev. A. 74, 042306 (2006).
[19] J. Eisert and M. M. Wolf, quant-ph/0505151.
[20] Paulina Marian and T. A. Marian, Gaussian entanglement of formation for two-mode Gaussian
states, talk given at CEWQO, Palermo 2007.
[21] Iulia Ghiu, M. Bourennane, and A. Karlsson, Phys. Lett. A 287, 12 (2001).
[22] Iulia Ghiu and A. Karlsson, Phys. Rev. A 72, 032331 (2005).
[23] Iulia Ghiu, Phys. Rev. A 67, 012323 (2003).
[24] T.Hiroshima, Phys. Rev. A 73, 012330 (2006).
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OBJECTIVES OF THE PROJECT
1. Scientific objectives
Most of the work within this proposal lies at the boundary of quantum optics and information theory.
In fact, most of the experiments that have been performed in the area of quantum information
processing (teleportation, cryptography) have been with light – either single photons, or beams of laser
light. Therefore, when speaking about the continuous-variable quantum information we think at the
processing of the electromagnetic field. Methods of quantum optics are thus of great importance in
describing nonclassical properties such inseparability of states. The two-mode field is the prototype of
a bipartite continuous-variable system. Needless to say, the continuous-variable quantum channels
(bosonic channels) were much studied [19] and interesting recent results were obtained [10,24]. The
main scientific objectives of this proposal grow from many recent results obtained by the team. They
can be summarized as follows:
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*)
**)
Give final results on the Uhlmann fidelity between two arbitrary two-mode Gaussian states
(TMGS)*
Give a Gaussian distance-type measure of entanglement for an arbitrary TMGS*
Purifications of one-mode states. The validity of the Gaussian approach
Examination of a possible relation between the entanglement of formation of a TMGS and the
capacity of a certain Gaussian channel. Implications.
Additivity and multiplicativity properties of Gaussian channels with classical noise
Evaluation of capacities for general one-mode and two-mode Gaussian channels
Other measures for capability of information processing through Gaussian channels: fidelity
(one-mode case) and entanglement fidelity (two-mode case).
Non-Gaussian channels of teleportation**
Finite-dimensional systems. Measures of entanglement for states under symmetry.
Finite-dimensional systems. Teleportation of entanglement.
to complete work published in [16].
to continue work published in [18].
2. Training objectives
The first training goal of the project is to introduce in the field of quantum optics and information
theory a young physicist just graduated from the Department of Physics, University of Bucharest.
The second objective is to consolidate the quantum physics expertize of the two post-doctorale
researchers Iulia Ghiu and Madalina Boca.
At the Centre for Advanced Quantum Physics of the University of Bucharest there is a tradition of
producing young researchers who have mastered scientific and intellectual abilities that are not
commonly found in the disciplinary curricula of Romanian Physics departments, such as quantum
optics, quantum information description and processing .
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