Quantum Secret Sharing Protocol Using Maximally Entangled Multi-qudit States

Abstract

The purpose of this paper is to develop a (N − k) threshold quantum secret sharing (QSS) scheme by using entangled multi-qudit states shared between N qudits such that (k ≤ N − k). We introduce first multi-qudit separable states of a Hilbert space associated with a disconnected multi-qudit system. The entangled multi-qudit states are obtained from disconnected states by means of a unitary interaction operator governing the evolution of the multi-qudit system, where the pairwise interaction establishes links between qudits. The generated entangled states are chosen to be maximally entangled with respect to a specific bi-partition ($A_{2} \bigcup A_{1}$) with k = |A₂|≤|A₁| = (N − k) of the whole system such that the von Neumann entropy $S(\rho _{A_{2}})$ is maximal. The maximally entanglement property with respect to the splitting ($A_{2} \bigcup A_{1}$) of this N-qudit entangled states will be used by a dealer (D) to share an encoded quantum secret with (N − 1) other players, such that at least the (N − k) specified players belonging to A₁ have to cooperate jointly to get the complete information about the secret.

1 Introduction

Quantum entanglement is the most peculiar phenomenon in quantum mechanics that has no classical analog. It is a promising resource for many applications in quantum information processing, quantum computing, quantum cryptography, and quantum communication (for review, see [1,2,3,4]). Multiqudit entangled states arise when qudits of a multiqudit quantum system are allowed to interact. The main features of entangled multipartite states are widely investigated (for review, see [5,6,7,8,9]). The k-maximally mixed states (see [10,11,12,13]) are a special kind of entangled states that present particular genuine multipartite entanglement and manifest many encouraging usages in the area of quantum secret communication and quantum computation.

There are many purposes why one would like to communicate with a legitimate recipient while making sure at the same time that no one is spying. The unconditional security of classical protocols cannot be demonstrated and is based on the complexity of the task to break them. In contrast, quantum protocols are based on the laws of quantum physics and consequently provide unconditional security. One of the highly secured quantum protocols is quantum secret sharing protocol, which was first introduced by Hillery et al in [14] using a three-particle GHZ state. The threshold QSS scheme is realized in [15,16,17]. Various quantum secret sharing schemes have been explained in the literature (see for example [18,19,20,21,22,23,24,25]). The quantum secret sharing scheme using graph states [26] was constructed from qubits [27] and qudits [28]. In the works [29, 30], quantum secret sharing protocols were developed by using absolutely maximally entangled states. QSS was experimentally validated and confirmed in several works (see for instance [31,32,33,34,35,36,37,38,39]). QSS schemes maintain numerous real applications in different areas of quantum information and are of prominent importance in quantum computation [15]. The principal goal of quantum secret sharing protocol is to share an encoded quantum state (secret) with a high level of protection amongst N players such that only particular subsets of players can get access to the original secret.

The main aim of this work is to build (N − k) threshold quantum secret sharing schemes with maximally entangled states shared between N qudits. Our approach is based upon the fact that qudits can be described via deformed bosonic algebras possessing finite-dimensional representations. The algebraic description of qudit systems allows us to define a disconnected quantum state as a initial state of a collection of non-interacting qudits. The stabilizer entangled states are obtained from disconnected states by assuming that the dynamics of the whole system is governed by a special unitary interaction operator expressed in terms of the number operators of the individual qudit-algebras. The stabilizer entangled states are chosen to be maximally entangled with respect to the specific bi-partition $\left [A_{2} = \{ n_{1},..,n_{k} \} \right ]$ $\bigcup \left [A_{1} = \{ n_{k+1},..,n_{N} \} \right ]$. We denote these states as (k, N)-maximally entangled states which are not k-uniform states as defined in ([10,11,12]). In this work, the dealer selects the legal players by choosing the appropriate bipartition and its corresponding maximally entangled state. Hereafter, he builds (N − k) threshold quantum secret sharing schemes such that at least the specified players have to work together to get back the original secret.

This paper is organized as follows. In Section 2, we introduce a special form of generalized Weyl–Heisenberg algebra possessing finite-dimensional representations which is appropriate to describe quantum physical systems of d-level. We give the consistent way to introduce the so-called stabilizer maximally entangled states and the corresponding stabilizer codes. The stabilizer states are maximally entangled with respect to the specific bipartition $\left [A_{2} = \{ n_{1},..,n_{k} \} \right ] \bigcup \left [A_{1} = \{ n_{k+1},..,n_{N} \} \right ]$ of the whole system. In Section 3, we explain the more general case of QSS schemes using stabilizer (k, N)-maximally entangled states shared between N qudits. Then, we build some interesting example of QSS schemes with some particular (k, N)-maximally entangled states. We finish with concluding remarks.

