Open access peer-reviewed chapter

Can We Entangle Entanglement?

By Mrittunjoy Guha Majumdar

Reviewed: May 24th 2021Published: July 16th 2021

DOI: 10.5772/intechopen.98535

Downloaded: 89

Abstract

In this chapter, nested multilevel entanglement is formulated and discussed in terms of Matryoshka states. The generation of such states that contain nested patterns of entanglement, based on an anisotropic XY model has been proposed. Two classes of multilevel-entanglement- the Matryoshka Q-GHZ states and Matryoshka generalised GHZ states, are studied. Potential applications of such resource states, such as for quantum teleportation of arbitrary one, two and three qubits states, bidirectional teleportation of arbitrary two qubit states and probabilistic circular controlled teleportation are proposed and discussed, in terms of a Matryoshka state over seven qubits. We also discuss fractal network protocols, surface codes and graph states as well as generation of arbitrary entangled states at remote locations in this chapter.

Keywords

  • Quantum Computation
  • Multipartite Entanglement
  • Quantum State Sharing

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 alfirst 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 alshowed 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 alinvestigated 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 Matryoshkastates: Matryoshka Generalised GHZstates, Matryoshka GHZ-BellStates and Matryoshka Q-GHZStates.

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

τρA1A2+τρA1A3++τρA1AnτρA1A2AnE1

where τρA1A2Andenotes the bipartite quantum entanglement measured by the tangle across the bipartition A1:A2A3An. 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:

EdiEdj,di>djE2

where Ediis 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=2multiqubit systems, which are as follows:

  1. Matryoshka Generalised GHZ states

ψMGHz=k=1LλkGHZd1ak,d1,±GHZdNak,dN,±E3
GHZdiak,di,±GHZdiak,di,±=δkkiE4

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

ψMGHzB=k=1LλkGHZd1ak,d1,±Bd2ak,d2,±BdNak,dN,±E5

where Bsignifies a Bell state.

GHZd1ak,d1,±GHZd1ak,d1,±=δkkiE6
Bdiak,di,±Bdiak,di,±=δkkiE7

  1. Matryoshka Q-GHZ states

ψMExG=k=1LλkA1kGHZd2ak,d2,±GHZdNak,dN,±E8
GHZdiak,di,±GHZdiak,di,±=δkki,A1kA1k=δkkE9

where Aare orthogonal states that are eigenstates in the Z-basis for all qubits in the state. Here the subscript ‘di’ in GHZdiak,didenotes the number of qubits in the ithsubsystem, while ais 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+101can be created from GHZ=12000+111using I2×2σxI2×2, or in other words - we apply a qubit flip σxoperation on the second qubit, leaving the other qubits untouched. In the summation above, L=2nhwhere nhis 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 0to0+12and 1to012.

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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+GHZmN(with GHZN±=120N±1N), 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...0with total number of qubits being n=3kfor some finite, non-vanishing integer k.

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

Step 3: Application of CNOToperation with the 3n+1thqubits as the control for the corresponding 3n+2thqubits and 3n+3thqubits as target to give a state of the form GHZ123GHZ456GHZn2n1n.

Step 4: Application of composite operation of the form of i=0n/3P3i+1P3i+2P3i+3where Prepresents Pauli operations or combination of Pauli operations such as σxσzand

  1. P3i+1P3j+1foriji,jZ

  2. P3i+2P3j+2foriji,jZ

  3. P3i+3P3j+3foriji,jZ

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 Nspin-12particles, with each spin coupled to its nearest neighbours by the XY Hamiltonian

H=i=1N1JX,iX̂iX̂i+1+JY,iŶiŶi+1E10

where Jσ,iis the pairwise coupling constant with σ=X̂,Ŷ,Ẑbeing the Pauli operators. For the purposes of this chapter, we take Nto 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 X̂1to X̂Ndepends only on the set JY,1JX,2JY,N1and is independent of any other coupling rate in the spin-chain. Christandl et al[60, 61] showed that after a time t=π/λwith λbeing 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=λiNi), 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 Zioperators, say Ψ0=000...012N, we obtain an information flux towards symmetric two-site spin operators, and a final state of the form [58].

ψ0=0ci=0Mψ+2i+1,N2ii=1Mψ2i,N2i+1E11
ψ1=1ci=0Mψ2i+1,N2ii=1Mψ+2i,N2i+1E12

where clabels the central site of the spin-chain, M=N34and ψ±=1200±11. An illustration of the setup has been shown in Figure 1.

Figure 1.

Scheme for the generation of Matryoshka GHZ-Bell resource-states, where the effective spin–spin XY Hamiltonianan is obtained as an effective adiabatic Hamiltonian for a linear chain of optical cavities with each interacting with a three-level atomic system. The ground states of each atomic unit provide the computational space of each spin, and the dipole-forbidden transition between these states is realised as an (adiabatic) Raman transition through the excited state:eiwithi=1,2,,N. The cavity field drives off-resonantly the dipole-allowed channeljieiwith the Rabi frequencygj,j=0,1. Two lasers are also coupled to these atomic transitions with strengthΩjand detuningΛj.

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 MatryoshkaGHZ-Bell states in this chapter, which can be extended to other classes of Matryoshkastates. 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...112N, we would have obtained a final state of the form

ψ0=0ci=0Mψ2i+1,N2ii=1Mψ+2i,N2i+1E13
ψ1=1ci=0Mψ+2i+1,N2ii=1Mψ2i,N2i+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...00012Nand states in Eqs. (4) and (5) transform back to 111...1112N. 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=λi3i) 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=120c+1ci=0Mψ+2i+1,N2ii=1Mψ2i,N2i+1E15

We now perform the truncated subsystem time-evolution with the parameters Jtto give us the state

ψ0=12000+111c1,c,c+1i=0M1ψ+2i+1,N2ii=1Mψ2i,N2i+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.

Figure 2.

Schematic for all (three) classes of Matryoshka states ford=2levels of the quantum system, explored in this chapter. The triangular formations encapsulate the logical units of two/three qubits mediated by CNOT gates. Each of these triangular units are weakly coupled to each other (shown with light blue patches). In the case of theMatryoshka GHZ-Bellstates, we only have the black links, while for theMatryoshka Generalised GHZstates andMatryoshka Q-GHZstates, we also have the blue links.

