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Journal of Physics B Atomic Molecular and Optical Physics

From squeezing to Gaussian entanglement via beamsplitters

Shuangshuang Fu, Shunlong Luo, Yue Zhang

Published Feb 10, 2020• 3 citations
Publisher

Abstract

Abstract We investigate relations between squeezing of separable input states and entanglement of output states generated by a beamsplitter, both of which are valuable nonclassical resources for quantum information processing. Motivated by the seminal work of Asboth´ et al [2005 Phys. Rev. Lett. 94 173602], where a systematic study of nonclassicality (including squeezing) and beamsplitter-generated entanglement is conducted, we consider the situation when the two input states are all Gaussian and identify certain constraints on the squeezing parameters to generate entanglement. Contrary to naive expectation, large squeezing does not necessarily guarantee large entanglement. Depending on the phase relations between the input states and the beamsplitters, the conditions to generate entanglement vary, which may be regarded as a manifestation of interference. In two typical cases, it is the difference or sum of the squeezing amplitudes that plays a key role in generating entanglement. Analytic expressions of entanglement in terms of the difference or sum of the squeezing amplitudes are obtained, which are further shown to be monotonic in the relevant parameters.

DOI
10.1088/1361-6455/ab746b

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Abstract We investigate relations between squeezing of separable input states and entanglement of output states generated by a beamsplitter, both of which are valuable nonclassical resources for quantum information processing. Motivated by the seminal work of Asboth´ et al [2005 Phys. Rev. Lett. 94 173602], where a systematic study of nonclassicality (including squeezing) and beamsplitter-generated entanglement is conducted, we consider the situation when the two input states are all Gaussian and identify certain constraints on the squeezing parameters to generate entanglement. Contrary to naive expectation, large squeezing does not necessarily guarantee large entanglement. Depending on the phase relations between the input states and the beamsplitters, the conditions to generate entanglement vary, which may be regarded as a manifestation of interference. In two typical cases, it is the difference or sum of the squeezing amplitudes that plays a key role in generating entanglement. Analytic expressions of entanglement in terms of the difference or sum of the squeezing amplitudes are obtained, which are further shown to be monotonic in the relevant parameters.
While for single-mode bosonic fields, one talks about nonclassicality of quantum states, which plays an important role in various quantum information processing tasks, for composite systems, one talks about quantum correlations such as quantum entanglement and quantum discord, which are crucial and indispensable for implementation of quantum communication and computation. Both nonclassicality and quantum correlations are quantum features of physical systems and fundamental resources which can be exploited to bring supremacy over classical systems. A natural question arises on the relation and interconversion between these two resources. In this work, we investigate the conversion of single-mode nonclassicality into quantum correlations via beamsplitters. Our work is based on the idea of Asbó et al . ( Phys. Rev. Lett. , 94 (2005) 173602) of converting nonclassicality into quantum entanglement via a balanced beamsplitter. Through a paradigmatic setup with Gaussian states fed into the beamsplitters, we demonstrate that quantum correlations between the output states, as exhibited by either entanglement or quantum discord, increase monotonically with nonclassicality of the input Gaussian states, as quantified by an information-theoretic measure in terms of the Wigner-Yanase skew information.
Abstract Beam splitters are routinely used for generating entanglement. Their entangling properties have been studied extensively, with nonclassicality of the input states a prerequisite for entanglement at the output. Here we quantify the amount of entanglement generated by weakly-reflecting beam splitters, and look for nonclassical states that are not entangled by general beam splitters. We extend the known class of results to mixed and non-Gaussian states, finding that inputting highly nonclassical combinations of unpolarized states that are squeezed and displaced onto a beam splitter can still yield separable output states. This is important in light of the challenge of characterizing mixed state entanglement. We further identify a parallel between SU(2) unpolarized states and the two-mode vacuum. Our result is crucial for understanding the generation of modal entanglement by beam splitters.
In quantum optics a pure state is considered classical, relative to the statistics of photodetection, if and only if it is a coherent state. A different and newer notion of nonclassicality is based on modal entanglement. One example that relates these two notions is the Hong-Ou-Mandel effect, where modal entanglement is generated by a beamsplitter from the nonclassical photon-number state $|1\ensuremath{\rangle}\ensuremath{\bigotimes}|1\ensuremath{\rangle}$. This suggests that beamsplitters or, more generally, linear-optical networks are mediators of the two notions of nonclassicality. In this Brief Report, we show the following: Given a nonclassical pure-product-state input to an NNN-port linear-optical network, the output is almost always mode entangled; the only exception is a product of squeezed states, all with the same squeezing strength, input to a network that does not mix the squeezed and antisqueezed quadratures. Our work thus gives a necessary and sufficient condition for a linear network to generate modal entanglement from pure-product inputs, a result that is of immediate relevance to the boson-sampling problem.
We study an experimental scheme to generate Gaussian two-mode entangled states via beam splitter. Specifically, we consider a nonclassical Gaussian state (squeezed state) and a thermal state as two input modes, and evaluate the degree of entanglement at the output. Experimental conditions to generate entangled outputs are completely identified and the critical thermal noise to destroy entanglement is analytically obtained. By doing so, we discuss the possibility to link the resistance to noise in entanglement generation with the degree of single-mode nonclassicality.
A classical analysis of two equal intensity coherent fields incident on a beamsplitter shows that the difference-phase noise of the output depends only on the noise in the difference of the amplitudes of the input fields and not on their phase noise. This suggests that in the quantum mechanical case squeezing in the amplitudes of the input beams can lead to squeezing in the phase difference of the output beams. We show that this is true. We also find the phase properties of the output when the input consists of two number states with an equal number of photons. The difference phase distribution consists of two narrow peaks, at and . States with small phase difference noise should be useful in the measurement of phase shifts.
Continuous-variable beam-splitter (BS)-generated entanglement from single-mode optical states generated by a single nonclassicality (NC)-inducing operation has been found to be immensely important in several information processing tasks. There exists a broader class of optical states, generated from successive action of multiple different NC-inducing operations, which show many intriguing nonclassical properties; however, the BS conversion of the NC for such states remains unexplored. In this work we have critically analyzed the BS-generated entanglement from such nonclassical optical states at input. Here we present a scenario where BS output entanglement becomes nonmonotonic with the input NC parameters, accessible experimentally (e.g., number of photon excitation and squeezing strength), in contrast to the previous results with states comprising a single NC-inducing operation. We explain this counterintuitive feature in terms of the competition between these two NC-inducing operations as manifest in the contours of the QQQ functions associated with these states.