Quantum repeaters using coherent-state communication

We investigate quantum repeater protocols based upon atomic qubit-entanglement distribution through optical coherent-state communication. Various measurement schemes for an optical mode entangled with two spatially separated atomic qubits are considered in order to nonlocally prepare conditional two-qubit entangled states. In particular, generalized measurements for unambiguous state discrimination enable one to completely eliminate spin-flip errors in the resulting qubit states, as they would occur in a homodyne-based scheme due to the finite overlap of the optical states in phase space. As a result, by using weaker coherent states, high initial fidelities can still be achieved for larger repeater spacing, at the expense of lower entanglement generation rates. In this regime, the coherent-state-based protocols start resembling single-photon-based repeater schemes.

Figure 1

(Color online) (a) Three steps for the generation of spin entanglement between two qubits at neighboring repeater stations: the first interaction results in an entangled state between the atomic qubit 1 and the optical qubus; after transmission, the qubus interacts with the atomic qubit 2, leading to a tripartite entangled state between the two qubits and the qubus; finally, a measurement on the qubus mode conditionally creates the two-qubit entanglement. (b) Example of a possible measurement scheme for discriminating between the conditionally phase-rotated coherent probe beams in the hybrid quantum repeater; the LO pulse is a sufficiently strong local oscillator used for homodyne detection [19].

Figure 2

(Color online) The entanglement of formation of the qubit-1-qubus states as a function of the qubus amplitude $\alpha$ (square root of the qubus photon number) for different channel transmissions, i.e., different distances of qubus propagation; photon loss is assumed to be $0.18\mathrm{dB}\text{per}\mathrm{km}$. The phase $\theta$ is always 0.01.

Figure 3

(Color online) Phase-rotated coherent states to be discriminated for entangled-state preparation; the unrotated state belongs to the preferred qubit subspace (even parity); the two rotated states are correlated with the odd subspace of the two qubits.

Figure 4

(Color online) The optimal failure probability for unambiguous state discrimination of the two density operators in Eqs. (15) as a function of the fidelity of the desired entangled two-qubit state in Eq. (14). The regions below each curve are quantum mechanically inaccessible. Note that for a transmission $\eta =1∕2$ $\left(17\mathrm{km}\right)$, the functional dependence is linear. Photon loss is again assumed to be $0.18\mathrm{dB}\text{per}\mathrm{km}$.

Figure 5

(Color online) Unambiguous state discrimination of a zero-phase coherent state from two phase-rotated coherent states via linear optics, phase-space displacements, and photon detection. The circuit $U\left(3\right)$ represents a linear three-port device, corresponding to a sequence of two beam splitters [see Eq. (20)].

Figure 6

(Color online) Failure probabilities as functions of the final two-qubit maximally entangled-state fidelities for different distances. The regions below USD bound are quantum mechanically inaccessible. The curves USD correspond to those linear-optics implementations in which all conclusive patterns for both even and odd subspaces are combined (choosing a beam-splitter parameter $\lambda =0.4$). The plots for even $(\lambda =0.7)$ and odd $(\lambda =0.01)$ describe those measurement schemes where only a single detection pattern is used in order to project onto the respective two-qubit subspaces (∣click, click, no click⟩ for even and ∣no click, no click, click⟩ for odd). Photon loss is again $0.18\mathrm{dB}\text{per}\mathrm{km}$.

Figure 7

(Color online) Purifying two copies of a light-matter hybrid entangled pair into one copy with higher fidelity (purity) through local operations on the qubits and the light modes.

Figure 8

(Color online) Hybrid entanglement swapping through unambiguous Bell-state measurements on the joint systems of qubit and light mode.