Oktay Pashaev | İZMİR INSTITUTE OF TECHNOLOGY (original) (raw)

Papers by Oktay Pashaev

Теоретическая и математическая физика, 2007

Springer Proceedings in Mathematics & Statistics

By introducing the hierarchy of Fibonacci divisors and corresponding quantum derivatives, we deve... more By introducing the hierarchy of Fibonacci divisors and corresponding quantum derivatives, we develop the golden calculus, hierarchy of golden binomials and related exponential functions, translation operator and infinite hierarchy of Golden analytic functions. The hierarchy of Golden periodic functions, appearing in this calculus we relate with the method of images in planar hydrodynamics for incompressible and irrotational flow in bounded domain. We show that the even hierarchy of these functions determine the flow in the annular domain, bounded by concentric circles with the ratio of radiuses in powers of the Golden ratio. As an example, complex potential and velocity field for the set of point vortices with Golden proportion of images are calculated explicitly.

The reaction-di#usion system realizing a particular gauge fixing condition of the Jackiw--Teitelb... more The reaction-di#usion system realizing a particular gauge fixing condition of the Jackiw--Teitelboim gravity is represented as a coupled pair of Burgers equations with positive and negative viscosity. For acoustic metric in the Madelung fluid representation the space-time points where dispersion change the sign correspond to the event horizon, while shock soliton solutions to the black holes, creating under collision the resonance states. 1

Journal of Physics: Conference Series, 2016

From two circle theorem described in terms of q-periodic functions, in the limit q → 1 we have de... more From two circle theorem described in terms of q-periodic functions, in the limit q → 1 we have derived the strip theorem and the stream function for N vortex problem. For regular N-vortex polygon we find compact expression for the velocity of uniform rotation and show that it represents a nonlinear oscillator. We describe q-dispersive extensions of the linear and nonlinear Schrödinger equations, as well as the q-semiclassical expansions in terms of Bernoulli and Euler polynomials. Different kind of q-analytic functions are introduced, including the pq-analytic and the golden analytic functions.

Journal of Physics: Conference Series, 2016

We extend the concept of q-analytic function in two different directions. First we find expansion... more We extend the concept of q-analytic function in two different directions. First we find expansion of q-binomial in terms of q-Hermite polynomials, analytic in two complex arguments. Based on this representation, we introduce a new class of complex functions of two complex arguments, which we call the double q-analytic functions. As another direction, by the hyperbolic version of q-analytic functions we describe the q-analogue of traveling waves, which is not preserving the shape during evolution. The IVP for corresponding q-wave equation we solved in the q-D'Alembert form.

New Journal of Physics, 2012

Motivated by the Möbius transformation for symmetric points under the generalized circle in the c... more Motivated by the Möbius transformation for symmetric points under the generalized circle in the complex plane, the system of symmetric spin coherent states corresponding to antipodal qubit states is introduced. In terms of these states, we construct the maximally entangled complete set of two-qubit coherent states, which in the limiting cases reduces to the Bell basis. A specific property of our symmetric coherent states is that they never become unentangled for any value of ψ from the complex plane. Entanglement quantifications of our states are given by the reduced density matrix and the concurrence determinant, and it is shown that our basis is maximally entangled. Universal one-and twoqubit gates in these new coherent state basis are calculated. As an application, we find the Q symbol of the X Y Z model Hamiltonian operator H as an average energy function in maximally entangled two-and three-qubit phase space. It shows regular finite-energy localized structure with specific local extremum points. The concurrence and fidelity of quantum evolution with dimerization of double periodic patterns are given.

