Generation of coherence via Gaussian measurements

We address measurement-based generation of quantum coherence in continuous variable systems. We consider Gaussian measurements performed on Gaussian states and focus on two scenarios: In the first one, we assume an initially correlated bipartite state shared by two parties and study how correlations may be exploited to remotely create quantum coherence via measurement back action. In particular, we focus on conditional states with zero first moments, so as to address coherence due to properties of the covariance matrix. We consider different classes of bipartite states with incoherent marginals and show that the larger the measurement squeezing, the larger the conditional coherence. Homodyne detection is thus the optimal Gaussian measurement to remotely generate coherence. We also show that for squeezed thermal states there exists a threshold value for the generated coherence which separates entangled and separable states at a fixed energy. Finally, we briefly discuss the tripartite case and the relationship between tripartite correlations and the conditional two-mode coherence. In the second scenario, we address the steady-state coherence of a system interacting with an environment which is continuously monitored. In particular, we discuss the dynamics of an optical parametric oscillator in order to investigate how the coherence of a Gaussian state may be increased by means of time-continuous Gaussian measurement on the interacting environment.

Figure 1

Relative entropy of coherence $C_S$ (left) and Gaussian relative entropy of coherence $C_S^G$ (right) for the conditional state after Gaussian measurement of one mode of a symmetric STS. Coherence is shown as a function of the STS thermal photons $N$ and the measurement squeezing $r_m$ for a fixed value of the STS squeezing parameter $r=1$.

Figure 2

Gaussian relative entropy of coherence $C_S^G$ (upper, solid curves) and relative entropy of coherence $C_S$ (lower, dotted lines) as a function of the covariance matrix term $c$ for a symmetric STS and homodyne measurement ($s\to \infty$). From left to right the sets of curves represent $a=1.5$ (red), $a=2$ (blue), and $a=2.5$ (yellow); vertical lines are drawn at the physicality bound $c=\sqrt{a^2-1}$.

Figure 3

Relative entropy of coherence (left) and Gaussian relative entropy of coherence (right) for homodyne measurement on mode $B$ of a symmetric STS, as a function of $a$. The $5\times {10}^4$ random symmetric STSs are generated by sampling uniformly the parameters $a$ and $c$. The solid red (dashed blue) curve at the top (in the middle) represents the physicality (separability) threshold. Entangled (separable) states correspond to red (blue) points above (below) the separability threshold.

Figure 4

Relative entropy of coherence (left) and Gaussian relative entropy of coherence (right) for homodyne measurements on mode $B$ of asymmetric STSs, as a function of the parameters $a$ and $b$. The $5\times {10}^4$ random asymmetric STSs are generated by sampling uniformly the parameters $a$, $b$, and $c$. The upper red (lower blue) surface corresponds to states on the physicality (separability) threshold. Entangled (separable) states correspond to red (blue) points above (below) the separability threshold.

Figure 5

Relative entropy of coherence (left) and Gaussian relative entropy of coherence (right) for optimal homodyne measurements on mode $B$ of normal-form two-mode Gaussian states, as a function of the parameters $a$ and $b$. The $5\times {10}^4$ (left) and $5\times {10}^5$ (right) random states in normal form are generated by sampling uniformly the parameters $a$, $b$, $c_1$, and $c_2$. The upper, red surface corresponds to states on the physicality threshold with maximal $|c_1|$, while the lower, blue surface corresponds to states on the physicality threshold with $c_2=0$. Entangled (separable) states correspond to lighter red (darker blue) points; only entangled states are above the lower, blue surface.

Figure 6

Coherence and discord of the two-mode conditional state obtained by measuring mode $A$ of the feasible three-mode state, (30). In both panels solid red curves represent $C_S^G$; dashed blue curves, $C_S$; and dotted orange curves, the quantum discord $D(B:C)$. In the left panel, the quantities are shown as a function of the measurement squeezing $r_m$ at a fixed $N_B=1$ and $N_C=2$; horizontal lines correspond to the same figures of merit computed for the marginal state (and are thus independent of measurement squeezing). In the right panel, the quantities are shown as a function of $N_A$ by fixing $N_B=N_C=N_A/2$, with $r_m=5$; the lower curves correspond to the same figures of merit computed for the marginal state.

Figure 7

Steady-state coherence (at zero first moments). In the left panel we have the relative entropy of coherence and in the right panel its Gaussian counterpart, both as a function of the measurement squeezing $r_m$ and the mean number of excitations of the environmental mode $N$, with fixed $\tilde{\chi }=0.4$. Dashed red lines represent the threshold values $r_m^{\mathrm{th}}$ for which the coherence of the monitored state is equal to that of the unconditional dynamics. In the region above the red curve we have more coherence than for unconditional dynamics; in the region below the curve, vice versa.

Figure 8

Value of the threshold squeezing as a function of $\tilde{\chi }$. The solid red curve is obtained for $C_S$, and the dashed blue for $C_S^G$, both for $N=0.1$; the dotted yellow curve represents both $C_S$ and $C_S^G$ for $N=5$.

Figure 9

Remote coherence as a function of the outcome angle $\theta$ for a symmetric STS with $N=1$ and $r=1$. From bottom to top we have $|{\mathbf{r}}_{\text{out}}|=1$ (red lines), $|{\mathbf{r}}_{\text{out}}|=2$ (blue lines), $|{\mathbf{r}}_{\text{out}}|=4$ (yellow lines), and $|{\mathbf{r}}_{\text{out}}|=6$ (green lines). Dashed lines represent the relative entropy of coherence $C_S$; solid lines, the Gaussian counterpart $C_S^G$.

Figure 10

Relative entropy of coherence (top) and Gaussian relative entropy of coherence (bottom) as a function of $r_m$ for a symmetric STS with $N=1$ and $r=1$. The lowest, solid red curve is for zero outcome, $|{\mathbf{r}}_{\text{out}}|=0$. The blue, yellow, and green dashed curves in the middle represent $|{\mathbf{r}}_{\text{out}}|=1$ and $\theta =0$, $\frac{\pi }{4}$, and $\frac{\pi }{2}$ (from bottom to top), respectively. The blue, yellow, and green dotted curves at the top are for $|{\mathbf{r}}_{\text{out}}|=4$ and $\theta =0$, $\frac{\pi }{4}$, and $\frac{\pi }{2}$ (from bottom to top in the region $r_m\approx 2$), respectively.

Figure 11

The average remote Gaussian relative entropy of coherence ${\overline{C}}_S^G$ for fixed $r=1$ as a function of $r_m$ and $N$.

Figure 12

The average remote Gaussian relative entropy of coherence ${\overline{C}}_S^G$ for heterodyne measurements ($r_m=0$) and a symmetrical STS as a function of $N$. Different curves represent different values of the initial squeezing: $r=0.5$ (solid red line), $r=1$ (dashed blue line), and $r=1.5$ (dotted yellow line).