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arXiv:1308.6154 · Reviews in Theoretical Science

Entanglement in Classical Optics

Partha Ghose, Anirban Mukherjee

Published Jan 30, 2014• 80 citations
arXivHTMLPDF

Abstract

The emerging field of entanglement or nonseparability in classical optics is reviewed, and its similarities with and differences from quantum entanglement clearly pointed out through a recapitulation of Hilbert spaces in general, the special restrictions on Hilbert spaces imposed in quantum mechanics and the role of Hilbert spaces in classical polarization optics. The production of Bell-like states in classical polarization optics is discussed, and new theorems are proved to discriminate between separable and nonseparable states in classical wave optics where no discreteness is involved. The influence of the Pancharatnam phase on a classical Bell-like state is deived. Finally, to what extent classical polarization optics can be used to simulate quantum information processing tasks is also discussed. This should be of great practical importance because coherence and entanglement are robust in classical optics but not in quantum systems.

DOI
10.1166/rits.2014.1024

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The emerging field of entanglement or nonseparability in classical optics is reviewed, and its similarities with and differences from quantum entanglement clearly pointed out through a recapitulation of Hilbert spaces in general, the special restrictions on Hilbert spaces imposed in quantum mechanics and the role of Hilbert spaces in classical polarization optics. The production of Bell-like states in classical polarization optics is discussed, and new theorems are proved to discriminate between separable and nonseparable states in classical wave optics where no discreteness is involved. The influence of the Pancharatnam phase on a classical Bell-like state is deived. Finally, to what extent classical polarization optics can be used to simulate quantum information processing tasks is also discussed. This should be of great practical importance because coherence and entanglement are robust in classical optics but not in quantum systems.
In this review we discuss intriguing properties of apparently classical optical fields, that go beyond purely classical context and allow us to speak about quantum characteristics of such fields and about their applications in quantum technologies. We briefly define the genuinely quantum concepts of entanglement and steering. We then move to the boarder line between classical and quantum world introducing quantum discord, a more general concept of quantum coherence, and finally a controversial notion of classical entanglement. To unveil the quantum aspects of often classically perceived systems, we focus more in detail on quantum discordant correlations between the light modes and on nonseparability properties of optical vector fields leading to entanglement between different degrees of freedom of a single beam. To illustrate the aptitude of different types of correlated systems to act as quantum or quantum-like resource, entanglement activation from discord, high-precision measurements with classical entanglement and quantum information tasks using intra-system correlations are discussed. The common themes behind the versatile quantum properties of seemingly classical light are coherence, polarization and inter and intra-mode quantum correlations.
The nonclassicality of a macroscopic single-mode optical superposition state is potentially convertible into entanglement, when the state is mixed with the vacuum on a beam splitter. Considering light beams with polarization degree of freedom in Euclidean space as coherent product states in a bipartite Hilbert space, we propose a method to convert the two orthogonal polarizations into simultaneous entanglement and classical nonseparability through nonclassicality in the superpositions of coherent and displaced Fock states. Equivalent Bell state emerges from the resulted superpositions and the proportion of mixed entanglement and nonseparablity is determined by the displacement amplitudes along the polarization directions. We characterize the state nonclassicality via features in Wigner distributions and propose an experimental method for generating these states and measuring them via homodyne tomography.
The nonclassicality of a macroscopic single-mode optical superposition state is potentially convertible into entanglement, when the state is mixed with the vacuum on a beam splitter. Considering light beams with polarization degree of freedom in Euclidean space as coherent product states in a bipartite Hilbert space, we propose a method to convert the two orthogonal polarizations into simultaneous entanglement and classical nonseparability through nonclassicality in the superpositions of coherent and displaced Fock states. Equivalent Bell state emerges from the resulted superpositions and the proportion of mixed entanglement and nonseparablity is determined by the displacement amplitudes along the polarization directions. We characterize the state nonclassicality via features in Wigner distributions and propose an experimental method for generating these states and measuring them via homodyne tomography.
We explore the concepts of coherence and entanglement as they apply to both the classical and quantum natures of light. In the classical domain, we take inspiration from the tools and concepts developed in foundational quantum mechanics and quantum information science to gain a better understanding of classical coherence theory of light with multiple degrees of freedom (DoFs). First, we use polarization and spatial parity DoFs to demonstrate the notion of classical entanglement, and show that Bell's measure can serve as a useful tool in distinguishing between classical optical coherence theory. Second, we establish a methodical yet versatile approach called 'optical coherency matrix tomography' for reconstructing the coherency matrix of an electromagnetic beam with multiple DoFs. This technique exploits the analogy between this problem in classical optics and that of tomographically reconstructing the density matrix associated with multipartite quantum states in quantum information science. Third, we report the first experimental measurements of the 4 x 4 coherency matrix associated with an electromagnetic beam in which polarization and a spatial DoF are relevant, ranging from the traditional two-point Young's double slit to spatial parity and orbital angular momentum modes. In the quantum domain, we use the modal structure of classical fields to develop qubits and structure Hilbert spaces for use in quantum information processing. Advancing to three-qubit logic gates is an important step towards the success of optical schemes for quantum computing. We experimentally implement a variety of two- and three- qubit, linear and deterministic, single-photon, controlled, quantum logic gates using polarization and spatial parity qubits. Lastly, we demonstrate the implementation of two-qubit single-photon logic using polarization and orbital angular momentum qubits.
Abstract Quantum Mechanics and classical optics feature similar phenomena such as superposition, interference and even entanglement. Hence, techniques from optics can be used in quantum mechanics and vice versa. In this article I address the question: What can we learn from formulating optics in the language of quantum mechanics? It is argued that the solutions of the wave equations for the electromagnetic field form a tensor product of Hilbert spaces corresponding to the degrees of freedom of classical light. Therefore, it comprises non-separable solutions reminiscent of entanglement. Moreover, the two spatial degrees of freedom each carry non-commuting position and momentum variables forming a Heisenberg algebra like quantum particles moving in a two dimensional space. An analogy between the dynamics of a quantum harmonic oscillator and paraxial light propagating through a converging lens is drawn. This article presents a modern formulation of optics in the language of state vectors and operators (based on Dirac’s notation) along the lines of an earlier contribution [1], specifying and explaining its results.