2 Maximally Entangled Multi-qudit States

2.1 Multi-qudit Algebra

Following Wu and Lidar [40], qubits (two level systems (d = 2)) are objects which exhibits simultaneously both bosonic and fermionic properties and cannot be described by Bose-like or Fermi-like operators. Qudits are generalizations of qubits and can be implemented by using quantum physical systems of d-level (d > 2). In this work, we consider the qudit algebra ${\mathscr{H}}$ spanned by the operators {Q⁺, Q⁻, M} and the identity $\mathbb {I}$ satisfying the structure relations [41,42,43,44,45]

$$ [ Q^{-} , Q^{+} ] = (d-1)\mathbb{I} - 2M , \qquad [Q^{+} , M] = Q^{+}, \qquad [Q^{-} , M ] = - Q^{-}. $$

(1)

The qudit-algebra ${\mathscr{H}}$ is a variant form of the non linear generalized Weyl-Heisenberg algebras $\mathcal {A}_{\kappa }$ discussed in [43]. More concretely, if we set (see [41])

$$ \begin{array}{@{}rcl@{}} B^{\pm} = \frac{1}{\sqrt{d-1}} Q^{\pm}. \end{array} $$

(2)

Then, we show that the operators B⁻, B⁺, and M span the Weyl-Heisenberg algebra $\mathcal {A}_{\kappa }$

$$ \begin{array}{@{}rcl@{}} [B^{-} , B^{+}] = \mathbb{I} + 2 \kappa M, \quad [M , B^{\pm}] = \pm B^{\pm}, \quad (B^{-})^{\dagger} = B^{+}, \quad M^{\dagger} = M, \end{array} $$

(3)

where the parameter κ is

$$ \begin{array}{@{}rcl@{}} \kappa = - \frac{1}{d-1}. \end{array} $$

(4)

Furthermore, since $- \frac {1}{d-1} < 0$, the algebra $\mathcal {A}_{\kappa }$ admits finite-dimensional representations (see [43]). Then the corresponding representation space of the qudit algebra ${\mathscr{H}}$ is d-dimensional. It is given by the set {|n〉, n = 0,1,…d − 1} spanned by the eigenstates of the number operator M (M|n〉 = n|n〉). The actions of the operators Q⁻ and Q⁺ on the basis of the Hilbert space ${\mathscr{H}}$ are given by [43]

$$ Q^{+} \vert n \rangle = \sqrt{F(n+1)}~ \vert n+1 \rangle, \qquad Q^{-} \vert n \rangle = \sqrt{F(n)}~ \vert n-1 \rangle $$

(5)

$$ Q^{-} \vert 0 \rangle = 0 \qquad Q^{+} \vert d-1 \rangle = 0. $$

(6)

with F(n) is the structure function defined by F(n) = n(d − n). The creation and annihilation operators satisfy the following nilpotency conditions

$$ (Q^{-})^{d} = 0 \qquad (Q^{+})^{d} = 0, $$

(7)

which extends the Pauli exclusion principle (see for instance [44]) for ordinary qubits.

Now we consider a set of N qudit-algebras associated to N quantum qudit-systems. We denote as $\{ Q_{i}^{+}, Q_{i}^{-} , M_{i};\quad i=1,2,\cdots ,N\}$ the raising, lowering, and number operators associated with N qudits. They satisfy the commutation relations

$$ \begin{array}{@{}rcl@{}} \left[ {Q}_{i}^{-}, {Q}_{j}^{+}\right ] = \left( (d-1)\mathbb{I} - 2 M_{i}\right)~ \delta_{ij}, \quad \left[ {Q}_{i}^{+}, {Q}_{j}^{+}\right ] = \left[ {Q}_{i}^{-}, {Q}_{j}^{-}\right ]=0,\\ \left[ {Q}_{i}^{\pm}, M_{j} \right ] = \pm \delta_{ij} {Q}_{i}^{\pm} \end{array} $$

(8)

The corresponding multi-qudit Hilbert space ${\mathscr{H}}^{\otimes N}$ is given as tensor product of N copies of single qudit Hilbert space ${\mathscr{H}}$ with the orthonormal basis {|n₁, n₂,⋯ , n_i,⋯ , n_N〉;n_i = 0,1,…, d − 1}. The actions of the number operators M_i on the computational basis are defined by

$$ M_{i}\vert n_{1},\cdots,n_{N} \rangle = n_{i}\vert n_{1},\cdots,n_{N} \rangle; \quad M_{i} M_{j} = M_{j} M_{i}. $$

(9)

Hence, an arbitrary normalized multi-qudit state of ${\mathscr{H}}^{\otimes N}$ takes the following form

$$ \vert \phi \rangle = \sum\limits_{n_{1},n_{2},\cdots,n_{N} =0}^{d-1} \alpha_{n_{1},n_{2},\cdots,n_{N}} \vert n_{1},n_{2},\cdots,n_{N} \rangle. $$

(10)

with ${\sum }_{n_{1},n_{2},\cdots ,n_{N} } |\alpha _{n_{1},n_{2},\cdots ,n_{N}}|^{2} = 1$. In this paper, we consider a particular case of the general state (10) where all the coefficients are equal to a unique non-zero value ($\alpha _{n_{1},n_{2},\cdots ,n_{N}} = \frac {1}{\sqrt {d^{N}}})$. This particular state of the multi-qudit system corresponds to the situation, where all qudits occupying the sites are disconnected and prepared in $(\vert + \rangle _{i} = \frac {1}{\sqrt {d}}({\sum }_{n_{j}=0}^{d-1} \vert n_{j} \rangle _{i} )$. The initial state of the multi-qudit system denoted now (|ϕ₀〉) can be considered as the N-qudit state without any connection between qudits and it is given by

$$ \vert \phi_{0} \rangle= \vert +, +,...,+ \rangle = \frac{1}{\sqrt{d^{N}}}\sum\limits_{n_{1},n_{2},\cdots,n_{N} =0}^{d-1} \vert n_{1},n_{2},\cdots,n_{N} \rangle. $$

(11)

