Production of entanglement with highly mixed states

We study production of entanglement with highly mixed states. We find that entanglement between highly mixed states can be generated via a direct unitary interaction even when both states have purities arbitrarily close to zero. This indicates that purity of a subsystem is not required for entanglement generation. This result is in contrast to previous studies where the importance of the subsystem purity was emphasized.

Figure 1

Negativity of partial transpose (NPT) of the projected density matrix in Eq. (1) versus (interaction strength $g$)$\times$(interaction time $t$) for entanglement between an atom and a thermal state discussed in the text. The projection basis on the field mode is $\{\left|n\right.〉\left.〈n\right|,\left|n+1\right.〉\left.〈n+1\right|\}$ with $n=0$ (thick), $n=10$ (dashed), and $n=100$ (thin). The temperature of the field is assumed to be infinite for panels (a)–(c) and the purity $p$ of the atomic state is (a) 1, (b) 0.9, and (c) 0.8. The atom is assumed to be maximally mixed ($p=1/2$) for panels (d)–(f) and $\lambda$ of the field is (d) 0, (e) 0.1, and (f) 0.2 in order of decreasing purity.

Figure 2

NPT of the projected density matrix for Eq. (4) against displacement $d$ of the displaced thermal field. We choose the amplitude $\gamma =2$ for the projection basis ${\left|\pm \right.〉}_{\gamma }$ with several values of $V$, and the normalized purity $r$ is (a) 1 and (b) 0.1. Unless the microscopic superposition is totally mixed as $r=0$, entanglement is observed for $d\gg \sqrt{V}$ for any values of $V$.

Figure 3

NPT of locally projected state of ${\rho }^{\text{BS}(+)}$ with (a) $r=1$ and (b) $r=0.1$. We used $\gamma =2$ for the projection basis. Entanglement is always observed for large values of $V\gg 1$ even when $d=0$ unless $r=0$.

Figure 4

NPT of the locally projected state of ${\rho }^{TT(+)}$ versus the displacement $d$ with (a) $r=1$ and (b) $r=0.1$. The amplitude of the projection basis is $\gamma =2$ as in Figs. 2 and 3. Each curve corresponds to $V=10$ (solid), 500, (dashed), and 1000, (dotted). Nonzero values of $d$ much larger than $\sqrt{V}$ are required to observe entanglement compared to ${\rho }^{\text{BS}(+)}$ cases.

Figure 5

NPT of ${\rho }_{\gamma }^{\psi }$ with $\gamma =2$. In this case $V$ is the only parameter that adjusts the purities of both parties simultaneously. We can always obtain the entanglement even in $V\to \infty$ limit with sufficient displacement $d\gg \sqrt{V}$. This indicates that nonzero purity of the system is sufficient to generate entanglement.