An Exact Solution of the Nonlinear Differential Equation .. y &amp;nbsp; + p(t)y = q m (t)/y 2m-1

Reid, James L.

doi:10.2307/2037261

Cited by 13 publications

(22 citation statements)

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“…Experimentally, at variance with the case of Bell tests, a demonstration of steering free of detection and locality loopholes is in reach [19][20][21][22], which makes one-sided device-independent QKD appealing for current technology and quantum steering a practically useful concept. EPR steering also plays an operative role in channel discrimination [23] and teleamplification [24].Several experiments have been already performed, demonstrating steering and its asymmetry [21,22,[25][26][27][28][29][30][31], and a number of recent studies have been devoted to improving our understanding of quantum steerability, ranging from the development of better criteria to detect steerable states [32][33][34][35][36], to the analysis of the distribution of steering among multiple parties [37][38][39][40]. However, unlike entanglement, for which a variety of operationally motivated measures exist [6,41], there is still a surprisingly scarce literature addressing the fundamental question of quantifying how steerable a given quantum state is [11,23,42].…”

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Quantification of Gaussian Quantum Steering

et al. 2015

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Einstein-Podolsky-Rosen steering incarnates a useful nonclassical correlation which sits between entanglement and Bell nonlocality. While a number of qualitative steering criteria exist, very little has been achieved for what concerns quantifying steerability. We introduce a computable measure of steering for arbitrary bipartite Gaussian states of continuous variable systems. For two-mode Gaussian states, the measure reduces to a form of coherent information, which is proven never to exceed entanglement, and to reduce to it on pure states. We provide an operational connection between our measure and the key rate in one-sided device-independent quantum key distribution. We further prove that Peres' conjecture holds in its stronger form within the fully Gaussian regime: namely, steering bound entangled Gaussian states by Gaussian measurements is impossible. Steering is the quantum mechanical phenomenon that allows one party, Alice, to change (i.e., to "steer") the state of a distant party, Bob, by exploiting their shared entanglement. This phenomenon, fascinatingly discussed by Schrödinger [8,9], was already noted by Einstein, Podolksy, and Rosen (EPR) in their famous 1935 paper [12], and is at the heart of the so-called EPR paradox [13]. There it was argued that steering implied an unacceptable "action at a distance," which led EPR to claim the incompleteness of quantum theory. The EPR expectations for local realism were mostly extinguished by Bell's theorems [14,15], which showed that no locally causal theory can reproduce all the correlations observed in nature [16]. The first experimental criterion for the demonstration of the EPR paradox, i.e., for the detection of quantum steering, was later proposed by Reid [17], but it was not until 2007 that the particular type of nonlocality captured by the concept of steering [8,9,12] was in fact formalized [10,18].From a quantum information perspective [10], steering corresponds to the task of verifiable entanglement distribution by an untrusted party. If Alice and Bob share a state which is steerable in one way, say from Alice to Bob, then Alice is able to convince Bob (who does not trust Alice) that their shared state is entangled, by performing local measurements and classical communication [10]. Notice that steering, unlike entanglement, is an asymmetric property: a quantum state may be steerable from Alice to Bob, but not vice versa. On the operational side, it has been recently realized that steering provides security in one-sided device-independent quantum key distribution (QKD) [19], where the measurement apparatus of one party only is untrusted. These protocols are less demanding than totally device-independent ones, for which Bell nonlocality is known to be necessary [3]. Experimentally, at variance with the case of Bell tests, a demonstration of steering free of detection and locality loopholes is in reach [19][20][21][22], which makes one-sided device-independent QKD appealing for current technology and quantum steering a practically useful concept. EPR ste...

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Quantification of Gaussian Quantum Steering

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“…Various nonlinear differential equations whose solutions can be found by solving linear differential equations were cited in [18,21,15,14,19,20,16]. However, each of these papers overlooked the equations (1.1) that satisfy Appell's condition.…”

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The differential equation 𝑄=0 in which 𝑄 is a quadratic form in 𝑦”,𝑦’,𝑦 having meromorphic coefficients

Chalkley¹

1992

Proc. Amer. Math. Soc.

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Abstract.A simple necessary and sufficient condition is given for the solutions of Q = 0 to be free of movable branch points. And, when the condition is satisfied, all the solutions of Q = 0 can be obtained by solving linear differential equations of order < 2. There are four mutually exclusive cases. We shall relate Case 4 to less convenient conditions P. Appell had introduced. We shall also show how Cases 3 and 4 together motivated our discovery of an identity that is essential for a satisfactory theory of relative invariants for homogeneous linear differential equations.

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“…A limited number of nonlinear differential equations have exact solutions which are constructed in terms of u and v. The equation derived by Pinney [1] can easily be extended to include the first derivative term, i.e., where the Wronskian W=uv -vit is a function oft and the constants a and c axe arbitrary. Likewise the results of [2] are readily modified to include the case r(z);¿0 and a¿¿l, the generalization being…”

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“…For this case Q' reduces to the constant (ac-b2/4) and the Wronskian also is constant. Indeed, when the coefficient r(t) is identically zero the Wronskian is a constant in all the equations above, and it can be incorporated into the solutions as was done in [1], [2], and [5].…”

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