Rajiah Simon - Academia.edu (original) (raw)

Papers by Rajiah Simon

Pramana, 1995

We present a utilitarian review of the family of matrix groups Sp(2n,), in a form suited to vario... more We present a utilitarian review of the family of matrix groups Sp(2n,), in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the socalled unitary metaplectic representation of Sp(2n,). Global decomposition theorems, interesting subgroups and their generators are described. Turning to n-mode quantum systems, we define and study their variance matrices in general states, the implications of the Heisenberg uncertainty principles, and develop a U (n)-invariant squeezing criterion. The particular properties of Wigner distributions and Gaussian pure state wavefunctions under Sp(2n,) action are delineated.

Journal of The Optical Society of America A-optics Image Science and Vision, 2004

It is shown that Hermite-Gaussian beams, Laguerre-Gaussian beams, and certain linear combinations... more It is shown that Hermite-Gaussian beams, Laguerre-Gaussian beams, and certain linear combinations thereof are the only finite-energy coherent beams that propagate, on free propagation, in a shape-invariant manner. All shape-invariant beams have Gouy phase of the universal c arctan(z/zR) form, with quantized values for the prefactor c. It is also shown that, as limiting cases, even two- and three-dimensional nondiffracting beams belong to this class when the Rayleigh distance goes to infinity. The results are deduced from the transport-of-intensity equations, by elementary means as well as by use of the Iwasawa decomposition. A pivotal role in the analysis is the finding that the only possible change in the phase front of a shape-invariant beam from one transverse plane to another is quadratic.

Since any measure of non-Gaussianity is necessarily an attempt at making a quantitative statement... more Since any measure of non-Gaussianity is necessarily an attempt at making a quantitative statement on the departure of the shape of the QQQ function from Gaussian, any good measure of non-Gaussianity should be invariant under transformations which do not alter the shape of the QQQ functions, namely displacements, passage through passive linear systems, and uniform scaling of all the phase space variables: Q(alpha)tolambda2nQ(lambdaalpha)Q(\alpha)\to \lambda^{2n}Q(\lambda \alpha)Q(alpha)tolambda2nQ(lambdaalpha). Our measure which meets this `shape criterion' is computed for a few families of states, and the results are contrasted with existing measures of non-Gaussianity. The shape criterion implies, in particular, that the non-Gaussianity of the photon-added thermal states should be independent of temperature.

Since any measure of non-Gaussianity is necessarily an attempt at making a quantitative statement... more Since any measure of non-Gaussianity is necessarily an attempt at making a quantitative statement on the departure of the shape of the QQQ function from Gaussian, any good measure of non-Gaussianity should be invariant under transformations which do not alter the shape of the QQQ functions, namely displacements, passage through passive linear systems, and uniform scaling of all the phase space variables: Q(alpha)tolambda2nQ(lambdaalpha)Q(\alpha)\to \lambda^{2n}Q(\lambda \alpha)Q(alpha)tolambda2nQ(lambdaalpha). Our measure which meets this `shape criterion' is computed for a few families of states, and the results are contrasted with existing measures of non-Gaussianity. The shape criterion implies, in particular, that the non-Gaussianity of the photon-added thermal states should be independent of temperature.

Operator-sum or Kraus representations for single-mode Bosonic Gaussian channels are developed, an... more Operator-sum or Kraus representations for single-mode Bosonic Gaussian channels are developed, and several of their consequences explored. Kraus operators are employed to bring out the manner in which the unphysical matrix transposition map when accompanied by injection of a threshold classical noise becomes a physical channel. The action of the quantum-limited attenuator and amplifier channels as simply scaling maps on suitable quasi-probabilities in phase space is examined in the Kraus picture. Consideration of cumulants is used to examine the issue of fixed points. In the cases of entanglement-breaking channels a description in terms of rank one Kraus operators is shown to emerge quite simply. In contradistinction, it is shown that there is not even one finite rank operator in the entire linear span of Kraus operators of the quantum-limited amplifier or attenuator families, an assertion far stronger than the statement that these are not entanglement breaking channels. A characterization of extremality in terms of Kraus operators, originally due to Choi, is employed to show that all quantum-limited Gaussian channels are extremal. The fact that every noisy Gaussian channel can be realised as product of a pair of quantum-limited channels is used to construct a discrete set of linearly independent Kraus operators for noisy Gaussian channels, including the classical noise channel, and these Kraus operators have a particularly simple structure.

