Generation and Application of Nested Entanglement in Matryoshka Quantum Resource-States

Multipartite entanglement is a resource for application in disparate protocols, of computing, communication and cryptography. Nested entanglement provides resource-states for quantum information processing. In this paper, Matryoshka quantum resource-states, which contain nested entanglement patterns, has been studied. A novel scheme for the generation of such quantum states has been proposed using an anisotropic XY spin-spin interaction-based model. The application of the Matryoshka GHZ-Bell states for n-qubit teleportation is reviewed and an extension to more general Matryoshka ExhS-Bell states is posited. An example of Matryoshka ExhS-Bell states is given in the form of the genuinely entangled seven-qubit Xin-Wei Zha state. Generation, characterisation and application of this seven-qubit resource state in theoretical schemes for quantum teleportation of arbitrary one, two and three qubits states, bidirectional teleportation of arbitrary two qubit states and probabilistic circular controlled teleportation are presented.

Entanglement has also been key in quantum communication protocols [53][54][55], with Quantum Repeaters and Entanglement Purification being the subject of interest lately [56][57][58]. For information transfer using entanglement between parties, one needs an established entangled channel-state and means of classical communication. For a large number of parties, multipartite entanglement and entangled multiqubit states play the preeminent role, with states varying from GHZ-and Wstates to clusters states [59,60]. Teleportation of an arbitrary single qubit state using a channel comprising of an EPR pair was first demonstrated by Bennett et al [14]. Lately, W-GHZ composite states have been used for teleportation, remote state sharing as well as superdense coding of arbitrary quantum states [61]. These composite (Matryoshka) quantum states contain nested entanglement, which can be used for applications in quantum information processing. In this paper, the generation, characterisation and application of Matryoshka GHZ-Bell and GHZ-ExhS quantum resource-states will be explored.

Generation of Matryoshka GHZ-Bell States.
The primary resource state used as a resource in this study is a GHZ-Bell state. The concept of Matryoshka states was first given by Di Franco et al [62], with the name 'Matryoshka' coming from the Russian word for 'nesting doll'. In such states, we have a nested entanglement pattern. In this paper, we will be looking at the Matryoshka generation of the GHZ-Bell states over arbitrary number of qubits. We consider N spin-1 2 particles, with each spin coupled to its nearest neighbors by the XY Hamiltonian where J σ,i is the pairwise coupling constant with σ = X,Ŷ ,Ẑ being the Pauli operators. For the purposes of this paper, we take N to be odd. Franco et al [62] showed that it is sufficient to state that the information flux between theX (Ŷ ) operators of the first and last qubits in the spin-chain depends on an alternating set of coupling strengths. For example, the information flux fromX 1 tô X N depends only on the set {J Y,1 , J X,2 , ..., J Y,N −1 } and is independent of any other coupling rate in the spinchain. Christandl et al [63,64] showed that after a time t * = π/λ with λ being a scaling constant (as mentioned in the definition of the case of a perfect state transfer in a linear spin-chain given by weighted coupling strengths: J σ,i = λ i(N − i)), the state of the first qubit in the spin-chain can be perfectly transferred to the last qubit. We see that by preparing the initial state of this spinchain in an completely separable eigenstate of the tensorial product of Z i operators, say |Ψ(0) = |000...0 12...N , we obtain an information flux towards symmetric twosite spin operators, and a final state of the form [62] (3) where c labels the central site of the spin-chain, M = N −3 4 and |ψ ± = 1 √ 2 (|00 ± |11 ). An illustration of the setup has been shown in Figure 1.
The critical step in the creation of the Matryoshka GHZ-Bell state is the evolution of the central and two neighbouring qubits to the GHZ state, without disturbing the rest of the spin-chain. For this, we need to switch off all the interactions except for those connecting the central qubit to the neighbouring ones. A point to note here is that had we started with |Ψ(0) = |111...1 12...N , we would have obtained a final state of the form (5) We use this principle and the idea that after evolution over time t * , the states in equations (2) and (3) transform back to |000...000 12...N and states in equations (4) and (5) transform back to |111...11 12...N . We can utilise this concept, by taking the state in equation (2) and evolving it, for the truncated subsystem comprising of the central qubit and the adjoining qubits. A point to note here is that due to only coupling that connects to the central qubits, the coupling strength (J σ,i = λ i(3 − i)) and time of evolution (t" = π/λ ) vary accordingly. Before carrying out this evolution, we perform a Hadamard operation on the central qubit to give (6) We now perform the truncated subsystem time-evolution with the parameters (J , t") to give us the state (7) Therefore, we can obtain a Matryoshka GHZ-Bell state using nearest spin-spin interactions in a spin-chain.
Teleportation of arbitrary n-qubit quantum states using Matryoshka GHZ-Bell States. The teleportation of an arbitrary n-qubit state can be performed using Matryoshka GHZ-Bell States. Saha et al proposed a scheme for teleportation of a multiqubit state using the following resource state [65]: To achieve the teleportation of an n-qubit state, we start with a 2n + 1 qubit state of the form given in equation (8), with Alice having n + 1 qubits and Bob having n qubits. Let us say the arbitrary n-qubit state Alice wants to teleport to Bob is: where a i denotes the binary representation of i. The combined state |ψ c can then be written in terms of subsystems possessed by Alice and Bob, where |ω i A ∀i ∈ [0, 2 2n − 1] constitute a mutually orthogonal basis. It is seen that for even n, |ω 0 1,4,3,2,7,...,2n,2n−3,2n−2,2n+1 = |GHZ + |ψ + ...|ψ + and for odd n, |ω 0 1,2,5,4,3,...,2n,2n−3,2n−2,2n+1 = |GHZ + |ψ + ...|ψ + , where |GHZ + = 1 √ 2 (|000 + |111 ) and |ψ + = 1 √ 2 (|00 + |11 ) [65]. We can obtain the other |ω i A from this where i is the decimal representation of the string b 2n ......b 2 b 1 with a general b k being 0 or 1. After measurement is done using these orthogonal states, Alice's state evolves into one of the states |ω i . If her state is |ω i then Bob must apply ⊗ n k=1 (Z k ) b k (X k ) b k+n on his n-qubit system to obtain the unknown state, where i is the decimal representation of the string b 2n ......b 2 b 1 with a general b k being 0 or 1.

