Entanglement-induced sub-Planck phase-space structures

We show that entanglement of two superposed coherent states produces sub-Planck structures in phase space. The origin of these structures depends entirely on entanglement, unlike those in Zurek [Nature (London) 412, 712 (2001)], where the effect of entanglement was not considered. The constituent states are sensitive enough to carry out the Heisenberg sensitive measurements. However, their usefulness is limited in providing high sensitivity in either the position $\left(x\right)$ direction or in the momentum $\left(p\right)$ direction of the phase-space. Interestingly, the entanglement causes the proposed states to be sensitive in both $x$ and $p$ directions. This state offers a better sensitivity compared to the single partite compass state in Zurek [Nature (London) 412, 712 (2001)] in carrying out precision measurements. It is also argued that such states are easy to create and are better suited for carrying out precision measurements.

Figure 1

(Color online) Cross-sectional view of the Wigner function of an entangled wave function with $A_1=A_2=1∕\sqrt{2}$ and $B_1=-B_2=1∕\sqrt{2}$ in (a) the $\left(x_1{p}_1\right)$ plane, with $x_2=0$, $p_2=0$ and (b) the $\left(x_2{p}_2\right)$ plane with $x_1=0$, $p_1=0$.

Figure 2

(Color online) Cross-sectional view of the Wigner function with parameter values as in Fig. 1 except the entanglement has been turned off by setting the parameter $B=0$.