H=Jxyi=13SixSi+1x+SiySi+1yJzi=13SizSi+1z+HzE17

here the three spins Si, with S = 1/2, are located at the corners i = 1, 2, 3, and S1=S4. Jxyand Jzare the in-plane and out-of-plane exchange coupling constants respectively, and HZ=i=13bi.Sidenotes the Zeeman coupling of the spins Sito the externally applied magnetic fields biat the sites i. If we consider isotropic exchange couplings: Jxy=Jz=J>0(ferromagnetic coupling) and bi=0i, we have a ground-state qudruplet that is spanned by the GHZ states: 12000+111and 12000111, 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/3with respect to each other. Going by the schematic in Figure 2, we can write the hamiltonian

H=Jxyi=13SixSi+1x+SiySi+1yJzi=13SizSi+1z+Hz+il=1NlJxyili=13SixSi+1x+SiySi+1yJzili=13SizSi+1z+Hzilir=1NrJxyiri=13SixSi+1x+SiySi+1yJziri=13SizSi+1z+Hzir+il=1Nlλilil+1lSinr.Si+1nl+ir=0Nr1λirir+1rSinr.Si+1nlE18

where the superscripts iland irdenote 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. Nland Nrdenote the number of units on the left and right side of the central triangular unit. In principle, we can have an asymmetric case where NlNr. In the fourth line, the term SNl+1and S0refer to the spins in the central triangular unit connected to the adjacent left and right triangular units respectively. Moreover, both λilil+1land λirir+1lare coupling constants between adjacent triangular units that are numerically negligible with respect to Jbut are non-zero, to account for inter-unit coupling. Sinrand Sinlare right and left connecting nodes of the ithtriangular unit.

An important point here is the condition: GHZdiak,di,±GHZdiak,di,±=δkki,A1kA1k=δkkin 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+111σx212010+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.

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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 Applicationsection of this chapter. An interesting future direction of pursuing this line of research would be in squeezed baths, which Zippilli et alstudied 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.

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.

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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

0Ln=120Ln10Ln10Ln1+1Ln11Ln11Ln1E19
1Ln=120Ln10Ln10Ln11Ln11Ln11Ln1E20

As you can see, these are special cases of Matryoshka Generalised GHZ states, with the superscript ndefining the layer of the network. A point to note here is that n=1is the layer with physical qubits, and so 0L1=0and 1L1=1. 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, 65, 66, 67, 68]. Mhalla and Perdrix [69] proved that the application of measurements in the (X, Z) plane, with one-qubit measurement as per the basis

cosθ0+sinθ1sinθ0cosθ1E21

for some θover 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 cos2θZ+sin2θX. For a given simple undirected graph G=VEof order n, where Vrepresent vertices and Eedges, the graph state Gis the unique quantum state such that for any vertex uV,

XuZNuG=GE22

The Pauli operators constitute a group acting on a set Vof nqubits is generated by Xu,Zu,i.IuV, where Iis the identity, Xuand Zuare operators that act as identity on the neighbourhood of uand with the following action on vertex u

X:xx¯E23
Z:x1xx¯E24

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: 12200c0+11c100c0+11c100c0+11c1, with the subscript cdenoting the physical qubits adjacent to each other and that are projected to a single subspace. If we initialise an ancilla qubit in the state +=120+1and use the conditional rotation gate

Uγ=1000010000cosγ2sinγ200sinγ2cosγ2E25

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

ψMGHzB=k=1LλkGHZmak,m,±Bd1ak,d1,±Bdnak,dn,±E26

where Bsignifies a Bell state, GHZmak,m,±GHZmak,m,±=δkkiand Bdiak,di,±Bdiak,di,±=δkki. 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.

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).

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 statecan also assist in the recovery of quantum network operability upon node failure, based on the formalism given by Guha Majumdar and Srinivas Garani[77].

Figure 5.

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

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(000135ψ+24ψ+67+001135ϕ24ϕ+67+010135ψ24ϕ67+011135ϕ+24ψ67+100135ϕ+24ϕ+67+101135ψ24ψ+67+110135ϕ24ψ67+111135ψ+24ϕ67)E27

This state is a specific form of the Q-GHZ State defined in Eq. (6), with λkk=122and A1k000001010011100101110111. 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=I4×4σzσz.

Figure 6.

Quantum circuit for the generation of the seven-qubit genuinely entangled state, onIBM Quantum Experience. HereCX gateis the CNOT gate,cZ gateis the CPHASE gate andH gateis the Hadamard gate.

This state has marginal density matrices for subsystems over one or two qubits that are completely mixed, with πij=Trijρij2=14i,j1,2,3,4,5,6,7,i<j, πi=Triρi2=12i1,2,3,4,5,6,7. For three-qubit subsystems, some of the partitions have mixed marginal density matrices: πijk=Trijkρijk2=18i,j1,2,3,4,5,6,7,i<j<kijk127,367,457and π127=π367=π457=14.

The seven-qubit genuinely entangled resource state Γ7can be used for a number of applications, such as quantum secret sharing (Supplementary MaterialA.1, A.2 and A.3), the perfect linear teleportation of an arbitrary one-qubit state (Supplementary MaterialB.1.1), probabilistic circular teleportation of arbitrary one-qubit states (Supplementary MaterialB.1.2), perfect linear teleportation of an arbitrary two-qubit state (Supplementary MaterialB.2.1), bidirectional teleportation of arbitrary two-qubit states (Supplementary MaterialB.2.2) and perfect linear teleportation of an arbitrary three-qubit state (Supplementary MaterialB.3).

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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 statesfor n-qubit teleportation is reviewed and an extension of this formalism to more general classes of Matryoshka statesis posited. An example of a state close to a perfect Matryoshka Q-GHZ stateis 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.

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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.

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Quantum Secret Sharing (QSS) is a procedure for splitting a message into several parts so that no single subset of parts is sufficient to read the message, but the entire set is. This can also naturally be extended to Quantum Operation Sharing (QOS). In this section, quantum secret sharing using the 7 qubit XZW resource-state is proposed, with three proposals for the same.

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A.1 Proposal 1

Let us consider the situation in which Alice possesses the 1st qubit, Bob possesses qubits 2, 3, 4, 5, 6 and Charlie possesses the 7th qubit. Alice has an unknown qubit α0+β1which she wants to share with Bob and Charlie.

