arXiv:1301.5310v2 [quant-ph] 4 Feb 2013 Dual quantum information splitting with degenerate graph states Akshata Shenoy H, 1, ∗ R. Srikanth, 2, 3, † and T. Srinivas 1 1 Applied Photonics Lab, ECE Dept., IISc, Bangalore, India 2 Poornaprajna Institute of Scientific Research, Bengaluru, India 3 Raman Research Institute, Bengaluru, India. We propose a protocol for secret sharing, called dual quantum information splitting (DQIS), that reverses the roles of state and channel in standard quantum information splitting. In this method, a secret is shared via teleportation of a fiducial input state over an entangled state that encodes the secret in a graph state basis. By performing a test of violation of a Bell inequality on the encoded state, the legitimate parties determine if the violation is sufficiently high to permit distilling secret bits. Thus, the code space must be maximally and exclusively nonlocal. To this end, we propose two ways to obtain code words that are degenerate with respect to a Bell operator. The security of DQIS comes from monogamy of nonlocal correlations, which we illustrate by means of a simple single-qubit attack model. The nonlocal basis of security of our protocol makes it suitable for security in general monogamous theories and in the more stringent, device-independent cryptographic scenario. I. INTRODUCTION Quantum entanglement enables tasks in communica- tion and cryptography not possible in the classical world, e.g., quantum teleportation [1], dense coding and un- conditionally secure key distribution [2]. Experimental breakthroughs have enabled practical creation and ma- nipulation of entanglement [3], an achievement duly rec- ognized by the 2012 Nobel prizes in physics. Several teleportation-based protocols with multi-particle chan- nels have been proposed [4–12]. In particular, entangle- ment can be used for quantum secret sharing (QSS), the quantum version of classical secret sharing [13]. QSS in- volves a secret dealer splitting information, representing the secret quantum state |Ψ〉, among a number of agents, such that only authorized subsets of them can reconstruct the secret. A protocol for splitting quantum information, and tele- porting it to more than one party over an entangled chan- nel, such that a subset of agents sharing the entangle- ment, is able to reconstruct the information, was first proposed in Ref. [14], further studied by various au- thors [15–19], and also implemented experimentally [20– 22] (the last employing only sequential measurements on a single qubit). We will refer to such teleportation-based QSS as quantum information splitting (QIS). Both QSS and QIS can be used to share both quantum and classical secrets. An important resource of entanglement are graph states that are useful in quantum error correction [23], one-way quantum computing [24] and cryptography [10, 25–27]. They have been studied extensively theoretically, and been realized experimentally recently [28, 29]. Given a graph G =(V,E) defined by the set V of n vertices, and set E of edges, we denote by N (j ), the set of vertices with which vertex j is connected by an edge * Electronic address: akshata@ece.iisc.ernet.in † Electronic address: srik@poornaprajna.org (the neighborhood). Corresponding to each vertex j , one can associate a stabilizer operator: g j = X j  k∈N (j) Z k , (1) where Z k and X k , along with Y k , denote Pauli ma- trices acting on qubit k. We define the graph state basis by the 2 n common eigenstates |G x 〉 ≡ |G x1x2···xn 〉 =  j (Z j ) xj |G 000···0 〉, with (x j ∈{0, 1}) of the n commuting operators g j , where g j |G x1x2···xn 〉 = (−1) xj |G x1x2···xn 〉. In particular, the canonical n-qubit graph state |G〉≡|G 00···0 〉 is characterized by n indepen- dent perfect correlations of the form g j |G〉 = |G〉. (2) The set of all 2 n products (h k ) of the g j ’s forms the sta- bilizer group S . It follows from Eq. (2) that h j |G〉 = |G〉 for all h j ∈S . Graph states are robust against decoher- ence [30], which enhances their practical value. An alternate equivalent definition of graph states, based on their generation via an Ising type of interac- tion, is as follows: |G〉 =Π (j,k)∈E C {j,k} Z |+〉, (3) where C Z is the controlled-phase gate. Here we use the usual notation Z |0〉 = |0〉,Z |1〉 = −|1〉,X |±〉 = ±|±〉. A special class of graph states are the linear cluster states, which correspond to a linear graph. An n-qubit cluster state is given by: |φ N 〉 = 1 2 n/2  j (|0〉 + |1〉 j Z j+1 ), (4) with Z n+1 ≡ 1. For example, |φ 4 〉 = 1 2 (|+0+0〉 + |+0−1〉 + |−1−0〉 + |−1+1〉) , (5) where we use the notation |+0+0〉 = |+〉|0〉|+〉|0〉, etc. It should be noted that different graphs may lead to the