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 Einstein-Podolsky-Rosen (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 multi-parties.
- Probabilistic quantum teleportation:
We have found a new protocol where by using non-maximally 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 non-maximally 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 non-orthogonal 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 non-orthogonal 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 ``no-deleting'' theorem in quantum information theory. Like the famous no-cloning 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 Deutsch-Jozsa 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.
- 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 no-cloning and the no-anticloning 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.
- Non-Existence of universal constructor:
Recently, we have addressed another important question: Can a quantum system self-replicate? 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 no-cloning theorem. Fifty years back, in the classical context, Von Neumann showed that a `universal constructor' can exist which can self-replicate 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 self-replication 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.
- No-Partial 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.
- No-hiding theorem in quantum information:
Recently, we have proved the no-hiding 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 semi-classical 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 non-adiabatic, non-cyclic, non-unitary and even non-Schrodinger evolutions of quantum systems. Using gauge-invariant reference section I have provided a connection form for non-cyclic, non-unitary and non-Schrodinger evolutions. I have introduced another metric structure called ``reference-distance'' in addition to usual Fubini-Study distance in studying geometry of quantum states.
- Adiabatic Berry phase for open paths and semi-classical limit:
I have defined the open-path 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 semi-classical limits of the open-path Berry phase.
- Berry phase and response function in many-body system:
The Berry phase has found important applications in trying to understand response function in many-body systems. We have shown that the Berry phase during cyclic and non-cyclic 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 non-unitary 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 non-additive 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.
