(a) Schematic of one step of the nonlinear quantum protocol. U and P denote the entangling two-qubit transformation and the projective measurement, respectively. (b) The convergence regions of the corresponding complex map f on the complex plane, where red (blue) color represents convergence to the asymptotic state |+ x (|−− x ), and the lighter the shading the more iterations are needed to reach the respective state. The white line represents the Julia set of the map.

Contexts in source publication

Context 1

... is to implement a measurement-induced nonlinear quantum transformation [8] on photonic qubits. This can be realized on one member of a pair of qubits, initially in the same quantum state, via a controlled two-qubit unitary transformation on the composite system and a subsequent post-selective measurement on the other member of the pair (shown in Fig. 1(a)). For the two qubits, we consider two two-level systems: one encoded by the polarizations {|H = |0 p , |V = |1 p } and the other by the spatial modes {|D = |0 s , |U = |1 s } of single photons. Note that the subscripts p and s refer to the two types of degrees of freedom, ...

Context 2

... fast. There is a set of points which do not converge to any of the attractive fixed points when iterating the map f and these form the so-called Julia set of the complex map (the third fixed point of the map z 3 = 0, which is repelling, is also a member of the Julia set). The Julia set of the map f is the imaginary axis on the complex plane (see Fig. 1(b)) or equivalently, the great circle which intersects the y axis on the Bloch sphere, while the two superattractive fixed points correspond to the orthogonal quantum states pointing in the +x and −x directions on the Bloch sphere, respectively. It can be seen in Fig. 1(b) that initial states which can be described by a complex number z ...

Context 3

... The Julia set of the map f is the imaginary axis on the complex plane (see Fig. 1(b)) or equivalently, the great circle which intersects the y axis on the Bloch sphere, while the two superattractive fixed points correspond to the orthogonal quantum states pointing in the +x and −x directions on the Bloch sphere, respectively. It can be seen in Fig. 1(b) that initial states which can be described by a complex number z that has a positive (negative) real part, all converge to the asymptotic state |+ x (|−− x ), as represented by the coloring. Initial states which lie closer to the border of these convergence regions (i.e., the Julia set) need more iterations to approach the respective ...

Context 4

... demonstrate that the nonlinear protocol effectively orthogonalizes initially close quantum states [31][32][33][34][35][36], the step presented in Fig. 1(a) has to be iterated, i.e., the initial state of the input qubits of the second step has to be equal to the output state |ψ 1 of the first step. In order to implement this, we use quantum state tomography to determine the output state after each step via a PBS, a QWP and a HWP with the setting angles θ M Q and θ M H , respectively, ...

Context 5

... is to implement a measurement-induced nonlinear quantum transformation [8] on photonic qubits. This can be realized on one member of a pair of qubits, initially in the same quantum state, via a controlled two-qubit unitary transformation on the composite system and a subsequent post-selective measurement on the other member of the pair (shown in Fig. 1(a)). For the two qubits, we consider two two-level systems: one encoded by the polarizations {|H = |0 p , |V = |1 p } and the other by the spatial modes {|D = |0 s , |U = |1 s } of single photons. Note that the subscripts p and s refer to the two types of degrees of freedom, ...

Context 6

... fast. There is a set of points which do not converge to any of the attractive fixed points when iterating the map f and these form the so-called Julia set of the complex map (the third fixed point of the map z 3 = 0, which is repelling, is also a member of the Julia set). The Julia set of the map f is the imaginary axis on the complex plane (see Fig. 1(b)) or equivalently, the great circle which intersects the y axis on the Bloch sphere, while the two superattractive fixed points correspond to the orthogonal quantum states pointing in the +x and −x directions on the Bloch sphere, respectively. It can be seen in Fig. 1(b) that initial states which can be described by a complex number z ...

Context 7

... The Julia set of the map f is the imaginary axis on the complex plane (see Fig. 1(b)) or equivalently, the great circle which intersects the y axis on the Bloch sphere, while the two superattractive fixed points correspond to the orthogonal quantum states pointing in the +x and −x directions on the Bloch sphere, respectively. It can be seen in Fig. 1(b) that initial states which can be described by a complex number z that has a positive (negative) real part, all converge to the asymptotic state |+ x (|−− x ), as represented by the coloring. Initial states which lie closer to the border of these convergence regions (i.e., the Julia set) need more iterations to approach the respective ...

Context 8

... demonstrate that the nonlinear protocol effectively orthogonalizes initially close quantum states [31][32][33][34][35][36], the step presented in Fig. 1(a) has to be iterated, i.e., the initial state of the input qubits of the second step has to be equal to the output state |ψ 1 of the first step. In order to implement this, we use quantum state tomography to determine the output state after each step via a PBS, a QWP and a HWP with the setting angles θ M Q and θ M H , respectively, ...