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CRC Press eBooks

Two-Mode Schrödinger Cats, Entanglement and Teleportation

Takayoshi Kobayashi

Published Oct 15, 2025• 0 citations
Publisher

Abstract

Two-mode Schrödinger cats (SCs) consisting of two and four coherent-state terms are considered. It is shown that they are closely related to the quadrature-entangled Einstein, Podolsky, and Rosen pair. The properties of their wave functions are investigated from the aspects of entanglement. It is shown that at certain parameters the entangled properties of the two-mode SCs allow for good teleportation fidelity. A theoretical generalization to the teleportation of two qubits is also discussed.

DOI
10.1201/9781003618997-16

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Two-mode Schrödinger cats (SCs) consisting of two and four coherent-state terms are considered. It is shown that they are closely related to the quadrature-entangled Einstein, Podolsky, and Rosen pair. The properties of their wave functions are investigated from the aspects of entanglement. It is shown that at certain parameters the entangled properties of the two-mode SCs allow for good teleportation fidelity. A theoretical generalization to the teleportation of two qubits is also discussed.
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Quantum superpositions of coherent states are produced both in microwave and optical domains, and are considered realizations of the famous “Schroedinger cat” state. The recent progress shows an increase in the number of components and the number of modes involved. Our work gives a theoretical treatment of multicomponent two-mode Schroedinger cat states. We consider a class of single-mode states, which are superpositions of N coherent states lying on a circle in the phase space. In this class we consider an orthonormal basis created by rotationally-invariant circular states (RICS). A two-mode extension of this basis is created by splitting a single-mode RICS on a balanced beam-splitter. After performing a symmetric (Loewdin) orthogonalization of the sets of coherent states in both modes we obtain the Schmidt decomposition of the two-mode state, and therefore an analytic expression for its entanglement. We show that the states obtained by splitting a RICS are generalizations of Bell states of two qubits to the case of N -level systems encoded into superpositions of coherent states on the circle, and we propose for them the name of generalized quasi-Bell states. We show that an exact probabilistic teleportation of arbitrary superposition of coherent states on the circle is possible with such a state used as shared resource.
The fidelity of quantum teleportation of a single-mode squeezed state of light is calculated based on the general theory of quantum-mechanical measurement in the Schrödinger picture. It is shown that the criterion for the nonclassical state teleportation is different from that for coherent state. F = 1/2 is no longer the rigorous boundary between classical and quantum teleportation for a squeezed state of light. When the quantum entanglement of an Einstein–Podolsky–Rosen (EPR) beam used for teleportation and the parameters of the system are given, the fidelity depends on the squeezing of the input squeezed state. The higher the squeezing is, the smaller the fidelity is, and the lower the classical limitation of fidelity is. The dependence of the optimum gain for teleporting a squeezed vacuum state upon the EPR entanglement is also calculated. The results obtained provide important references for designing experimental systems of teleporting a non-classical state and judging the quality of the teleported quantum state.
We investigate the entanglement swapping of the continuous variable states by taking the pair Schrödinger cat states as the input entangled states, in which the two-mode squeezed vacuum and the pair cat states serve as the quantum channel, respectively. The entanglement of the initial states as well as the final states is analyzed by adopting the logarithmic negativity as the measure of entanglement. The quantum teleportation task by exploiting the swapped states as the quantum channel is also considered, where a coherent state serves as the target state and the average fidelity is examined.