Quantum Secret Sharing Among Four Players Using Multipartite Bound Entanglement of an Optical Field

Secret sharing is a conventional technique for realizing secure communications in information networks, where a dealer distributes to $n$ players a secret, which can only be decoded through the cooperation of $k$ ($n/2<k\le n$) players. In recent years, quantum resources have been employed to enhance security of secret sharing, which has been named quantum secret sharing (QSS). A multipartite bound entanglement (BE) state of an optical field, due to its special entanglement features, can be used in quantum networks to improve security and flexibility of communication. We design and experimentally demonstrate a QSS protocol, where the dealer modulates a secret on a four-partite BE state and then distributes the submodes of the BE state to four spatially separated players. The presented QSS scheme has the capability to protect secrets from eavesdropping and dishonest players, because a nonlocal and deterministic BE state is shared among four authorized players.

Figure 1

Schematics of the experimental principle and setup for QSS using a four-partite BE state. (a) Diagram for the QSS. (b) Experimental setup for generating a four-partite BE state and implementing the QSS scheme. MC1 and MC2: Mode cleaner cavity. HWP: Half-wave plate. PBS: Polarization beam splitter. DBS: Dichroic beam splitter. BS: $50:50$ beam splitter. HR: Mirror with high reflection. NOPA: Nondegenerate optical parametric amplifier. AM1–AM4: Amplitude modulator. PM1–PM4: Phase modulator. BHD1–BHD4: Balanced homodyne detector.

Figure 2

The calculated dependences of the secure key rate $K$ on the strength of a modulated signal for different correlation factors $r$. The blue and red traces are the corresponding traces for the collaboration of four players and of any three players, respectively. The dashed, solid, and dotted traces correspond to correlation factors $r$ of 1.5, 0.93 and 0.5, respectively, where $r=0.93$ corresponds to our experimental condition.

Figure 3

The measured noise power spectra with different combinations of players for QSS. (a) The measured noises of quadrature amplitude. (b) The measured noises of the quadrature phase. The blue traces are the noise with the collaboration of four players $\{1,2,3,4\}$. The red traces are the noise with the collaboration of any three players. The black traces correspond to the noise with the collaboration of $\{1,2\}$ as well as $\{3,4\}$, and the pink traces correspond to the noise with the collaboration of $\{1(2),3(4)\}$. The green traces correspond to the noise measured by any individual player. The value of the peak at the frequency of 2.25 MHz represents the strength of the measured signal power. The measurement parameters are as follows for the spectrum analyzer (SA): Resolution bandwidth (RBW): 10 kHz. Video bandwidth (VBW): 100 Hz.

Figure 4

Schematic of a secret string communication. (a) Modulated secret string by dealer. (b) and (c) The measured results by the collaboration of four players and by any three players, respectively. The measurement parameters are as follows for the SA: RBW: 300 kHz. VBW: 100 kHz.