Elsevier

Physics Letters A

Volume 382, Issue 1, 5 January 2018, Pages 27-32
Physics Letters A

A tighter steering criterion using the Robertson–Schrödinger uncertainty relation

https://doi.org/10.1016/j.physleta.2017.10.036Get rights and content

Highlights

  • •

    Reid-steering criterion fails to detect steerability for many non-Gaussian states.

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    We ask a question: Is Reid-Inequality based on variances tight?

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    A tighter steering criterion using the Robertson–Schrodinger uncertainty relation is proposed.

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    Proposed inequality is able to detect steerability of non-Gaussian states.

Abstract

We consider quantum steering by non-Gausssian entangled states. The Reid steering criterion based on the Heisenberg uncertainty relation fails to detect steerability for many categories of such states. Here, we derive a tighter steering criterion using the Robertson–Schrödinger uncertainty relation. We show that our steering condition is able to detect steerability of several classes of non-Gaussian states such as entangled eigenstates of the two-dimensional harmonic oscillator, the photon subtracted squeezed vacuum state and the NOON state.

Introduction

Based on two assumptions, viz., locality and realism, Einstein–Podolsky–Rosen (EPR) [1] in 1935 argued that the quantum mechanical description of a physical system is incomplete. In the same year Schrödinger published a work [2] in response to the EPR paper, pointing out the fact that an experimenter by a suitable choice of measurements on one part of a composite system, can control the state of the other spatially separated part without directly interfering with that part. The word “steering” was coined by Schrödinger to describe this non classical feature of quantum mechanics, as well as the word “entanglement” to describe the correlations of such spatially separated systems.

An experimental formulation of the EPR steering was first proposed by Reid [3] and Drummond in the context of continuous variable systems using the position-momentum uncertainty relation, based upon inferred variances of observables and the Heisenberg uncertainty Relation. They established [4] the non classical correlations present in the quadrature amplitudes of the output beams and demonstrated the EPR scenario through violations of the inferred Heisenberg uncertainty principle. Later Ou et al. demonstrated the EPR paradox using spatially separated and correlated light modes generated by non-degenerate parametric amplification [5]. A stronger violation of the Reid inequality for two-mode squeezed vacuum states has been experimentally reported recently [6]. However, in systems with correlations having higher than the second order moment, e.g., the non-Gaussian states, the Reid criterion for the EPR experiment in the continuous variable scenario failed to reveal steerability, even though they exhibit Bell-non locality [7], [8].

The concept of EPR-Schrödinger steering has been recently further developed in the information theoretic context by Wiseman et al. [9]. They showed using similar formulations in terms of nonlocal tasks for entanglement as well as Bell non-locality, that a clear distinction between these three types of correlation is possible using joint probability distributions. EPR-steering stems form a correlation that is strictly intermediate between quantum entanglement and Bell non-locality [10]. The connection between the concept of steering as an information theoretic task formalized by Wiseman et al. [9], and the experimental criterion for demonstration of EPR introduced by Reid [3], has also been clearly established [11]. Subsequently, Brunner et al. [12] have showed the inequivalence between entanglement, steering and Bell-nonlocality for the general measurement scenario in bipartite qubit systems. The experimental demonstration of these three types of correlations has been obtained, as well [13]. A loop-hole free EPR steering experiment has been also performed [14]. Quantum steering is fundamentally linked with quantum uncertainty, and hence, other versions of uncertainty relations have also been employed to obtain correspondingly different steering relations such as the entropic [7] and the fine-grained [15] steering inequalities.

Reid's inequality has been employed to detect EPR steering for several continuous variable quantum systems. However, a number of entangled continuous variable non-Gaussian states do not violate the inferred variance inequality proposed by Reid. In recent developments in quantum information theory, non-Gaussian states have applications in several protocols [16]. Extensions of the entanglement criterion for non-Gaussian states have been developed [17], [18], and Bell-violations have been studied for such states too [19]. It is thus relevant to study steering by non-Gaussian entangled states. Walborn et al. [7] raised a question as to whether such states violate some higher order EPR-steering inequality. It was also pointed out that the Reid inequality based on the variances is unable to capture the correlations which are of higher than second order in the tested observables. The entropic steering inequality [7] is able to reveal the steerability of several categories of non-Gaussian states [8]. In the present work we ask a somewhat different though related question regarding the steerability of pure non-Gaussian entangled states: Is the Reid's inequality based on variances tight enough to reveal steerability for various categories of non-Gaussian states? The steering bound proposed by Reid is based on the Heisenberg uncertainty relation for two conjugate observables. A more generalized form of variance based uncertainty relation was derived by Robertson and Schrödinger for any two hermitian observables [20]. In order to address the question posed above, in this paper we investigate the Reid criterion in context of the Robertson–Schrödinger uncertainty relation for the purpose of studying steerability of non-Gaussian states.

The plan of the paper is as follows. In the section 2, we present a brief review of the concepts of the EPR paradox and the demonstration of steering through the Reid criterion. The main purpose of this section is to discuss the formulation of steering criterion based on Heisenberg's uncertainty relation proposed by Reid and recall its applicability for a Gaussian state, e.g., the two-mode squeezed vacuum state. We next review briefly the development of the Robertson–Schrödinger uncertainty relation. In the section 3, we first construct a new steering inequality based on the Robertson–Schrödinger uncertainty relation. The steerability of several non-Gaussian states is then studied based upon our proposed condition that is tighter than the Reid criterion. Here we consider examples of experimentally realizable non-Gaussian states such as the entangled eigenstates of the two-dimensional harmonic oscillator given by Laguerre–Gaussian wave functions, the photon subtracted squeezed vacuum state [21], and the NOON state [22]. In section 4, we provide a summary of our main results.

Section snippets

The Einstein–Podolsky–Rosen–Schrödinger argument and the Reid criterion

The EPR argument is based on the notion of local realism. The sufficient condition of reality defined by EPR [1] states, “If without any way disturbing a system we can predict with certainty (with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity”. The locality assumption keeps consistency with the idea of the special theory of relativity, i.e., nothing can travel faster than light, or, “no

Proposed steering criterion and applications on non-Gaussian states

We use Robertson–Schrödinger uncertainty relation to obtain the lower bound of the average errors of the inferred values of the quadrature amplitudes Xθ1 and Xθ2. Here we consider the coherent state to evaluate the RHS of the equation (18) as it gives the minimum uncertainty, given by|α〉=e−|α|22∑αnn!|n〉 For particular θ and ϕ the quadrature amplitudes can be expressed in terms of the Bosonic operatorsX1ˆ=aˆ+a†ˆ2andX2ˆ=−i(aˆ−a†ˆ)2 where the commutator [X1ˆ,X2ˆ]=[a†ˆ,aˆ]=−1 and the

Conclusions

In the present work we have proposed a new steering criterion based on the Robertson–Schrödinger uncertainty relation. Our steering condition is tighter than the Reid criterion based on the Heisenberg uncertainty relation. We have studied EPR-Schrödinger steering by several examples of non-Gaussian entangled pure states, such as the two-dimensional harmonic oscillator states (LG beams), the photon subtracted squeezed vacuum state and the NOON state which do not demonstrate steering through the

Acknowledgements

This work has been supported by the INSPIRE program, Department of Science and Technology, Govt of India.

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