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The Majorana representation for generalized GHZ states |gGHZ ≡ a|000 + b|111, and generalized W states |gW ≡ c|001 + d|010 + e|100. Here, a ≡ cos ϑ, b ≡ sin ϑ, and ϑ = π/10.
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Majorana stars, the 2j spin coherent states that are orthogonal to a spin-j state, offer a visualization of general quantum states and may disclose deep structures in quantum states and their evolutions. In particular, the genuine tripartite entanglement—the three-tangle of a symmetric three-qubit state, which can be mapped to a spin-3/2 state—is m...
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... to the equator-the closer to the equator, the higher the entanglement. As an example, a generalized Greenberger-Horne-Zeilinger (GHZ) state |gGHZ ≡ a|000 + b|111 with a three-tangle 4a 2 b 2 is already symmetrized and hence is represented by three distinct Majorana stars distributed evenly on the southern hemisphere with the same latitude [see Fig. ...
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... root. Hence, a generalized W state is represented by three Majorana stars on unit sphere-one is located at the south pole and two degenerate ones are located at the north pole. It shows that the appearance of a pair of degenerate Majorana stars on the unit sphere indicates the vanishing of the three-tangle for a general three-qubit state [see Fig. ...
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Citations
... Apart from indisputable mathematical advantages [22], this picture builds a bridge between the abstract Hilbert space (where the states live) and the simple geometry of the Bloch sphere. Consequently, this representation rapidly meets the increasing interest in high-dimensional quantum systems and, several decades after its conception, is being used in fields as diverse as polarization [23][24][25][26][27], spinor Bose gases [28][29][30][31], multiqubit systems [32][33][34][35][36][37][38], metrology [39][40][41][42][43], geometric phases [44][45][46][47][48][49][50][51], non-Hermitian lattices [52], and algebraic quantum models such as the Lipkin-Meshkov-Glick model [53,54]. The distribution of Majorana stars conveys complete information and can be directly computed when the quantum state is known. ...
Majorana stars, the 2S spin coherent states that are orthogonal to a spin-S state, offer an elegant method to visualize quantum states, disclosing their intrinsic symmetries. These states are naturally described by the corresponding multipoles. These quantities can be experimentally determined and allow for an SU(2)-invariant analysis. We investigate the relationship between Majorana constellations and state multipoles, thus providing insights into the underlying symmetries of the system. We illustrate our approach with some relevant and informative examples.
... However, the problem of creating analogues of more general entanglements in classical continuum systems continues to present challenges [10,17,18]. Here, by building on ideas underlying the Poincaré-Bloch sphere [19,20] and Majorana star [21][22][23][24] representations of quantum states, we exhibit a family of formal mappings between n-qubit states and 2D hydrodynamic flow fields. The topological defects and stagnation points of these flow fields encode properties of the corresponding n-qubit state. ...
We describe a general procedure for mapping arbitrary $n$-qubit states to two-dimensional (2D) vector fields. The mappings use complex rational function representations of individual qubits, producing classical vector field configurations that can be interpreted in terms of 2D inviscid fluid flows or electric fields. Elementary qubits are identified with localized defects in 2D harmonic vector fields, and multi-qubit states find natural field representations via complex superpositions of vector field products. In particular, separable states appear as highly symmetric flow configurations, making them both dynamically and visually distinct from entangled states. The resulting real-space representations of entangled qubit states enable an intuitive visualization of their transformations under quantum logic operations. We demonstrate this for the quantum Fourier transform and the period finding process underlying Shor's algorithm, along with other quantum algorithms. Due to its generic construction, the mapping procedure suggests the possibility of extending concepts such as entanglement or entanglement entropy to classical continuum systems, and thus may help guide new experimental approaches to information storage and non-standard computation.
... This representation provides an intuitive tool to study the many-body boson system and symmetric multiqubit system. By using it, the symmetric related feature of these system, such as Berry phase [2][3][4][5][6], entanglement [7][8][9][10][11][12][13][14][15], geometric structure and operations [16][17][18][19][20][21][22][23][24][25][26]. can be efficiently interpreted by the structures and trajectories of stars on the sphere. ...
The Majorana representation, which provides an intuitive way to represent the quantum state by stars on the Bloch sphere, has drawn considerable attention in the context of many body systems or systems with high-dimensional Hilbert space. In this work, we study Majorana representation for an interacting spin-1/2 composite system and its relation with the quantum correlation between the subsystems. Similar with the Majorana representation, we can define a new intuitive representation for the composite system by defining three stars which represents the dynamics and correlation of the two subsystems. The results show that, these two representations are the same for the separable case. The Berry phase has been found to be directly related to the concurrence between the two subsystems. By study the cases of no correlation and large correlation, the quantum entanglement and the Berry phases of the subsystem and their two representations on the Bloch sphere are discussed. These representations by stars provides a intuitive way to study the dynamics, entanglement and Berry phase composite system and its subsystems.