Annals of Physics, 1996

In pseudo-Euclidean metrics the Chern Simons gauge theory in the infrared region is found to be a... more In pseudo-Euclidean metrics the Chern Simons gauge theory in the infrared region is found to be associated with dissipative dynamics. In the infrared limit the Lagrangian of (2+1)dimensional pseudo-euclidean topologically massive electrodynamics has indeed the same form as the Lagrangian of the damped harmonic oscillator. On the hyperbolic plane a set of two damped harmonic oscillators, each time-reversed from the other, is shown to be equivalent to a single undamped harmonic oscillator. The equations for the damped oscillators are proven to be the same as the ones for the Lorentz force acting on two particles carrying opposite charge in a constant magnetic field and in the electric harmonic potential. This provides an immediate link with Chern Simons-like dynamics of Bloch electrons in solids propagating along the lattice plane with a hyperbolic energy surface. The symplectic structure of the reduced theory is finally discussed in the Dirac constrained canonical formalism and in the Faddeev Jackiw symplectic formalism.

ii ACKNOWLEDGEMENTS First of all, I particularly would like to thank my advisor, Assist. Prof. Dr... more ii ACKNOWLEDGEMENTS First of all, I particularly would like to thank my advisor, Assist. Prof. Dr. Bedrettin SUBA ILAR and co-adviser Assoc. Prof. Dr. Orhan ÖZTÜRK, for their continuous guidance, support and vision that they never hesitated to share, without which this study would not become a reality. I also owe a great deal of thanks to Prof. Dr. Gürbüz ATAGÜNDÜZ and Prof. Dr.Ali GÜNGÖR, I have worked together with throughout the period of this exciting and wonderful study, for sharing their experience with me and for their never ending technical and personal support. I would like to take this opportunity to express my gratitude to Prof. Dr. Oktay PASHAEV and Assist. Prof. Dr. Gülden GÖKÇEN for agreeing to be in my thesis defense committee. I also would like to take this opportunity to thank Prof. Dr. Rasim

Springer Proceedings in Mathematics & Statistics

The Schrödinger cat states, constructed from Glauber coherent states and applied for description ... more The Schrödinger cat states, constructed from Glauber coherent states and applied for description of qubits are generalized to the kaleidoscope of coherent states, related with regular n-polygon symmetry and the roots of unity. This quantum kaleidoscope is motivated by our method of classical hydrodynamics images in a wedge domain, described by q-calculus of analytic functions with q as a primitive root of unity. First we treat in detail the trinity states and the quartet states as descriptive for qutrit and ququat units of quantum information. Normalization formula for these states requires introduction of specific combinations of exponential functions with mod 3 and mod 4 symmetry, which are known also as generalized hyperbolic functions. We show that these states can be generated for an arbitrary n by the Quantum Fourier transform and can provide in general, qudit unit of quantum information. Relations of our states with quantum groups and quantum calculus are discussed.

Theoretical and Mathematical Physics

The problem of Hadamard quantum coin measurement in n trials, with arbitrary number of repeated c... more The problem of Hadamard quantum coin measurement in n trials, with arbitrary number of repeated consecutive last states is formulated in terms of Fibonacci sequences for duplicated states, Tribonacci numbers for triplicated states and N-Bonacci numbers for arbitrary N-plicated states. The probability formulas for arbitrary position of repeated states are derived in terms of Lucas and Fibonacci numbers. For generic qubit coin, the formulas are expressed by Fibonacci and more general, N-Bonacci polynomials in qubit probabilities. The generating function for probabilities, the Golden Ratio limit of these probabilities and Shannon entropy for corresponding states are determined. By generalized Born rule and universality of n-qubit measurement gate, we formulate problem in terms of generic n-qubit states and construct projection operators in Hilbert space, constrained on the Fibonacci tree of the states. The results are generalized to qutrit and qudit coins, described by generalized Fibonacci-N-Bonacci sequences.

arXiv: High Energy Physics - Theory, 1997

We establish the isomorphism between a nonlinear �-model and the abelian gauge theory on an arbit... more We establish the isomorphism between a nonlinear �-model and the abelian gauge theory on an arbitrary curved background, which allows us to derive integrable models and the corresponding Lax representations from gauge theoretical point of view. In our approach the spectral parameter is related to the global degree of freedom asso- ciated with the conformal or Galileo transformations of the spacetime. The Backlund transformations are derived from Chern-Simons theory where the spectral parameter is defined in terms of the additional compactified space dimension coordinate.

