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arXiv:2004.11782 · Physical Review A

Relating the entanglement and optical nonclassicality of multimode states of a bosonic quantum field

Anaelle Hertz, Nicolas J. Cerf, Stephan De Bièvre

Published Sep 21, 2020• 18 citations
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Abstract

The quantum nature of the state of a bosonic quantum field may manifest itself in its entanglement, coherence, or optical nonclassicality. Each of these distinct properties is known to be a resource for quantum computing or metrology and can be evaluated via a variety of measures, witnesses, or monotones. Here, we provide quantitative and computable bounds relating, in particular, some entanglement measures with optical nonclassicality measures. Overall, these bounds capture the fact that strongly entangled states must necessarily be strongly optically nonclassical. As an application, we infer strong bounds on the entanglement that can be produced with an optically nonclassical state impinging on a beam splitter. Then, focusing on Gaussian states, we analyze the link between the logarithmic negativity and a specific nonclassicality measure called quadrature coherence scale.

DOI
10.1103/physreva.102.032413

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The quantum nature of the state of a bosonic quantum field may manifest itself in its entanglement, coherence, or optical nonclassicality. Each of these distinct properties is known to be a resource for quantum computing or metrology and can be evaluated via a variety of measures, witnesses, or monotones. Here, we provide quantitative and computable bounds relating, in particular, some entanglement measures with optical nonclassicality measures. Overall, these bounds capture the fact that strongly entangled states must necessarily be strongly optically nonclassical. As an application, we infer strong bounds on the entanglement that can be produced with an optically nonclassical state impinging on a beam splitter. Then, focusing on Gaussian states, we analyze the link between the logarithmic negativity and a specific nonclassicality measure called quadrature coherence scale.
Gaussian states with nonclassical properties such as squeezing and entanglement serve as crucial resources for quantum information processing. Accurately quantifying these properties within multimode Gaussian states has posed some challenges. To address this, we introduce a unified quantification: the "classical-nonclassical polarity," represented by P. For a single mode, a positive value of P captures the reduced minimum quadrature uncertainty below the vacuum noise, while a negative value represents an enlarged uncertainty due to classical mixtures. For multimode systems, a positive P indicates bipartite quantum entanglement. We show that the sum of the total classical-nonclassical polarity is conserved under arbitrary linear optical transformations for any two-mode and three-mode Gaussian states. For any pure multimode Gaussian state, the total classical-nonclassical polarity equals the sum of the mean photon number from single-mode squeezing and two-mode squeezing. Our results provide a new perspective on the quantitative relation between single-mode nonclassicality and entanglement, which may find applications in a unified resource theory of nonclassical features.
We introduce, for each state of a bosonic quantum field, its quadrature coherence scale (QCS), a measure of the range of its quadrature coherences. Under coupling to a thermal bath, the purity and QCS are shown to decrease on a timescale inversely proportional to the QCS squared. The states most fragile to decoherence are therefore those with quadrature coherences far from the diagonal. We further show a large QCS is difficult to measure since it induces small scale variations in the state's Wigner function. These two observations imply a large QCS constitutes a mark of "macroscopic coherence." Finally, we link the QCS to optical classicality: optical classical states have a small QCS and a large QCS implies strong optical nonclassicality.
Abstract Quantum optics connects fundamental phenomena such as entanglement and superposition with practical applications like metrology and communication. Throughout, there is an interplay between information theoretic concepts such as entropy and physical considerations such as quantum system design, noise, and loss. Therefore, a result at the heart of these fields can have numerous ramifications in development of applications and advancing our understanding of quantum physics. Our recent proof for the entanglement properties of states interfering with the vacuum on a beam splitter led to monotonicity and convexity properties for quantum states undergoing photon loss [Lupu-Gladstein et al., PRA 112 , 033722 (2025)] by breathing life into a decades-old conjecture. In this work, we extend these fundamental properties to measures of similarity between states, provide inequalities for creation and annihilation operators beyond the Cauchy–Bunyakovsky–Schwarz inequality, prove a conjecture [Hertz et al., PRA 110 , 012408 (2024)] dictating that nonclassicality through the quadrature coherence scale is uncertifiable beyond a loss of 50%, place constraints on quasiprobability distributions of all physical states, and explore entanglement generation optimized over all input states and linear optical networks. These ideas can now be used throughout quantum optics.
Photon addition and subtraction render Gaussian states non-Gaussian. We provide a quantitative analysis of the change in nonclassicality produced by these processes by analyzing the Wigner negativity and quadrature coherence scale (QCS) of the resulting states. The QCS is a recently introduced measure of nonclassicality [Phys. Rev. Lett. 122, 080402 (2019); Phys. Rev. Lett. 124, 090402 (2020)], which we show to undergo a relative increase under photon addition and subtraction that can be as large as 200%. This implies that the degaussification and the concomitant increase of nonclassicality come at a cost. Indeed, the QCS is proportional to the decoherence rate of the state so that the resulting states are considerably more prone to environmental decoherence. Our results are quantitative and rely on explicit and general expressions for the characteristic and Wigner functions of photon-added and -subtracted single- and multimode Gaussian states for which we provide a simple and straightforward derivation. These expressions further allow us to certify the quantum non-Gaussianity of the photon-subtracted states with positive Wigner function.
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.