... Multipartite symmetric states with maximal entropy of entanglement were identified by using the symmetry of the Majorana stars [41,42]. The genuine tripartite entanglement of non-symmetric three-qubit states was found represented by three distinct Majorana stars [43]. ...
Majorana stars, the antipodal directions associated with the coherent states that are orthogonal to a spin state, provide a visualization and a geometric understanding of the structures of general quantum states. For example, the Berry phase of a spin-1/2 is given by half the solid angle enclosed by the close path of its Majorana star. It is conceivable that the Berry phase of higher spins may also be related to the geometry of the Majorana constellation. We find that for a spin-1 state, besides the expected contributions from the solid angles enclosed by the close paths of the two Majorana stars, the Berry phase includes a term related to the twist of the relative position vector around the barycenter vector of the two Majorana stars, i.e., the self-rotation of the constellation. Interestingly, if the spin-1 state is taken as a symmetrized two-qubit state, the extra contribution to the Berry phase is given by the self-rotation of the Majorana constellation weighted by the quantum entanglement of the two qubits. This discovery alludes to the relevance of the Majorana stellar geometry in representing the deep structures of quantum states and of quantum entanglement.
... These 2J points are called Majorana stars (MSs) of the state. Consequently, this representation becomes an efficient method to study the geometric related properties of multipartite or high-dimensional state such as entanglement [2][3][4][5][6][7][8][9][10][11], Berry phase [12][13][14][15][16][17][18], as well as their dynamics, geometric structures and response to geometric transformations (e.g. rotations and inversions) [19][20][21][22][23][24][25][26][27][28][29]. ...
... Since the MR can represent all the symmetric states, a generic two-qubit pure state can be consequently decomposed by (2) asym (11) with two MSs u 1 = (θ 1 , φ 1 ) and u 2 = (θ 2 , φ 2 ) for |Ψ (2) sym on the Bloch sphere S M which can be determined by the roots x j ≡ tan θ j 2 e iφ j of star equation (see appendix A for details) ...
... With this Majorana decomposition (11), and equation (4), the Berry phase accumulated by an adiabatic evolution process of |ψ can be derived by ...
The Majorana representation (MR), which represents a quantum state and its evolution with the Majorana stars (MSs) on the Bloch sphere, provides us an intuitive way to study symmetric multiqubit pure states physical system with SU(2) symmetry. In this work, we propose a method to extend the MR for generic two-qubit pure states. By adding an additional reference star (RS), which represents the symmetry of the qubits, on the symmetric two qubit state with two base stars, we give a more general MR for both symmetric, anti-symmetric and generic two-qubit states. Using this method, we study the entanglement, Berry phase and their relation by this Majorana decomposition. Furthermore, the symmetries for different type of two-qubit states are also discussed via the RSs and correlation between the two MSs.
... Those two representations are linked by the Majorana polynomial, whose roots determine the associated system of points on the sphere by the stereographic projection [19]. The Majorana representation is especially convenient for studying SLOCC transformations, which are represented by the well-studied Möbius transformations [19,20], and has become a useful tool to study permutation-symmetric states [21][22][23]. ...
We show that the invariance of an entanglement measure under Stochastic Local Operations with Classical Communication (SLOCC) expands to its convex roof extension. SLOCC may be represented by M\"obius transformations on the roots of the entanglement measure on the Bloch sphere. We derive efficient conditions for n-qubit states to be SLOCC-equivalent. We apply our approach on 4-qubit states and show that roots of the 3-tangle measure classify 4-qubit generic states. Moreover, we propose a method to obtain the normal form of a 4-qubit state which bypasses the possibly infinite iterative procedure.
... In theoretical quantum information, the Majorana representation has enabled the geometric study of symmetric multiqubit states [9,10], the construction of geometrically mutually unbiased bases and symmetric informationally complete positive operator valued measures [11], the calculation of the spectrum of the Lipkin-Meshkov-Glick model [12,13], the study of Berry phases [14,15], and the classification of high-* shruti.dogra@aalto.fi dimensional entanglement [10,16,17]. It has also inspired the search for alternative geometric representations of quantum states [18][19][20][21][22][23]. ...
