Spherical designs and anticoherent spin states

Crann, Jason; Pereira, Rajesh; Kribs, David W.

doi:10.1088/1751-8113/43/25/255307

Cited by 22 publications

(49 citation statements)

References 18 publications

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“…This representation has been widely used to deal with various problems about spin states [7,8,18,20,30,31]. We then review the conditions for t-anticoherence in the light of this mapping in terms of reduced density matrices.…”

Section: B Majorana Representation Of Spin-j Statesmentioning

confidence: 99%

“…In particular, for g = 1, the state (60) is connected by a rotation to the spin-2 tetrahedron state (2). Note that it was shown in [7] that any state with a tetrahedral symmetry of its Majorana points is 2-anticoherent. While there exists a 2-anticoherent state for j = 2, 3, Fig.…”

Section: -Anticoherent Statesmentioning

confidence: 99%

“…Their Majorana representation corresponds to points at the apex of an octahedron (octahedral symmetry is known to imply anticoherence to order 3 [7]), see Fig. 7.…”

Section: -Anticoherent Statesmentioning

confidence: 99%

“…Such states have been studied in the literature, e.g. in the contexts of anticoherent spin states [6][7][8][9][10], completely entangled spin states [11,12] or multiphoton polarization states [13][14][15][16]. They also appear as some of the most non-classical spin states, where classicality of a spin state refers to the possibility of expressing it as a statistical mixture of spincoherent states with positive weights [17,18].…”

Section: Introductionmentioning

confidence: 99%

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Anticoherence measures for pure spin states

Baguette

Martin

2017

Phys. Rev. A

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The set of pure spin states with vanishing spin expectation value can be regarded as the set of the less coherent pure spin states. This set can be divided into a finite number of nested subsets on the basis of higher order moments of the spin operators. This subdivision relies on the notion of anticoherent spin state to order t: A spin state is said to be anticoherent to order t if the moment of order k of the spin components along any directions are equal for k = 1, 2, . . . , t. Most spin states are neither coherent nor anticoherent, but can be arbitrary close to one or the other. In order to quantify the degree of anticoherence of pure spin states, we introduce the notion of anticoherence measures. By relying on the mapping between spin-j states and symmetric states of 2j spin-1/2 (Majorana representation), we present a systematic way of constructing anticoherence measures to any order. We briefly discuss their connection with measures of quantum coherence. Finally, we illustrate our measures on various spin states and use them to investigate the problem of the existence of anticoherent spin states with degenerated Majorana points.

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Section: B Majorana Representation Of Spin-j Statesmentioning

confidence: 99%

Section: -Anticoherent Statesmentioning

confidence: 99%

“…Their Majorana representation corresponds to points at the apex of an octahedron (octahedral symmetry is known to imply anticoherence to order 3 [7]), see Fig. 7.…”

Section: -Anticoherent Statesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Anticoherence measures for pure spin states

Baguette

Martin

2017

Phys. Rev. A

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“…Such states have been studied in various contexts, such as quantum polarization of light (see e.g. [28,29]), spherical designs [30], as well as in the search for maximally entangled symmetric states [31]. We believe that our representation should prove useful in many of the contexts where the spin formalism is used.…”

mentioning

confidence: 99%

Tensor Representation of Spin States

et al. 2015

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We propose a generalization of the Bloch sphere representation for arbitrary spin states. It provides a compact and elegant representation of spin density matrices in terms of tensors that share the most important properties of Bloch vectors. Our representation, based on covariant matrices introduced by Weinberg in the context of quantum field theory, allows for a simple parametrization of coherent spin states, and a straightforward transformation of density matrices under local unitary and partial tracing operations. It enables us to provide a criterion for anticoherence, relevant in a broader context such as quantum polarization of light. The concept of spin is ubiquitous in quantum theory and all related fields of research, such as solidstate physics, molecular, atomic, nuclear or high-energy physics [1][2][3][4][5]. It has profound implications for the structure of matter as a consequence of the celebrated spinstatistics theorem [6]. The spin of a quantum system, be it an electron, a nucleus or an atom, has also been proven to be a key resource for many applications such as in spintronics [7], quantum information theory [8] or nuclear magnetic resonance [9]. Simple geometrical representations of spin states [10] allow one to develop physical insight regarding their general properties and evolution. Particularly well studied is the case of a single twolevel system, formally equivalent to a spin-1/2. In this case, the geometric representation is particularly simple. Indeed, the density matrix can be expressed in a basis formed of Pauli matrices and the identity matrix, leading to a parametrization in terms of a vector in R 3 . Pure states correspond to points on a unit sphere, the so-called Bloch sphere, and mixed states fill the inside of the sphere, the "Bloch ball". The simplicity of this representation help visualize the action and geometry of all possible spin-1/2 quantum channels [11]. For arbitrary pure spin states, another nice geometrical representation has been developed by Majorana in which a spin-j state is visualized as 2j points on the Bloch sphere [12]. This so-called Majorana or stellar representation has been exploited in various contexts (see, e. g., [11,[13][14][15][16][17]), but cannot be generalized to mixed spin states.Given the importance of geometrical representations, there have been numerous attempts to extend the previous representations to arbitrary mixed states. The former rely on a variety of sophisticated mathematical concepts such as su(N )-algebra generators [10,18,19], polarization operator basis [20][21][22] [26]. In the present Letter we propose an elegant generalisation to arbitrary spin-j of the spin-1/2 Bloch sphere representation based on matrices introduced by Weinberg in the context of relativistic quantum field theory [27]. The main result of the paper is theorem 2, which allows us to express any spin-j density matrix as a linear combination of matrices with convenient properties. The remarkable features of our representation are especially reflected in the simple coo...

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Geometric Entanglement of Symmetric States and the Majorana Representation

Aulbach

Markham

Murao

2011

Theory of Quantum Computation, Communication, and Cryptography

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Permutation-symmetric quantum states appear in a variety of physical situations, and they have been proposed for quantum information tasks. This article builds upon the results of [New J. Phys. 12, 073025 (2010)], where the maximally entangled symmetric states of up to twelve qubits were explored, and their amount of geometric entanglement determined by numeric and analytic means. For this the Majorana representation, a generalization of the Bloch sphere representation, can be employed to represent symmetric n qubit states by n points on the surface of a unit sphere. Symmetries of this point distribution simplify the determination of the entanglement, and enable the study of quantum states in novel ways. Here it is shown that the duality relationship of Platonic solids has a counterpart in the Majorana representation, and that in general maximally entangled symmetric states neither correspond to anticoherent spin states nor to spherical designs. The usability of symmetric states as resources for measurement-based quantum computing is also discussed.

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Spherical designs and anticoherent spin states

Cited by 22 publications

References 18 publications

Anticoherence measures for pure spin states

Anticoherence measures for pure spin states

Tensor Representation of Spin States

Geometric Entanglement of Symmetric States and the Majorana Representation

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