Optimal state for keeping reference frames aligned and the platonic solids

Kolenderski, Piotr; Demkowicz-Dobrzański, Rafał

doi:10.1103/physreva.78.052333

Cited by 62 publications

(80 citation statements)

References 10 publications

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“…Majorana representation, among other applications [14][15][16][17], has proven very useful to the study and classification of entanglement in symmetric states [2][3][4][5][6][7]18]. There are two ways to classify entanglement or in other words, to regroup states in classes according to their entanglement properties.…”

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confidence: 99%

Entanglement classification of pure symmetric states via spin coherent states

2014

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We prove that the vast majority of symmetric states of qubits (or spin 1/2) can be decomposed in a unique way into a superposition of spin 1/2 coherent states. For the case of two qubits, the proposed decomposition reproduces the Schmidt decomposition and therefore, in the case of a higher number of qubits, can be considered as its generalization. We analyze the geometrical aspects of the proposed representation and its invariant properties under the action of local unitary and local invertible transformations. As an application, we identify the most general classes of entanglement and representative states for any number of qubits in a symmetric state.Symmetric states under permutations have drawn lately a lot of attention in the field of quantum information. The essential reason is that the number of parameters needed for the description of a state in a symmetric subspace scales just linearly with the number of parties. This simplification makes symmetric states a good testground for complex quantum information tasks such as the description of multipartite entanglement [1-8] and quantum tomography [9].There are two well known representations for symmetric states, the Dicke basis [10] and Majorana representation [11]. In the present work, we introduce a novel representation whose forms resembles strongly to a generalized Schmidt decomposition and that presents advantages with respect to the previous ones as regards the readability of the entanglement properties of the state. The structure of the proposed representation remains invariant under the action of local unitary operations which leave the state in the symmetric subspace. We associate to the proposed decomposition a geometric representation of the states that has the interest of displaying this invariance. We proceed by identifying invariant forms of the representation under the action of local unitary and local invertible transformations. In this way we arrive to an exact classification of entanglement for symmetric states. With the well studied example of three qubits we establish a first connection among the suggested representation and measures of multipartite entanglement. Furthermore, an immediate consequence of our methods is a straightforward estimation of a well established measure of entanglement, the so called Schmidt measure [12] for the symmetric states.The structure of this paper is the following. We start by recalling some basic properties of the Dicke and Majorana representation and their connection to entanglement classification. We then move to the presentation of the novel decomposition that is the main result of the paper. Finally we employ the properties of the decomposition in order to arrive at a classification of entanglement applicable to the vast majority of symmetric states.Every symmetric state of N qubits can be expressed in a unique way over the Dicke basis formed by the N + 1 joined eigenstates {|N/2, m } of the collective operatorŝ This projection leads to a polynomial of N th order on the complex parameter α, the so-calle...

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confidence: 99%

Entanglement classification of pure symmetric states via spin coherent states

2014

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“…[5] for a recent review). The problem of estimating rotations around a known axis is well-understood [6][7][8][9], while rotation measurement around unknown axes is a relatively new endeavour [10][11][12]. Measurement precision can be enhanced using special quantum input states in the known-axis case [13,14]; here we investigate quantum enhancements for simultaneously estimating rotation angles and rotation axes.…”

Section: Introductionmentioning

confidence: 99%

Quantum-limited Euler angle measurements using anticoherent states

Goldberg

James

2018

Phys. Rev. A

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Many protocols require precise rotation measurement. Here we present a general class of states that surpass the shot noise limit for measuring rotation around arbitrary axes. We then derive a quantum Cramér-Rao bound for simultaneously estimating all three parameters of a rotation (e.g., the Euler angles), and discuss states that achieve Heisenberg-limited sensitivities for all parameters; the bound is saturated by "anticoherent" states [Zimba, Electron. J. Theor. Phys. 3, 143 (2006)] (we are reluctant to use "anticoherent" to describe the states, but the name has become commonplace over the last decade). Anticoherent states have garnered much attention in recent years, and we elucidate a geometrical technique for finding new examples of such states. Finally, we discuss the potential for divergences in multiparameter estimation due to singularities in spherical coordinate systems. Our results are useful for a variety of quantum metrology and quantum communication applications.

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“…Another possible direction of investigation concerns quantum metrology. As anticoherent states have been shown to be optimal in detecting rotations [20] and for reference frame alignment [21], it would be worth investigating the connections between our measures of anticoherence and the efficiency of a state for such tasks.…”

Section: Discussionmentioning

confidence: 99%

“…Surprisingly, the transition probability between a spin-2 state and the state obtained from it by a rotation has been shown to be minimized by (2) for a large range of angles, making it an optimal state in detecting rotations [20]. The state (2) was also shown to be optimal for reference frame alignment [21].…”

Section: A Examples and Propertiesmentioning

confidence: 99%

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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Optimal state for keeping reference frames aligned and the platonic solids

Cited by 62 publications

References 10 publications

Entanglement classification of pure symmetric states via spin coherent states

Entanglement classification of pure symmetric states via spin coherent states

Quantum-limited Euler angle measurements using anticoherent states

Anticoherence measures for pure spin states

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