Can We Entangle Entanglement?

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,

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, 9, 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, 24, 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, 38, 39, 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, 42, 43, 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, 47, 48, 49, 50, 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]:

where τρA1A2…An denotes the bipartite quantum entanglement measured by the tangle across the bipartition A1:A2A3…An. In this chapter, we discuss the weak coupling between near-maximally entangled (sub)states due to the constraint placed by entanglement monogamy [54, 55, 56, 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:

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=2 multiqubit systems, which are as follows:

1. Matryoshka Generalised GHZ states

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

where ∣B⟩ signifies a Bell state.

1. Matryoshka Q-GHZ states

where ∣A⟩ are orthogonal states that are eigenstates in the Z-basis for all qubits in the state. Here the subscript ‘di’ in ∣GHZdiak,di⟩ denotes the number of qubits in the ith 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 ∣GHZ2,+⟩=12010⟩+101⟩ can be created from ∣GHZ⟩=12000⟩+111⟩ using I2×2⊗σx⊗I2×2, or in other words - we apply a qubit flip σx operation on the second qubit, leaving the other qubits untouched. In the summation above, L=2nh 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: ∣ψ±⟩=12000⟩±111⟩. A Hadamard Operator is a quantum logical gate that acts on a single qubit and maps the basis state ∣0⟩to∣0⟩+∣1⟩2 and ∣1⟩to∣0⟩−∣1⟩2.

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: ∣ϕC⟩=12GHZm+⟩⊗N+GHZm−⟩⊗N (with ∣GHZN±⟩=120⟩⊗N±1⟩⊗N), 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...0⟩ with total number of qubits being n=3k for some finite, non-vanishing integer k.

Step 2: Application of a Hadamard gate on the 3n+1th qubits to give ∣+00+...0⟩ where ∣+⟩=120⟩+1⟩.

Step 3: Application of CNOT operation with the 3n+1th qubits as the control for the corresponding 3n+2th qubits and 3n+3th qubits as target to give a state of the form ∣GHZ123⟩∣GHZ456⟩…∣GHZn−2n−1n⟩.

Step 4: Application of composite operation of the form of ∑i=0n/3P3i+1P3i+2P3i+3 where P represents Pauli operations or combination of Pauli operations such as σxσz and

1. P3i+1≠P3j+1fori≠j∀i,j∈Z

2. P3i+2≠P3j+2fori≠j∀i,j∈Z

3. P3i+3≠P3j+3fori≠j∀i,j∈Z

2.1 Generation of Matryoshka states using spin systems in condensed matter physics

In this chapter, the generation of Matryoshka states will be explored in spin systems in condensed matter physics. Unlike in the case of the aforementioned algorithm, instead of composite operators, in this case we have localised generation and minimal interactions between different GHZ and GHZ-like states to create the Matryoshka states. In this case, we consider N spin-12 particles, with each spin coupled to its nearest neighbours by the XY Hamiltonian

H=∑i=1N−1JX,iX̂iX̂i+1+JY,iŶiŶi+1E10

where Jσ,i is the pairwise coupling constant with σ=X̂,Ŷ,Ẑ being the Pauli operators. For the purposes of this chapter, we take N to be odd. Franco et al [58] showed that it is sufficient to state that the information flux between the X̂ (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 from X̂1 to X̂N depends only on the set JY,1JX,2…JY,N−1 and is independent of any other coupling rate in the spin-chain. Christandl et al [60, 61] 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: Jσ,i=λiN−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 spin-chain in an completely separable eigenstate of the tensorial product of Zi operators, say ∣Ψ0⟩=∣000...0⟩12…N, we obtain an information flux towards symmetric two-site spin operators, and a final state of the form [58].