Theoretical and Mathematical Physics, 2021

The problem of Hadamard quantum coin measurement in n trials, with arbitrary number of repeated c... more The problem of Hadamard quantum coin measurement in n trials, with arbitrary number of repeated consecutive last states is formulated in terms of Fibonacci sequences for duplicated states, Tribonacci numbers for triplicated states and N -Bonacci numbers for arbitrary N -plicated states. The probability formulas for arbitrary position of repeated states are derived in terms of Lucas and Fibonacci numbers. For generic qubit coin, the formulas are expressed by Fibonacci and more general, N -Bonacci polynomials in qubit probabilities. The generating function for probabilities, the Golden Ratio limit of these probabilities and Shannon entropy for corresponding states are determined. By generalized Born rule and universality of n-qubit measurement gate, we formulate problem in terms of generic n-qubit states and construct projection operators in Hilbert space, constrained on the Fibonacci tree of the states. The results are generalized to qutrit and qudit coins, described by generalized F...

Abstract. We present several ideas in direction of physical interpretation of qand f -oscillators... more Abstract. We present several ideas in direction of physical interpretation of qand f -oscillators as a nonlinear oscillators. First we show that an arbitrary one dimensional integrable system in action-angle variables can be naturally represented as a classical and quantum f -oscillator. As an example, the semi-relativistic oscillator as a descriptive of the Landau levels for relativistic electron in magnetic field is solved as an f -oscillator. By using dispersion relation for q-oscillator we solve the linear qSchrödinger equation and corresponding nonlinear complex q-Burgers equation. The same dispersion allows us to construct integrable q-NLS model as a deformation of cubic NLS in terms of recursion operator of NLS hierarchy. Peculiar property of the model is to be completely integrable at any order of expansion in deformation parameter around q = 1. As another variation on the theme, we consider hydrodynamic flow in bounded domain. For the flow bounded by two concentric circles ...

concurrence determinant methods, it is shown that our basis is maximally entangled. In addition w... more concurrence determinant methods, it is shown that our basis is maximally entangled. In addition we find that the average of spin operators in these states vanish, as it must be according to another, operational definition of completely entangled states. Universal one qubit and two qubit gates in this new basis are calculated and time evolution of these states for some spin systems is derived. We find that the average energy for XYZ model in two qubit case (Q symbol of H) shows regular finite energy localized structure with characteristic extremum points, and appears as a soliton in maximally entangled two qubit phase space. Generalizations to three and higher qubit states are discussed.

New Broer-Kaup type systems of hydrodynamic equations are derived from the derivative reaction-di... more New Broer-Kaup type systems of hydrodynamic equations are derived from the derivative reaction-diffusion systems arising in SL(2,R) Kaup-Newell hierarchy, represented in the non-Madelung hydrodynamic form. A relation with the problem of chiral solitons in quantum potential as a dimensional reduction of 2+1 dimensional Chern-Simons theory for anyons is shown. By the Hirota bilinear method, soliton solutions are constructed and the resonant character of soliton interaction is found.

The Schrodinger cat states, constructed from Glauber coherent states and applied for description ... more The Schrodinger cat states, constructed from Glauber coherent states and applied for description of qubits are generalized to the kaleidoscope of coherent states, related with regular n-polygon symmetry and the roots of unity. This quantum kaleidoscope is motivated by our method of classical hydrodynamics images in a wedge domain, described by q-calculus of analytic functions with q as a primitive root of unity. First we treat in detail the trinity states and the quartet states as descriptive for qutrit and ququat units of quantum information. Normalization formula for these states requires introduction of specific combinations of exponential functions with mod 3 and mod 4 symmetry, which are known also as generalized hyperbolic functions. We show that these states can be generated for an arbitrary n by the Quantum Fourier transform and can provide in general, qudit unit of quantum information. Relations of our states with quantum groups and quantum calculus are discussed.

