Quantum teleportation with identical particles

We study teleportation with identical massive particles. Indistinguishability imposes that the relevant degrees of freedom to be teleported are not particles, but rather addressable orthogonal modes. We discuss the performances of teleportation under the constraint of conservation of the total number of particles. The latter inevitably decreases the teleportation fidelity. Moreover, even though a phase reference, given by the coupling to a reservoir, circumvents the constraint, it does not restore perfect deterministic teleportation. The latter is only achievable with some special resource entangled states and when the number of particles tends to infinity. Interestingly, some of such states are the many-particle atomic coherent states and the ground state of cold atoms loaded into a double well potential, which are routinely prepared in experiments.

Figure 1

Fidelity $f_{1/2,0}$, Eq. (22), of the teleportation protocol with the resource state being the SU(2) symmetric coherent state $|\xi =1/2,\vartheta =0\rangle$, Eq. (34). Panel (a) shows the teleportation fidelity $f_{1/2,0}$, the inset of (a) shows the difference $f_{\mathrm{sep}}-f_{1/2,0}$ with respect to the fidelity with separable resource states (27), and panel (b) is the difference $f_{\max \mathrm{ent}}-f_{1/2,0}$ with respect to the fidelity with the maximally entangled resource state (30), for $N=1$ (continuous line), $N=5$ (dotted line), and $N=10$ (dashed line). The inset of (b) shows the cases where the fidelity with the SU(2) symmetric coherent resource state is larger than the fidelity with the maximally entangled resource state (30) even for large $\nu$: $N=1$ (continuous line), $N=2$ (dotted line), $N=3$ (dashed line).

Figure 2

Average final entanglement $E_{1/2,0}$, Eq. (25), of the teleportation protocol with the resource state being the SU(2) symmetric coherent state ${|\xi =1/2,\vartheta =0\rangle }_{34}$, Eq. (34). Panel (a) shows the average final entanglement $E_{1/2,0}$, the inset of (a) shows the difference $\frac{\pi N}{8}-E_{1/2,0}$ with respect to the maximum average final entanglement, and panel (b) is the difference $E_{\max \mathrm{ent}}-E_{1/2,0}$ with respect to the average final entanglement with the maximally entangled resource state (30), for $N=1$ (continuous line), $N=5$ (dotted line), and $N=10$ (dashed line). The inset of (b) shows the cases where the average final entanglement with the SU(2) symmetric coherent resource state is larger than the average final entanglement with the maximally entangled resource state (30) even for large $\nu$: $N=1$ (continuous line), $N=2$ (dotted line), $N=3$ (dashed line).

Figure 3

Fidelity $f_{\mathrm{coh}}$, Eq. (22) and panel (a), and average final entanglement $E_{\mathrm{coh}}$, Eq. (25) and panel (b), of the teleportation protocol with the resource state being the SU(2) coherent state $|\xi ,\vartheta \rangle$, Eq. (34), for $N=10$ and $\nu =100$. The fidelity $f_{\mathrm{coh}}$ is maximized by the symmetric coherent state ${|\xi =1/2,\vartheta =0\rangle }_{34}$, and quickly decays with $\vartheta$, while $E_{\mathrm{coh}}$ is maximized by $\xi =1/2$ for any phase $\vartheta$. Nevertheless, phases $\vartheta \ne 0$ can be compensated by local unitary operations (see the Appendix).

Figure 4

Fidelity $f_{\mathrm{BH},-0.5}$, Eq. (22), of the teleportation protocol with the resource state being the ground state of the Hamiltonian (35) in the single Gaussian regime for $\gamma =-0.5$, Eqs. (36) and (38). Panel (a) shows the teleportation fidelity $f_{\mathrm{BH},-0.5}$, the inset of (a) shows the difference $f_{\mathrm{sep}}-f_{\mathrm{BH},-0.5}$ with respect to the fidelity with separable resource states (27), and panel (b) is the difference $f_{\max \mathrm{ent}}-f_{\mathrm{BH},-0.5}$ with respect to the fidelity with the maximally entangled resource state (30), for $N=1$ (continuous line), $N=6$ (dotted line), and $N=10$ (dashed line). The inset of (b) shows the cases where the fidelity with the Gaussian resource state (38) is larger than the fidelity with the maximally entangled resource state (30) even for large $\nu$: $N=1$ (crosses), $N=2$ (continuous line), $N=3$ (dotted line), $N=4$ (dashed line), $N=5$ (dash-dotted line).