The corresponding separable density matrices denoted σ writes as

$$ \sigma = \vert \phi_{0} \rangle \langle \phi_{0} \vert = \left[ \frac{1}{d} \sum\limits_{n_{i},n_{i}^{\prime}=0}^{d-1} \vert n_{i} \rangle \langle {n}_{i}^{\prime} \vert \right]^{\otimes N}. $$

(12)

2.2 Evolution of Multi-qudit States

The evolution of the disconnected multi-qudit system writes

$$ U(t)~\sigma~ U^{\dagger}(t) = e^{-itH} ~\sigma~ e^{+itH} \equiv \sigma_{t}. $$

(13)

We assume that is governed by an Hamiltonian of Heisenberg type involving qudit-qudit interaction of the form

$$ H = \sum\limits_{i<j}^{N} a_{ij} M_{i} M_{j}. $$

(14)

The physical parameters a_ij entering in the expression of the Hamiltonian describing the multi-qubit system are reals and symmetric (a_ij = a_ji). The unitary operator e^−itH is diagonal in the computational basis

$$ e^{-itH} \vert n_{1},\cdots,n_{N} \rangle = e^{-it[{\sum}_{i<j}^{N} a_{ij} n_{i}n_{j}]}\vert n_{1},\cdots,n_{N} \rangle $$

(15)

In the special situation where the parameter t takes the following discrete values

$$ t = \frac{2\pi}{d}~(d-q) \quad \text{with}\quad q \in \mathbb{Z}/d\mathbb{Z} $$

(16)

the resulting labelled evolved density matrices denoted now ϱ ≡ σ_t writes as

$$ \varrho = \frac{1}{{{d^{N}}}} \sum\limits_{\underset{n^{\prime}_{1},\dots,n^{\prime}_{N}}{n_{1},\dots,n_{N}}} \omega^{({\sum}_{i<j} p_{ij}(n_{i} n_{j}- {n}_{i}^{\prime} {n}_{j}^{\prime})) } |n_{1},\dots,n_{N} \rangle\langle {n}_{1}^{\prime},\dots, {n}_{N}^{\prime}| $$

(17)

where the coupling parameters $\{ p_{ij}= p a_{ij} \in \mathbb {Z}/d\mathbb {Z} \}$ encode the pairwise interaction between the different components of the quantum multi-qudit system and denote the number of connections linking two sites i and j. The entangled density matrices ϱ can be expressed as

$$ \varrho = \vert{\Phi}_{N} \rangle\langle {\Phi}_{N} \vert, $$

(18)

where the entangled multi-qudit vector state |Φ_N〉 is given by

$$ \vert{\Phi}_{N}\rangle = \frac{1}{{{\sqrt{d^{N}}}}} \sum\limits_{n_{1},\dots,n_{N}} \omega^{{\sum}_{i < j} p_{ij} n_{i} n_{j}} \vert n_{1},n_{2},\cdots,n_{N} \rangle. $$

(19)

To study the bipartite entanglement of the resulting entangled multi-qudit system, we split the entire system into two subsets $\left [A_{2} = \{ n_{1},..,n_{k} \} \right ] \bigcup \left [A_{1} = \{ n_{k+1},..,n_{N} \} \right ]$. The matrix representing the connections between the qudits living in A₁ and the qudits of A₂ is given by

$$ \begin{array}{@{}rcl@{}} C_{k}= \left( \begin{array}{llllll} \quad p_{1(k+1)} &\quad p_{1(k+2)} & {\ldots} &{\ldots} &\quad p_{1(N-1)} & p_{1N}\\ \quad p_{2(k+1)} & \quad p_{2(k+2)} & {\ldots} &{\ldots} & \quad p_{2(N-1)} & p_{2N}\\ \quad \quad {\vdots} & \quad \ldots& \ldots&{\ldots} & \quad \quad {\ldots} & \quad {\vdots} \\ \quad \quad {\vdots} & \quad \ldots& \ldots&{\ldots} & \quad \quad {\ldots} & \quad {\vdots} \\ p_{(k-1)(k+1)} & p_{(k-1)(k+2)} & {\ldots} &{\ldots} & p_{(k-1)(N-1)} & p_{(k-1)N}\\ \quad p_{k(k+1)} &\quad p_{k(k+2)} & {\ldots} &{\ldots} & \quad p_{k(N-1)} & \quad p_{kN} \end{array}\right) \end{array} $$

The density matrix of the subsystem A₂ is given by

$$ \begin{array}{@{}rcl@{}} \rho_{A_{2}}=Tr_{A_{1}}[|{\Phi}_{N}\rangle\langle{\Phi}_{N}|]. \end{array} $$

(20)

The entangled state |Φ_N〉 is a maximally entangled state, with respect to the bipartition of the entire system into the two parts $[A_{2}=\{n_{1},\ldots ,n_{k}\}]\bigcup [A_{1}=\{n_{k+1},\ldots , n_{N}\}]$, that is, the von Neumann entropy is maximal

$$ S(\rho_{A_{2}})= k ~ log_{2}~ d, $$

if and only if the k vectors (p_i(k+ 1), p_i(k+ 2),..., p_iN) with (1 ≤ i ≤ k) are linearly independent (see [29, 30]). In the following, a maximally entangled state with respect to the bipartition $\left [A_{2} = \{ n_{1},..,n_{k} \} \right ] \bigcup \left [A_{1} = \{ n_{k+1},..,n_{N} \} \right ]$, termed (k, N)-maximally entangled state will be denoted |Φ_{(k, N)}〉 and is expressed by

$$ \vert {\Phi}_{(k,N)} \rangle = \frac{1}{{{\sqrt{d^{N}}}}} \sum\limits_{n_{1},\dots,n_{N}} \omega^{{\sum}_{i < j} p_{ij} n_{i} n_{j}} \vert n_{1},n_{2},\cdots,n_{N} \rangle. $$

