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Single-mode nonclassicality measure from Simon-Peres-Horodecki criterion

Mehmet Emre Taşgın

Published Feb 3, 2015• 11 citations
OpenAlex

Abstract

Nonclassicality of a single-mode field is equivalent to the rank of two-mode entanglement this field generates at the output of a beam-splitter [Phys. Rev. A 89, 052302 (2014)]. We derive a measure for the nonclassicality of a single-mode field using the two-mode inseparability measure of Simon-Peres-Horodecki. Degree of the single-mode nonclassicality requires the knowledge of only $\langle \hat{a}^2\rangle$ and $\langle\hat{a}^\dagger\hat{a}\rangle$. This condition/measure is both a necessary and sufficient condition for Gaussian single-mode states. We show that this measure increases with squeezing strength and display a jump at the critical coupling for a superradiant phase transition. Additionally, we derive a simple analytical condition, $|\langle \hat{a}^2\rangle|>\langle\hat{a}^\dagger\hat{a}\rangle$, for nonclassicality from Duan-Giedke-Cirac-Zoller criterion. In a forthcoming study, we use the derived measure to show the following. Non-causal linear optical response of an optomechanical cavity emerges at the same critical cavity-mechanical coupling where output field becomes nonclassical.

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Nonclassicality of a single-mode field is equivalent to the rank of two-mode entanglement this field generates at the output of a beam-splitter [Phys. Rev. A 89, 052302 (2014)]. We derive a measure for the nonclassicality of a single-mode field using the two-mode inseparability measure of Simon-Peres-Horodecki. Degree of the single-mode nonclassicality requires the knowledge of only ⟨a^2⟩\langle \hat{a}^2\rangle⟨a^2⟩ and ⟨a^†a^⟩\langle\hat{a}^\dagger\hat{a}\rangle⟨a^†a^⟩. This condition/measure is both a necessary and sufficient condition for Gaussian single-mode states. We show that this measure increases with squeezing strength and display a jump at the critical coupling for a superradiant phase transition. Additionally, we derive a simple analytical condition, ∣⟨a^2⟩∣>⟨a^†a^⟩|\langle \hat{a}^2\rangle|>\langle\hat{a}^\dagger\hat{a}\rangle∣⟨a^2⟩∣>⟨a^†a^⟩, for nonclassicality from Duan-Giedke-Cirac-Zoller criterion. In a forthcoming study, we use the derived measure to show the following. Non-causal linear optical response of an optomechanical cavity emerges at the same critical cavity-mechanical coupling where output field becomes nonclassical.
Abstract Nonclassicality of a single-mode state can be quantified by the entanglement (potential) the state creates at the output of a beam splitter. Thus, beam splitter transformation can be utilized for obtaining single-mode nonclassicality conditions (quantifications) from two-mode entanglement criteria (measures). Here, (i) we employ this relation to quantify the single-mode nonclassicality (SMNc) of a nonlinear crystal field generating second harmonic signals. (ii) We also manage to witness the SMNc of superradiant emission, a non-Gaussian state, in a Dicke phase transition. Entanglement potential, at the beam splitter output, is obtained in terms of measurable single-mode quantities. Thus, presence of a beam splitter in an experiment is not necessary. Employing the beam splitter method, one can also utilize hybrid entanglement criteria as SMNc conditions working well in different classes of single-mode states. (iii) We also work on essentials of the entanglement-SMNc relation: one obtains the generalized squeezing condition ⟨ â † â ⟩ < |⟨ â 2 ⟩| as the SMNc condition when an optimized form of Duan–Giedke–Cirac–Zoller criterion [PRL 84, 2722] is used at the output of the beam splitter.
We examine the internal structure of two-mode entanglement criteria for quadrature- and number-phase-squeezed states. For criteria obtained from the partial transpose of the Schrödinger--Robertson inequality, we show that the additional covariance term effectively performs an optimization over the intra-mode rotations entering the criterion. We demonstrate this both for quadrature variables and for number-phase-squeezed states. We further show that Simon's criterion carries out this optimization automatically, which motivates a Simon-like criterion for number-phase-squeezed states that performs the optimization directly in the number-phase plane. We also analyze entanglement in terms of the product of the noises of the two modes, which we call the noise area. Analytically and numerically, we explore whether widely used entanglement criteria can be interpreted as searches for a noise area below unity. In particular, for the product form of the Duan--Giedke--Cirac--Zoller criterion, we show numerically that the minimum noise area obtained after optimization over intra-mode rotations equals the input nonclassicality that a beam splitter needs to generate the same amount of entanglement as the state under consideration. Finally, for Gaussian states we introduce an alternative entanglement measure that can also be extended to multimode settings, and we outline several open questions, including a simpler definition of entanglement depth for number-phase-squeezed states.
