My research is mostly in the area of Quantum Information and
Quantum Computation, Theory of Geometric phases and its applications,
mathematical and
fundamentals aspects of Quantum physics.
Quantum Information and Computation:
Quantum Information and Quantum Computation has undergone explosive growth
in recent years and emerged as one of the most important area of research.
This field has left its impact in quantum theory, information theory,
computer science, complex systems,
My interests are quantum algorithms for continuous variable systems,
quantum communication protocols,
quantum cloning, quantum deleting, quantum entanglement
and related issues. I am exploring the
possibility of generating new effects using shared entangled states, local
operations and classical communications.
Also, I have been investigating the boundaries between quantum and classical
world, especially, what are possible and impossible operations in quantum
information theory.
One of the main motivation in this area is to discover fundamental
principles of quantum
information, limitations on quantum information and what we can do with it.
 Remote state preparation :
I have proposed a new
protocol called remote state preparation (RSP). Here, the aim is to prepare
a known state at a distant location without physically sending the object.
The exact remote state preparation (RSP) protocol for special
class of qubits has been proposed.
It was shown that using one pair of EinsteinPodolskyRosen (EPR) entangled
state one can remotely prepare a qubit (from special ensemble) using one
classical bit of communication.
Also, it has been shown that Alice can ask Bob to simulate any single particle
measurement outcome on an arbitrary qubit using one
EPR pair and one classical bit at a remote location. This is called
Remote state
measurement (RSM) protocol. This has opened up a new topic of research in
quantum information theory. The RSP protocol that I had proposed for special
class of qubits has been recently tested using NMR devices and photon entangled
states. Recently, we have also generalized exact RSP of qubits and higher
dimensional quantum systems for multiparties.
 Probabilistic quantum teleportation:
We have found a new protocol where by using nonmaximally entangled basis
as the measurement basis, one can use a general pure entangled state
as a resource for quantum teleportation. Using our protocol one can teleport
a qubit with unit fidelity but with a probability that is less than unit.
We have described
the probabilistic teleportation using the language of quantum operations
and calculated the success and failure probabilities. This scheme could be of
use in real experiments which may be tested in near future.

Probabilistic superdense coding:
We have proposed a protocol to perform super dense coding using any pure
nonmaximally
entangled state as a shared resource. Our scheme works in a probabilistic
manner.
Interestingly, this problem is related to the problem of distinguishing a
set of
nonorthogonal states in quantum theory. We have obtained a tight bound
on the
success probability of performing dense coding.

Quantum cloning:
In quantum world we cannot clone a quantum state. However, we can have an
approximate clone by a deterministic process or an exact clone by a
probabilistic method. We have asked the question if we can have a linear
superposition of multiple clones of an unknown state in quantum theory.
Ideally it is not
possible. It was proved that
unitarity allows us to create a linear
superposition of multiple clones of nonorthogonal states along with a
failure branch if and only if they are linearly independent.
Further, it has been
shown that probabilistic and deterministic cloning machines are special
cases of our
novel cloning machine. This proposal has been pursued by other groups in
details.

Quantum deleting:
We have discovered the ``nodeleting'' theorem in
quantum information theory.
Like the famous nocloning theorem, it is another fundamental limitation
on quantum information. It states that given two copies of an unknown
quantum state we cannot delete a copy against the other using any physical
operation. This is different from Landauer's erasure principle. Erasure
of quantum information is possible but deletion is impossible. Moreover,
the deleting machine we have studied is not reverse of cloning machine.
Also we have shown that deleting an arbitrary states would imply super
luminal signalling. In future, we would like to construct an optimal
universal quantum deleting machine.
This discovery had featured in the News and Views column of {\bf NATURE}
and also in many leading news papers all over the world.
Quantum deleting has also opened up a new avenue of research
in quantum information. Recently, it has been shown by many groups that
probabilistic deletion of linearly independent quantum states is possible.
Also, a universal quantum deleting machine has been proposed in the literature.

Quantum computation, algorithm for continuous variables:
We have generalized Grover's algorithm for continuous
variable systems. Using continuous analog of Hadamard transformation and
inversion operator we have constructed a search operator which gives square
root speed up. Quantum searching with continuous variables may be useful for
arbitrarily large data base search.
We have also generalized DeutschJozsa algorithm for continuous variable
systems. It was shown that if one replaces the unitary transformations
in the logic circuit of discrete case with their continuous variable
analog then it works perfectly.

Quantum computation and entanglement:
In quantum computation one important question
has been whether we always need entanglement in achieving the desired speed
compared to classical computers. Though the answer in general is
not known, we have found that for Grover algorithm one needs the presence
of entanglement. This is true for pure as well as pseudopure state
implementations (like NMR devices). We have argued that in the absence of
quantum entanglement the computation cannot be called true quantum
computation. In future, we would like to see to what extent entanglement
plays an important role in general quantum mechanical algorithms.

General impossible theorems in quantum information:
I have proved a general impossible theorem which suggests some new
limitations on quantum information. I have argued that the nocloning and
the noanticloning are special cases of the general limitations. In
addition we have found that one cannot design a Hadamard gate for an
unknown qubit. Though we can design a Hadamard gate for a qubit
in computational basis, a similar gate cannot be designed for an arbitrary
qubit. The implications of these limitations on designing universal
logic gates for quantum computation has been discussed and it is argued
that quantum computers are inherently personal.