With a product state of the form rhoin=rhoaotimes∣0>bb<0∣\rho_{in} = \rho_a\otimes|0>_b_b< 0|rhoin=rhoaotimes∣0>bb<0∣ as input, the output two-... more With a product state of the form rhoin=rhoaotimes∣0>bb<0∣\rho_{in} = \rho_a\otimes|0>_b_b< 0|rhoin=rhoaotimes∣0>bb<0∣ as input, the output two-mode state rhormout\rho_{{\rm out}}rhormout, of the beam splitter is shown to be NPT whenever the photon number distribution (PND) statistics p(na)\{p(n_a) \}p(na) associated with the possibly mixed state rhoa\rho_arhoa of the a_mode is antibunched or otherwise nonclassical, i.e., if p(na)\{p(n_a)\}p(na) fails to respect any one of an infinite of classicality conditions.

The entanglement of formation (EOF) is computed for arbitrary two-mode Gaussian states. Apart fro... more The entanglement of formation (EOF) is computed for arbitrary two-mode Gaussian states. Apart from a conjecture, our analysis rests on two main ingredients. The first is a four-parameter canonical form we develop for the covariance matrix, one of these parameters acting as a measure of EOF, and the second is a generalisation of the EPR correlation, used in the work of Giedke {\em et al} [Phys. Rev. Lett. {\bf 91}, 107901 (2003)], to noncommuting variables. The conjecture itself is in respect of an extremal property of this generalized EPR correlation.

Every dXd bipartite system is shown to have a large family of undistillable states with nonpositi... more Every dXd bipartite system is shown to have a large family of undistillable states with nonpositive partial transpose (NPPT). This family subsumes the family of conjectured NPPT bound entangled Werner states. In particular, all one-copy undistillable NPPT Werner states are shown to be bound entangled.

We exploit results from the classical Stieltjes moment problem to bring out the totality of all t... more We exploit results from the classical Stieltjes moment problem to bring out the totality of all the information regarding phase insensitive nonclassicality of a state as captured by the photon number distribution pn. Central to our approach is the realization that n! pn constitutes the sequence of moments of a (quasi) probability distribution, notwithstanding the fact that pn can by itself be regarded as a probability distribution. This leads to classicality restrictions on pn that are local in n involving pn's for only a small number of consecutive n's, enabling a critical examination of the conjecture that oscillation in pn is a signature of nonclassicality.

Resonance, 2011

We describe the connection between continuous symmetries and conservation laws in classical mecha... more We describe the connection between continuous symmetries and conservation laws in classical mechanics. This is done at successively more sophisticated levels, bringing out important features at each level: the Newtonian, the Euler-Lagrange, and the Hamiltonian phase-space forms of mechanics. The role of the Action Principle is emphasised, and many examples are given.

A new operator based condition for distinguishing classical from non-classical states of quantise... more A new operator based condition for distinguishing classical from non-classical states of quantised radiation is developed. It exploits the fact that the normal ordering rule of correspondence to go from classical to quantum dynamical variables does not in general maintain positivity. It is shown that the approach naturally leads to distinguishing several layers of increasing nonclassicality, with more layers as the number of modes increases. A generalisation of the notion of subpoissonian statistics for two-mode radiation fields is achieved by analysing completely all correlations and fluctuations in quadratic combinations of mode annihilation and creation operators conserving the total photon number. This generalisation is nontrivial and intrinsically two-mode as it goes beyond all possible single mode projections of the two-mode field. The nonclassicality of pair coherent states, squeezed vacuum and squeezed thermal states is analysed and contrasted with one another, comparing the generalised subpoissonian statistics with extant signatures of nonclassical behaviour.

