Locally optimal control of continuous-variable entanglement

We consider a system of two bosonic modes each subject to the dynamics induced by a thermal Markovian environment and we identify instantaneous, local symplectic controls that minimize the loss of entanglement in the Gaussian regime. By minimizing the decrease of the logarithmic negativity at every instant in time, it will be shown that a nontrivial, finite amount of local squeezing helps to counter the effect of decoherence during the evolution. We also determine optimal control routines in the more restrictive scenario where the control operations are applied on only one of the two modes. We find that applying an instantaneous control only at the beginning of the dynamics, i.e., preparing an appropriate initial state, is the optimal strategy for states with symmetric correlations and when the dynamics is the same on both modes. More generally, even in asymmetric cases, the delayed decay of entanglement resulting from the optimal preparation of the initial state with no further action turns out to be always very close to the optimized control where multiple operations are applied during the evolution. Our study extends directly to “monosymmetric” systems of any number of modes, i.e., to systems that are invariant under any local permutation of the modes within any one partition, as they are locally equivalent to two-mode systems.

Figure 1

Logarithmic negativity of states which are initially locally equivalent to a pure two-mode squeezed state with squeezing parameter $r=ln[\sqrt{2}+1]/2$, evolving in a thermal environment with $\chi /\gamma =2$ (left panel) and $\chi /\gamma =1.000013$, i.e., room temperature at $450\mathrm{THz}$ (right panel); the two modes have the same loss rate $\gamma =100\mathrm{kHz}$. The blue (solid) curve refers to an initial state in standard form and the orange (dashed) curve refers to an initial state where one mode is locally squeezed by the squeezing transformation $\overline{Z}=\mathrm{diag}(2,1/2)$, while the green (dotted) curve refers to a state where both modes have been squeezed by $\overline{Z}$.

Figure 2

Logarithmic negativity of states which are initially locally equivalent to a pure two-mode squeezed state with squeezing parameter $r=ln[\sqrt{2}+1]/2$. Left panel: the two modes are evolving in two thermal environments with ${\chi }_1/\gamma =1$ and ${\chi }_2/\gamma =2$, while the loss rate $\gamma =100\mathrm{kHz}$ is the same for both modes. Right panel: the two modes have different loss rates ${\gamma }_1=100\mathrm{kHz}$ and ${\gamma }_2=10\mathrm{kHz}$, but the same parameter $\chi ={\gamma }_{1}1.000013$. The blue (solid) curve refers to an initial state in standard form, the orange (dashed) curve refers to an initial state where the first mode is locally squeezed by the squeezing transformation $\overline{Z}=\mathrm{diag}(2,1/2)$, and the green (dotted) curve refers to a state where the second mode is squeezed by $\overline{Z}$, while the red (dot-dashed) curve refers to a state where both modes have been squeezed by $\overline{Z}$.

Figure 3

Logarithmic negativity for initial states locally equivalent to a state with normal form parameters $a=4.5$, $b=3.5$, $c_{+}=2.2$, and $c_{-}=-3.5$, evolving in a thermal environment with $\chi /\gamma =1.000013$ and loss rate $\gamma =100\mathrm{kHz}$. The blue (solid) curve pertains to a state that was optimally adjusted on both modes, the orange (short dashed) curve pertains to a state that was optimally adjusted on one mode, the green (dotted) curve pertains to an initial state in normal form, and the red (dot-dashed) curve pertains to an initial state whose normal form was altered by the squeezing transformation $\overline{Z}=\mathrm{diag}(2,1/2)$ on one mode, while the purple (large dashed) curve pertains to a state further altered locally by a phase plate $R_{\frac{\pi }{4}}$ (besides the local squeezing $\overline{Z}$).

Figure 4

Difference between the logarithmic negativity with repeated controls and the one with a single initial control, for an initial state in normal form with parameters $a=4.5$, $b=3.5$, $c_{+}=2.2$, and $c_{-}=-3.5$, evolving in thermal environments with ${\chi }_1/\gamma =1.14769$, ${\chi }_2/\gamma =1.02956$, and same loss rate $\gamma =100\mathrm{kHz}$. The blue (solid) curve represents one additional control operation after the first one, the orange (short dashed) curve three additional control operations, the green (dotted) curve represents a total of ten control operations, and the red (dot-dashed) curve represents a total of 100 control operations, while the purple (large dashed) curve is obtained when the control is applied at every time step. Inset: the slight delay of entanglement “sudden death” due to repeated controls. The time step used in the plots is $2\times {10}^{-3}\mu \mathrm{s}$ and $3\times {10}^{-4}\mu \mathrm{s}$ for the main plot and the inset, respectively.

Figure 5

Difference between the logarithmic negativity with repeated controls and the one without control, for an initial state with normal form parameters $a=5$, $b=6$, $c_{+}=5.2$, and $c_{-}=-4.8$, evolving in two thermal environments with ${\chi }_1/\gamma =1.14769$, ${\chi }_2/\gamma =1.02956$, and the same loss rate $\gamma =100\mathrm{kHz}$. For each color, the solid lines always represent the application of an initial control operation, while the dashed curves represent controls applied at each time of the evolution (the time step chosen for the plot is ${10}^{-2}\mu \mathrm{s}$). The green curves above all the others represent control operations on both modes and they are practically superimposed at this scale, the dashed one attaining slightly higher values of $\mathrm{\Delta }{E}_{\mathcal{N}}$. The blue curves in the middle (in the region $t\approx 2\mu \mathrm{s}$) represent a control operation on subsystem 1, while the orange curves at the bottom (again, in the region $t\approx 2\mu \mathrm{s}$) pertain to control operations on subsystem 2 only.

Figure 6

Difference in the separability parameter $\tilde{\mathrm{\Sigma }}$ between optimal local control minimizing ${\tilde{\nu }}_{-}$ and optimal local control minimizing $\tilde{\mathrm{\Sigma }}$ for an initial state in normal form with $a=4.5$, $b=3.5$, $c_{+}=2.2$, and $c_{-}=-3.5$ evolving in an environment with equal thermal noises with $\chi /\gamma =2$ and loss rates $\gamma =1\mathrm{MHz}$. As apparent, after about $30\mathrm{ns}$, the local minimization of ${\tilde{\nu }}_{-}$ offers an advantage also in minimizing $\tilde{\mathrm{\Sigma }}$, whose time-local minimization is thus shown to be globally suboptimal. The evolving state is entangled up to a time of around $90\mathrm{ns}$.