Now, Alice combines the unknown qubit with Ψ7and performs a Bell measurement, and conveys her outcome to Charlie by two classical bits. For instance if Alice measures in the Φ+basis, then the Bob-Charlie system evolves into the entangled state.

α100001α000100α000111α001001+α001010+α010101α010110α011000+α011011+α100010+α101100+α101111α110011+α111101α111110+β000000+β000011+β001101+β001110+β010001β010010+β011100β011111β100101β000000β100110+β101000+β101011+β110100β110111β111001+β111010EA1

Now, Bob can perform a five-qubit measurement and convey his outcome to Charlie through a classical channel. Having known the outcome of both their measurement, Charlie will obtain a certain single qubit quantum state. The outcome of the measurement performed by Bob is correlated with the state obtained by Charlie. If Bob measures A±then Charlie obtains the state α0±β1, while if Bob measures the state B±then Charlie obtains the state β0±α1, where

A±=00010+001010101101100+10001+1011011111±(00001+00110+010000111110010+101011101111100)EA2
B±=±(100000001100100+01010+01101+1011111001+11110)+00000+0011101001+011100000010011+10100+11010+11101EA3
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A.2 Proposal 2

Let us consider the situation in which Alice possesses the qubits 1 and 2, Bob possesses qubits 3, 4, 5 and 6 and Charlie possesses the 7th qubit. Alice has an unknown qubit α0+β1which she wants to share with Bob and Charlie. Now Alice can measure in a particular basis. Suppose she measures in the GHZ Basis. Now, Bob can perform a four-qubit measurement and convey his outcome to Charlie through a classical channel. Having known the outcome of both their measurement, Charlie will obtain a certain single qubit quantum state. The outcome of the measurement performed by Bob and the state obtained by Charlie is given as follows: if Bob measures states x±, Charlie obtains states α0±β1, while if Bob measures states Y±then Charlie obtains the states β0±α1, where x±=14α0000+α0111+α1001+α1110±β1001+β0000+β1110β0111and Y±=14α0001+α0110+α1000α1111±β1000+β0001β1111β0110

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A.3 Proposal 3

Let us consider the situation in which Alice possesses the qubits 1, 2, 3 and 4, Bob possesses qubits 5 and 6 and Charlie possesses the 7th qubit. Alice has an unknown qubit α0+β1which she wants to share with Bob and Charlie. Based on the state Alice measures Aii1,2,3,4,5,6,7,8, Bob and Charlie obtain a corresponding state BCi, where

A1=140111101011+10010+11001+11100+1110111000A2=1401111+01011100101100111100+1110111000A3=1401111+01011+10010+11001+1110011101+11000A4=14011110101110010110011110011101+11000A5=141111111011+00010+01001+01100+0110101000A6=1411111+11011000100100101100+0110101000A7=1411111+11011+00010+01001+0110001101+01000A8=14111111101100010010010110001101+01000

and

BC1=α1Φ+α0Ψ+β0Φ++β1Ψ+BC2=α1Φα0Ψβ0Φ++β1Ψ+BC3=α1Φα0Ψ+β0Φ+β1Ψ+BC4=α1Φ+α0Ψβ0Φ+β1Ψ+BC5=β1Φ+β0Ψ+α0Φ++α1Ψ+BC6=β1Φβ0Ψα0Φ++α1Ψ+BC7=β1Φβ0Ψ+α0Φ+α1Ψ+BC8=β1Φ+β0Ψα0Φ+α1Ψ+

Bob can now perform a Bell measurement on his particles, and Charlie can obtain a particular resultant state by applying the appropriate unitary operation.

For example, if the joint-state obtained by Bob and Charlie is β1Φ+β0Ψα0Φ+α1Ψ+, one can see that Charlie will obtain the state Ci,i=1,2,3,4corresponding to the state measured by Bob Bi, where B1=1201,B2=1210,B3=1211,B4=1200and C1=α0+β1,C2=α0β1,C3=α1+β0,C4=α1β0

B.1 Quantum teleportation of arbitrary one-qubit state

B.1.1 Linear teleportation scheme

To begin with, an arbitrary single qubit state can be teleported using the resource state Γ7will 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 ψ1=α0+β1to Bob. The combined state of the system is Γ71=ψ1Γ7. Alice measures the seven qubits in her possession via the seven qubit orthonormal states:

ξ±=0000ΨGHZ00001ΨGHZ3+0010ΨGHZ7+0011ΨGHZ40100ΨGHZ50101ΨGHZ6+0110ΨGHZ2+0111ΨGHZ1±(1000ΨGHZ21001ΨGHZ11010ΨGHZ51011ΨGHZ6+1100ΨGHZ7+1101ΨGHZ4+1110ΨGHZ01111ΨGHZ3)EA4
ν±=1000ΨGHZ01001ΨGHZ3+1010ΨGHZ7+1011ΨGHZ41100ΨGHZ51101ΨGHZ6+1110ΨGHZ2+1111ΨGHZ1±(0000ΨGHZ20001ΨGHZ10010ΨGHZ50011ΨGHZ6+0100ΨGHZ7+0101ΨGHZ4+0110ΨGHZ00111ΨGHZ3)EA5

where ΨGHZ0,1=12000±111,ΨGHZ2,3=12001±110,ΨGHZ4,5=12010±101and ΨGHZ6,7=12100±011.

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,σx,iσy,σzto recover the original state, sent by Alice. In this way, one can teleport an arbitrary single-qubit state using the state Γ7.

B.1.2 Probabilistic circular teleportation scheme for arbitrary one-qubit states

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 we 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 ψA=αA0A+βA1A,ψB=αB0B+βB1Band ψC=αC0C+βC1C. Then, the composite state is given by ψAψBψCΓ7TA1B1C1A1B1C1, where Γ7Tis the control qubit. We apply a CNOT gate using the qubits A,Band Cof 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 Γ7T=0.