We discuss questions pertaining to the definition of 'momentum', 'momentum space', 'phase space' ... more We discuss questions pertaining to the definition of 'momentum', 'momentum space', 'phase space' and 'Wigner distributions'; for finite dimensional quantum systems. For such systems, where traditional concepts of 'momenta' established for continuum situations offer little help, we propose a physically reasonable and mathematically tangible definition and use it for the purpose of setting up Wigner distributions in a purely algebraic manner. It is found that the point of view adopted here is limited to odd dimensional systems only. The mathematical reasons which force this situation are examined in detail.

We exploit classical results on the Stieltjes moment problem to obtain completely explicit necess... more We exploit classical results on the Stieltjes moment problem to obtain completely explicit necessary and sufficient conditions for the photon number distribution of a radiation field mode to be classical. These conditions are given in two formsrespectively local and global in the individual photon number probabilities. Equivalence of the two approaches is demonstrated. Detailed quantitative statements on oscillations in the photon number probabilities are also presented.

Nonclassicality in the sense of quantum optics is a prerequisite for entanglement in multi-mode r... more Nonclassicality in the sense of quantum optics is a prerequisite for entanglement in multi-mode radiation states. In this work we bring out the possibilities of passing from the former to the latter, via action of classicality preserving systems like beamsplitters, in a transparent manner. For single mode states, a complete description of nonclassicality is available via the classical theory of moments, as a set of necessary and sufficient conditions on the photon number distribution. We show that when the mode is coupled to an ancilla in any coherent state, and the system is then acted upon by a beamsplitter, these conditions turn exactly into signatures of NPT entanglement of the output state. Since the classical moment problem does not generalize to two or more modes, we turn in these cases to other familiar sufficient but not necessary conditions for nonclassicality, namely the Mandel parameter criterion and its extensions. We generalize the Mandel matrix from one-mode states to the two-mode situation, leading to a natural classification of states with varying levels of nonclassicality. For two--mode states we present a single test that can, if successful, simultaneously show nonclassicality as well as NPT entanglement. We also develop a test for NPT entanglement after beamsplitter action on a nonclassical state, tracing carefully the way in which it goes beyond the Mandel nonclassicality test. The result of three--mode beamsplitter action after coupling to an ancilla in the ground state is treated in the same spirit. The concept of genuine tripartite entanglement, and scalar measures of nonclassicality at the Mandel level for two-mode systems, are discussed. Numerous examples illustrating all these concepts are presented.

Resonance, 2011

In Part 1 of this two-part article we have spelt out, in some detail, the link between symmetries... more In Part 1 of this two-part article we have spelt out, in some detail, the link between symmetries and conservation principles in the Lagrangian and Hamiltonian formulations of classical mechanics (CM). In this second part, we turn our attention to the corresponding question in quantum mechanics (QM). The generalization we embark upon will proceed in two directions: from the classical formulation to the quantum mechanical one, and from a single (infinitesimal) symmetry to a multi-dimensional Lie group of symmetries. Of course, we always have some definite physical system in mind. We also assume that the reader is familiar with the elements of quantum mechanics at the level of a standard first course on the subject. Operators will be denoted with an overhead caret, e.g., \( \hat A,\hat G,\hat U \) , etc., while \( [\hat A,\hat B] = \hat A\hat B - \hat B\hat A \) is the commutator of \( \hat A \) and \( \hat B \) .

Applications to several examples involving SU(2),SU(2),SU(2), SU(3),SU(3),SU(3), and the Heisenberg-Weyl group are pr... more Applications to several examples involving SU(2),SU(2),SU(2), SU(3),SU(3),SU(3), and the Heisenberg-Weyl group are presented, showing that there are simple examples of generalized coherent states which do not meet these conditions. Our results are relevant for phase-space description of quantum mechanics and quantum state reconstruction problems.

We present a new class of entangled bipartite states. These remain positive under partial transpo... more We present a new class of entangled bipartite states. These remain positive under partial transposition, and hence live outside the jurisdiction of the Peres-Horodecki separability criterion. Their inseparability is demonstrated using Choi-type indecomposable positive maps. It is shown that these states are not isolated, but occupy a finite volume in the bipartite state space. It is further shown that in the space of positive maps there exists a neighborhood around the Choi map with the property that every map in this neighborhood is indecomposable.

