**3. Creating tesselated networks of Matryoshka states**

The Matryoshka Generalised GHZ states can also be oriented in a tesselated manner, as shown in **Figure 3(a)** for the case of symmetric 3-qubit GHZ triangular units. The Matryoshka GHZ-Bell states, a specific form of these states, can even be oriented in an emanatory manner, as shown in **Figure 3(b)**. These two orientations can be used for tessellation in three-dimensions, as in the case of the spherical configuration shown in **Figure 3(c)**, which shows the method of lattice surgery (discussed later in the chapter). More complex forms such as the hexagonal-pentagonal tiling with 6-qubit and 5-qubit GHZ states can be used for forms such as truncated icosahedrons. Lastly, we can also have higher GHZ-forms in a self-similar, fractal manner, as shown in **Figure 3(d)**. Each of these configurations will be studied in the *Application* section of this chapter. An interesting future direction of pursuing this line of research would be in squeezed baths, which *Zippilli et al* studied and showed that a squeezed bath, which acts on the central element of a harmonic chain, could drive the entire system to a steady state that features a series of nested entangled pairs of oscillators [63]. This series ideally covers the entire chain regardless of its size. Extending this result to higher number of nearest neighbour interactions is non-trivial.

### **4. Where can we use entangled entanglement?**

Matryoshka states have a second level of entanglement (nesting) and have additional protection against loss of coherence under local transformations.

#### **4.1 Fractal network protocol**

In this chapter, a new quantum communication architecture is being proposed, whereby there are levels of entanglement which underly a distributed network. If we have

#### **Figure 3.**

*The various tesselation patterns possible with the GHZ triangular units in (a) generalised GHZ states in a planar tesselated format, (b) GHZ-Bell states with an emanatory geometry, (c) spherical pattern created by planar codes, along with illustration of lattice surgery with projective measurements, and (d) hierarchical GHZ-state levels, where we have a self-similar nature of the tesselation. A point to note here is that each node in the diagram has three physical qubits (one from each GHZ triangular unit) in the generalised GHZ states and two physical qubits in the GHZ-Bell states.*

$$|\mathbf{0}\rangle\_{L}^{n} = \frac{1}{\sqrt{2}}\left(|\mathbf{0}\_{L}^{n-1}\mathbf{0}\_{L}^{n-1}\mathbf{0}\_{L}^{n-1}\rangle + |\mathbf{1}\_{L}^{n-1}\mathbf{1}\_{L}^{n-1}\mathbf{1}\_{L}^{n-1}\rangle\right) \tag{19}$$

$$|\mathbf{1}\rangle\_{L}^{n} = \frac{1}{\sqrt{2}}\left(|\mathbf{0}\_{L}^{n-1}\mathbf{0}\_{L}^{n-1}\mathbf{0}\_{L}^{n-1}\rangle - |\mathbf{1}\_{L}^{n-1}\mathbf{1}\_{L}^{n-1}\mathbf{1}\_{L}^{n-1}\rangle\right) \tag{20}$$

As you can see, these are special cases of Matryoshka Generalised GHZ states, with the superscript *n* defining the layer of the network. A point to note here is that *<sup>n</sup>* <sup>¼</sup> 1 is the layer with physical qubits, and so <sup>∣</sup>0i<sup>1</sup> *<sup>L</sup>* <sup>¼</sup> <sup>∣</sup>0<sup>i</sup> and <sup>∣</sup>1i<sup>1</sup> *<sup>L</sup>* ¼ ∣1i. This effectively creates layers of *entangled entanglement*. This is highly useful in providing multiple levels of protection in quantum network encoding. The key point here is the heralded nature in which we can access levels from the highest to the lowest, with a projective measurement onto the basis logical qubits of the just-lower level of entanglement to pass through a level of entanglement-enabled security and robustness.

#### **4.2 Surface codes, graph states and cluster states**

We can define effective surface codes with Matryoshka states, with triangular units. The primary operation proposed to be utilised in this regard is that of lattice surgery and merging. Topological encoding of quantum data facilitates information processing to be protected from the effects of decoherence on physical qubits, by having a logical qubit encoded in the entangled state of many physical qubits. Among the various codes used for this purpose, the surface code has the highest tolerance of component error, when implemented on a two-dimensional lattice of spin-qubits with nearest-neighbour interactions [64–68]. Mhalla and Perdrix [69] proved that the application of measurements in the (X, Z) plane, with one-qubit measurement as per the basis

$$\left\{ \cos \theta |0\rangle + \sin \theta |1\rangle, \sin \theta |0\rangle - \cos \theta |1\rangle \right\} \tag{21}$$

for some *θ* over graph states that are represented by triangular grids, is a universal model of quantum computation. A point to note here is that, for any *θ*, the observable associated with the measurement in this basis is cos 2*θZ* þ sin 2*θX*. For a given simple undirected graph *G* ¼ ð Þ *V*, *E* of order *n*, where *V* represent vertices and *E* edges, the graph state ∣*G*i is the unique quantum state such that for any vertex *u*∈*V*,

