**1. Introduction**

Quantum Entanglement is a fundamental non-classical aspect of entities in the quantum realm, which disallows a reductionist description of a composite system in terms of the state and properties of its quantum constituents. Erwin Schrodinger once famously said,

"*Thus one disposes provisionally until the entanglement is resolved by actual observation of only a common description of the two in that space of higher dimension. This is the reason that knowledge of the individual systems can decline to the scantiest, even to zero, while that of the combined system remains continually maximal. The best possible knowledge of a whole does not include the best possible knowledge of its parts—and this is what keeps coming back to haunt us*"

Albert Einstein, Boris Podolsky and Nathan Rosen, famously known as EPR, and Schrödinger, who called it *Verschränkung*, highlighted the intrinsic order of statistical relations between the constituents of a compound quantum system, first recognised what they called a 'spooky' feature of the quantum world. John Bell showed that it is entanglement which irrevocably rules out the possibility of ascribing values to physical quantities of entangled systems prior to measurement. He accepted the EPR conclusion around the quantum description of nature not being 'complete', with the principles of 'realism' (measurement results are determined by properties that the particles carry prior to, and are independent of, the measurement), 'locality' (measurements obtained at one location are independent of any actions performed at another point that is spacelike separated) and 'free will' (settings of a local apparatus are independent of what EPR called 'hidden variables' that determine the local results) being primary in this discussion. Bell showed that if one were to assume these principles, then one obtains constraints in the form of certain inequalities, called Bell's Inequalities, on the statistical correlations in the measured values of properties of the systems, and that the probabilities of the outcomes of a measurement performed on constituents of an entangled system violate the Bell inequality. In this manner, it was shown that entanglement makes it impossible to simulate quantum correlations within the classical manner of thinking. Greenberger, Horne, and Zeilinger (GHZ) went beyond two particles in showing entanglement of quantum particles leads to contradictions with *Local Hidden Variables Models (LVHM)* for non-statistical predictions of quantum systems. During his doctoral studies at Université d'Orsay, Alain Aspect performed the first experimental realisation of the Bell's Inequalities.

Today, entanglement is instrumental in the formulation of information processing tasks in the quantum realm. It has been used in applications such as superdense coding and teleportation. *Bennett et al* first proposed a scheme for quantum teleportation, wherein a genuinely entangled Bell state was used to transmit an arbitrary single qubit [1]. Many different kinds of entangled quantum states have been used to teleport arbitrary quantum states since then, including Bell states [2, 3], GHZ states [4, 5], W states [6, 7] and multiqubit states [8–10]. There have been hop-by-hop and multi-hop quantum teleportation schemes proposed since then as well as schemes to teleport GHZ-like states using two types of four-qubit states [11, 12]. Teleportation has been proposed in two-copy quantum teleportation scheme [13], using cluster states [14], in higher dimensions [15] and also shown to be possible over atmospheric channels [16]. More recently, various derivatives of the standard teleportation scheme have been proposed, including those used for bidirectional teleportation [15, 17, 18], controlled teleportation [19, 20], quantum operation sharing [21, 22], quantum secret sharing [23–25] and arbitrated quantum teleportation [26, 27]. For multiple participants in a quantum information processing task, entangled multiqubit states and multipartite entanglement play the preeminent role, with multiqubit resource states varying from GHZ- and W-states to clusters states [28]. Lately, W-GHZ composite states have been used for remote state preparation, teleportation and superdense coding of arbitrary quantum states [29, 30]. *Shuai et al* showed how GHZ-GHZ channels can be used for bidirectional quantum communication [31]. The physical realisation of such composite systems have been explored in a number of physical platforms such as using cavity QED [32]. Properties of spin squeezing when multi-qubit GHZ state and W state are superposed have also been studied [33]. These composite quantum states contain varying degrees of multilevel and genuine multipartite entanglement, which can be used for applications in quantum information processing [34, 35]. *Yang et al* investigated the feasibility of experimentally creating GHZ states comprising of three logical qubits in a decoherence-free subspace, by using superconducting transmon qutrits coupled to a co-planar waveguide resonator [36].