Schmidt decomposition theorem for three particle entangled state:
For bipartite systems in general there always exist Schmidt decomposition
for all wave functions such that the reduced density matrices have equal
spectrum. For tripartite systems this does not hold.
I have provided a necessary and sufficient condition for the existence of
Schmidt decomposition for tripartite systems.
This shows that one can use von Neumann entropy of
the partial density matrix as a measure of entanglement for such tripartite
systems.

NonExistence of universal constructor:
Recently, we have addressed another important question:
Can a quantum system selfreplicate?
We know that an arbitrary quantum state cannot be copied.
In fact, to make a copy we must
provide complete information about the system. However, our question
is not answered by the nocloning theorem.
Fifty years back, in the classical context, Von Neumann showed that
a `universal constructor'
can exist which can selfreplicate an arbitrary system, provided that it had
access to instructions for making copy of the system. We have questioned the
existence of a universal constructor that may allow for the selfreplication
of an arbitrary quantum system. We have proved that there is no deterministic
universal quantum constructor which can operate with finite resources.
We have delineated conditions under which such a universal constructor
can be designed to operate deterministically and probabilistically.
This work has been featured in the News of the Week column of
{\bf SCIENCE}, 9th May 2003.

NoPartial erasure of quantum information:
We have introduced a new process called partial erasure where we would like to
forget one or more parameters of a quantum state keeping the rest intact. We
have found that it is impossible to have partial erasure of quantum
information even by irreversible operation, though one can completely erase.
This suggests that quantum information is a `whole' entity. We cannot
forget part of it, we have to deal it as a whole. Thus, this result gives a new
meaning to quantum information, namely, it is an indivisible entity.

Nohiding theorem in quantum information:
Recently, we have proved the nohiding theorem for
quantum information, its robustness to imperfect hiding process and its
application to black hole information loss. This says that if the original
quantum information is missing from the one subsystem then it must be found in
the reminder of the subsystem and moreover this cannot be hidden in the
correlations. According to present understanding
one possible resolution of black hole information paradox that might
avoid both unitarity violation and causality
violation for macroscopic black holes is to suppose that the information
is neither retained below the horizon, nor contained solely within the
radiation emitted by the black hole. Instead, the quantum information is
"hidden" in the correlations between the state of the radiation and the
internal state of the black hole. However, our paper rigorously eliminates
this possibility by showing that quantum information cannot reside
purely in the spooky correlations.
It is not a question of choosing between unitarity violation and the
breakdown of the semiclassical
approximation, but our paper does prove that these are the only two
possibilities.
Geometric Phases and Applications:
Classical system in general cannot remember its past
whereas quantum system can remember its history. The way it does is
reflected in the geometric phase of the state of a quantum system. In the
last two decades the geometric phase has been extensively studied and applied
in almost all areas of physics.

Generalization of geometric phase:
I have generalized the geometric phase for
most general situations such as nonadiabatic, noncyclic, nonunitary and even
nonSchrodinger evolutions of quantum systems.
Using gaugeinvariant reference section I have provided a
connection form for noncyclic, nonunitary and
nonSchrodinger evolutions. I have introduced another metric structure called
``referencedistance'' in addition to usual FubiniStudy distance in studying
geometry of quantum states.

Adiabatic Berry phase for open paths and semiclassical limit:
I have defined the openpath Berry phase for quantum systems. Using a
generalized gauge potential, we can obtain this phase as an integral along
the open path. I have introduced the Hannay angle for open paths. Also,
I have studied the classical and semiclassical limits of the openpath
Berry phase.

Berry phase and response function in manybody system:
The Berry phase has found
important applications in trying to understand response function in manybody
systems. We have shown that the Berry phase during cyclic and noncyclic evolutions
of a finite fermi system is directly related to the response function of the system.
Our result relates effective single particle property to bulk property of the system
and explains damping of collective excitations in finite fermi systems.

Geometric phase for mixed states:
The notion of relative phase for mixed states was an elusive concept
and has been a great challenge since development of quantum theory.
Recently, we have introduced the notion of relative phase shift for
mixed states and generalized the concept of
geometric phase for mixed states undergoing unitary time evolution.
Our approach is based on quantum interferometric method which can be
easily tested. This may have potential application in geometric
quantum computation using NMR techniques. This work has triggered new
research in the area of
mixed state geometric phases in the last six years. In fact, very recently
our mixed state phase has been observed experimentally by several groups.

Geometric phase for completely positive maps:
We have generalized the concept of
mixed state phase when the system undergoes an evolution described by
a completely positive map. Thus, one can define the notion of mixed state
geometric phase during nonunitary evolutions and open systems.
I have introduced the notion of
`in phase' quantum channel and shown that geometric phase during a sequence of
quantum operation will be nonadditive in nature. Thus, the geometric phase may
be used to associate `memory' to a quantum channel.
In future, we would like to
apply our definition to a qubit undergoing decoherence and study its effect
on the geometric phase.