∣ψ0⟩=∣0⟩c⊗i=0M∣ψ+⟩2i+1,N−2i⊗i=1M∣ψ−⟩2i,N−2i+1E11

∣ψ1⟩=∣1⟩c⊗i=0M∣ψ−⟩2i+1,N−2i⊗i=1M∣ψ+⟩2i,N−2i+1E12

where c labels the central site of the spin-chain, M=N−34 and ∣ψ±⟩=1200⟩±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. This is a key result around the generation of Matryoshka GHZ-Bell states in this chapter, which can be extended to other classes of Matryoshka states. 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 ∣Ψ0⟩=∣111...1⟩12…N, we would have obtained a final state of the form

∣ψ0⟩=∣0⟩c⊗i=0M∣ψ−⟩2i+1,N−2i⊗i=1M∣ψ+⟩2i,N−2i+1E13

∣ψ1⟩=∣1⟩c⊗i=0M∣ψ+⟩2i+1,N−2i⊗i=1M∣ψ−⟩2i,N−2i+1E14

We use this principle and the idea that after evolution over time t∗, the states in Eqs. (2) and (3) transform back to ∣000...000⟩12…N and states in Eqs. (4) and (5) transform back to ∣111...11⟩12…N. We can utilise this concept, by taking the state in Eq. (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 (Jσ,i′=λ′i3−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

∣ψ0⟩=120⟩c+1⟩c⊗i=0M∣ψ+⟩2i+1,N−2i⊗i=1M∣ψ−⟩2i,N−2i+1E15

We now perform the truncated subsystem time-evolution with the parameters J′t″ to give us the state

∣ψ0⟩=12000⟩+111⟩c−1,c,c+1⊗i=0M−1∣ψ+⟩2i+1,N−2i⊗i=1M∣ψ−⟩2i,N−2i+1E16

Therefore, we can obtain a Matryoshka GHZ-Bell state using nearest spin–spin interactions in a spin-chain. A similar generation protocol can be defined for the other two classes of Matryoshka states. The teleportation of an arbitrary n-qubit state can be performed using Matryoshka GHZ-Bell States [62].

Given the triangular three-qubit configuration, we can also consider the anisotropic Heisenberg Hamiltonian, which describes the interaction between three spins that are located at the corners of an equilateral triangle lying in the xy-plane, as shown in Figure 2.

H=−Jxy∑i=13SixSi+1x+SiySi+1y−Jz∑i=13SizSi+1z+HzE17

here the three spins Si, with S = 1/2, are located at the corners i = 1, 2, 3, and S1=S4. Jxy and Jz are the in-plane and out-of-plane exchange coupling constants respectively, and HZ=∑i=13bi.Si denotes the Zeeman coupling of the spins Si to the externally applied magnetic fields bi at the sites i. If we consider isotropic exchange couplings: Jxy=Jz=J>0 (ferromagnetic coupling) and bi=0∀i, we have a ground-state qudruplet that is spanned by the GHZ states: 12000⟩+111⟩ and 12000⟩−111⟩, along with the W- and spin-flipped W-states. A set of appropriately chosen magnetics fields will allow us to split off an approximate GHZ state from this degenerate eigenspace. If we find a set of magnetic fields that, in classical spin systems, shall result in exactly two degenerate minima for the configurations ∣000⟩, representing the ↓↓↓ spin configuration, and ∣111⟩, representing the ↑↑↑ spin configuration, with an energy barrier in between, quantum mechanical tunnelling shall yield the desired states. The magnetic fields must be of the same strength, in-plane and sum to zero, with a convenient additional choice being that of the field pointing radially outward. Therefore, the successive directions of the magnetic fields have to differ by an angle of 2π/3 with respect to each other. Going by the schematic in Figure 2, we can write the hamiltonian

H=−Jxy∑i=13SixSi+1x+SiySi+1y−Jz∑i=13SizSi+1z+Hz+∑il=1Nl−Jxyil∑i=13SixSi+1x+SiySi+1y−Jzil∑i=13SizSi+1z+Hzil∑ir=1Nr−Jxyir∑i=13SixSi+1x+SiySi+1y−Jzir∑i=13SizSi+1z+Hzir+∑il=1Nlλilil+1lSinr.Si+1nl+∑ir=0Nr−1λirir+1rSinr.Si+1nlE18

where the superscripts il and ir denote the left and right branches respectively of the schematic arounnd a central triangular unit. For il=1, we have the leftmost triangular unit and for ir=Nr, we have the rightmost triangular unit. Nl and Nr denote the number of units on the left and right side of the central triangular unit. In principle, we can have an asymmetric case where Nl≠Nr. In the fourth line, the term SNl+1 and S0 refer to the spins in the central triangular unit connected to the adjacent left and right triangular units respectively. Moreover, both λilil+1l and λirir+1l are coupling constants between adjacent triangular units that are numerically negligible with respect to J but are non-zero, to account for inter-unit coupling. Sinr and Sinl are right and left connecting nodes of the ith triangular unit.