We study the entanglement properties of two-mode Gaussian light emerging from a generic SU (1,1) interferometer. Our tool is the two-mode characteristic function which is determined by the 4 × 4 covariance matrix. For an initial product of two mixed single-mode Gaussian states we investigate the output two-mode covariance matrix. Its structure displays the noise properties of the reduced states as well as the correlations between modes. Classicality of the output two-mode state is characterized by the existence of the Glauber-Sudarshan P representation. For testing separability we apply the Peres-Simon criterion requiring preservation of the positivity of the density matrix under partial transposition. Since inseparability entails nonclassicality, the threshold gain above which nonclassicality of the output state becomes manifest is lower than that allowing for its inseparability. We find that only for a thermal input do nonclassicality and inseparability of the output have the same threshold gain.
Nonclassicality of optical fields are the key resources in quantum information processing.Recently, Agarwal [New J. Phys., 13, 073008(2011)] proposed a method for engineering of a two-mode vortex state, a non-Gaussian state, by subtracting a photon from a two-mode squeezed state produced by down-converter from idler mode via a beam splitter with low reflectivity and detection of one photon by the avalanche photo diode (APD), or even better by a single-photon detector.We study the nonclassicalities of this state in terms of squeezing, antibunching, photon number correlation and distribution of photon number.In addition, we study higher-order nonclassical properties, viz., sum squeezing.We find that the non-Gaussian state shows strong nonclassical effects.
The behavior of general nonclassical two-mode Gaussian states at a beam splitter is investigated. Single-mode nonclassicality as well as two-mode entanglement of both input and output states are analyzed suggesting their suitable quantifiers. These quantifiers are derived from local and global invariants of linear unitary two-mode transformations such that the sum of input (or output) local nonclassicality measures and entanglement measure gives a global invariant. This invariant quantifies the global nonclassicality resource. Mutual transformations of local nonclassicalities and entanglement induced by the beam splitter are analyzed considering incident noisy twin beams, single-mode noisy squeezed vacuum states, and states encompassing both squeezed states and twin beams. A rich tapestry of interesting nonclassical output states is predicted.
Nonclassicality in the sense of quantum optics is a prerequisite for entanglement in multimode radiation states. In this work we bring out the possibilities of passing from the former to the latter, via action of classicality preserving systems like beam splitters, in a transparent manner. For single-mode states, a complete description of nonclassicality is available via the classical theory of moments, as a set of necessary and sufficient conditions on the photon number distribution. We show that when the mode is coupled to an ancilla in any coherent state, and the system is then acted upon by a beam splitter, these conditions turn exactly into signatures of negativity under partial transpose (NPT) entanglement of the output state. Since the classical moment problem does not generalize to two or more modes, we turn in these cases to other familiar sufficient but not necessary conditions for nonclassicality, namely the Mandel parameter criterion and its extensions. We generalize the Mandel matrix from one-mode states to the two-mode situation, leading to a natural classification of states with varying levels of nonclassicality. For two-mode states we present a single test that can, if successful, simultaneously show nonclassicality as well as NPT entanglement. We also develop a test for NPT entanglement after beam-splitter action on a nonclassical state, tracing carefully the way in which it goes beyond the Mandel nonclassicality test. The result of three-mode beam-splitter action after coupling to an ancilla in the ground state is treated in the same spirit. The concept of genuine tripartite entanglement, and scalar measures of nonclassicality at the Mandel level for two-mode systems, are discussed. Numerous examples illustrating all these concepts are presented.