(21)

2.3 Maximally Entangled Multi-qudit States Stabilizer

To construct maximally entangled states stabilizer, we introduce first a positive map denoted $\mathcal {G}_{\vec {m}}$ that maps a (k, N)-maximally entangled state |Φ_{(k, N)}〉 to a labelled (k, N)-maximally entangled state $\vert {\Phi }_{(k,N)}^{\vec {m}} \rangle$ parameterized by the sets of parameters $\{ \vec {m} \equiv (m_{1},m_{2},\cdots ,m_{n}); m_{i} \in \mathbb {Z}/d \mathbb {Z} \}$ and obtained as

$$ \vert {\Phi}_{(k,N)}^{\vec{m}} \rangle = ~\mathcal{G}_{\vec{m}}~(\vert{\Phi}_{(k,N)} \rangle) = \left( {\bigotimes}_{i=1}^{n} {Z}_{i}^{m_{i}}\right) \vert{\Phi}_{(k,N)} \rangle. $$

(22)

The labelled maximally entangled state with respect to the bipartition $\left [A_{2} = \{ n_{1},..,n_{k} \} \right ] \bigcup \left [A_{1} = \{ n_{k+1},..,n_{N} \} \right ]$, given by (22) are expressed by means of the local unitary operations

$$ Z_{i} = \sum\limits_{n_{i}=0}^{d-1} \omega^{n_{i}} \vert n_{i} \rangle \! \langle n_{i} \vert,\quad $$

which do not affect the entanglement present in the initial (k, N)-maximally entangled states |Φ_{(k, N)}〉. It follows that the pair of N-qudit entangled states $||{\Phi }_{(k,N)}^{\vec {m}}\rangle \rangle$ and |Φ_{(k, N)}〉 maximally entangled, with respect to the bipartition $[A_{2}=\{n_{1},\ldots ,n_{k}\}]\bigcup [A_{1}=\{n_{k+1},\ldots , n_{N}\}]$, have the same value for monotone entanglement measures. We note also that the (k, N)-maximally entangled states |Φ_{(k, N)}〉 are stabilizer states such that

$$ \mathcal{S}_{i} |{\Phi}_{(k,N)}\rangle = |{\Phi}_{(k,N)}\rangle, $$

(23)

where the corresponding n commuting stabilizer generators $\{\mathcal {S}_{i},i=1,2,...,N \}$ are given by:

$$ \mathcal{S}_{i} = X_{i} \prod\limits_{j=1,j\neq i}^{N} {Z}_{j}^{p_{ij}}. $$

(24)

X_i is the generalized Pauli operator referred as shift operator acting on a single d-dimensional Hilbert space ${\mathscr{H}}_{i}$ corresponding to the i th qudit and is given by

$$ X_{i} = \sum\limits_{n_{i}=0}^{d-1} \vert n_{i} \rangle \! \langle n_{i} \oplus 1 \vert, $$

(25)

with ⊕ is the addition modulo d. The N stabilizer operators $\{ \mathcal {S}_{i}, 1 \leq i \leq N \}$ span the so-called stabilizer group denoted as $\mathcal {S}_{N}$. The stabilizer group $\mathcal {S}_{N}$ is defined as an abelian subgroup of the multiplicative Pauli group $\mathcal {P}_{N}$ whose elements are operators of the form:

$$ P = {X}_{1}^{m_{1}} {Z}_{1}^{l_{1}} {X}_{2}^{m_{2}} {Z}_{2}^{l_{2}} {\cdots} {X}_{N}^{m_{N}} {Z}_{N}^{l_{N}}, $$

(26)

with the parameters m_i and l_i are integers in the range 0 to d − 1. The action of the N stabilizer operators $\{ \mathcal {S}_{i} \}$ on the labelled maximally entangled states with respect to the bipartition $A_{2}\bigcup A_{1}$,

$$ \vert {\Phi}_{(k,N)}^{\vec{m}} \rangle = \frac{1}{\sqrt{d^{N}}}\sum\limits_{n_{1},n_{2},\cdots,n_{N}} \omega^{m_{1} n_{1}}\omega^{m_{2} n_{2}}{\cdots} \omega^{m_{N} n_{N}} \omega^{{\sum}_{i < j} p_{ij} n_{i} n_{j}} \vert n_{1},n_{2},\cdots,n_{N} \rangle $$

(27)

writes as

$$ \mathcal{S}_{i} \vert {\Phi}_{(k,N)}^{\vec{m}} \rangle = \omega^{-m_{i}}\vert {\Phi}_{(k,N)}^{\vec{m}} \rangle. $$

(28)

3 Quantum Secret Sharing with (k, N)-Maximally Entangled Multi-qudit States

In the following, we briefly describe the threshold quantum secret sharing schemes associated with a (k, N)-maximally entangled state shared between N qudits {n₁,..., n_N}. For that, we accentuate that the role of the distributor (Dealer) D is assigned to one of the qudits of the multi-qudit system. The secret is defined by an arbitrary normalized qudit state expressed by

$$ |S\rangle=\sum\limits_{i=0}^{d-1} \alpha_{i} |i\rangle; \text{with} \quad \sum\limits_{i=0}^{d-1} |\alpha_{i}|^{2}=1. $$