Journal of Physics: Conference Series

Journal of Physics: Conference Series

A representation of one qubit state by points in complex plane is proposed, such that the computa... more A representation of one qubit state by points in complex plane is proposed, such that the computational basis corresponds to two fixed points at a finite distance in the plane. These points represent common symmetric states for the set of quantum states on Apollonius circles. It is shown that, the Shannon entropy of one qubit state depends on ratio of probabilities and is a constant along Apollonius circles. For two qubit state and for three qubit state in Apollonius representation, the concurrence for entanglement and the Cayley hyperdeterminant for tritanglement correspondingly, are constant on the circles as well. Similar results are obtained also for n-tangle hyperdeterminant with even number of qubit states. It turns out that, for arbitrary multiple qubit state in Apollonius representation, fidelity between symmetric qubit states is also constant on Apollonius circles. According to these, the Apollonius circles are interpreted as integral curves for entanglement characteristics. The bipolar and the Cassini representations for qubit state are introduced, and their relations with qubit coherent states are established. We proposed the differential geometry for qubit states in Apollonius representation, defined by the metric on a surface in conformal coordinates, as square of the concurrence. The surfaces of the concurrence, as surfaces of revolution in Euclidean and Minkowski spaces are constructed. It is shown that, curves on these surfaces with constant Gaussian curvature becomes Cassini curves.

Journal of Physics: Conference Series

The set of mod n functions associated with primitive roots of unity and discrete Fourier transfor... more The set of mod n functions associated with primitive roots of unity and discrete Fourier transform is introduced. These functions naturally appear in description of superposition of coherent states related with regular polygon, which we call kaleidoscope of quantum coherent states. Displacement operators for kaleidoscope states are obtained by mod n exponential functions with operator argument and non-commutative addition formulas. Normalization constants, average number of photons, Heinsenberg uncertainty relations and coordinate representation of wave functions with mod n symmetry are expressed in a compact form by these functions.

Теоретическая и математическая физика, 2007

Springer Proceedings in Mathematics & Statistics

By introducing the hierarchy of Fibonacci divisors and corresponding quantum derivatives, we deve... more By introducing the hierarchy of Fibonacci divisors and corresponding quantum derivatives, we develop the golden calculus, hierarchy of golden binomials and related exponential functions, translation operator and infinite hierarchy of Golden analytic functions. The hierarchy of Golden periodic functions, appearing in this calculus we relate with the method of images in planar hydrodynamics for incompressible and irrotational flow in bounded domain. We show that the even hierarchy of these functions determine the flow in the annular domain, bounded by concentric circles with the ratio of radiuses in powers of the Golden ratio. As an example, complex potential and velocity field for the set of point vortices with Golden proportion of images are calculated explicitly.

The reaction-di#usion system realizing a particular gauge fixing condition of the Jackiw--Teitelb... more The reaction-di#usion system realizing a particular gauge fixing condition of the Jackiw--Teitelboim gravity is represented as a coupled pair of Burgers equations with positive and negative viscosity. For acoustic metric in the Madelung fluid representation the space-time points where dispersion change the sign correspond to the event horizon, while shock soliton solutions to the black holes, creating under collision the resonance states. 1

Journal of Physics: Conference Series, 2016

From two circle theorem described in terms of q-periodic functions, in the limit q → 1 we have de... more From two circle theorem described in terms of q-periodic functions, in the limit q → 1 we have derived the strip theorem and the stream function for N vortex problem. For regular N-vortex polygon we find compact expression for the velocity of uniform rotation and show that it represents a nonlinear oscillator. We describe q-dispersive extensions of the linear and nonlinear Schrödinger equations, as well as the q-semiclassical expansions in terms of Bernoulli and Euler polynomials. Different kind of q-analytic functions are introduced, including the pq-analytic and the golden analytic functions.