Pramana, 1995

We present a utilitarian review of the family of matrix groups Sp(2n,), in a form suited to vario... more We present a utilitarian review of the family of matrix groups Sp(2n,), in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the socalled unitary metaplectic representation of Sp(2n,). Global decomposition theorems, interesting subgroups and their generators are described. Turning to n-mode quantum systems, we define and study their variance matrices in general states, the implications of the Heisenberg uncertainty principles, and develop a U (n)-invariant squeezing criterion. The particular properties of Wigner distributions and Gaussian pure state wavefunctions under Sp(2n,) action are delineated.

Journal of The Optical Society of America A-optics Image Science and Vision, 2004

It is shown that Hermite-Gaussian beams, Laguerre-Gaussian beams, and certain linear combinations... more It is shown that Hermite-Gaussian beams, Laguerre-Gaussian beams, and certain linear combinations thereof are the only finite-energy coherent beams that propagate, on free propagation, in a shape-invariant manner. All shape-invariant beams have Gouy phase of the universal c arctan(z/zR) form, with quantized values for the prefactor c. It is also shown that, as limiting cases, even two- and three-dimensional nondiffracting beams belong to this class when the Rayleigh distance goes to infinity. The results are deduced from the transport-of-intensity equations, by elementary means as well as by use of the Iwasawa decomposition. A pivotal role in the analysis is the finding that the only possible change in the phase front of a shape-invariant beam from one transverse plane to another is quadratic.

Since any measure of non-Gaussianity is necessarily an attempt at making a quantitative statement... more Since any measure of non-Gaussianity is necessarily an attempt at making a quantitative statement on the departure of the shape of the QQQ function from Gaussian, any good measure of non-Gaussianity should be invariant under transformations which do not alter the shape of the QQQ functions, namely displacements, passage through passive linear systems, and uniform scaling of all the phase space variables: Q(alpha)tolambda2nQ(lambdaalpha)Q(\alpha)\to \lambda^{2n}Q(\lambda \alpha)Q(alpha)tolambda2nQ(lambdaalpha). Our measure which meets this `shape criterion' is computed for a few families of states, and the results are contrasted with existing measures of non-Gaussianity. The shape criterion implies, in particular, that the non-Gaussianity of the photon-added thermal states should be independent of temperature.

Since any measure of non-Gaussianity is necessarily an attempt at making a quantitative statement... more Since any measure of non-Gaussianity is necessarily an attempt at making a quantitative statement on the departure of the shape of the QQQ function from Gaussian, any good measure of non-Gaussianity should be invariant under transformations which do not alter the shape of the QQQ functions, namely displacements, passage through passive linear systems, and uniform scaling of all the phase space variables: Q(alpha)tolambda2nQ(lambdaalpha)Q(\alpha)\to \lambda^{2n}Q(\lambda \alpha)Q(alpha)tolambda2nQ(lambdaalpha). Our measure which meets this `shape criterion' is computed for a few families of states, and the results are contrasted with existing measures of non-Gaussianity. The shape criterion implies, in particular, that the non-Gaussianity of the photon-added thermal states should be independent of temperature.

Operator-sum or Kraus representations for single-mode Bosonic Gaussian channels are developed, an... more Operator-sum or Kraus representations for single-mode Bosonic Gaussian channels are developed, and several of their consequences explored. Kraus operators are employed to bring out the manner in which the unphysical matrix transposition map when accompanied by injection of a threshold classical noise becomes a physical channel. The action of the quantum-limited attenuator and amplifier channels as simply scaling maps on suitable quasi-probabilities in phase space is examined in the Kraus picture. Consideration of cumulants is used to examine the issue of fixed points. In the cases of entanglement-breaking channels a description in terms of rank one Kraus operators is shown to emerge quite simply. In contradistinction, it is shown that there is not even one finite rank operator in the entire linear span of Kraus operators of the quantum-limited amplifier or attenuator families, an assertion far stronger than the statement that these are not entanglement breaking channels. A characterization of extremality in terms of Kraus operators, originally due to Choi, is employed to show that all quantum-limited Gaussian channels are extremal. The fact that every noisy Gaussian channel can be realised as product of a pair of quantum-limited channels is used to construct a discrete set of linearly independent Kraus operators for noisy Gaussian channels, including the classical noise channel, and these Kraus operators have a particularly simple structure.