(29)

In this paper, we choose {n₁} to play the role of the dealer. Next, he has to prepare first the suitable (k, N)-maximally entangled state |Φ_{(k, N)}〉 with respect to a given bipartition $[A_{2}=\{n_{1},\ldots ,n_{k}\}]\bigcup [A_{1}=\{n_{k+1},\ldots , n_{N}\}]$, where the interaction parameters, p_ij, connecting the qudits living in A₁ and the qudits of A₂ are chosen to specify the legal players in this QSS scheme. Thereupon, the dealer combines a quantum secret |S〉 with the prepared (k, N)-maximally entanled state to form a (N + 1)-qudit entangled state, given by

$$ \begin{array}{@{}rcl@{}} |{{\Psi}},N+1\rangle&=& |S\rangle\otimes|{\Phi}_{(k,N)}\rangle. \end{array} $$

Then, the dealer distributes each share |n_i〉 to each other player and performs on his two qudits a projective measurement onto the Bell basis.

$$ \begin{array}{@{}rcl@{}} |\mathcal{B}_{mn}\rangle\langle\mathcal{B}_{mn}|{\Psi},N+1\rangle= |\mathcal{B}_{mn}\rangle\frac{1}{\sqrt{d^{2}}} \sum\limits_{j} \alpha_{j} \omega^{-jn} |\phi_{j}\rangle. \end{array} $$

where the Bell basis is given by

$$ |\mathcal{B}_{mn}\rangle=\frac{1}{\sqrt{d}}\sum\limits_{j=0}^{d-1} \omega^{jn} |j\rangle|j+m\rangle, \quad 0\leq m,n \leq d-1. $$

(30)

If the dealer informs the players of his measurement outcome (m, n), then only a subset with (N − k) players or more can apply a correction operator to recover the secret shared by the dealer.

$$ \mathcal{S}_{r}^{-n p^{-1}_{1r}} \mathbb{Z}_{k+1}^{-mp_{1(k+1)}} {\ldots} \mathbb{Z}_{N}^{-mp_{1N}}\left( \sum\limits_{j} \alpha_{j} \omega^{-jn} |\phi_{j}\rangle \right)=\sum\limits_{j} \alpha_{j} |\tilde{\phi}_{j}\rangle;\quad r \in \{k+1,...,N\} $$

(31)

where $\mathcal {S}_{r}$ is a stabilizer operator defined in (24). It is noted that this scheme requires the collaboration of at least (N − k) players specified by the dealer to access the secret. In the next section we give the explicit construction of quantum secret sharing schemes by using (k, N)-maximally entangled states shared between N qudits. More concretely, we build QSS schemes with some particular (k, N)-maximally entangled states which allow us to investigate the effect of changing N and also changing k for a fixed N on the construction of the QSS schemes.

3.1 QSS with (1,3) Maximally Entangled State

In the case of three qudits system N = 3, we consider the following maximally entangled state

$$ |{\Phi}_{(1,3)}\rangle=\frac{1}{\sqrt{d^{3}}} \sum\limits_{n_{1},n_{2},n_{3}} \omega^{n_{1} n_{2}} \omega^{n_{1} n_{3}} \omega^{n_{2} n_{3}} |n_{1},n_{2},n_{3}\rangle \quad \{ p_{12}= 1, p_{13} =1, p_{23} =1 \} $$

(32)

The entangled state |Φ_(1,3)〉 as given in (32) is a maximally entangled state with respect to the bipartition of the system into two sets A₂ ∪ A₁ ≡{n₁}∪{n₂, n₃}, that is the reduced density matrix $\rho _{A_{2}}$ is obtained as

$$ \begin{array}{@{}rcl@{}} \rho_{A_{2}} = \sum\limits_{n_{2},n_{3}} \langle n_{2},n_{3} |{\Phi}_{(1,3)}\rangle\langle {\Phi}_{(1,3)} |n_{2},n_{3}\rangle = \frac{1}{d} \mathbb{I}_{d}. \end{array} $$

(33)

Thus, it is worth noticing that the maximally entangled state |Φ_(1,3)〉 rewrites in the Schmidt form as follows:

$$ |{\Phi}_{(1,3)}\rangle=\frac{1}{\sqrt{d}} \sum\limits_{n_{1}} |n_{1}\rangle |\varphi(n_{1})\rangle, $$

(34)

where the states |φ(n₁)〉 are given by

$$ |\varphi(n_{1})\rangle =\frac{1}{\sqrt{d^{2}}} \sum\limits_{n_{2},n_{3}} \omega^{n_{1} n_{2}} \omega^{n_{1} n_{3}} \omega^{n_{2} n_{3}} |n_{2},n_{3}\rangle $$

(35)

are such that

$$ \langle \varphi(n_{1})| \varphi({n}_{1}^{\prime})\rangle =\frac{1}{d^{2}} \sum\limits_{n_{2},n_{3}} \omega^{(n_{1} -{n}_{1}^{\prime})n_{2}} \omega^{(n_{1} - {n}_{1}^{\prime}) n_{3}} = \delta_{n_{1},{n}_{1}^{\prime}}. $$

(36)

