Random coding for sharing bosonic quantum secrets

We consider a protocol for sharing quantum states using continuous variable systems. Specifically we introduce an encoding procedure where bosonic modes in arbitrary secret states are mixed with several ancillary squeezed modes through a passive interferometer. We derive simple conditions on the interferometer for this encoding to define a secret sharing protocol and we prove that they are satisfied by almost any interferometer. This implies that, if the interferometer is chosen uniformly at random, the probability that it may not be used to implement a quantum secret sharing protocol is zero. Furthermore, we show that the decoding operation can be obtained and implemented efficiently with a Gaussian unitary using a number of single-mode squeezers that is at most twice the number of modes of the secret, regardless of the number of players. We benchmark the quality of the reconstructed state by computing the fidelity with the secret state as a function of the input squeezing.

Figure 1

A sketch of the encoding procedure with $n=2, m=2$, followed by decoding from the share of the access party consisting of the first, second, and fourth mode, while the third mode, corresponding to an adversary, is discarded. As shown in Sec. 4, the secret can be recovered from any $m+\lceil \frac{n}{2}\rceil =3$ out of the four modes. The decoded state ${\rho }_{\mathrm{out}}$ converges to the secret state ${\rho }_{\mathrm{s}}$ as the input squeezing increases and the last mode can also be discarded after the decoding (see Sec. 5).

Figure 2

Quality of the reconstructed state for several interferometers. The blue solid (red dashed) lines correspond to the access party the decoding of which results in the worst (best) reconstruction of the secret. In the plots of ${\nu }_{\max }$, the upper white region ${\nu }_{\max }>1$ corresponds to the encoding-decoding channel being entanglement breaking, while the dark gray region $\nu <0.5$ corresponds to the access party having the best possible copy of the secret allowed by optimal cloning. The plots (a) and (b) were obtained for two different interferometers and assuming two out of three players are trying to reconstruct a single-mode secret. The plot in (c) is for three out of five players and a single-mode secret, while (d) is for three out of four players trying to reconstruct a two-modes state (the scheme in Fig. 1). The matrices representing the corresponding interferometers are reported in Appendix pp5.