Genuine secret-sharing states

Quantum secret sharing allows each player to have classical information for secret sharing in quantum mechanical ways. In this work, we construct a class of quantum states on which players can quantumly perform secret-sharing secure against dishonest players as well as eavesdropper. We here call them the genuine secret-sharing states. In addition, we show that if N players share an N -party genuine secret-sharing state, then arbitrary M players out of the total players can share an M -party genuine secret-sharing state by means of local operations and classical communication on the state, where $$N > M \ge 2$$ N > M ≥ 2 . We also define the distillable rate with respect to the genuine secret-sharing state and explain the connection between the distillable rate and the relative entropy of entanglement.

[1]  A. Zeilinger,et al.  Going Beyond Bell’s Theorem , 2007, 0712.0921.

[2]  Michal Horodecki,et al.  General Paradigm for Distilling Classical Key From Quantum States , 2009, IEEE Transactions on Information Theory.

[3]  R. Cleve,et al.  HOW TO SHARE A QUANTUM SECRET , 1999, quant-ph/9901025.

[4]  M. Plenio,et al.  Quantifying Entanglement , 1997, quant-ph/9702027.

[5]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[6]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[7]  Wayne C. Myrvold,et al.  Bell’s Theorem , 2011 .

[8]  Jaewan Kim,et al.  Teleportation capability, distillability, and nonlocality on three-qubit states , 2007 .

[9]  G. Vidal,et al.  Computable measure of entanglement , 2001, quant-ph/0102117.

[10]  Adi Shamir,et al.  How to share a secret , 1979, CACM.

[11]  V. Buzek,et al.  Quantum secret sharing , 1998, quant-ph/9806063.

[12]  V. Vedral,et al.  Entanglement measures and purification procedures , 1997, quant-ph/9707035.

[13]  Travis Norsen,et al.  Bell's theorem , 2011, Scholarpedia.

[14]  Jeong San Kim,et al.  Quantum states for perfectly secure secret sharing , 2008 .

[15]  J. Oppenheim,et al.  Secure key from bound entanglement. , 2003, Physical Review Letters.

[16]  A. Winter,et al.  Distillation of secret key and entanglement from quantum states , 2003, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[17]  R. Jozsa,et al.  A Complete Classification of Quantum Ensembles Having a Given Density Matrix , 1993 .

[18]  Gerardo Adesso,et al.  Unconditional security of entanglement-based continuous-variable quantum secret sharing , 2016, 1603.03224.

[19]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.