Let us now measure the first qubits of Alice, Bob and Charlie in the Z-basis. Let us say Γ7A1B1C1=010, then we have the state ψ=14(001010αAαBαC0A0B0C+100111αAαBβC0A0B1C+000+011αAβBαC0A1B0C+101+110αAβBβC0A1B1C+(100111)βAαBαC1A0B0C+001+010βAαBβC1A0B1C+(101110)βAβBαC1A1B0C+000+011βAβBβC1A1B1C). We can now measure the control qubits in the X-basis. So, let us say, we have QAQBQC=+AB+C, then we obtain the state C1A1B1+χB3χ1B4B2+A2B1+χB3+χ1B4+B2+C2A1B1χB3χ1B4+B2+A2B1χB3+χ1B4B2+C3A1B4χB2χ1B1B3+A2B4χB2+χ1B1B3+C4A1B4+χB2χ1B1B3+A2B4+χB2+χ1B1+B3, where C1=βC0+αC1,C2=βC0αC1,C3=βC1+αC0,C4=βC1αC0,B1=αB1+βB0,B2=αB1βB0,B3=αB0+βB1,B4=αB0βB1,χ=a2a1with a1=βA+αA,a2=αAβA,A1=a10+a21,A1=a10a21. Therefore I see that the users can obtain states derived from the original state of the users next to them (AliceBob 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.

B.2 Quantum teleportation of arbitrary two-qubit state

B.2.1 Linear teleportation scheme

Similarly, an arbitrary two qubit quantum state can be teleported using the resource-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 ψ2=α00+μ10+γ01+β11to Bob. The combined state of the system is Γ72=ψ2Γ7,

Γ72=αA0000+A0101+A1010+A1111+μB0000+B0101+B1010+B1111+γC0000+C0101+C1010+C1111+βD0000+D0101+D1010+D1111EA6

where

A00=0000ΨGHZ0+0001ΨGHZ40010ΨGHZ50011ΨGHZ7,A11=0000ΨGHZ1+0001ΨGHZ50010ΨGHZ20011ΨGHZ7,A01=0000ΨGHZ60001ΨGHZ2+0010ΨGHZ4+0011ΨGHZ0,A10=0000ΨGHZ70001ΨGHZ3+0010ΨGHZ5+0011ΨGHZ1,B00=1000ΨGHZ0+1001ΨGHZ41010ΨGHZ2+1011ΨGHZ6,B11=1000ΨGHZ0+1001ΨGHZ51010ΨGHZ31011ΨGHZ7,B01=1000ΨGHZ61001ΨGHZ2+1010ΨGHZ4+1011ΨGHZ0,B10=1000ΨGHZ71001ΨGHZ3+1010ΨGHZ5+1011ΨGHZ1,C00=0100ΨGHZ0+0101ΨGHZ40110ΨGHZ3+0111ΨGHZ6,C11=0100ΨGHZ1+0101ΨGHZ50110ΨGHZ30111ΨGHZ7,C01=0100ΨGHZ60101ΨGHZ20110ΨGHZ4+0111ΨGHZ0,C10=0100ΨGHZ70101ΨGHZ3+0110ΨGHZ5+0111ΨGHZ1,D00=1100ΨGHZ0+1101ΨGHZ41110ΨGHZ2+1111ΨGHZ6,D11=1100ΨGHZ1+1101ΨGHZ51110ΨGHZ31111ΨGHZ7,D01=1100ΨGHZ61101ΨGHZ2+1110ΨGHZ4+1111ΨGHZ0,D10=1100ΨGHZ61101ΨGHZ3+1110ΨGHZ5+1111ΨGHZ1,

where ΨGHZ0,1=12000±111,ΨGHZ2,3=12001±110,ΨGHZ4,5=12010±101and ΨGHZ6,7=12100±011. Now, Bob can carry out a combination of unitary operations, according to the given table, to obtain the original state teleported by Alice.

State Obtained by AliceUnitary Operation by Bob
A01+B11+C00+D01Iσx
A01+B11C00D01σzσx
A01B11+C00D01Iiσy
A01B11C00+D01σziσy
A11+B01+C10+D00σxσx
A11B01+C10D00σxiσy
A11+B01C10D00iσyσx
A11B01C10+D00iσyiσy
A00+B10+C01+D11II
A00B10+C01D11Iσz
A00+B10C01D11σzI
A00B10C01+D11σzσz
A10+B11+C00+D01σxI
A10B11+C00D01σxσz
A10+B11C00D01iσyI
A10B11C00+D01iσyσz

B.2.2 Bidirectional teleportation of arbitrary two-qubit states

The resource-state can also be used for bidirectional quantum teleportation. Bidirectional Controlled Quantum Teleportation (BCQT) protocols have been proposed for multi-qubit resource states, such as five-qubit [81], six-qubit [82, 83], seven-qubit [84, 85, 86] and eight-qubit states [87]. Bidirectional Controlled Quantum Teleportation can teleport arbitrary states between two users under the supervision of a third party. Zha et al proposed the first scheme for BCQT of single qubit states using a maximally entangled seve-qubit quantum state [85]. There have been schemes proposed for BCQT that utilise states with the same number of qubits as the quantum channel being used, and thereby realise bidirectional teleportation of arbitrary single-and two-qubit states under the controller Charlie [84, 86].

Let us say Alice and Bob would like to teleport two-qubit states to each other by utilizing the seven-qubit genuinely entangled resource state. We assume the form of the two-qubit states to be

ϕA1A2=α000+α101+α210+α311EA7
ϕB1B2=β000+β101+β210+β311EA8

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 A1in the bell basis.

  • Bob measures qubit 2 of the resource state and B1in the bell basis.

  • Charlie, Alice and Bob measure their qubits in the Zbasis.

  • Alice and Bob measure their qubits A2and B2in 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.

We will now be looking more closely at these steps with a specific one instance to illustrate each step.

Step 1: Alice measures qubit 7 of the resource state and A1in the bell basis. If Alice measures ψ+, the remainder state is

142(000ψ+α0060A2+α1061A2+α2160A2+α3161A22+001ϕα0160A2+α1161A2+α2060A2+α3061A22+010ψα0160A2α1161A2α2060A2α3061A22+011ϕ+α0060A2+α1061A2α2160A2α3161A22+100ϕ+α0160A2+α1161A2+α2060A2+α3061A22+101ψα0060A2α1061A2α2160A2α3161A22+110ϕα0060A2α1061A2+α2160A2+α3161A22+111ψ+α0160A2α1161A2+α2060A2+α3061A22)ϕB.

