Tripartite Quantum State Sharing

We demonstrate a multipartite protocol to securely distribute and reconstruct a quantum state. A secret quantum state is encoded into a tripartite entangled state and distributed to three players. By collaborating, any two of the three players can reconstruct the state, while individual players obtain nothing. We characterize this $(2,3)$ threshold quantum state sharing scheme in terms of fidelity, signal transfer, and reconstruction noise. We demonstrate a fidelity averaged over all reconstruction permutations of $0.73\pm 0.04$, a level achievable only using quantum resources.

Figure 1

Schematic of the $(2,3)$ quantum state sharing scheme. ${\psi }_{\mathrm{in}}$, secret quantum state; OPA, optical parametric amplifier; $G$, electronic gain; AM, amplitude modulator; LO, optical local oscillator; $\mathrm{x}\mathrm{:}\mathrm{y}$, beam splitter with reflectivity $\mathrm{x}/(\mathrm{x}+\mathrm{y})$ and transmitivity $\mathrm{y}/(\mathrm{x}+\mathrm{y})$.

Figure 2

Experimental results for the $\{1,2\}$ access structure. (a) and (b) show the spectra of the amplitude and phase quadrature variances for the secret (input, blue/dark grey) and reconstructed (output, green/light grey) quantum states. $\Delta f$ is the offset from the signal frequency at 6.12 MHz. Resolution bandwidth $=1\mathrm{kHz}$ and video bandwidth $=30\mathrm{Hz}$. (c) Standard deviation contours of Wigner functions of the secret (blue/dark grey) and reconstructed (green/light grey) quantum states. (d) Measured fidelity as a function of gain deviation $r^2=(⟨{\widehat{X}}_{\mathrm{out}}^{+}⟩-⟨{\widehat{X}}_{\mathrm{in}}^{+}⟩{)}^2+(⟨{\widehat{X}}_{\mathrm{out}}^{-}⟩-⟨{\widehat{X}}_{\mathrm{in}}^{-}⟩{)}^2$. (d) Grey area highlights the accessible fidelity region. Points plotted are from six different experimental runs.

Figure 3

Experimental results for the $\{2,3\}$ access structure. (a) and (b) show the spectra of the amplitude and phase quadrature variances for the secret (input, blue/dark grey) and reconstructed (output, green/light grey) quantum states. (c) Standard deviation contours of Wigner functions of the secret (blue/dark grey) and reconstructed (green/light grey) quantum states. The dashed circle represents the quantum state $\delta {\widehat{X}}_{\mathrm{para}}^{\pm }=(\sqrt{3}{)}^{\mp 1}\delta {\widehat{X}}_{\mathrm{out}}^{\pm }$ after the a posteriori local unitary parametric operation.

Figure 4

Experimental fidelity for the $\{2,3\}$ access structure as a function of the product of $g^{+}{g}^{-}$. Dashed line: calculated theoretical curve with squeezing of $-4.5\mathrm{dB}$, added noise of $+3.5\mathrm{dB}$, electronic noise of $-13\mathrm{dB}$ with respect to the quantum noise limit, and feed forward detector efficiency of $0.93$. Solid line and dotted lines: experimental fidelity for the adversary structure and error bar, respectively. Grey area highlights the classical boundary for the access structure.

Figure 5

Experimental signal transfer ($\mathcal{T}$) and additional noise ($\mathcal{V}$) for the $\{2,3\}$ access structure (green/light grey circles), and the adversary structure (red/dark grey diamonds). Solid line: calculated theoretical curve for varying gain with same parameters as in Fig. 4. Triangle: unitary gain point for the $\{2,3\}$ access structure. Square: calculated theoretical point for the adversary structure. Grey area: the classical region for the $\{2,3\}$ access structure. Inset: experimental $\mathcal{T}$ and $\mathcal{V}$ for the $\{1,2\}$ access structure (green/light grey circles) and the theoretical point (black circles).