To share a quantum secret between n₂ and n₃, the dealer D = {n₁} builds first a 4-qudit quantum state |Ψ,4〉, given by

$$ \begin{array}{@{}rcl@{}} |{\Psi},4\rangle= |S\rangle |{\Phi}_{(1,3)}\rangle=\frac{1}{\sqrt{d^{3}}} \sum\limits_{i,n_{1},n_{2},n_{3}} \alpha_{i} \omega^{n_{1} n_{2}} \omega^{n_{1} n_{3}} \omega^{n_{2} n_{3}} |i,n_{1},n_{2},n_{3}\rangle. \end{array} $$

(37)

The dealer distributes the two qudits {n₂} and {n₃} to the two other players, and performs a Bell measurement of his two qudits in the generalized Bell basis $|{\mathscr{B}}_{mn}\rangle$, as

$$ \begin{array}{@{}rcl@{}} |\mathcal{B}_{mn}\rangle\langle \mathcal{B}_{mn}|{\Psi},4 \rangle=|\mathcal{B}_{mn}\rangle \frac{1}{\sqrt{d^{2}}} \sum\limits_{j} \alpha_{j} \omega^{-jn} |\phi_{j}\rangle, \end{array} $$

(38)

with

$$ |\phi_{j}\rangle=\frac{1}{\sqrt{d^{2}}} \sum\limits_{n_{2},n_{3}} \omega^{n_{2} (j+m)} \omega^{(j+m) n_{3}} \omega^{n_{2} n_{3}} |n_{2},n_{3}\rangle. $$

Therefore, the two players {n₂} and {n₃} must cooperate to recover the secret. For that end, they apply the following operations:

$$ \begin{array}{@{}rcl@{}} \mathcal{S}_{k}^{-n } \mathbb{Z}_{2}^{-m } \mathbb{Z}_{3}^{-m } \left( \sum\limits_{j} \alpha_{j} \omega^{-jn} |\phi_{j}\rangle\right)= \sum\limits_{j} \alpha_{j} |\tilde{\phi}_{j}\rangle, \end{array} $$

(39)

with

$$ |\tilde{\phi}_{j}\rangle =\frac{1}{\sqrt{d^{2}}} \sum\limits_{n_{2},n_{3}} \omega^{n_{2} j} \omega^{j n_{3}} \omega^{n_{2} n_{3}} |n_{2},n_{3}\rangle. $$

The players {n₂} and {n₃} can apply the stabilizer operators, $\mathcal {S}_{2}$ or $\mathcal {S}_{3}$, which are defined respectively as $\mathcal {S}_{2} = X_{2} Z_{3}$, $\mathcal {S}_{3} = X_{3} Z_{2}$. The residual states $|\tilde {\phi }_{j}\rangle$ are basis states $\langle \tilde {\phi }_{j}|\tilde {\phi }_{j}^{\prime }\rangle =\delta _{j,j^{\prime }},$ and are labelled maximally entangled with respect to the bipartition $\{n_{2}\} \bigcup \{n_{3}\}$,

$$ \begin{array}{@{}rcl@{}} \sum\limits_{n_{2}} \langle n_{2} |\tilde{\phi}_{j}\rangle\langle \tilde{\phi}_{j} |n_{2}\rangle = \frac{1}{d} \mathbb{I}_{d}. \end{array} $$

(40)

Therefore, neither the player {n₂} nor the player {n₃} can get any information about the secret, but by collaborating, they will be able to decode and restore the secret.

3.2 QSS with (3,9)-Maximally Entangled States

We set up a ((3,9))-QSS scheme using a maximally entangled state shared between 9-qudits. We consider the following maximally entangled state with respect to the splitting A₁ ∪ A₂ where A₂ = {n₁, n₂, n₃} and A₁ = {n₄, n₅, n₆, n₇, n₈, n₉}.

$$ \begin{array}{@{}rcl@{}} |{\Phi}_{(3,9)}\rangle&=&\frac{1}{\sqrt{d^{9}}} \sum\limits_{n_{1},\dots,n_{9}} \omega^{n_{1}(n_{4}+n_{5}+ n_{6}+ 2 n_{7}+n_{8})} \omega^{n_{2}(n_{4}+n_{6}+n_{9})} \omega^{n_{3}(n_{4}+n_{5}+n_{7})} \\ &&\times \omega^{(n_{4} n_{5}+n_{4} n_{9}+n_{5} n_{6}+n_{6} n_{7}+ n_{7} n_{8}+n_{8} n_{9})} \omega^{(n_{5} n_{9}+n_{6} n_{9}+n_{7} n_{9})}|n_{1},{\ldots} ,n_{9}\rangle. \end{array} $$

The dealer {n₁} uses the state |Φ_(3,9)〉 to share the quantum secret {|S〉} with the other players {n₂, n₃, n₄, n₅, n₆, n₇, n₈, n₉}. For this purpose, he builds the 10-qudit entangled state |Ψ,10〉 = |S〉|Φ_(3,9)〉 expressed explicitly by

$$ \begin{array}{@{}rcl@{}} |{\Psi},10\rangle&=&\frac{1}{\sqrt{d^{9}}} \sum\limits_{i, n_{1},\dots,n_{9}=0}^{d-1} \alpha_{i} \omega^{n_{1}(n_{4}+n_{5}+n_{6}+2 n_{7}+n_{8})} \omega^{n_{2}(n_{4}+n_{6}+n_{9})} \omega^{n_{3}(n_{4}+n_{5}+n_{7})} \\ &&\times \omega^{(n_{4} n_{5}+n_{4} n_{9}+n_{5} n_{6}+n_{6} n_{7}+ n_{7} n_{8}+n_{8} n_{9})} \omega^{(n_{5} n_{9}+n_{6} n_{9}+n_{7} n_{9})} |i,n_{1},{\ldots} ,n_{9}\rangle. \end{array} $$