Journal of Physics: Conference Series, 2016

We extend the concept of q-analytic function in two different directions. First we find expansion... more We extend the concept of q-analytic function in two different directions. First we find expansion of q-binomial in terms of q-Hermite polynomials, analytic in two complex arguments. Based on this representation, we introduce a new class of complex functions of two complex arguments, which we call the double q-analytic functions. As another direction, by the hyperbolic version of q-analytic functions we describe the q-analogue of traveling waves, which is not preserving the shape during evolution. The IVP for corresponding q-wave equation we solved in the q-D'Alembert form.

New Journal of Physics, 2012

Motivated by the Möbius transformation for symmetric points under the generalized circle in the c... more Motivated by the Möbius transformation for symmetric points under the generalized circle in the complex plane, the system of symmetric spin coherent states corresponding to antipodal qubit states is introduced. In terms of these states, we construct the maximally entangled complete set of two-qubit coherent states, which in the limiting cases reduces to the Bell basis. A specific property of our symmetric coherent states is that they never become unentangled for any value of ψ from the complex plane. Entanglement quantifications of our states are given by the reduced density matrix and the concurrence determinant, and it is shown that our basis is maximally entangled. Universal one-and twoqubit gates in these new coherent state basis are calculated. As an application, we find the Q symbol of the X Y Z model Hamiltonian operator H as an average energy function in maximally entangled two-and three-qubit phase space. It shows regular finite-energy localized structure with specific local extremum points. The concurrence and fidelity of quantum evolution with dimerization of double periodic patterns are given.

Annals of Physics, 1996

In pseudo-Euclidean metrics the Chern Simons gauge theory in the infrared region is found to be a... more In pseudo-Euclidean metrics the Chern Simons gauge theory in the infrared region is found to be associated with dissipative dynamics. In the infrared limit the Lagrangian of (2+1)dimensional pseudo-euclidean topologically massive electrodynamics has indeed the same form as the Lagrangian of the damped harmonic oscillator. On the hyperbolic plane a set of two damped harmonic oscillators, each time-reversed from the other, is shown to be equivalent to a single undamped harmonic oscillator. The equations for the damped oscillators are proven to be the same as the ones for the Lorentz force acting on two particles carrying opposite charge in a constant magnetic field and in the electric harmonic potential. This provides an immediate link with Chern Simons-like dynamics of Bloch electrons in solids propagating along the lattice plane with a hyperbolic energy surface. The symplectic structure of the reduced theory is finally discussed in the Dirac constrained canonical formalism and in the Faddeev Jackiw symplectic formalism.

ii ACKNOWLEDGEMENTS First of all, I particularly would like to thank my advisor, Assist. Prof. Dr... more ii ACKNOWLEDGEMENTS First of all, I particularly would like to thank my advisor, Assist. Prof. Dr. Bedrettin SUBA ILAR and co-adviser Assoc. Prof. Dr. Orhan ÖZTÜRK, for their continuous guidance, support and vision that they never hesitated to share, without which this study would not become a reality. I also owe a great deal of thanks to Prof. Dr. Gürbüz ATAGÜNDÜZ and Prof. Dr.Ali GÜNGÖR, I have worked together with throughout the period of this exciting and wonderful study, for sharing their experience with me and for their never ending technical and personal support. I would like to take this opportunity to express my gratitude to Prof. Dr. Oktay PASHAEV and Assist. Prof. Dr. Gülden GÖKÇEN for agreeing to be in my thesis defense committee. I also would like to take this opportunity to thank Prof. Dr. Rasim

Springer Proceedings in Mathematics & Statistics

The Schrödinger cat states, constructed from Glauber coherent states and applied for description ... more The Schrödinger cat states, constructed from Glauber coherent states and applied for description of qubits are generalized to the kaleidoscope of coherent states, related with regular n-polygon symmetry and the roots of unity. This quantum kaleidoscope is motivated by our method of classical hydrodynamics images in a wedge domain, described by q-calculus of analytic functions with q as a primitive root of unity. First we treat in detail the trinity states and the quartet states as descriptive for qutrit and ququat units of quantum information. Normalization formula for these states requires introduction of specific combinations of exponential functions with mod 3 and mod 4 symmetry, which are known also as generalized hyperbolic functions. We show that these states can be generated for an arbitrary n by the Quantum Fourier transform and can provide in general, qudit unit of quantum information. Relations of our states with quantum groups and quantum calculus are discussed.