Alice communicates her result to Bob using a classical channel.

Step 2: Bob measures qubit 2 of the resource state and B1in the bell basis. If Bob Measures ψ+, the remainder state is

122(000α0060A2+α1061A2+α2160A2+α3161A22×12β000+β101+β210+β3114,B2+001×α0160A2+α1161A2+α2060A2+α3061A22×12β000+β101+β210+β3114,B2+010×α0060A2+α1061A2α2160A2α3161A22×12β000+β101β210β3114,B2+011×α0060A2+α1061A2α2160A2α3161A22+12β010+β111β200β3014,B2+100×α0160A2+α1161A2+α2060A2+α3061A22×12β010+β111+β200+β3014,B2+101×α0060A2α1061A2α2160A2α3161A22
×12β000+β101β210β3114,B2+110×α0060A2α1061A2+α2160A2+α3161A22×12β010+β111β210β3014,B2+111×α0160A2α1161A2+α2060A2+α3061A22×12β000+β101+β210+β3114,B2).

Bob communicates his result via a classical channel to Alice.

Step 3: Charlie, Alice and Bob measure their qubits in the Z-basis. Let us say they all measure 0, we have the

12(000α0060A2+α1061A2+α2160A2+α3161A22×12β000+β101+β210+β3114,B2)EA9

Step 4:Let Alice apply a CNOT with A2as control and qubit 1 as target, and let Bob apply a CNOT with and B2as control and qubit 3 as target, to get

122(0α0000+α1101A2+α2010+α31111,6,A22×12β0000+β1101+β2010+β31113,4,B2EA10

Step 4:Alice and Bob measure their qubits A2and B2in the X-basis. Let us say they obtain the state +=120+1, then the composite state is given by

12(0α000+α110A2+α201+α3111,62×12β000+β110+β201+β3113,4EA11

Step 5: 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. In this instance, the unitary transformation is simply IIIIwith Ibeing the identity matrix.

B.3 Quantum teleportation of arbitrary three-qubit state

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 ψ3=a000+b001+c010+d011+e100+f101+g110+h111to 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=12,b=c=d=e=f=g=0and the W state for b=c=e=13,a=d=f=g=h=0.

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

Γ73=ψ3Γ7=aA000000+aA001001+aA010010+aA011011+aA100100+aA101101+aA110110+aA111111+bB000000+bB001001+bB010010+bB011011+bB100100+bB101101+bB110110+bB111111+cC000000+cC001001+cC010010+cC011011+cC100100+cC101101+cC110110+cC111111+dD000000+dD001001+dD010010+dD011011+dD100100+dD101101+dD110110+dD111111+eE000000+eE001001+eE010010+eE011011+eE100100+eE101101+eE110110+eE111111+fF000000+fF001001+fF010010+fF011011+fF100100+fF101101+fF110110+fB111111+gG000000+gG001001+gG010010+gG011011+gG100100+gG101101+gG110110+gG111111+hH000000+hH001001+hH010010+hH011011+hH100100+hH101101+hH110110+hH111111EA12

with

A000=0000000+00001010001011+0001110A001=0000010+0001001+00011000000111A010=00001110000010+0001001+0001100A011=0000000+0000101+00010110001110A100=0000011+00001100001000+0001101A101=00000010000100+0001010+0001111A110=0000001000010000010100001111A111=0001101000001100001100001000
B000=0010000+00101010011011+0011110B001=0010010+0011001+00111000010111B010=00101110010010+0011001+0011100B011=0010000+0010101+00110110011110B100=0010011+00101100011000+0011101B101=00100010010100+0011010+0011111B110=0010001001010000110100011111B111=0011101001001100101100011000
C000=0100000+01001010101011+0101110C001=0100010+0101001+01011000100111C010=01001110100010+0101001+0101100C011=0100000+0100101+01010110101110C100=0100011+01001100101000+0101101C101=01000010100100+0101010+0101111C110=0100001010010001010100101111C111=0101101010001101001100101000
D000=0110000+01101010111011+0111110D001=0110010+0111001+01111000110111D010=01101110110010+0111001+0111100D011=0110000+0110101+01110110111110D100=0110011+01101100111000+0111101D101=01100010110100+0111010+0111111D110=0110001011010001110100111111D111=0111101011001101101100111000
E000=1000000+10001011001011+1001110E001=1000010+1001001+10011001000111E010=10001111000010+0001001+0001100E011=1000000+1000101+10010111001110E100=1000011+10001101001000+1001101E101=10000011000100+1001010+1001111E110=1000001100010010010101001111E111=1001101100001110001101001000
F000=1010000+10101011011011+1011110F001=1010010+1011001+10111001010111F010=10101111010010+1011001+1011100F011=1010000+1010101+10110111011110F100=1010011+10101101011000+1011101F101=10100011010100+1011010+1011111F110=1010001101010010110101011111F111=1011101101001110101101011000
G000=1100000+11001011101011+1101110G001=1100010+1101001+11011001100111G010=11001111100010+1101001+1101100G011=1100000+1100101+11010111101110G100=1100011+11001101101000+1101101G101=11000011100100+1101010+1101111G110=1100001110010011010101101111G111=1101101110001100001100001000
H000=1110000+11101011111011+1111110H001=1110010+1111001+11111001110111H010=11101111110010+1111001+1111100H011=1110000+1110101+11110111111110H100=1110011+11101101111000+1111101H101=11100011110100+1111010+1111111H110=1110001111010011110101111111H111=1111101111001111101101111000

An arbitrary three qubit state can be decomposed in terms of these basis-vectors,

(a000+b001+c010+d011+e100+f101+g110+h111)Ψ7=permutations(1I1Aa1a2a3+1I2Bb1b2b3+1I3Cc1c2c3+1I4Dd1d2d3+1I5Ee1e2e3+1I6Ff1f2f3+1I7Gg1g2gs+1I8Hh1h2h3)(1I1aa1a2a3+1I2bb1b2b3+1I3cc1c2c3+1I4dd1d2d3+1I5ee1e2e3+1I6ff1f2f3+1I7gg1g2g3+1I8hh1h2h3)EA13

where Iii=1,2,3,4,5,6,7,8can take values 0or 1 independently, and LjL=abcdefghj=123can take values 0or 1 independently. The summation is over all possible permutation states obtained.