$$X\_u Z\_{\mathcal{N}(u)}|G\rangle = |G\rangle \tag{22}$$

The Pauli operators constitute a group acting on a set *V* of *n* qubits is generated by *Xu*, *Zu*, *i:Iu*∈*<sup>V</sup>*, where *I* is the identity, *Xu* and *Zu* are operators that act as identity on the neighbourhood of *u* and with the following action on vertex *u*

$$X: |\mathfrak{x}\rangle \to |\overline{\mathfrak{x}}\rangle \tag{23}$$

$$Z: \left| \mathbf{x} \right> \to (-\mathbf{1})^{\mathbf{x}} \left| \overline{\mathbf{x}} \right>\tag{24}$$

In our circuit, we will have to project three physical qubits from three adjacent triangular units to a single subspace for implementing this model. If we consider the state: <sup>1</sup> <sup>2</sup> ffiffi 2 <sup>p</sup> ðj00*c*0iþj11*c*1iÞ jð 00*c*0iþj11*c*1iÞ jð Þ 00*c*0iþj11*c*1i , with the subscript *c* denoting the physical qubits adjacent to each other and that are projected to a single subspace. If we initialise an ancilla qubit in the state <sup>∣</sup>þi ¼ <sup>1</sup>ffiffi 2 <sup>p</sup> ð Þ j0iþj1i and use the conditional rotation gate

$$U\_{\gamma} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos\frac{\gamma}{2} & \sin\frac{\gamma}{2} \\ 0 & 0 & -\sin\frac{\gamma}{2} & \cos\frac{\gamma}{2} \end{pmatrix} \tag{25}$$

and apply this sequentially with the three adjacent physical qubits (with subscript 'c') and the ancilla as target, we project the ancilla to a unique state that can be retained for the graph state that is thereby defined, by going over the entire tessellated lattice of triangular GHZ-units.

### **4.3 Establishing multiparticle entanglement between nodes of a quantum communication network**

We can use the unique form of the asymmetric Matryoshka Generalised GHZ states to establish multipartite entanglement between nodes of a quantum

communication network. The important part about this protocol is the role of projection measurements on a central terminal. Considering a Matryoshka GHZ-Bell state with an *m*-particle GHZ state and *n*-terminals in a quantum network

$$|\boldsymbol{\mu}\_{\text{MGHz}\boldsymbol{B}}\rangle = \sum\_{k=1}^{L} \lambda\_{k} |\text{GHZ}\_{m}^{a\_{k}, \pm}\rangle |\boldsymbol{B}\_{d\_{1}}^{a\_{k}, \pm}\rangle \dots |\boldsymbol{B}\_{d\_{n}}^{a\_{k, d\_{n}}, \pm}\rangle \tag{26}$$

where <sup>∣</sup>*B*<sup>i</sup> signifies a Bell state, *GHZak*,*m*,� *<sup>m</sup>* <sup>j</sup>*GHZak*<sup>0</sup>,*m*,� *m* D E <sup>¼</sup> *<sup>δ</sup>kk*<sup>0</sup>∀*<sup>i</sup>* and *B ak*,*di* ,� *di* j*B ak*<sup>0</sup>,*di* ,� *di* D E <sup>¼</sup> *<sup>δ</sup>kk*<sup>0</sup>∀*i*. Each user has one particle of a Bell-state, while the other particle of the Bell-state is with the central terminal. Measuring the particles of the Bell-pairs at the central terminal in a basis defined by maximally entangled states over *n*-qubits will project the distant qubits into maximally *n*-qubit entangled states as well. In fact, it need not only be one *n*-qubit maximally entangled state at the spatially distant nodes but could be multiple (partially or maximally) entangled states of varying number of qubits connecting different permutations of endterminals, depending on the projective measurement performed on the central terminal. Some examples of such remote establishment of entanglement have been shown in **Figure 4**.

### **4.4 Quantum networks, repeater protocols and quantum communication**

Quantum networks can facilitate the realisation of quantum technologies such as distributed quantum computing [70], secure communication schemes [71] and quantum metrology [72–75]. In our formalism for GHZ-based network protocols, the key element is that of being able to merge GHZ triangular units, which is done by projecting states at adjacent nodes into a single subspace (as shown in **Figure 5**), as has been tried on atomic systems previously [76]. A *generalised GHZ-GHZ Matryoshka state* can also assist in the recovery of quantum network operability upon node failure, based on the formalism given by *Guha Majumdar and Srinivas Garani* [77].

