Optimal Squeezing and Entanglement from Noisy Gaussian Operations

We investigate the creation of squeezing via operations subject to noise and losses and ask for the optimal use of such devices when supplemented by noiseless passive operations. Both single and repeated uses of the device are optimized analytically and it is proven that in the latter case the squeezing converges exponentially fast to its asymptotic optimum, which we determine explicitly. For the case of multiple iterations we show that the optimum can be achieved with fixed intermediate passive operations. Finally, we relate the results to the generation of entanglement and derive the maximal two-mode entanglement achievable within the considered scenario.

Figure 1

Various scenarios for the optimization of squeezing: (a) single iteration case: from a given input ${\gamma }_{\mathrm{in}}$, we want to generate as much squeezing as possible by properly choosing $K$ and using the noisy device $\mathcal{E}$; (b) multiple iteration case: the device can be applied repeatedly, and we have to determine the $K_i$’s such as to maximize the finally obtained squeezing; (c) circular setup: with identical $K_i=K$.

Figure 2

Illustration of the single iteration optimality proof. Left: the input state is decomposed into a coherent kernel $s\mathbb{1}$ and into some extra noise $N$ with a null eigenvector $|\nu ⟩$. Right: after the application of the operation $\mathcal{E}$, the coherent kernel is squeezed in the direction $|\varphi ⟩$. By choice of $K$, one can achieve $K(X|\varphi ⟩)\propto |\nu ⟩$; i.e., the noise $N$ does not lead to a contribution in the most squeezed direction.

Figure 3

The function $s^{\prime }=f_{\mathcal{E}}(s)$ describes the optimal output squeezing $s^{\prime }$ which can be obtained from input squeezing $s$ by an operation $\mathcal{E}$. The straight line is the identity, and the gray zigzag line describes the evolution of $f$ when using the optimal strategy for multiple iterations; cf. Fig. 1b.

Figure 4

Optimization results for a physical device ${\mathcal{E}}_t$ which is given by applying the master equation for a time $t$, with $H=\frac{3}{4}{a}^{†}a-\frac{1}{4}{a}^{†}{a}^{†}+\mathrm{H}.\mathrm{c}.$ and $\nu =0.1$ (cf. text). The dotted line shows the squeezing obtained by simply applying ${\mathcal{E}}_t$ to a coherent state, while the solid line gives the asymptotically attainable optimum as derived in the paper. The dashed line shows the rotation angle which leads to the optimal value.