Since not all forms of entanglement are relevant for distinct information processing applications, the determination of resource states for specific information processing tasks is of paramount importance. This, along with any characteristic protection or resilience against noise and decoherence provided by a resource state, forms the underlying principle of quantum resource theories [37–40]. In the latter pursuit, decoherence-free subspaces provide a natural solution and associated resources to produce quantum resource-states that are not easily decohered [41–44]. Stabiliser codes are a resource that constitutes a crucial ingredient for effective quantum error correction [45], while cluster states are resource states that are used for measurement-based quantum computation and error corrections [46–51]. Certain realisations of a standard resource-state have more resilience against decoherence,

such as in the case of cluster states generated with Ising-type interactions, wherein the entanglement in the state persisted upto a fairly large number of measurements on the qubits to disentangle them [52]. These resource state display various distinct forms of entanglement: some are maximally entangled, such as resource-states used for teleportation, while others are partially entangled, such as in the case of cluster states. In the case of cluster states, the partial entanglement is a resource in itself, since the one requires a specific protection of the 'quantumness' and correlations in the segments of the state against perturbations or measurements of other segments of the state. If the resource-state were maximally entangled, such a measurement or perturbation of one segment will collapse the state of the remaining segments to a specific state, thereby not maintaining the system as a viable quantum resource for further cluster operations. If we were to generalise and extend this idea to conceptualise states that maintain near maximal entanglement in segments of the state while maintaining weak correlations between the segments, we could have interesting resource-states and associated applications of such states. This is the central idea and motivation behind generalising the concept of *Matryoshka* states: *Matryoshka Generalised GHZ* states, *Matryoshka GHZ-Bell* States and *Matryoshka Q-GHZ* States.

In multi-qubit quantum states, an important property is that entanglement is monogamous - quantum entanglement cannot be shared freely among various parties. *Osborne and Verstraete* showed that the entanglement for bipartitions over an n-qubit system follows a monogamy relation [53]:

$$\begin{aligned} \left(\pi(\rho\_{A\_1A\_2}) + \pi(\rho\_{A\_1A\_3}) + \dots + \pi(\rho\_{A\_1A\_n})\right) \\ \leq \pi\left(\rho\_{A\_1(A\_2\dots A\_n)}\right) \end{aligned} \tag{1}$$

where *τ ρ<sup>A</sup>*1ð Þ *<sup>A</sup>*<sup>2</sup> … *An* � � denotes the bipartite quantum entanglement measured by the tangle across the bipartition *A*<sup>1</sup> : *A*2*A*<sup>3</sup> … *An*. In this chapter, we discuss the weak coupling between near-maximally entangled (sub)states due to the constraint placed by entanglement monogamy [54–57]. The concept of Matryoshka states was first given by *Di Franco et al* [58], with the name 'Matryoshka' coming from the Russian word for 'nesting doll'. The underlying concept of a Matryoshka state is genuine entanglement in multilevel systems, with the entanglement in higher level systems being more than or equal to the entanglement in the lower level constituents:

$$\mathcal{E}\_{d\_i} \ge \mathcal{E}\_{d\_j}, d\_i > d\_j \tag{2}$$

where *Edi* is the entanglement measure of the level *di*. In this chapter, we will discuss the characteristics and applications of two classes of Matryoshka states for *d* ¼ 2 multiqubit systems, which are as follows:

1.*Matryoshka Generalised GHZ states*

$$|\psi\_{MGHx}\rangle = \sum\_{k=1}^{L} \lambda\_k |\text{GHZ}\_{d\_1}^{a\_{k,d\_1}, \pm}\rangle \dots |\text{GHZ}\_{d\_N}^{a\_{k,d\_N}, \pm}\rangle \tag{3}$$

$$
\left\langle \text{GHZ}\_{d\_i}^{a\_{k\ell\_i}, \pm} \middle| \text{GHZ}\_{d\_i}^{a\_{k'\ell\_i}, \pm} \right\rangle = \delta\_{kk'} \forall i \tag{4}
$$