#### **4.5 Teleportation and superdense coding**

Let us look at the applications of such nested entanglement with the example of a state close to a Matryoshka Q-GHZ state: the Xin-Wei Zha (XZW) State. *Xin-Wei Zha et al* [78] discovered a genuinely entangled seven-qubit state through a numerical optimization process, following the path taken by *Brown et al* [79] and *Borras et al* [80] to find genuinely entangled five-qubit and six-qubit states:

**Figure 4.**

*Illustration of networks for entanglement generation in remote nodes in (a) triangular format (b) rectangular format (c) polyhedra (dodecagon) format, with distinct patterns of entanglement generated at the periphery depending on the projective measurements at the central terminal(s).*

#### **Figure 5.**

*Network repeater protocol with three-qubit projective measurements at nodes to create higher-distance entangled networks.*

$$\begin{split} |\Psi\_{7}\rangle &= \frac{1}{2\sqrt{2}} (|000\rangle\_{135}|\mu\_{+}\rangle\_{24}|\mu\_{+}\rangle\_{67} + |001\rangle\_{135}|\phi\_{-}\rangle\_{24}|\phi\_{+}\rangle\_{67} \\ &+ |010\rangle\_{135}|\mu\_{-}\rangle\_{24}|\phi\_{-}\rangle\_{67} + |011\rangle\_{135}|\phi\_{+}\rangle\_{24}|\nu\_{-}\rangle\_{67} \\ &+ |100\rangle\_{135}|\phi\_{+}\rangle\_{24}|\phi\_{+}\rangle\_{67} + |101\rangle\_{135}|\nu\_{-}\rangle\_{24}|\nu\_{+}\rangle\_{67} \\ &+ |110\rangle\_{135}|\phi\_{-}\rangle\_{24}|\nu\_{-}\rangle\_{67} + |111\rangle\_{135}|\nu\_{+}\rangle\_{24}|\phi\_{-}\rangle\_{67} \end{split} \tag{27}$$

This state is a specific form of the Q-GHZ State defined in Eq. (6), with *<sup>λ</sup>k*∀*<sup>k</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> ffiffi 2 p and ∣*Ak* <sup>1</sup> ∈ f g j000i, j001i, j010i, j011i, j100i, j101i, j110i, j111i . Another point to note here is that the GHZ states here are for *d* ¼ 2, thereby effectively being the Bell states. This resource state can be used for teleportation of arbitrary single, double and triple qubit states. The 3 (Q State)-2 (Bell State)-2 (Bell State) structure of the resourcestate, given in Eq. (17), helps us in devising a quantum circuit to generate the state, as shown in **Figure 6** and realised on *IBM Quantum Experience*. To obtain the resourcestate, we apply a unitary operator on qubits 1, 3 and 5: *U* ¼ *I*<sup>4</sup>�<sup>4</sup> ⊕ ð Þ *σ<sup>z</sup>* ⊗ *σ<sup>z</sup>* .

This state has marginal density matrices for subsystems over one or two qubits that are completely mixed, with *<sup>π</sup>ij* <sup>¼</sup> *Trijρ*<sup>2</sup> *ij* <sup>¼</sup> <sup>1</sup> <sup>4</sup> ∀ *i*, *j*∈f g 1, 2, 3, 4, 5, 6, 7 , *i* <*j*, *π<sup>i</sup>* ¼ *Triρ*<sup>2</sup> *<sup>i</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> ∀ *i* ∈f g 1, 2, 3, 4, 5, 6, 7 . For three-qubit subsystems, some of the partitions have mixed marginal density matrices: *<sup>π</sup>ijk* <sup>¼</sup> *Trijkρ*<sup>2</sup> *ijk* <sup>¼</sup> <sup>1</sup> <sup>8</sup> ∀ *i*, *j*∈ f g 1, 2, 3, 4, 5, 6, 7 , *<sup>i</sup>* <sup>&</sup>lt;*j*<*k*∧ð Þ *ijk* 6¼ ð Þ <sup>127</sup> , 367 ð Þ, 457 ð Þ and *<sup>π</sup>*<sup>127</sup> <sup>¼</sup> *<sup>π</sup>*<sup>367</sup> <sup>¼</sup> *<sup>π</sup>*<sup>457</sup> <sup>¼</sup> <sup>1</sup> 4.

The seven-qubit genuinely entangled resource state ∣Γ7i can be used for a number of applications, such as quantum secret sharing (*Supplementary Material* A.1, A.2 and A.3), the perfect linear teleportation of an arbitrary one-qubit state (*Supplementary Material* B.1.1), probabilistic circular teleportation of arbitrary one-

**Figure 6.**

*Quantum circuit for the generation of the seven-qubit genuinely entangled state, on* IBM Quantum Experience*. Here* CX gate *is the CNOT gate,* cZ gate *is the CPHASE gate and* H gate *is the Hadamard gate.*

qubit states (*Supplementary Material* B.1.2), perfect linear teleportation of an arbitrary two-qubit state (*Supplementary Material* B.2.1), bidirectional teleportation of arbitrary two-qubit states (*Supplementary Material* B.2.2) and perfect linear teleportation of an arbitrary three-qubit state (*Supplementary Material* B.3).