The relevant transformations for the three-qubit teleportation are given in terms of the following basic operations:

Projection of ithcomponent Pi:

P1=1000P2=0001EA14

Flip and Projection of ithcomponent Fi:

F1=0100F2=0010EA15

State Obtained by AliceShort-Hand Form of Transformation
A000+B001+C010+D011+E100+F101+G110+H111I2I2I2
A000B001+C010+D011+E100+F101+G110+H111I2I2P2+σzP1P1+I2P2P1
A000+B001+C010D011+E100+F101+G110+H111I2I2P2+I2P1P1+σzP2P1
A000+B001+C010+D011+E100F101+G110+H111I2I2P1+σzP1P2+I2P2P2
A000+B001+C010+D011+E100+F101+G110H111I2I2P1+σzP2P2+I2P2P2
A000B001+C010D011+E100+F101+G110+H111I2I2P2+σzP1P1+σzP2P1
A000B001+C010+D011+E100F101+G110+H111σzP1P1+I2P2P1+σzP1P2+I2P2P2
A000B001+C010D011+E100F101+G110+H111σzP1P1+σzP2P1+σzP1P2+I2P2P2
A000+B001+C010D011+E100F101+G110+H111I2P1P1+σzP2P1+σzP1P2+I2P2P2
A000B001+C010+D011+E100+F101+G110H111σzP1P1+I2P2P1+P2P1P2+σzP2P2
A000+B001+C010D011+E100+F101+G110H111I2P1P1+σzP2P1+I2P1P2+σzP2P2
A000+B001+C010+D011+E100F101+G110H111I2P1P1+I2P2P1+σzP1P2+σzP2P2
A000B001+C010+D011+E100F101+G110H111σzP1P1+I2P2P1+σzP1P2+σzP2P2
A000+B001+C010+D011+E100F101+G110H111I2P1P1+I2P2P1+σzP1P2+σzP2P2
A000B001+C010D011+E100+F101+G110H111σzP1P1+σzP2P1+I2P1P2+σzP2P2
A000B001+C010D011+E100F101+G110H111σzP1P1+σzP2P1+σzP1P2+σzP2P2
A000+B001C010+D011+E100+F101+G110+H111I2P1P1σzP2P1+I2P1P2+I2P2P2
A000+B001C010+D011+E100+F101G110+H111I2P1P1σzP2P1+I2P1P2σzP2P2
A000+B001C010D011+E100+F101+G110+H111I2P1P1I2P2P1+I2P1P2+I2P2P2
A000+B001C010+D011+E100+F101G110+H111I2P1P1σzP2P1+I2P1P2σzP2P2
A000+B001C010+D011+E100+F101+G110H111I2P1P1σzP2P1+I2P1P2+σzP2P2
A000+B001+C010D011+E100+F101G110+H111I2P1P1+σzP2P1+I2P1P2+σzP2P2
A000+B001+C010+D011+E100+F101G110H111I2P1P1+I2P2P1+I2P1P2+σzP2P2
A000+B001C010D011+E100+F101G110+H111I2P1P1I2P2P1+I2P1P2σzP2P2
A000+B001C010D011+E100+F101+G110H111I2P1P1I2P2P1+I2P1P2+σzP2P2
A000+B001+C010D011+E100+F101G110H111I2P1P1+σzP2P1+I2P1P2I2P2P2
A000+B001C010+D011+E100+F101G110H111I2P1P1σzP2P1+I2P1P2I2P2P2
A000+B001C010D011+E100+F101G110H111I2P1P1I2P2P1+I2P1P2I2P2P2
A000+B001+C010+D011E100+F101+G110+H111I2P1P1+I2P2P1σzP1P2+I2P2P2
A000+B001C010+D011E100F101+G110+H111σzP1P1σzP2P1I2P1P2+I2P2P2
A000+B001+C010+D011E100F101+G110+H111I2P1P1+I2P2P1I2P1P2+I2P2P2
A000+B001+C010+D011E100+F101+G110H111I2P1P1+I2P2P1σzP1P2+σzP2P2
A000+B001+C010+D011+E100F101G110+H111I2P1P1+I2P2P1+σzP1P2σzP2P2
A000+B001+C010+D011+E100+F101G110H111I2P1P1+I2P2P1+I2P1P2I2P2P2
A000+B001+C010+D011+E100F101G110+H111I2P1P1+I2P2P1+σzP1P2σzP2P2
A000+B001+C010+D011E100+F101G110H111I2P1P1+I2P2P1σzP1P2I2P2P2
A000+B001+C010+D011+E100F101G110H111I2P1P1+I2P2P1+σzP1P2I2P2P2
A000+B001+C010+D011E100F101G110H111I2P1P1+I2P2P1I2P1P2I2P2P2
A001+B000+C011+D010+E101+F100+G111+H110σxI2I2
A001+B000+C011+D010+E101+F100+G111+H110σxP1P2+σxP2P2+IσyP1P1
A001+B000C011+D010+E101+F100+G111+H110σxP1I2+IσyP2P1+σxI2P2
A001+B000+C011+D010E101+F100+G111+H110σxI2P1+IσyP1P2+σxP2P2
A001+B000+C011+D010+E101+F100G111+H110σxI2P1+σxP1P2+IσyP2P2
A001+B000C011+D010+E101+F100+G111+H110IσyI2P1+σxI2P2
A001+B000+C011+D010E101+F100+G111+H110IσyP1P1+σxP2P1+IσyP1P2+σxP2P2
A001+B000+C011+D010+E101+F100G111+H110IσyP1P1+σxP2P1+σxP1P2+IσyP2P2
A001+B000C011+D010E101+F100+G111+H110σxP1P1+IσyP2P1+IσyP1P2+σxP2P2
A001+B000C011+D010+E101+F100G111+H110σxP1P1+IσyP2P1+σxP1P2+IσyP2P2
A001+B000+C011+D010E101+F100G111+H110σxP1P1+σxP2P1+IσyP1P2+IσyP2P2
A001+B000C011+D010E101+F100+G111+H110σxP1P1+IσyP2P1+IσyP1P2+σxP2P2