And he sends the qudits {|n₂〉,|n₃〉,|n₄〉,|n₅〉,|n₆〉,|n₇〉,|n₈〉,|n₉〉} to the other players and measures his two qudits

$$ \begin{array}{@{}rcl@{}} |\mathcal{B}_{mn}\rangle\langle \mathcal{B}_{mn}|{\Psi},10 \rangle= |\mathcal{B}_{mn}\rangle\frac{1}{\sqrt{d^{2}}} \sum\limits_{j} \alpha_{j} \omega^{-jn} |\phi_{j}\rangle, \end{array} $$

(41)

with

$$ \begin{array}{@{}rcl@{}} |\phi_{j}\rangle&=&\frac{1}{\sqrt{d^{8}}} \sum\limits_{n_{2},\dots,n_{9}=0}^{d-1} \omega^{(j+ m)(n_{4}+n_{5}+n_{6}+2n_{7}+n_{8})} \omega^{n_{2}(n_{4}+n_{6}+n_{9})} \omega^{n_{3}(n_{4}+n_{5}+n_{7})}\\ &&\times \omega^{(n_{4} n_{5} +n_{4} n_{9}+n_{5} n_{6}+n_{6} n_{7}+ n_{7} n_{8}+n_{8} n_{9})} \omega^{(n_{5} n_{9}+n_{6} n_{9}+n_{7} n_{9})} |n_{2},{\ldots} ,n_{9}\rangle. \end{array} $$

The players of {A₁} must apply the following operation

$$ \mathcal{S}_{8}^{-n} \mathbb{Z}_{4}^{-m} \mathbb{Z}_{5}^{-m} \mathbb{Z}_{6}^{-m} \mathbb{Z}_{7}^{-2m} \mathbb{Z}_{8}^{-m} \left( \sum\limits_{j} \alpha_{j} \omega^{-jn} |\phi_{j}\rangle \right)=\sum\limits_{j} \alpha_{j} |\tilde{\phi}_{j}\rangle, $$

(42)

where the stabilizer operator is expressed by

$$ \mathcal{S}_{8}=X_{8} Z_{7} Z_{9}. $$

(43)

The resulting states $|\tilde {\phi }_{j}\rangle$ are obtained as follows

$$ \begin{array}{@{}rcl@{}} |\tilde{\phi}_{j}\rangle&=&\frac{1}{\sqrt{d^{8}}} \sum\limits_{n_{2},\dots,n_{9}=0}^{d-1} \omega^{j(n_{4}+n_{5}+n_{6}+2n_{7}+n_{8})} \omega^{n_{2}(n_{4}+n_{6}+n_{9})} \omega^{n_{3}(n_{4}+n_{5}+n_{7})} \\ &&\times \omega^{(n_{4} n_{5}+n_{4} n_{9}+n_{5} n_{6}+n_{6} n_{7}+ n_{7} n_{8}+n_{8} n_{9})} \omega^{(n_{5} n_{9}+n_{6} n_{9}+n_{7} n_{9})} |n_{2},{\ldots} ,n_{9}\rangle. \end{array} $$

(44)

The states $|\tilde {\phi }_{j}\rangle$ are maximally entangled states with respect to the bipartition $\{A^{\prime }_{2}=[n_{2},n_{3}]\}\bigcup \{A^{\prime }_{1}=[n_{4},n_{5},n_{6},n_{7},n_{8},n_{9}]\}$. Then, the players {n₄, n₅, n₆, n₇, n₈, n₉} cannot obtain any information about the secret until they collaborate.

3.3 QSS with (4,9)-Maximally Entangled States

The same as in the previous section, we develop a ((4,9))-QSS scheme using a maximally entangled state shared between 9-qudits. We prepare the following maximally entangled state with respect the bipartition A₁ ∪ A₂ where A₂ = {n₁, n₂, n₃, n₄} and A₁ = {n₅, n₆, n₇, n₈, n₉}.

$$ \begin{array}{@{}rcl@{}} |{\Phi}_{(4,9)}\rangle&=&\frac{1}{\sqrt{d^{9}}} \sum\limits_{n_{1},\dots,n_{9}} \omega^{n_{1}(n_{5}+n_{6}+n_{7}+ n_{8})} \omega^{n_{2}(n_{5}+n_{7}+n_{9})} \omega^{n_{3}(n_{5}+n_{6})} \omega^{n_{4} n_{9}} \\ &&\times \omega^{(n_{5} n_{6}+n_{6} n_{7}+ n_{7} n_{8} + n_{5} n_{9} + n_{6} n_{9}+ n_{7} n_{9} + n_{8} n_{9})}|n_{1},{\ldots} ,n_{9}\rangle. \end{array} $$

(45)

