**2. Localised correlation generation: how can we generate entangled entanglement?**

Matryoshka states can be generated in various physical platforms, such as in spin systems and in trapped ions. Fröwis and Dür [59] studied the stability of superpositions of macroscopically distinct quantum states under decoherence, wherein they looked at realising concatenated-GHZ states: <sup>∣</sup>*ϕC*i ¼ <sup>1</sup>ffiffi 2 <sup>p</sup> j*GHZ*<sup>þ</sup> *<sup>m</sup>*<sup>i</sup> <sup>⊗</sup> *<sup>N</sup>*þj*GHZ*� *<sup>m</sup>*<sup>i</sup> <sup>⊗</sup> *<sup>N</sup>* � � (with ∣*GHZ*� *<sup>N</sup>*i ¼ <sup>1</sup>ffiffi 2 <sup>p</sup> <sup>j</sup>0<sup>i</sup> <sup>⊗</sup> *<sup>N</sup>*�j1<sup>i</sup> <sup>⊗</sup> *<sup>N</sup>* � �), which is a Matryoshka generalised state state, in trapped ion systems. The underlying principle to realise entangled entanglement is to have localised and intra-level correlation generation, which begins with creation of entanglement in one level, thereafter entanglement of this entangled structure over higher-level basis states and so on. For the purposes of this chapter, we will be considering the GHZ and GHZ-like states as the primary unit of entanglement.

The algorithm for generating entangled entanglement in a system comprising of GHZ and GHZ-like states as the units of entanglement is given by

*Step 1*: Creation of a ground state ∣0000*:::*0i with total number of qubits being *n* ¼ 3*k* for some finite, non-vanishing integer *k*.

*Step 2*: Application of a Hadamard gate on the 3ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> th qubits to give <sup>∣</sup> <sup>þ</sup> <sup>00</sup> <sup>þ</sup> *:::*0<sup>i</sup> where <sup>∣</sup>þi ¼ <sup>1</sup>ffiffi 2 <sup>p</sup> ð Þ j0iþj1i .

*Step 3*: Application of *CNOT* operation with the 3ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> th qubits as the control for the corresponding 3ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>2</sup> th qubits and 3ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>3</sup> th qubits as target to give a state of the form <sup>∣</sup>*GHZ*ð Þ <sup>123</sup> <sup>i</sup>∣*GHZ*ð Þ <sup>456</sup> <sup>i</sup> … <sup>∣</sup>*GHZ*ð Þ *<sup>n</sup>*�2,*n*�1,*<sup>n</sup>* <sup>i</sup>.

*Step 4*: Application of composite operation of the form ofP*n=*<sup>3</sup> *<sup>i</sup>*¼0*P*3*i*þ1*P*3*i*þ2*P*3*i*þ<sup>3</sup> where *P* represents Pauli operations or combination of Pauli operations such as *σxσ<sup>z</sup>* and