Extension of Matryoshka formalism to ExhS-Bell
States. We can extend the idea of nested entanglement (Matryoshka formalism) to the case of ExhS-Bell States, which comprise of the exhaustive set over the basis for the qubit states in the ExhS subsystem alongwith the maximally entangled Bell-states for the remainder subsystem. Such states are found to be less entangled in the three-qubit subsystem but have greater nested entangled resources.  (11) It is seen that this state can be used for teleportation of arbitrary single, double and triple qubit states. The 3-2-2 structure of the resource-state, given in equation (11), helps us in devising a quantum circuit to generate the state, as shown in Figure 2  To obtain the resource-state, we apply a unitary operator on qubits 1, 3 and 5: U = I 4×4 ⊕ (σ z ⊗ σ z ). This state has marginal density matrices for subsystems over one or two qubits that are completely mixed, with 2, 3, 4, 5, 6, 7}, i < j (13) For three-qubit subsystems, some of the partitions have mixed marginal density matrices: π 127 = π 367 = π 457 = 1 4 (16) Linear, Bidirectional and Circular Quantum Teleportation using the XZW Resource State. The seven-qubit genuinely entangled resource state |Γ 7 can be used for a number of applications, such as the teleportation of arbitrary one, two and three qubit states. To begin with, an arbitrary single qubit state can be teleported using the resource state |Γ 7 will be considered. In this case Alice possesses qubits 1, 2, 3, 4, 5, 6 and the 7th particle belongs to Bob. Alice wants to transport an arbitrary state |ψ (1) = α|0 + β|1 to Bob. The combined state of the system is |Γ (1) 7 = |ψ (1) ⊗ |Γ 7 . Alice measures the seven qubits in her possession via the seven qubit orthonormal states: Alice then conveys the outcome of the measurement results to Bob via two classical bits. Bob then applies a suitable unitary operation from the set I, σ x , iσ y , σ z to recover the original state, sent by Alice.
Let us say Alice and Bob would like to teleport two-qubit states to each other by utilizing the sevenqubit genuinely entangled resource state. We assume the form of the two-qubit states to be For the resource-state, let Alice have the qubits 1,4 and 7, while Bob has the qubits 2, 3 and 6 and Charlie has the qubit 5.
The steps for the scheme are as follows: • Alice measures qubit 7 of the resource state and A 1 in the bell basis.
• Bob measures qubit 2 of the resource state and B 1 in the bell basis.
• Charlie, Alice and Bob measure their qubits in the Z-basis.
• Alice and Bob measure their qubits A 2 and B 2 in the X-basis.
• We apply unitary transformations to the composite state to now get Alice's initial arbitrary state in Bob's terminal and Bob's initial arbitrary state in Alice's terminal.
Due to the special 3-2-2 form of the resource-state, we can also use entanglement swapping to perform bidirectional teleportation [75].
The seven-qubit resource state can be used for the perfect linear teleportation of an arbitrary three qubit state. In this case, Alice possesses qubits 1, 2, 3, 4 and 5, and the 6th and 7th particles belong to Bob. Alice wants to transport an arbitrary state |ψ (3) = a|000 + b|001 + c|010 + d|011 + e|100 + f |101 + g|110 + h|111 to Bob. Using the decomposition given in Supplementary Material, the states possessed by, and the unitary transforms to be performed by, Bob have been recorded, to accomplish the teleportation of an arbitrary three-qubit state. A point to note here is that we get the GHZ state for a = h = 1 Not only is the seven-qubit resource state useful for linear and bidirectional teleportation but can also facilitate the probabilistic teleportation of an arbitrary single-qubit states in a circular manner between three network-nodes (users). Let us say I have Alice, Bob and Charlie in the system, with the first qubit used as a control qubit, qubits 1 and 4 given to Alice, qubits 2 and 6 given to Bob and qubits 3 and 7 given to Charlie. Let us say the arbitrary states are where |Γ 7 T is the control qubit. We apply a CNOT gate using the qubits A, B and C of the arbitrary states as the control-qubits and the first qubits of each user as the target-qubit. Let us for simplicity only consider the case where |Γ 7 T = |0 . Let us now measure the first qubits of Alice, Bob and Charlie in the Z-basis. Let us say |Γ 7 A1B1C1 = |010 , then we have the state |ψ We can now measure the control qubits in the X-basis. So, let us say, I have . Therefore I see that the users can obtain states derived from the original state of the users next to them (Alice → Bob → Charlie → Alice). However, as you can see, this can be done in a probabilistic manner with one of the users not quite obtaining the original state but rather a derivative-state based on the original.

