Description of my research

Two different kinds of quantum systems, namely, discrete variable (DV) and continuous variable (CV) systems, can be used for encoding, manipulating, and decoding information in quantum information processing (QIP). 

DV system

If the system observables used for encoding information have discrete eigenvalues, the corresponding system is termed as a DV system.  Quantum bit or Qubit, a system with two distinguishable states, is the simplest example of a DV system. There are many examples of such systems, including electron spin, photon polarization, superconducting qubits, and nucleus spin.  

CV system

If the system observables used for encoding information have continuous eigenvalues, the corresponding system is termed as a CV system. Quantum harmonic oscillator is one the most important example of CV systems. Others examples of such systems include vibrational modes of solids, atomic ensembles, quantized electromagnetic field, and Josephson junctions. The Hilbert spaces associated with CV systems are infinite dimensional. 

QIP based on DV and CV systems are called DV and CV QIP, respectively.  

Quantized electromagnetic field

Different systems are suitable for different quantum tasks, for instance, atomic or solid-state based CV systems are promising candidates for quantum computation.  Similarly, in the case of quantum communication, quantized electromagnetic field  is a feasible option.  The classical electromagnetic field is described as a collection of modes, which are solutions to the Maxwell equations and upon quantization each mode behaves like a harmonic oscillator.

The quadrature amplitudes are the relevant degrees of freedom of the quantized electromagnetic field, which are mathematically equivalent to the position and momentum variables of the harmonic oscillator.  

Gaussian states, operations, and measurements

Most of the studies in CV systems have revolved around Gaussian states, for instance, coherent states, thermal states, and squeezed states.  Similarly, most of the research is based on Gaussian operations, which are described by Hamiltonians  up to second order in the quadrature operators.  Examples include displacement operations, phase change operations, single-mode squeezers, and two-mode beam splitters. These Gaussian operations transform a Gaussian state into another Gaussian state. Similarly, Gaussian measurements such as homodyne and heterodyne measurement schemes are employed in CV systems. 

This enhanced interest in the Gaussian states and the Gaussian operations is due to their elegant mathematical framework and ease in experimental implementations.  From the mathematical point of view, the effect of Gaussian operations on Gaussian states is particularly easy as Gaussian states are completely specified by the first and second moments of the quadrature operators. Further, Gaussian states are easy to prepare as for any physical system with quadratic Hamiltonian, the ground and thermal states are Gaussian.

Non-Gaussian states generated via photon number measurements

There are certain tasks, where non-Gaussian operations are required such as quantum entanglement distillation, quantum error correction and universal quantum computation. Similarly, Gaussian measurements based Bell inequalities can detect nonlocality only in non-Gaussian states. Further, several investigations have found that non-Gaussian operations, like photon addition, photon subtraction, and photon catalysis enhance nonclassicality as well as entanglement content. These non-Gaussian states have been shown to enhance the performance of various quantum protocols including quantum teleportation, quantum metrology and quantum key distribution. Therefore, the study of non-Gaussian states, which has gained relevance recently is important.

Practical scheme for the implementation of various non-Gaussian operations on a single mode system

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