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arXiv:1909.09896 · Journal of Russian Laser Research

Superposition Principle for Qubit States in the Spin-Projection Mean Representation

Andrey Yu. Fedorov, Vladimir I. Man’ko

Published Jan 1, 2022• 1 citation
arXivHTMLPDF
DOI
10.1007/s10946-022-10025-6

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A non-abelian spin liquid in triangular lattice spin-1 systems was recently formulated in the form of continuum field theory [T. Grover, and T. Senthil, Phys. Rev. Lett. 107, 077203 (2011); Cenke Xu, A.W.W. Ludwig, arXiv:1012.5671]. It has spin-1/2 bosonic spinons coupled to emergent quaternion gauge fields, and can be obtained by quantum disordering a non-collinear spin nematic order hypothesized to describe NiGa_2S_4 [H. Tsunetsugu, and M. Arikawa, J. Phys. Soc. Jpn. 75, 083701 (2006)], However a microscopic lattice description, e.g. the lattice spinon (mean-field) Hamiltonian and the spin wavefunction, has been missing, and it has been noted that the standard Schwinger boson or bosonic triplon representations of spin-1 cannot describe this spin liquid. In this paper a two-orbital Schwinger boson representation for spin-1 systems is developed and used to construct a mean-field description of this quaternion spin liquid. Projecting the mean-field state produces a prototype wavefunction, which is a superposition of close-packed AKLT loop configurations with nontrivial amplitudes. This new formalism and related wavefunctions may be generalized to higher spin systems and can possibly produce spin liquid states with even richer emergent gauge structures.
Coherent photons have both spin and orbital angular momentum, and the superposition state among orthogonal states can describe a qubit. Here, we show the internal structure of coherent photons with spin and orbital angular momentum can be described by a representation theory of Lie algebra and Lie group. As an application of this theory, we have experimentally demonstrated the controlled-NOT (CNOT) operation, using a standard green laser diode. We have used beam splitters and waveplates to generate the macroscopic entangled light source with amplitudes and phases, controlled by rotating the waveplates. By applying the Bell projection to the entangled light, the far-field image of the light shows an expected dipole shape in the superposition state of left and right vortices, allowing to identify the state from the image. Then, we have used a polarisation dependent beam splitter to split the spin state, and we applied a NOT operation, made of a pair of cylindrical lenses, only for a vortex with a chosen polarisation, and recombined the beams after the operation. The proposed architecture could be used for a platform to manipulate qubits, made of macroscopically coherent photons.
The superposition states of two qubits including entangled Bell states are considered in the probability representation of quantum mechanics. The superposition principle formulated in terms of the nonlinear addition rule of the state density matrices is formulated as a nonlinear addition rule of the probability distributions describing the qubit states. The generalization of the entanglement properties to the case of superposition of two-mode oscillator states is discussed using the probability representation of quantum states.