A001+B000+C011+D010E101+F100G111+H110IσyP1P1+σxP2P1+IσyP1P2+IσyP2P2
A001+B000C011+D010+E101+F100G111+H110IσyP1P1+IσyP2P1+σxP1P2+IσyP2P2
A001+B000C011+D010E101+F100G111+H110σxP1P1+IσyP2P1+IσyP1P2+IσyP2P2
A001+B000C011+D010E101+F100G111+H110IσyP1P1+IσyP2P1+σxP1P2+IσyP2P2
A001+B000+C011D010+E101+F100+G111+H110σxP1P1IσyP2P1+σxP1P2+σxP2P2
A001+B000+C011+D010+E101+F100+G111H110σxP1P1+σxP2P1+σxP1P2IσyP2P2
A001+B000C011+D010+E101+F100+G111H110σxP1P1+IσyP2P1+σxP1P2+IσyP2P2
A001+B000C011D010+E101+F100+G111+H110σxP1P1σxP2P1+σxP1P2+σxP2P2
A001+B000+C011D010+E101+F100G111+H110σxP1P1IσyP2P1+σxP1P2+IσyP2P2
A001+B000+C011D010+E101+F100+G111H110σxP1P1IσyP2P1+σxP1P2IσyP2P2
A001+B000+C011+D010+E101+F100G111H110σxP1P1+σxP2P1+IσyP1P2σxP2P2
A001+B000C011D010+E101+F100G111+H110σxP1P1σxP2P1+σxP1P2+IσyP2P2
A001+B000C011D010+E101+F100+G111H110σxP1P1σxP2P1+σxP1P2IσyP2P2
A001+B000C011+D010+E101+F100G111H110σxP1P1σxP2P1+σxP1P2IσyP2P2
A001+B000+C011D010+E101+F100G111H110σxP1P1IσyP2P1+σxP1P2σxP2P2
A001+B000C011D010+E101+F100G111H110σxP1P1σxP2P1+σxP1P2σxP2P2
A001+B000+C011+D010+E101F100+G111+H110σxP1P1+σxP2P1IσyP1P2+σxP2P2
A001+B000+C011+D010E101F100+G111+H110σxP1P1+σxP2P1+IσyP1P2+σxP2P2
A001+B000+C011+D010E101+F100G111+H110σxP1P1+σxP2P1+IσyP1P2+IσyP2P2
A001+B000+C011+D010E101+F100+G111H110σxP1P1+σxP2P1+IσyP1P2IσyP2P2
A001+B000+C011+D010+E101F100G111+H110σxP1P1+σxP2P1IσyP1P2+IσyP2P2
A001+B000+C011+D010+E101F100+G111H110σxP1P1+σxP2P1IσyP1P2IσyP2P2
A001+B000+C011+D010E101F100G111+H110σxP1P1+σxP2P1σxP1P2+IσyP2P2
A001+B000+C011+D010E101+F100G111H110σxP1P1+σxP2P1+IσyP1P2σxP2P2
A001+B000+C011+D010+E101F100G111H110σxP1P1+σxP2P1IσyP1P2σxP2P2
+A001+B000+C011+D010E101F100G111H110σxP1P1+σxP2P1σxP1P2σxP2P2
A010B011+C000+D001+E110+F111+G100+H101σzF1P1+I2F2P1+I2F1P2+I2P2P2
A010+B011+C000D001+E110+F111+G100+H101I2F1P1+σzF2P1+I2F1P2+I2P2P2
A010+B011+C000+D001+E110F111+G100+H101I2F1P1+I2F2P1+σzF1P2+I2P2P2
A010+B011+C000+D001+E110+F111+G100H101σzF1P1+I2F2P1+I2F1P2+σzP2P2
A010B011+C000D001+E110+F111+G100+H101σzF1P1+σzF2P1+I2F1P2+I2P2P2
A010B011+C000+D001+E110F111+G100+H101σzF1P1+I2F2P1+σzF1P2+I2P2P2
A010B011+C000+D001+E110+F111+G100H101σzF1P1+I2F2P1+I2F1P2+σzP2P2
A010+B011+C000+D001+E110+F111+G100+H101I2F1P1+I2F2P1+I2F1P2+I2P2P2
A010+B011+C000D001+E110F111+G100+H101I2F1P1+σzF2P1+σzF1P2+I2P2P2
A010+B011+C000D001+E110+F111+G100H101I2F1P1+σzF2P1+I2F1P2+σzP2P2
A010+B011+C000+D001+E110F111+G100H101I2F1P1+I2F2P1+σzF1P2+σzP2P2
A010B011+C000D001+E110F111+G100+H101σzF1P1+σzF2P1+I2F1P2+I2P2P2
A010B011+C000D001+E110+F111+G100H101σzF1P1+σzF2P1+I2F1P2+σzP2P2
A010+B011+C000D001+E110F111+G100H101I2F1P1+σzF2P1+σzF1P2+σzP2P2
A010B011+C000+D001+E110F111+G100H101σzF1P1+I2F2P1+σzF1P2+σzP2P2
A010B011+C000D001+E110F111+G100H101σzF1P1+σzF2P1+σzP2P2+σzP2P2
A010+B011+C000+D001+E110+F111+G100+H101σzF1P1+I2F2P1+I2F1P2+I2P2P2
A010+B011+C000+D001E110+F111+G100+H101I2F1P1+I2F2P1σzF1P2+I2P2P2
A010B011+C000+D001+E110+F111+G100+H101I2F1P1+I2F2P1+I2F1P2+I2P2P2
A010+B011+C000+D001E110+F111+G100+H101σzF1P1+I2F2P1σzF1P2+I2P2P2
A010+B011+C000+D001+E110F111+G100+H101σzF1P1+I2F2P1+σzF1P2+I2P2P2
A010B011+C000+D001E110+F111+G100+H101σzF1P1+I2F2P1σzF1P2+I2P2P2
A010+B011+C000+D001E110F111+G100+H101I2F1P1+I2F2P1I2F1P2+I2P2P2
A010B011+C000+D001E110+F111+G100+H101I2F1P1+I2F2P1σzF1P2+I2P2P2
A010+B011+C000+D001E110F111+G100+H101σzF1P1+I2F2P1I2F1P2+I2P2P2
A010B011+C000+D001E110F111+G100+H101σzF1P1+I2F2P1I2F1P2+I2P2P2