... We define the distance between two Majorana points as η = cos −1 (Ŝ 1 · S 2 ) = cos −1 [sin θ 1 sin θ 2 cos(ϕ 1 − ϕ 2 ) + cos θ 1 cos θ 2 ], where the Majorana starsŜ k have respective spherical coordinates (θ k , ϕ k ), k ∈ {1, 2}. This distance can be interpreted as a direct measure of entanglement between two qubits in the symmetrized-state representation of the qutrit [16,17]. Explicitly, any spin-1 state-and, by the identification above, any qutrit state Eq. ...
We show that stimulated Raman adiabatic passage (STIRAP) and its superadiabatic version (saSTIRAP) have a natural geometric two-star representation on the Majorana sphere. In the case of STIRAP, we find that the evolution is confined to a vertical plane. A faster evolution can be achieved in the saSTIRAP protocol, which employs a counterdiabatic Hamiltonian to nullify the nonadiabatic excitations. We derive this Hamiltonian in the Majorana picture, and we observe how, under realistic experimental parameters, the counterdiabatic term corrects the trajectory of the Majorana stars toward the dark state. We also introduce a spin-1 average vector and present its evolution during the two processes, demonstrating that it provides a measure of nonadiabaticity. We show that the Majorana representation can be used as a sensitive tool for the detection of process errors due to ac Stark shifts and nonadiabatic transitions. Finally, we provide an extension of these results to mixed states and processes with decoherence.
... This concept has been proved useful in various experimental contexts, such as for representing states of polarization of light [5], for characterizing the symmetry of the order parameter in spinor Bose-Einstein condensates [6,7], and for the decomposition of quantum gates used in nuclear magnetic resonance into experimentally implementable pulses [8]. In theoretical quantum information, the Majorana representation has enabled the geometric study of symmetric multi-qubit states [9,10], the construction of geometrically mutually unbiased bases and symmetric informationally complete positive operator valued measures [11], the study of Berry phases [12,13], and the classification of high-dimensional entanglement [10,14,15]. It has also inspired the search for alternative geometric representations of quantum states [16][17][18][19][20][21]. ...
... We define the distance between two Majorana points as η = cos −1 (Ŝ 1 .Ŝ 2 ) = cos −1 [sin θ 1 sin θ 2 cos(ϕ 1 − ϕ 2 ) + cos θ 1 cos θ 2 ], where the Majorana starsŜ k have respective spherical coordinates (θ k , ϕ k ), k ∈ {1, 2}. This distance can be interpreted as a direct measure of entanglement between two qubits in the symmetrized-state representation of the qutrit [14,15]. Explicitly, any spin-1 state -and, by the identification above, any qutrit state Eq. ...
We show that stimulated Raman adiabatic passage (STIRAP) and its superadiabatic version (saSTIRAP) have a natural geometric two-star representation on the Majorana sphere. In the case of STIRAP, we find that the evolution is confined to a vertical plane. A faster evolution can be achieved in the saSTIRAP protocol, which employs a counterdiabatic Hamiltonian to nullify the non-adiabatic excitations. We observe how, under realistic experimental parameters, the counterdiabatic term corrects the trajectory of the Majorana stars toward the dark state. We also introduce a spin-1 average vector and present its evolution during the two processes. We show that the Majorana representation can be used as a sensitive tool for the detection of process errors due to ac Stark shifts and non-adiabatic transitions.
We show that a single polynomial entanglement measure is enough to verify equivalence between generic n-qubit states under stochastic local operations with classical communication (SLOCC). SLOCC operations may be represented geometrically by Möbius transformations on the roots of the entanglement measure on the Bloch sphere. Moreover, we show how the roots of the three-tangle measure classify four-qubit generic states, and propose a method to obtain the normal form of a four-qubit state, which bypasses the possibly infinite iterative procedure.
One of the key manifestations of quantum mechanics is the phenomenon of quantum entanglement. While the entanglement of bipartite systems is already well understood, our knowledge of entanglement in multipartite systems is still limited. This dissertation covers various aspects of the quantification of entanglement in multipartite states and the role of symmetry in such systems. Firstly, we establish a connection between the classification of multipartite entanglement and knot theory and investigate the family of states that are resistant to particle loss. Furthermore, we construct several examples of such states using the Majorana representation as well as some combinatorial methods. Secondly, we introduce classes of highly-symmetric but not fully-symmetric states and investigate their entanglement properties. Thirdly, we study the well-established class of Absolutely Maximally Entangled (AME) quantum states. On one hand, we provide construction of new states belonging to this family, for instance, an AME state of 4 subsystems with six levels each, on the other, we tackle the problem of equivalence of such states. Finally, we present a novel approach for the general problem of verification of the equivalence between any pair of arbitrary quantum states based on a single polynomial entanglement measure.