An important point here is the condition: GHZdiak,di,±GHZdiak′,di,±=δkk′∀i,A1kA1k′=δkk′ in Eqs. (4), (6) and (9). This is ensured by the additional application of single qubit gates on the nodes of the triangular units. For instance, 12000⟩+111⟩→σx212010⟩+101. Using combination of such single qubit operations, we can span the entire space of GHZ and GHZ-like states. The important point here is the synchronised timing of these operations, with the inter-unit coupling, so as to give us a superposition over orthogonal GHZ and GHZ-like states for all triangular units, as shown in Figure 2.

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.

4.3 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

∣ψMGHzB⟩=∑k=1Lλk∣GHZmak,m,±⟩∣Bd1ak,d1,±⟩…∣Bdnak,dn,±⟩E26

where ∣B⟩ signifies a Bell state, GHZmak,m,±GHZmak′,m,±=δkk′∀i and Bdiak,di,±Bdiak′,di,±=δkk′∀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 end-terminals, 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, 73, 74, 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:

∣ψ7⟩=122(∣000⟩135∣ψ+⟩24∣ψ+⟩67+∣001⟩135∣ϕ−⟩24∣ϕ+⟩67+∣010⟩135∣ψ−⟩24∣ϕ−⟩67+∣011⟩135∣ϕ+⟩24∣ψ−⟩67+∣100⟩135∣ϕ+⟩24∣ϕ+⟩67+∣101⟩135∣ψ−⟩24∣ψ+⟩67+∣110⟩135∣ϕ−⟩24∣ψ−⟩67+∣111⟩135∣ψ+⟩24∣ϕ−⟩67)E27

This state is a specific form of the Q-GHZ State defined in Eq. (6), with λk∀k=122 and ∣A1k∈000⟩001⟩010⟩011⟩100⟩101⟩110⟩111⟩. Another point to note here is that the GHZ states here are for d=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 resource-state, 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 resource-state, we apply a unitary operator on qubits 1, 3 and 5: U=I4×4⊕σz⊗σz.

This state has marginal density matrices for subsystems over one or two qubits that are completely mixed, with πij=Trijρij2=14∀i,j∈1,2,3,4,5,6,7,i<j, πi=Triρi2=12∀i∈1,2,3,4,5,6,7. For three-qubit subsystems, some of the partitions have mixed marginal density matrices: πijk=Trijkρijk2=18∀i,j∈1,2,3,4,5,6,7,i<j<k∧ijk≠127,367,457 and π127=π367=π457=14.

The seven-qubit genuinely entangled resource state ∣Γ7⟩ 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-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).

5. Conclusion

In this chapter, 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 of this formalism to more general classes of Matryoshka states is posited. An example of a state close to a perfect Matryoshka Q-GHZ state 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 is presented. This work should lay the groundwork for other studies into the area of nested entanglement, including forays into higher layers of nesting entanglement. Particularly, the problem of composite quantum states containing nested entanglement can be explored further, theoretically and experimentally, be it in surface codes, establishment of multipartite entanglement in quantum networks, teleportation, superdense coding and more broadly in quantum communication protocols. The main advantage of the model and method presented in this chapter is the accessibility of the condensed matter system presented, while the primary limitation of the model presented in this chapter is the need for fine-tuning of various interaction terms that have to be time-sequenced very carefully. The concept of entangled entanglement is the key result of the chapter, which can be implemented with other non-trivial combination of unitary transformations over multiple qubits.

Acknowledgments

I would like to acknowledge the guidance and contribution of Prof. Prasanta Panigrahi, IISER-Kolkata. This work was supported by the Homi Bhabha Centre for Science Education, Tata Institute of Fundamental Research, Mumbai, India.