Theoretical and Mathematical Physics

The problem of Hadamard quantum coin measurement in n trials, with arbitrary number of repeated c... more The problem of Hadamard quantum coin measurement in n trials, with arbitrary number of repeated consecutive last states is formulated in terms of Fibonacci sequences for duplicated states, Tribonacci numbers for triplicated states and N-Bonacci numbers for arbitrary N-plicated states. The probability formulas for arbitrary position of repeated states are derived in terms of Lucas and Fibonacci numbers. For generic qubit coin, the formulas are expressed by Fibonacci and more general, N-Bonacci polynomials in qubit probabilities. The generating function for probabilities, the Golden Ratio limit of these probabilities and Shannon entropy for corresponding states are determined. By generalized Born rule and universality of n-qubit measurement gate, we formulate problem in terms of generic n-qubit states and construct projection operators in Hilbert space, constrained on the Fibonacci tree of the states. The results are generalized to qutrit and qudit coins, described by generalized Fibonacci-N-Bonacci sequences.

arXiv: High Energy Physics - Theory, 1997

We establish the isomorphism between a nonlinear �-model and the abelian gauge theory on an arbit... more We establish the isomorphism between a nonlinear �-model and the abelian gauge theory on an arbitrary curved background, which allows us to derive integrable models and the corresponding Lax representations from gauge theoretical point of view. In our approach the spectral parameter is related to the global degree of freedom asso- ciated with the conformal or Galileo transformations of the spacetime. The Backlund transformations are derived from Chern-Simons theory where the spectral parameter is defined in terms of the additional compactified space dimension coordinate.

Theoretical and Mathematical Physics, 2021

The problem of Hadamard quantum coin measurement in n trials, with arbitrary number of repeated c... more The problem of Hadamard quantum coin measurement in n trials, with arbitrary number of repeated consecutive last states is formulated in terms of Fibonacci sequences for duplicated states, Tribonacci numbers for triplicated states and N -Bonacci numbers for arbitrary N -plicated states. The probability formulas for arbitrary position of repeated states are derived in terms of Lucas and Fibonacci numbers. For generic qubit coin, the formulas are expressed by Fibonacci and more general, N -Bonacci polynomials in qubit probabilities. The generating function for probabilities, the Golden Ratio limit of these probabilities and Shannon entropy for corresponding states are determined. By generalized Born rule and universality of n-qubit measurement gate, we formulate problem in terms of generic n-qubit states and construct projection operators in Hilbert space, constrained on the Fibonacci tree of the states. The results are generalized to qutrit and qudit coins, described by generalized F...

Abstract. We present several ideas in direction of physical interpretation of qand f -oscillators... more Abstract. We present several ideas in direction of physical interpretation of qand f -oscillators as a nonlinear oscillators. First we show that an arbitrary one dimensional integrable system in action-angle variables can be naturally represented as a classical and quantum f -oscillator. As an example, the semi-relativistic oscillator as a descriptive of the Landau levels for relativistic electron in magnetic field is solved as an f -oscillator. By using dispersion relation for q-oscillator we solve the linear qSchrödinger equation and corresponding nonlinear complex q-Burgers equation. The same dispersion allows us to construct integrable q-NLS model as a deformation of cubic NLS in terms of recursion operator of NLS hierarchy. Peculiar property of the model is to be completely integrable at any order of expansion in deformation parameter around q = 1. As another variation on the theme, we consider hydrodynamic flow in bounded domain. For the flow bounded by two concentric circles ...

