Optimal supplier of single-error-type entanglement via coherent-state transmission

Compared with entanglement with multiple types of noise, entanglement including only one type of error is a favorable fundamental resource not only for quantum communication but also for distributed quantum computation. We consider protocols that present single-error-type entanglement for distant qubits via coherent-state transmission over a lossy channel. Such a protocol is regarded as a subroutine to serve entanglement for larger protocol to yield a final output, such as ebits or pbits. In this paper, we provide a subroutine protocol which achieves the global optimal for typical jointly convex yield functions monotonically nondecreasing with respect to the singlet fraction, such as an arbitrary convex function of a singlet fraction, two-way distillable entanglement, and two-way distillable key. Entanglement generation based on remote nondestructive parity measurement protocol [K. Azuma et al., Phys. Rev. A 85, 062309 (2012)] is identified as such an optimal subroutine.

Figure 1

Scenario of single-error-type entanglement generation. ${|\varphi \rangle }_A:={\sum }_{j=0,1}\sqrt{q_j}{e}^{i{\mathrm{\Theta }}_j}{|j\rangle }_A$. An outcome $k$ corresponds to the application of a separable operator ${\widehat{M}}_k^A\otimes {\widehat{N}}_k^B$. If the final entanglement ${\widehat{\tau }}_k^{AB}$ includes only one type of error, Alice and Bob may declare the success of the protocol.

Figure 2

An imaginary protocol equivalent to the real protocol in Fig. 1. $|{\psi }^{\prime }{\rangle }_{Ab}:={\sum }_{j=0,1}\sqrt{q_j}{e}^{i{\mathrm{\Theta }}_j+i{(-1)}^j\phi }{|j\rangle }_A{|u_j\rangle }_b$, where $\phi :=arg\left[\langle v_1|v_0\rangle \right]/2$. Channel $a\to b$ becomes ideal at the expense of the application of a phase-flip channel ${\mathrm{\Lambda }}_v^A$ with $v=|\langle v_1|v_0\rangle |$.

Figure 3

Performance of the point-to-point optimal supplier of single-error-type entanglement over coherent-state transmission over a lossy channel with the transmittance $T$. Curve (a) describes the quantum and private capacity of the lossy channel as a reference. Curve (b) describes ${\overline{E}}_{\mathrm{D},\max }$ which is the two-way distillable entanglement of the single-error-type entanglement generated by the optimal supplier, per channel use. Curves (c) and (d) represent the success probabilities to obtain single-error-type entanglement with $F=99.4\%$ and with $F=99.8\%$ by using the optimal supplier, respectively.

Figure 4

Scenario of single-error-type entanglement generation by three parties. ${|\varphi \rangle }_A:={\sum }_{j=0,1}\sqrt{{q_j^A}^{\phantom{\int }}}{e}^{i{\mathrm{\Theta }}_j^A}{|j\rangle }_A$ and ${|\phi \rangle }_B:={\sum }_{j=0,1}\sqrt{{q_j^B}^{\phantom{\int }}}{e}^{i{\mathrm{\Theta }}_j^B}{|j\rangle }_B$. An outcome $k$ corresponds to the application of a separable operator ${\widehat{M}}_k^A\otimes {\widehat{N}}_k^B\otimes {}_C\langle O_k|$. If the final entanglement ${\widehat{\tau }}_k^{AB}$ includes only one type of error, Alice, Bob, and Claire may declare the success of the protocol.

Figure 5

An imaginary protocol equivalent to the real protocol in Fig. 4. $|{\xi }^{\prime }{\rangle }_{A{c}_a}:={\sum }_{j=0,1}\sqrt{q_j^A}{e}^{i{\mathrm{\Theta }}_j^A+i{(-1)}^j{\phi }_a}{|j\rangle }_A{|u_j^a\rangle }_{c_a}$ and $|{\zeta }^{\prime }{\rangle }_{B{c}_b}:={\sum }_{j=0,1}\sqrt{q_j^B}{e}^{i{\mathrm{\Theta }}_j^B+i{(-1)}^j{\phi }_b}{|j\rangle }_B{|u_j^b\rangle }_{c_b}$. Channels $a\to c_a$ and $b\to c_b$ become ideal at the expense of the application of phase-flip channels ${\mathrm{\Lambda }}_{v^a}^A$ with $v^a=\left|\langle v_1^a|v_0^a\rangle \right|$ and ${\mathrm{\Lambda }}_{v^b}^B$ with $v^b=\left|\langle v_1^b|v_0^b\rangle \right|$, respectively. If the protocol returns success outcome $k$, the effect of phase-flip channel ${\mathrm{\Lambda }}_{v^b}^B$ (${\mathrm{\Lambda }}_{v^a}^A$) is equivalent to that of ${\mathrm{\Lambda }}_{v^b}^A$ (${\mathrm{\Lambda }}_{v^a}^B$). We define virtual protocol $A \left(B\right)$ as the modified protocol where ${\mathrm{\Lambda }}_{v^b}^B$ (${\mathrm{\Lambda }}_{v^a}^A$) is converted to ${\mathrm{\Lambda }}_{v^b}^A$ (${\mathrm{\Lambda }}_{v^a}^B$).

Figure 6

Performance of the optimal supplier of single-error-type entanglement with a middle measuring station. This middle station is connected with Alice and Bob by a lossy channel with transmittance $\sqrt{T}$, respectively (i.e., $T_a=T_b=\sqrt{T}$). Curve (a) describes the quantum and private capacity of this network. Curve (b) describes ${\overline{E}}_{\mathrm{D},\max }$ which is the two-way distillable entanglement of the single-error-type entanglement generated by the optimal supplier, per use of the pair of channels between $AC$ and $CB$. Curves (c) and (d) represent the success probabilities to obtain single-error-type entanglement with $F=99.4\%$ and with $F=99.8\%$ by using the optimal supplier, respectively. Curve (e) describes the quantum and private capacity of a lossy channel with transmittance $T$ (which can directly connect Alice and Bob).