With a product state of the form rhoin=rhoaotimes∣0>bb<0∣\rho_{in} = \rho_a\otimes|0>_b_b< 0|rhoin=rhoaotimes∣0>bb<0∣ as input, the output two-... more With a product state of the form rhoin=rhoaotimes∣0>bb<0∣\rho_{in} = \rho_a\otimes|0>_b_b< 0|rhoin=rhoaotimes∣0>bb<0∣ as input, the output two-mode state rhormout\rho_{{\rm out}}rhormout, of the beam splitter is shown to be NPT whenever the photon number distribution (PND) statistics p(na)\{p(n_a) \}p(na) associated with the possibly mixed state rhoa\rho_arhoa of the a_mode is antibunched or otherwise nonclassical, i.e., if p(na)\{p(n_a)\}p(na) fails to respect any one of an infinite of classicality conditions.

The entanglement of formation (EOF) is computed for arbitrary two-mode Gaussian states. Apart fro... more The entanglement of formation (EOF) is computed for arbitrary two-mode Gaussian states. Apart from a conjecture, our analysis rests on two main ingredients. The first is a four-parameter canonical form we develop for the covariance matrix, one of these parameters acting as a measure of EOF, and the second is a generalisation of the EPR correlation, used in the work of Giedke {\em et al} [Phys. Rev. Lett. {\bf 91}, 107901 (2003)], to noncommuting variables. The conjecture itself is in respect of an extremal property of this generalized EPR correlation.

Every dXd bipartite system is shown to have a large family of undistillable states with nonpositi... more Every dXd bipartite system is shown to have a large family of undistillable states with nonpositive partial transpose (NPPT). This family subsumes the family of conjectured NPPT bound entangled Werner states. In particular, all one-copy undistillable NPPT Werner states are shown to be bound entangled.

We exploit results from the classical Stieltjes moment problem to bring out the totality of all t... more We exploit results from the classical Stieltjes moment problem to bring out the totality of all the information regarding phase insensitive nonclassicality of a state as captured by the photon number distribution pn. Central to our approach is the realization that n! pn constitutes the sequence of moments of a (quasi) probability distribution, notwithstanding the fact that pn can by itself be regarded as a probability distribution. This leads to classicality restrictions on pn that are local in n involving pn's for only a small number of consecutive n's, enabling a critical examination of the conjecture that oscillation in pn is a signature of nonclassicality.

Resonance, 2011

We describe the connection between continuous symmetries and conservation laws in classical mecha... more We describe the connection between continuous symmetries and conservation laws in classical mechanics. This is done at successively more sophisticated levels, bringing out important features at each level: the Newtonian, the Euler-Lagrange, and the Hamiltonian phase-space forms of mechanics. The role of the Action Principle is emphasised, and many examples are given.

A new operator based condition for distinguishing classical from non-classical states of quantise... more A new operator based condition for distinguishing classical from non-classical states of quantised radiation is developed. It exploits the fact that the normal ordering rule of correspondence to go from classical to quantum dynamical variables does not in general maintain positivity. It is shown that the approach naturally leads to distinguishing several layers of increasing nonclassicality, with more layers as the number of modes increases. A generalisation of the notion of subpoissonian statistics for two-mode radiation fields is achieved by analysing completely all correlations and fluctuations in quadratic combinations of mode annihilation and creation operators conserving the total photon number. This generalisation is nontrivial and intrinsically two-mode as it goes beyond all possible single mode projections of the two-mode field. The nonclassicality of pair coherent states, squeezed vacuum and squeezed thermal states is analysed and contrasted with one another, comparing the generalised subpoissonian statistics with extant signatures of nonclassical behaviour.

