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Physical Review A

Inseparability of photon-added Gaussian states

Hongrong Li, Fuli Li, Shi‐Yao Zhu

Published Jun 18, 2007• 9 citations
Publisher

Abstract

The inseparability of photon-added Gaussian states which are generated from two-mode Gaussian states by adding photons is investigated. According to the established inseparability conditions [New J. Phys. 7, 211 (2005); Phys. Rev. Lett. 96, 050503 (2006)], we find that even if a two-mode Gaussian state is separable, the photon-added Gaussian state becomes entangled when the purity of the Gaussian state is larger than a certain value. The lower bound of entanglement of symmetric photon-added Gaussian states is derived. The result shows that entanglement of the photon-added Gaussian states is involved with high-order moment correlations. We find that fidelity of teleporting coherent states cannot be raised by employing the photon-added Gaussian states as a quantum channel of teleportation.

DOI
10.1103/physreva.75.062318

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The inseparability of photon-added Gaussian states which are generated from two-mode Gaussian states by adding photons is investigated. According to the established inseparability conditions [New J. Phys. 7, 211 (2005); Phys. Rev. Lett. 96, 050503 (2006)], we find that even if a two-mode Gaussian state is separable, the photon-added Gaussian state becomes entangled when the purity of the Gaussian state is larger than a certain value. The lower bound of entanglement of symmetric photon-added Gaussian states is derived. The result shows that entanglement of the photon-added Gaussian states is involved with high-order moment correlations. We find that fidelity of teleporting coherent states cannot be raised by employing the photon-added Gaussian states as a quantum channel of teleportation.
Teleportation of a general two-mode Gaussian state through double two-mode squeezed state quantum channels is studied. It is found that the quantum channels must be enough entangled for the output state to be inseparable. The required minimum entanglement for the quantum channels to maintain entanglement in the output state depends on the entanglement of the input state when the input state is pure. The minimum entanglement depends not only the on entanglement but also on the global purity of the input state when the input state is mixed.
Quantum state engineering frequently employs non-Gaussian operations, particularly photon addition and subtraction, to tailor and improve the nonclassical characteristics of quantum states. We investigate how photon addition to a two-mode squeezed Fock state (TMSFS) influences its performance in the Braunstein-Kimble quantum teleportation protocol. We obtain the Wigner characteristic function of the photon-added TMSFS (PA-TMSFS) and evaluate the corresponding teleportation fidelities for both Gaussian and non-Gaussian input states. We further compare the teleportation performance of PA-TMSFS with that of its photon-subtracted counterpart by accessing their fidelity enhancements over the parent TMSFS and considering the probabilistic nature of both the operations. Our results demonstrate that photon subtraction provides superior teleportation performance compared with photon addition, with this advantage becoming more pronounced when the success probabilities of the corresponding non-Gaussian operations are considered. In particular, the symmetric (1,1)(1,1)(1,1) photon subtraction yields the optimal trade-off between teleportation fidelity and success probability among the configurations investigated. These findings highlight that photon subtraction retains its advantage even when the underlying entangled resource is already non-Gaussian.
The entanglement properties of non-Gaussian states are investigated, which are obtained by performing the photon addition, photon subtraction, photon-addition-then-subtraction, and photon-subtraction-then-addition operations on the two-mode squeezed vacuum state. We show that the partial von Neumann entropy of all the resulting states is greater than that of the original squeezed state, but only the photon-subtracted states and the photon-added-then-subtracted states have the stronger Einstein-Podolsky-Rosen correlation than the original squeezed state. Quantum teleportation of Braunstein and Kimble protocol is studied for coherent states, squeezed states, and mixed Gaussian states with the non-Gaussian entangled resources. For all the states to be teleported, the fidelity with the photon-subtracted and the photon-added-then-subtracted entangled resources is higher than that with the two-mode squeezed vacuum resource. Based on Bures fidelity, we find that quantum teleportation for mixed and classical single-mode Gaussian states is more faithful than for single-mode Gaussian states with high purity and nonclassicality.
Considering the propagation of a two-mode optical field that is initially in a squeezed vacuum state in a thermal environment, we obtain an intuitive expression for inseparability condition of the two-mode mixed state which is given in the coherent state representation. This condition shows that the two modes have quantum entanglement if and only if the coefficient of the correlation term between the two modes is larger than that of the off-diagonal term of each mode in the density matrix. We find that even if the quantum channel is dynamically coupled to the thermal environment, the fidelity for teleporting coherent states larger than 1/2 is still the criterion for quantum teleportation. We also show that the entanglement, squeezing and quantum teleportation conditions are always consistent with each other.
Entanglement swapping of two-mode Gaussian states is investigated by use of quantum teleportation protocol of continuous variables. In the entanglement swapping protocol, we take the two-mode squeezed vacuum which is exposed in a thermal environment as a mixed-quantum-channel state and a two-mode standard form Gaussian state as an input state to be teleported. When the input state is pure, we find that the swapped state is inseparable if and only if both the input state and the quantum-channel state are entangled. When the input state is mixed and entangled, we find that the swapped state is inseparable only if the quantum-channel state is sufficiently much entangled. We also notice that the required minimum entanglement in the mixed quantum channel to maintain entanglement in the swapped state strongly depends on the global purity of the input state.
We introduce photon-added entangled coherent states (unnormalized) : and investigate the entanglement by employing the entropy of entanglement. Then, we propose a scheme of probabilistic teleportation via photon-added entangled coherent states. It is shown that the mean fidelity of the scheme can be increased by exciting entangled coherent states in the case of nonmaximally entangled quantum channel. The violation of the Clauser?Horne?Shimony?Holt inequality for photon-added entangled coherent states is also examined.