A particular case of such states are the *Matryoshka GHZ-Bell states*

$$|\boldsymbol{\mu}\_{\text{MGHz}\boldsymbol{B}}\rangle = \sum\_{k=1}^{L} \lambda\_{k} |\text{GHZ}\_{d\_{1}}^{a\_{k,d\_{1}}, \pm}\rangle |\boldsymbol{B}\_{d\_{2}}^{a\_{k,d\_{2}}, \pm}\rangle \ldots |\boldsymbol{B}\_{d\_{N}}^{a\_{k,d\_{N}}, \pm}\rangle \tag{5}$$

where ∣*B*i signifies a Bell state.

$$
\left\langle \text{GHZ}\_{d\_1}^{a\_{k,d\_1}, \pm} \middle| \text{GHZ}\_{d\_1}^{a\_{k',d\_1}, \pm} \right\rangle = \delta\_{kk'} \forall i \tag{6}
$$

$$
\left\langle \mathcal{B}\_{d\_i}^{a\_{k,d\_i}, \pm} \vert \mathcal{B}\_{d\_i}^{a\_{k',d\_i}, \pm} \right\rangle = \delta\_{kk'} \forall i \tag{7}
$$

2.*Matryoshka Q-GHZ states*

$$|\psi\_{M\text{ExG}}\rangle = \sum\_{k=1}^{L} \lambda\_k |A\_1^k\rangle |\text{GHZ}\_{d\_2}^{a\_{k,d\_2,\pm}}\rangle \dots |\text{GHZ}\_{d\_N}^{a\_{k,d\_N,\pm}}\rangle \tag{8}$$

$$\left\langle \text{GHZ}\_{d\_i}^{a\_{k,d\_i,\pm}} \middle| \text{GHZ}\_{d\_i}^{a\_{k',d\_i,\pm}} \right\rangle = \delta\_{kk'} \forall i, \left\langle A\_1^k \middle| A\_1^{k'} \right\rangle = \delta\_{kk'} \tag{9}$$

where ∣*A*i are orthogonal states that are eigenstates in the Z-basis for all qubits in the state. Here the subscript '*di*' in <sup>∣</sup>*GHZak*,*di di* i denotes the number of qubits in the *i* th subsystem, while *a* is the decimal representation of the superposed term in the GHZ-like state that has the lowest decimal representation and � denotes the relative phase between the terms in superposition. GHZ-like states are the states that can be created from the GHZ state using local unitary operations. So, for instance, in a three-qubit system <sup>∣</sup>*GHZ*2,þi ¼ <sup>1</sup>ffiffi 2 <sup>p</sup> ð Þ j010iþj101i can be created from ∣*GHZ*i ¼ 1ffiffi 2 <sup>p</sup> ð Þ j000iþj111i using *I*<sup>2</sup>�<sup>2</sup> ⊗ *σ<sup>x</sup>* ⊗ *I*<sup>2</sup>�2, or in other words - we apply a qubit flip *σ<sup>x</sup>* operation on the second qubit, leaving the other qubits untouched. In the summation above, *<sup>L</sup>* <sup>¼</sup> <sup>2</sup>*nh* where *nh* is the number of qubits in the largest subsystem.

*Nomenclature and Acronyms Used*. GHZ state is a multipartite maximally entangled state, first defined for three qubits: <sup>∣</sup>*ψ*�i ¼ <sup>1</sup>ffiffi 2 <sup>p</sup> ð Þ j000i�j111i . A Hadamard Operator is a quantum logical gate that acts on a single qubit and maps the basis state <sup>∣</sup>0ito <sup>∣</sup>0iþ∣1<sup>i</sup> ffiffi 2 <sup>p</sup> and <sup>∣</sup>1ito <sup>∣</sup>0i�∣1<sup>i</sup> ffiffi 2 <sup>p</sup> .
