How to Share a Quantum Secret
Open accessAbstract
We investigate the concept of quantum secret sharing. In a $(k,n)$ threshold scheme, a secret quantum state is divided into $n$ shares such that any $k$ of those shares can be used to reconstruct the secret, but any set of $k\ensuremath{-}1$ or fewer shares contains absolutely no information about the secret. We show that the only constraint on the existence of threshold schemes comes from the quantum ``no-cloning theorem,'' which requires that $n<2k$, and we give efficient constructions of all threshold schemes. We also show that, for $k\ensuremath{\le}n<2k\ensuremath{-}1$, then any $(k,n)$ threshold scheme must distribute information that is globally in a mixed state.
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There exist encoded operators that have support only on two of the three qutrits, and sets of encoded operators that have the same action on the codespace but nontrivial support on different pairs of qutrits.
“The unique feature of this specific code is that there exist encoded operators (also referred to as logical operators in the literature) that have support only on two of the three qutrits that constitute the encoded space.Furthermore, it is then possible to identify sets of encoded operators that have the same action on the codespace, but have nontrivial support on different pairs of qutrits.”
From Holographic spacetime, black holes and quantum error correcting codes: a review (2022) · cites this paper as [177]
The 3-qutrit code's erasure-correcting property ensures that any three-qutrit state within the codespace can be reconstructed even if one of the three qutrits is lost or erased.
“The erasure-correcting property of this code ensures that any three-qutrit state within the codespace can be reconstructed even if one of the three qutrits is lost or erased.”
From Holographic spacetime, black holes and quantum error correcting codes: a review (2022) · cites this paper as [177]
In the three-qutrit code, from any two of the qutrits Bob can reconstruct the state.
“Secondly, from any two of the qutrits Bob can reconstruct the state.”
From Bulk locality and quantum error correction in AdS/CFT (2015) · cites this paper as [25]
In the three-qutrit code, for any encoded state, the reduced density matrix on any one of the qutrits is maximally mixed.
“First of all for any state | ˜ ψ 〉 , the reduced density matrix on any one of the qutrits is maximally mixed.”
From Bulk locality and quantum error correction in AdS/CFT (2015) · cites this paper as [25]
3 and 15 analyzer settings are respectively required to obtain the single- and two-photon Stokes parameters.
“Thus 3 and 15 analyzer settings are respectively required to obtain the single- [28] and two-photon Stokes parameters.”
From Nonmaximally Entangled States: Production, Characterization, and Utilization (1999) · cites this paper as [28]
The citation is attached to the single-photon part of the sentence. Our copy discusses quantum secret sharing and qubits (pp. 1–2), but the passages found when searching for Stokes parameters and analyzer settings do not address the three-setting requirement. (LLM, conf=0.77)
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Quantum Solution to the Byzantine Agreement Problem
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- Authors
- Cleve, R., Gottesman, D., & Lo, H.-K.
- Affiliations
- University of Calgary (CA)
- Los Alamos National Laboratory (US)
- Hewlett-Packard (United Kingdom) (GB)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 1999-07-19
- Volume
- 83
- Issue
- 3
- Pages
- 648-651
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2083019346
- Cited by
- 1,373
- References
- 25
- Citations by year
- 2026: 43 · 2025: 87 · 2024: 63 · 2023: 56 · 2022: 38 · 2021: 66 · 2020: 65 · 2019: 64 · 2018: 72 · 2017: 56 · 2016: 51 · 2015: 64 · 2014: 60 · 2013: 69 · 2012: 65
- FWCI
- 14.90
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Computer Science › Artificial Intelligence › Cryptography and Data Security
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Keywords
- quantum secret sharing, threshold schemes, no-cloning theorem, quantum information, secret reconstruction, mixed states, quantum cryptography, information distribution
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