Continuous-variable quantum-state sharing via quantum disentanglement

Quantum-state sharing is a protocol where perfect reconstruction of quantum states is achieved with incomplete or partial information in a multipartite quantum network. Quantum-state sharing allows for secure communication in a quantum network where partial information is lost or acquired by malicious parties. This protocol utilizes entanglement for the secret-state distribution and a class of “quantum disentangling” protocols for the state reconstruction. We demonstrate a quantum-state sharing protocol in which a tripartite entangled state is used to encode and distribute a secret state to three players. Any two of these players can collaborate to reconstruct the secret state, while individual players obtain no information. We investigate a number of quantum disentangling processes and experimentally demonstrate quantum-state reconstruction using two of these protocols. We experimentally measure a fidelity, averaged over all reconstruction permutations, of $\mathcal{F}=0.73\pm 0.02$. A result achievable only by using quantum resources.

Figure 1

Schematic of the dealer protocol for the (2, 3) quantum-state sharing scheme. ${\psi }_{\mathrm{in}}$: secret quantum state. OPA: optical parametric amplifier. $x:y:$ beam splitter with reflectivity $x∕(x+y)$ and transmitivity $y∕(x+y)$. ${\varphi }_i$: optical phase delays. $N^{\pm }$: additional Gaussian noise. $G_i$: electronic gains.

Figure 2

Schematic of the reconstruction protocols for the (2, 3) quantum-state sharing scheme. For the players 1 and 2 as the authorized group: (a) Mach-Zehnder reconstruction protocol. For the players 1 and 3 (or players 2 and 3) as the authorized group: (b) Phase-insensitive amplifier reconstruction protocol. (c) Two optical parametric amplifier reconstruction protocol. (d) Feed-forward reconstruction protocol. (e) Two feed-forward reconstruction protocols. ${\psi }_{\mathrm{out}}$: reconstructed quantum state, $G$: electronic gain, and AM: amplitude modulator. Switch symbol: represents the use of either player 1 or player 2 in the corresponding reconstruction protocols.

Figure 3

Signal transfer $\left(\mathcal{T}\right)$ and additional noise $\left(\mathcal{V}\right)$ for the single feed-forward reconstruction protocol, with no squeezing in the dealer protocol. (a) Accessible regions for the {2,3} (and {1,3}) authorized groups and (b) accessible points for the {1} (and {2}) adversary group, for increasing additional Gaussian noise in the dealer protocol of (i) $V_{\mathrm{N}}=0$, (ii) $V_{\mathrm{N}}=0.25$, (iii) $V_{\mathrm{N}}=1.13$, and (iv) $V_{\mathrm{N}}=3.06$, normalized to the quantum noise limit. Gray square: unity gain point for the authorized group reconstruction protocol. Gray circles: corresponding adversary group points.

Figure 4

Signal transfer $\left(\mathcal{T}\right)$ and additional noise $\left(\mathcal{V}\right)$ for the single feed-forward reconstruction protocol, with increasing squeezing in the dealer protocol. (a) Accessible regions for the {2,3} (and {1,3}) authorized groups and (b) accessible points for the {1} (and {2}) adversary group, for increasing squeezing in the dealer protocol of (i) $0\mathrm{dB}$, (ii) $-3\mathrm{dB}$, (iii) $-6\mathrm{dB}$, and (iv) $-9\mathrm{dB}$ below the quantum noise limit and with no additional Gaussian noise.

Figure 5

Experimental fidelity for the authorized groups. (a) Classical fidelity for {2,3} authorized group as a function of the optical gain product $g^{+}{g}^{-}$. (b) Fidelity for {2,3} authorized group with $4.5\mathrm{dB}$ of squeezing in the dealer protocol. Solid line: theoretical curve with $3.5\mathrm{dB}$ of additional Gaussian noise, $13\mathrm{dB}$ of electronic noise below the quantum noise limit, and a feed-forward detector efficiency of ${\eta }_{\mathrm{ff}}=0.93$. Gray area: classical region for the authorized group. Inset: fidelity for the {1,2} authorized group as a function of the optical gain product. Solid line: theoretical curve.

Figure 6

Experimental signal transfer $\left(\mathcal{T}\right)$ and additional noise $\left(\mathcal{V}\right)$ for the authorized groups. (a) Classical $\mathcal{T}$ and $\mathcal{V}$ for the {2,3} authorized group, for varying electronic feed-forward gain. (b) $\mathcal{T}$ and $\mathcal{V}$ for the {2,3} authorized group with $-4.5\mathrm{dB}$ of squeezing in the dealer protocol. Solid line: theoretical curve for authorized group. Gray area: classical region for the authorized group. Inset: experimental $\mathcal{T}$ and $\mathcal{V}$ for the {1,2} authorized group.

Figure 7

Experimental signal transfer $\left(\mathcal{T}\right)$ and additional noise $\left(\mathcal{V}\right)$ for the {1} adversary group, for increasing squeezing and additional Gaussian noise in the dealer protocol. Solid line: theoretical curve with increasing squeezing or additional Gaussian noise. (i) Experimental point with no squeezing or additional noise, (ii) experimental points with squeezing of additional noise varied around $-4.5\mathrm{dB}$ and $+3.5\mathrm{dB}$, respectively, and (iii) and experimental point with $-4.5\mathrm{dB}$ of squeezing and $+18.6\mathrm{dB}$ of additional noise with respect to the quantum noise limit