We discuss questions pertaining to the definition of 'momentum', 'momentum space', 'phase space' ... more We discuss questions pertaining to the definition of 'momentum', 'momentum space', 'phase space' and 'Wigner distributions'; for finite dimensional quantum systems. For such systems, where traditional concepts of 'momenta' established for continuum situations offer little help, we propose a physically reasonable and mathematically tangible definition and use it for the purpose of setting up Wigner distributions in a purely algebraic manner. It is found that the point of view adopted here is limited to odd dimensional systems only. The mathematical reasons which force this situation are examined in detail.

We exploit classical results on the Stieltjes moment problem to obtain completely explicit necess... more We exploit classical results on the Stieltjes moment problem to obtain completely explicit necessary and sufficient conditions for the photon number distribution of a radiation field mode to be classical. These conditions are given in two formsrespectively local and global in the individual photon number probabilities. Equivalence of the two approaches is demonstrated. Detailed quantitative statements on oscillations in the photon number probabilities are also presented.

Nonclassicality in the sense of quantum optics is a prerequisite for entanglement in multi-mode r... more Nonclassicality in the sense of quantum optics is a prerequisite for entanglement in multi-mode radiation states. In this work we bring out the possibilities of passing from the former to the latter, via action of classicality preserving systems like beamsplitters, in a transparent manner. For single mode states, a complete description of nonclassicality is available via the classical theory of moments, as a set of necessary and sufficient conditions on the photon number distribution. We show that when the mode is coupled to an ancilla in any coherent state, and the system is then acted upon by a beamsplitter, these conditions turn exactly into signatures of NPT entanglement of the output state. Since the classical moment problem does not generalize to two or more modes, we turn in these cases to other familiar sufficient but not necessary conditions for nonclassicality, namely the Mandel parameter criterion and its extensions. We generalize the Mandel matrix from one-mode states to the two-mode situation, leading to a natural classification of states with varying levels of nonclassicality. For two--mode states we present a single test that can, if successful, simultaneously show nonclassicality as well as NPT entanglement. We also develop a test for NPT entanglement after beamsplitter action on a nonclassical state, tracing carefully the way in which it goes beyond the Mandel nonclassicality test. The result of three--mode beamsplitter action after coupling to an ancilla in the ground state is treated in the same spirit. The concept of genuine tripartite entanglement, and scalar measures of nonclassicality at the Mandel level for two-mode systems, are discussed. Numerous examples illustrating all these concepts are presented.

Resonance, 2011

In Part 1 of this two-part article we have spelt out, in some detail, the link between symmetries... more In Part 1 of this two-part article we have spelt out, in some detail, the link between symmetries and conservation principles in the Lagrangian and Hamiltonian formulations of classical mechanics (CM). In this second part, we turn our attention to the corresponding question in quantum mechanics (QM). The generalization we embark upon will proceed in two directions: from the classical formulation to the quantum mechanical one, and from a single (infinitesimal) symmetry to a multi-dimensional Lie group of symmetries. Of course, we always have some definite physical system in mind. We also assume that the reader is familiar with the elements of quantum mechanics at the level of a standard first course on the subject. Operators will be denoted with an overhead caret, e.g., \( \hat A,\hat G,\hat U \) , etc., while \( [\hat A,\hat B] = \hat A\hat B - \hat B\hat A \) is the commutator of \( \hat A \) and \( \hat B \) .

Applications to several examples involving SU(2),SU(2),SU(2), SU(3),SU(3),SU(3), and the Heisenberg-Weyl group are pr... more Applications to several examples involving SU(2),SU(2),SU(2), SU(3),SU(3),SU(3), and the Heisenberg-Weyl group are presented, showing that there are simple examples of generalized coherent states which do not meet these conditions. Our results are relevant for phase-space description of quantum mechanics and quantum state reconstruction problems.

We present a new class of entangled bipartite states. These remain positive under partial transpo... more We present a new class of entangled bipartite states. These remain positive under partial transposition, and hence live outside the jurisdiction of the Peres-Horodecki separability criterion. Their inseparability is demonstrated using Choi-type indecomposable positive maps. It is shown that these states are not isolated, but occupy a finite volume in the bipartite state space. It is further shown that in the space of positive maps there exists a neighborhood around the Choi map with the property that every map in this neighborhood is indecomposable.