A010B011+C000+D001+E110F111+G100+H101I2F1P1+I2F2P1+σzF1P2+I2P2P2
A010B011+C000+D001E110F111+G100+H101I2F1P1+I2F2P1I2F1P2+I2P2P2
A010+B011+C000+D001+E110+F111G100+H101I2F1P1+I2F2P1+I2F1P2σzP2P2
A010+B011+C000+D001E110+F111G100+H101I2F1P1+I2F2P1σzF1P2σzP2P2
A010+B011+C000+D001E110+F111+G100H101I2F1P1+I2F2P1σzF1P2+σzP2P2
A010+B011+C000+D001+E110F111G100+H101I2F1P1+I2F2P1+σzF1P2σzP2P2
A010+B011+C000+D001+E110F111+G100H101I2F1P1+I2F2P1+σzF1P2+σzP2P2
A010+B011+C000+D001E110F111G100+H101I2F1P1+I2F2P1I2F1P2σzP2P2
A010+B011+C000+D001E110+F111G100H101I2F1P1+I2F2P1σzF1P2I2P2P2
A010+B011+C000+D001+E110F111G100H101I2F1P1+I2F2P1+σzF1P2I2P2P2
A010+B011+C000+D001E110F111+G100H101I2F1P1+I2F2P1I2F1P2+σzP2P2
A010+B011+C000+D001E110F111G100H101I2F1P1+I2F2P1I2F1P2I2P2P2
A011+B010+C001+D000E111F110G101H100σxσxP1σxσxP2
A011+B010+C001+D000+E111+F110+G101+H100σxσxI2
A011+B010+C001+D000+E111+F110+G101+H100IσyF1P1+σxF2P1+σxσxP2
A011+B010C001+D000+E111+F110+G101+H100σxF1P1+IσyF2P1IσyF1P2+σxF2P2
A011+B010+C001+D000E111+F110+G101+H100σxσxP1+IσyF1P2+σxF2P2
A011+B010+C001+D000+E111+F110G101+H100σxσxP1+σxF1P2+IσyF2P2
A011+B010C001+D000+E111+F110+G101+H100IσyσxP1+σxF1P2+σxF2P2
A011+B010+C001+D000E111+F110+G101+H100IσyF1P1+σxF2P1+IσyF1P2+σxF2P2
A011+B010+C001+D000+E111+F110G101+H100IσyF1P1+σxF2P1+σxF1P2+IσyF2P2
A011+B010C001+D000E111+F110+G101+H100σxF1P1+IσyF2P1+IσyF1P2+σxF2P2
A011+B010C001+D000+E111+F110G101+H100σxF1P1+IσyF2P1+σxF1P2+IσyF2P2
A011+B010+C001+D000E111+F110G101+H100σxσxP1+IσyF1P2+IσyF2P2
A011+B010C001+D000E111+F110+G101+H100IσyσxP1+IσyF1P2+σxF2P2
A011+B010C001+D000+E111+F110G101+H100IσyσxP1+σxF1P2+IσyF2P2
A011+B010+C001+D000E111+F110G101+H100IσyF1P1+σxF2P1+IσyσxP2
A011+B010C001+D000E111+F110G101+H100σxF1P1+IσyF2P1+IσyF1P2+IσyF2P2
A011+B010C001+D000E111+F110G101+H100IσyσxP1+IσyF1P2+IσyF2P2
A011B010+C001+D000+E111+F110+G101+H100IσyF1P1+σxF2P1+σxσxP2
A011+B010+C001+D000+E111F110+G101+H100σxσxP1IσyF1P2+σxF2P2
A011B010+C001+D000+E111+F110+G101+H100σxIσyP1+σxF1P2+σxF2P2
A011+B010+C001+D000+E111F110+G101+H100IσyF1P1+σxF2P1IσyF1P2+σxF2P2
A011B010+C001+D000E111+F110+G101+H100IσyF1P1+σxF2P1+IσyF1P2+σxF2P2
A011B010+C001+D000+E111F110+G101+H100IσyF1P1+σxF2P1IσyF1P2+σxF2P2
A011+B010+C001+D000E111F110+G101+H100σxσxP1+σxIσyP2
A011B010+C001+D000E111+F110+G101+H100σxIσyP1+IσyF1P2+σxF2P2
A011B010+C001+D000+E111F110+G101+H100σxIσyP1IσyF1P2+σxF2P2
A011+B010+C001+D000E111F110+G101+H100IσyF1P1+σxF2P1+σxIσyP2
A011B010+C001+D000E111F110+G101+H100IσyF1P1+σxF2P1+σxIσyP2
A011B010+C001+D000E111F110+G101+H100σxIσyP1+σxIσyP2
A011+B010+C001+D000+E111+F110+G101H100σxσxP1+σxF1P2IσyF2P2
A011+B010+C001+D000E111+F110+G101H100σxσxP1+σxIσyP2
A011+B010+C001+D000+E111F110+G101H100σxσxP1IσyσxP2
A011+B010+C001+D000+E111F110G101+H100σxσxP1+IσyIσyP2
A011+B010+C001+D000+E111+F110G101H100σxσxP1σxIσyP2
A011+B010+C001+D000E111F110G101+H100σxσxP1σxF1P2+IσyF2P2
A011+B010+C001+D000E111F110+G101H100σxσxP1σxF1P2IσyF2P2
A011+B010+C001+D000E111+F110G101H100σxσxP1+IσyF1P2σxF2P2
A011+B010+C001+D000+E111F110G101H100σxσxP1IσyF1P2σxF2P2
A100B101C110D111+E000+F001+G010+H011I2I2F1+I2I2F2
A100+B101+C110+D111+E000+F001+G010+H011I2I2F1+I2I2F2
A100B101+C110+D111+E000+F001+G010+H011σzP1F1+I2P2F1+I2I2F2
A100+B101+C110D111+E000+F001+G010+H011I2P1F1+σzP2F1+I2I2F2
A100+B101+C110+D111+E000F001+G010+H011I2I2F1+σzP1F2+I2P2F2
A100+B101+C110+D111+E000+F001+G010H011I2I2F1+I2P1F2+σzP2F2
A100B101+C110D111+E000+F001+G010+H011σzI2F1