Figure 5

Average final entanglement $E_{\mathrm{BH},-0.5}$, Eq. (25), of the teleportation protocol with the resource state being the ground state of the Hamiltonian (35) in the single Gaussian regime for $\gamma =-0.5$, Eqs. (36) and (38). Panel (a) shows the average final entanglement $E_{\mathrm{BH},-0.5}$, the inset of (a) shows the difference $\frac{\pi N}{8}-E_{\mathrm{BH},-0.5}$ with respect to the maximum average final entanglement, and panel (b) is the difference $E_{\max \mathrm{ent}}-E_{\mathrm{BH},-0.5}$ with respect to the average final entanglement with the maximally entangled resource state (30), for $N=1$ (continuous line), $N=5$ (dotted line), and $N=10$ (dashed line). The inset of (b) shows the cases where the average final entanglement with the Gaussian resource state (38) is larger than the average final entanglement with the maximally entangled resource state (30) even for large $\nu$: $N=1$ (crosses), $N=2$ (continuous line), $N=3$ (dotted line), $N=4$ (dashed line), $N=5$ (dash-dotted line).

Figure 6

Fidelity $f_{\mathrm{BH},-0.5}$, Eq. (22), of the teleportation protocol with the resource state being the ground state of the Hamiltonian (35) in the single Gaussian regime for $\nu =100$, Eqs. (36) and (38). Panel (a) shows the teleportation fidelity $f_{\mathrm{BH},-0.5}$, and panel (b) is the difference $f_{\max \mathrm{ent}}-f_{\mathrm{BH},-0.5}$ with respect to the fidelity with the maximally entangled resource state (30), for $N=1$ (continuous line), $N=5$ (dotted line), and $N=10$ (dashed line). The insets show the zoom of the critical region $|\gamma +1|\le {\nu }^{-2/3}$. Since the ground state of (35) is not known in the critical region, the resource state of the teleportation fidelity plotted in the above insets is not the ground state of (35). It is instead the continuation of the ground state from the outside region $|\gamma +1|\ge {\nu }^{-2/3}$.

Figure 7

Average final entanglement $E_{\mathrm{BH},-0.5}$, Eq. (25), of the teleportation protocol with the resource state being the ground state of the Hamiltonian (35) in the single Gaussian regime for $\nu =100$, Eqs. (36) and (38). Panel (a) shows the average finale entanglement $E_{\mathrm{BH},-0.5}$, and panel (b) is the difference $E_{\max \mathrm{ent}}-E_{\mathrm{BH},-0.5}$ with respect to the fidelity with the maximally entangled resource state (30), for $N=1$ (continuous line), $N=5$ (dotted line), and $N=10$ (dashed line). The insets show the zoom of the critical region $|\gamma +1|\le {\nu }^{-2/3}$. Since the ground state of (35) is not known in the critical region, the resource state of the teleportation fidelity plotted in the above insets is not the ground state of (35). It is instead the continuation of the ground state from the outside region $|\gamma +1|\ge {\nu }^{-2/3}$.

Figure 8

Fidelity $f_{m,\max \mathrm{ent}}$, Eq. (44), of the teleportation protocol as a function of the number of the first $m$ modes of the initial state (42). Panel (a) shows the fidelity $f_{m,\max \mathrm{ent}}$, and panel (b) the difference $f_{\max \mathrm{ent}}-f_{m,\max \mathrm{ent}}$ with $N=10$ particles in the initial state and $\nu =N$ (continuous line), $\nu =10N$ (dotted line), $\nu =100N$ (dashed line). The teleportation fidelity $f_{m,\max \mathrm{ent}}$ monotonically increases with $m$.