Using this maximally entangled state |Φ_(4,9)〉, the dealer {n₁} can cast the quantum secret {|S〉} with the players {n₂, n₃, n₄, n₅, n₆, n₇, n₈, n₉}. To do that, the dealer prepares the following 10-qudits entangled state |Ψ,10〉 = |S〉|Φ_(4,9)〉 given by

$$ \begin{array}{@{}rcl@{}} |{\Psi},10\rangle&=&\frac{1}{\sqrt{d^{9}}} \sum\limits_{i, n_{1},\dots,n_{9}=0}^{d-1} \alpha_{i} \omega^{n_{1}(n_{5}+n_{6}+ n_{7}+n_{8})} \omega^{n_{2}(n_{5}+n_{7}+n_{9})} \omega^{n_{3}(n_{5}+n_{6})} \omega^{n_{4} n_{9}} \\ &&\times \omega^{(n_{5} n_{6}+n_{6} n_{7}+ n_{7} n_{8}+ n_{5} n_{9} + n_{6} n_{9}+ n_{7} n_{9} + n_{8} n_{9})} |i,n_{1},{\ldots} ,n_{9}\rangle. \end{array} $$

(46)

The distributor delivers the qudits {|n₂〉,|n₃〉,|n₄〉,|n₅〉,|n₆〉,|n₇〉,|n₈〉,|n₉〉} to the other players and performs a measurement of his two qudits in the Bell basis

$$ \begin{array}{@{}rcl@{}} |\mathcal{B}_{mn}\rangle\langle \mathcal{B}_{mn}|{\Psi},10 \rangle= |\mathcal{B}_{mn}\rangle\frac{1}{\sqrt{d^{2}}} \sum\limits_{j} \alpha_{j} \omega^{-jn} |\phi_{j}\rangle, \end{array} $$

such that

$$ \begin{array}{@{}rcl@{}} |\phi_{j}\rangle&=&\frac{1}{\sqrt{d^{8}}} \sum\limits_{n_{2},\dots,n_{9}=0}^{d-1} \omega^{(j+m)(n_{5}+n_{6}+ n_{7}+n_{8})} \omega^{n_{2}(n_{5}+n_{7}+n_{9})} \omega^{n_{3}(n_{5}+n_{6})} \omega^{n_{4} n_{9}} \\ &&\times \omega^{(n_{5} n_{6}+n_{6} n_{7}+ n_{7} n_{8}+ n_{5} n_{9} + n_{6} n_{9}+ n_{7} n_{9} + n_{8} n_{9})} |n_{2},{\ldots} ,n_{9}\rangle. \end{array} $$

(47)

The players of {A₁} must act by the following operation

$$ \mathcal{S}_{8}^{-n} \mathbb{Z}_{5}^{-m} \mathbb{Z}_{6}^{-m} \mathbb{Z}_{7}^{-m} \mathbb{Z}_{8}^{-m} \left( \sum\limits_{j} \alpha_{j} \omega^{-jn} |\phi_{j}\rangle \right)=\sum\limits_{j} \alpha_{j} |\tilde{\phi}_{j}\rangle, $$

(48)

where the stabilizer operator is defined by

$$ \mathcal{S}_{8}=X_{8} Z_{7} Z_{9}, $$

(49)

and the states $|\tilde {\phi }_{j}\rangle$ are given by

$$ \begin{array}{@{}rcl@{}} |\tilde{\phi}_{j}\rangle&=&\frac{1}{\sqrt{d^{8}}} \sum\limits_{n_{2},\dots,n_{9}=0}^{d-1} \omega^{j(n_{5}+n_{6}+ n_{7}+n_{8})} \omega^{n_{2}(n_{5}+n_{7}+n_{9})} \omega^{n_{3}(n_{5}+n_{6})} \omega^{n_{4} n_{9}} \\ &&\times \omega^{(n_{5} n_{6}+n_{6} n_{7}+ n_{7} n_{8}+ n_{5} n_{9} + n_{6} n_{9}+ n_{7} n_{9} + n_{8} n_{9})} |n_{2},{\ldots} ,n_{9}\rangle, \end{array} $$

(50)

The states $|\tilde {\phi }_{j}\rangle$ are maximally entangled states with respect to the bipartition $\{A^{\prime }_{2}=[n_{2},n_{3},n_{4}]\}\bigcup \{A^{\prime }_{1}=[n_{5},n_{6},n_{7},n_{8},n_{9}]\}$. To restore the encoded secret, a collaboration of the players of A₁ is required.

4 Conclusion

We developed quantum secret sharing protocols using maximally entangled states derived from a dynamical evolution of the disconnected multi-qudit system {n₁,…, n_N}. The algebraic description of qudits is formulated via a generalized variant of Weyl–Heisenberg algebra possessing finite-dimensional representations. This description provides the appropriate tool to define consistently a separable quantum multi-qudit state as a initial state of a collection of non-interacting qudits. The evolution of the multi-qudit system, governed by a unitary interaction operator $U(t)= {\prod }_{i=1 < j,}^{N} \omega ^{p_{ij} M_{i}M_{j}}$, allows to generate entangled states from separable states of a multi-qudit algebra. The coupling parameters {p_ij} which encode the pairwise interaction between qudits of the evolved multi-qudit system are chosen such that the entangled states are maximally entangled states with respect to a specific bipartition of the entire system into the two parts $[A_{2}=\{n_{1},\ldots ,n_{k}\}]\bigcup [A_{1}=\{n_{k+1},\ldots , n_{N}\}]$ that is the entanglement entropy is maximal $S(\rho _{A_{2}}) = k~log_{2}~d$. We have also shown that these states allow the dealer to choose the legal players in threshold quantum secret sharing schemes. Therefore, the legal players must cooperate to retrieve the secret, and the illegal players cannot acquire any information. We think that our protocol can be implemented in many applications in quantum internet, quantum communication channels, and for building collaborative quantum networks. We are planning to extend the results of this paper to develop a collaborative quantum network in a forthcoming paper.