Quantum Decoherence of Gaussian Steering and Entanglement in Hawking Radiation and Thermal Bath


We study the effects of Hawking radiation and bath temperature on quantum steering and entanglement for a two-mode Gaussian state exposed in the background of a black hole and immersed in the two independent thermal baths. We find that both the effects can destroy the quantum steering and entanglement. Quantum steering always exists sudden death for any Hawking temperature and any bath temperature, but entanglement does not in zero-temperature thermal bath. Both the Hawking radiation and the asymmetry of thermal baths can induce the asymmetry of quantum steering, but the latter effect is much weaker than the former. An unintuitive result is that the observer who stays in the Hawking radiation or in the thermal bath with higher temperature has more stronger steerability than the other one. We also find that Hawking radiation and thermal noise can change the asymptotic behavior of steering and entanglement versus the squeezing parameter.

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This work is supported by the National Natural Science Foundation of China (Grant No. 11275064), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20124306110003), and the Construct Program of the National Key Discipline.

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Correspondence to Hao-Sheng Zeng.

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Here we present the solving process of master equation (16). We first consider the case of single mode bosonic field. Defining the covariance matrix

$$ \sigma(t) = \left( {\begin{array}{ll} \sigma_{xx}(t) & \sigma_{xp}(t) \\ \sigma_{px}(t) & \sigma_{pp}(t) \end{array}} \right), $$

for the quadrature operators \(\hat {x}\) and \(\hat {p}\) of the bosonic mode, we obtain from (16) the system of equations [51, 53, 54]

$$ \frac{\mathrm{d}\sigma (t)}{\mathrm{d}t}=W_{1}\sigma(t)+\sigma(t)W^{\mathrm{T}}_{1}+2D_{1}, $$


$$ W_{1} = \left( {\begin{array}{cc} -(\lambda_{1}-\mu_{1}) & 1 \\ -1 & -(\lambda_{1}+\mu_{1}) \end{array}} \right) $$

with dissipative constant λ1. The diffusion matrix D1 may be determined by the asymptotic condition, which leads to

$$ D_{1}=-\frac{1}{2}[W_{1}\sigma(\infty)+\sigma(\infty)W^{\mathrm{T}}_{1}]. $$

The solution of (23) can be written as

$$ \sigma(t)=X_{1}(t)\sigma(0)X^{\mathrm{T}}_{1}+Y_{1}(t), $$


$$ Y_{1}(t)=-X_{1}(t)\sigma(\infty)X^{\mathrm{T}}_{1}+\sigma(\infty). $$

Where the matrix \(X_{1}(t)=\exp (W_{1}t)\) fulfils the asymptotic condition \(\lim _{t\rightarrow \infty }X_{1}(t)=0\). In the underdamped case with ω1 > μ1, X1(t) is given by (19).

If the bosonic mode is finally in the thermal equilibrium at temperature T1, then the asymptotic state is described by the Gibbs state,

$$ \sigma(\infty) = \frac{1}{2}\coth\frac{1}{2T_{1}}\left( {\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}} \right). $$

For the two-mode Gaussian state (15) considered in the text, with each mode coupled to its own thermal bath, the evolution of the compound system is then given by (18), and the compound Gaibbs state is given by (20).

In our model, T1 denotes the temperature of the thermal bath in the flat spacetime, and T2 denotes the temperature of the thermal bath outside the horizon of the black hole. The bath temperature T2 would be enhanced by the Hawking radiation compared with the flat spacetime. Formally, T2 is determined by the total number of photons \(\bar {n}_{tot}\) through the expression \(\bar {n}_{tot}=[e^{h\varOmega /kT_{2}}-1]^{-1}\), where Ω is the frequency of the bosonic mode, and \(\bar {n}_{tot}=\bar {n}_{the}+\bar {n}_{rad}\), with \(\bar {n}_{the}\) being the thermal photon number irrelevant to Hawking radiation and \(\bar {n}_{rad}\) the photon number of Hawking radiation.

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Wu, S., Zeng, H. Quantum Decoherence of Gaussian Steering and Entanglement in Hawking Radiation and Thermal Bath. Int J Theor Phys (2020) doi:10.1007/s10773-019-04372-5

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  • Gaussian steering
  • Gaussian entanglement
  • Hawking radiation
  • Thermal bath