concurrence determinant methods, it is shown that our basis is maximally entangled. In addition w... more concurrence determinant methods, it is shown that our basis is maximally entangled. In addition we find that the average of spin operators in these states vanish, as it must be according to another, operational definition of completely entangled states. Universal one qubit and two qubit gates in this new basis are calculated and time evolution of these states for some spin systems is derived. We find that the average energy for XYZ model in two qubit case (Q symbol of H) shows regular finite energy localized structure with characteristic extremum points, and appears as a soliton in maximally entangled two qubit phase space. Generalizations to three and higher qubit states are discussed.

New Broer-Kaup type systems of hydrodynamic equations are derived from the derivative reaction-di... more New Broer-Kaup type systems of hydrodynamic equations are derived from the derivative reaction-diffusion systems arising in SL(2,R) Kaup-Newell hierarchy, represented in the non-Madelung hydrodynamic form. A relation with the problem of chiral solitons in quantum potential as a dimensional reduction of 2+1 dimensional Chern-Simons theory for anyons is shown. By the Hirota bilinear method, soliton solutions are constructed and the resonant character of soliton interaction is found.

The Schrodinger cat states, constructed from Glauber coherent states and applied for description ... more The Schrodinger cat states, constructed from Glauber coherent states and applied for description of qubits are generalized to the kaleidoscope of coherent states, related with regular n-polygon symmetry and the roots of unity. This quantum kaleidoscope is motivated by our method of classical hydrodynamics images in a wedge domain, described by q-calculus of analytic functions with q as a primitive root of unity. First we treat in detail the trinity states and the quartet states as descriptive for qutrit and ququat units of quantum information. Normalization formula for these states requires introduction of specific combinations of exponential functions with mod 3 and mod 4 symmetry, which are known also as generalized hyperbolic functions. We show that these states can be generated for an arbitrary n by the Quantum Fourier transform and can provide in general, qudit unit of quantum information. Relations of our states with quantum groups and quantum calculus are discussed.

Journal of Physics: Conference Series

Journal of Physics: Conference Series

A representation of one qubit state by points in complex plane is proposed, such that the computa... more A representation of one qubit state by points in complex plane is proposed, such that the computational basis corresponds to two fixed points at a finite distance in the plane. These points represent common symmetric states for the set of quantum states on Apollonius circles. It is shown that, the Shannon entropy of one qubit state depends on ratio of probabilities and is a constant along Apollonius circles. For two qubit state and for three qubit state in Apollonius representation, the concurrence for entanglement and the Cayley hyperdeterminant for tritanglement correspondingly, are constant on the circles as well. Similar results are obtained also for n-tangle hyperdeterminant with even number of qubit states. It turns out that, for arbitrary multiple qubit state in Apollonius representation, fidelity between symmetric qubit states is also constant on Apollonius circles. According to these, the Apollonius circles are interpreted as integral curves for entanglement characteristics. The bipolar and the Cassini representations for qubit state are introduced, and their relations with qubit coherent states are established. We proposed the differential geometry for qubit states in Apollonius representation, defined by the metric on a surface in conformal coordinates, as square of the concurrence. The surfaces of the concurrence, as surfaces of revolution in Euclidean and Minkowski spaces are constructed. It is shown that, curves on these surfaces with constant Gaussian curvature becomes Cassini curves.

Journal of Physics: Conference Series

The set of mod n functions associated with primitive roots of unity and discrete Fourier transfor... more The set of mod n functions associated with primitive roots of unity and discrete Fourier transform is introduced. These functions naturally appear in description of superposition of coherent states related with regular polygon, which we call kaleidoscope of quantum coherent states. Displacement operators for kaleidoscope states are obtained by mod n exponential functions with operator argument and non-commutative addition formulas. Normalization constants, average number of photons, Heinsenberg uncertainty relations and coordinate representation of wave functions with mod n symmetry are expressed in a compact form by these functions.