Secret Sharing of a Quantum State

Secret sharing of a quantum state, or quantum secret sharing, in which a dealer wants to share a certain amount of quantum information with a few players, has wide applications in quantum information. The critical criterion in a threshold secret sharing scheme is confidentiality: with less than the designated number of players, no information can be recovered. Furthermore, in a quantum scenario, one additional critical criterion exists: the capability of sharing entangled and unknown quantum information. Here, by employing a six-photon entangled state, we demonstrate a quantum threshold scheme, where the shared quantum secrecy can be efficiently reconstructed with a state fidelity as high as 93%. By observing that any one or two parties cannot recover the secrecy, we show that our scheme meets the confidentiality criterion. Meanwhile, we also demonstrate that entangled quantum information can be shared and recovered via our setting, which shows that our implemented scheme is fully quantum. Moreover, our experimental setup can be treated as a decoding circuit of the five-qubit quantum error-correcting code with two erasure errors.

Figure 1

Illustration of the experimental setup. An ultraviolet (UV) pulse passes through ${\mathrm{BBO}}_1$, ${\mathrm{BBO}}_2$, and ${\mathrm{BBO}}_3$ successively. After pumping on ${\mathrm{BBO}}_1$ (${\mathrm{BBO}}_2$), the UV pulse is refocused by lenses and directed to ${\mathrm{BBO}}_2$ (${\mathrm{BBO}}_3$) by mirrors (not shown here). The interference on ${\mathrm{PBS}}_1$, PDBS, and ${\mathrm{PBS}}_2$ is obtained by finely adjusting the path length of the two input photons. To achieve good visibility of interference, we filter the photons temporally by narrow band filters and spatially by single-mode fibers (SMFs). Only sixfold coincidence events, among paths 1, 2’, 3’, 4’, 5”, and 6’, are postselected, with rates of 75 counts per hour in the confidentiality test and 14 counts per hour in the reliability test. Polarizers (POL) are used for projection-valued measures.

Figure 2

Quantum secret recovery by three players. The column represents the corresponding fidelity of the recovered state when the initial state is prepared into $|H(V)⟩$, $|\pm ⟩=(|H⟩\pm |V⟩)/\sqrt{2}$, $|L(R)⟩=(|H⟩\pm i|V⟩)/\sqrt{2}$ and $|v(w)⟩=(|H⟩\pm \sqrt{3}|V⟩)/2$. The error bars are calculated by assuming that our experimental data follow a normal distribution. The red dashed line represents the classical limit of $2/3$.

Figure 3

Entanglement witness results as local measurements in the $Z$, $X$, and $Y$ bases. (a) Coincidence detections in the $Z$ basis, projecting to $H$ and $V$, $P(H_1,H_{2^{\prime }})=0.40(3)$, $P(H_1,V_{2^{\prime }})=0.10(2)$, $P(V_1,H_{2^{\prime }})=0.11(2)$ and $P(V_1,V_{2^{\prime }})=0.40(3)$. In Eq. (4), $⟨Z_1{Z}_{2^{\prime }}⟩=P(H_1,H_{2^{\prime }})-P(H_1,V_{2^{\prime }})-P(V_1,H_{2^{\prime }})+P(V_1,V_{2^{\prime }})=0.59$. (b) $X$ basis, projecting to $+$ and $-$, $P(+,{+}_{2^{\prime }})=0.40(2)$, $P({+}_1,{-}_{2^{\prime }})=0.11(1)$, $P({-}_1,{+}_{2^{\prime }})=0.12(1)$, and $P({-}_1,{-}_{2^{\prime }})=0.39(2)$. (c) $Y$ basis, projecting to $L$ and $R$, $P(L_1,L_{2^{\prime }})=0.03(1)$, $P(L_1,R_{2^{\prime }})=0.41(3)$, $P(R_1,L_{2^{\prime }})=0.52(3)$, and $P(R_1,R_{2^{\prime }})=0.05(1)$.

Figure 4

The geometric interpretation of the channel ${\mathcal{E}}_i$ on a Bloch sphere. Qubit states can be represented in a Bloch sphere. A pure state is on the sphere and a mixed state is in the ball. The ideal ${\mathcal{E}}_{\text{ideal}}$ of the quantum (3,3) threshold scheme maps each point on the surface (pure state) to the origin (the maximally mixed state $I/2$). (a) Geometric interpretation of the initial pure states possibly prepared by the dealer. [(b)–(d)] Geometric interpretations of the channel ${\mathcal{E}}_{\text{Alice}}$, ${\mathcal{E}}_{\mathrm{Bob}}$, and ${\mathcal{E}}_{\text{Charlie}}$, respectively. We observe that $F_{\text{Alice}}=0.90\pm 0.03$, $F_{\mathrm{Bob}}=0.97\pm 0.02$, and $F_{\text{Charlie}}=0.89\pm 0.03$, where the error bars are calculated by performing 500 runs of channel operator matrix reconstructions and assuming white noise.