Conclusion.
In this paper, the generation and application of nested entanglement in Matryoshka resource-states for quantum information processing was studied. A novel scheme for the generation of such quantum states has been proposed using an anisotropic XY spin-spin interaction-based model. The application of the Matryoshka GHZ-Bell states for n-qubit teleportation is reviewed and an extension to more general Matryoshka ExhS-Bell states is posited. An example of Matryoshka ExhS-Bell states was given in the form of the genuinely entangled seven-qubit Xin-Wei Zha state. Generation, characterisation and application of this seven-qubit resource state in theoretical schemes for quantum teleportation of arbitrary one, two and three qubits states, bidirectional teleportation of arbitrary two qubit states and probabilistic circular controlled teleportation were presented. This work should lay the groundwork for other studies into the area of nested entanglement.

Data Availability Statement.
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
Acknowledgement. I would like to acknowledge the guidance and contribution of Prof. Prasanta Panigrahi, IISER-Kolkata. This work was supported by the Trinity Barlow Scholarship and Nehru Bursary -Cambridge, and the Homi Bhabha Centre for Science Education, Tata Institute of Fundamental Research, Mumbai, India. The author acknowledges Prof. V. Singh of HBCSE for being a part of this effort.

Supplementary Materials
The teleportation of an arbitrary three-qubit state using our resource-state has as the initial composite state,