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arXiv:quant-ph/0307196 · International Journal of Quantum Information

Tutorial Notes on One-Party and Two-Party Gaussian States

Berthold‐Georg Englert, Krzysztof Wódkiewicz

Published Jun 1, 2003• 44 citations
arXivHTMLPDF

Abstract

Gaussian states — or, more generally, Gaussian operators — play an important role in Quantum Optics and Quantum Information Science, both in discussions about conceptual issues and in practical applications. We describe, in a tutorial manner, a systematic operator method for first characterizing such states and then investigating their properties. The central numerical quantities are the covariance matrix that specifies the characteristic function of the state, and the closely related matrices associated with Wigner's and Glauber's phase space functions. For pedagogical reasons, we restrict the discussion to one-dimensional and two-dimensional Gaussian states, for which we provide illustrating and instructive examples.

DOI
10.1142/s0219749903000206

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Gaussian states — or, more generally, Gaussian operators — play an important role in Quantum Optics and Quantum Information Science, both in discussions about conceptual issues and in practical applications. We describe, in a tutorial manner, a systematic operator method for first characterizing such states and then investigating their properties. The central numerical quantities are the covariance matrix that specifies the characteristic function of the state, and the closely related matrices associated with Wigner's and Glauber's phase space functions. For pedagogical reasons, we restrict the discussion to one-dimensional and two-dimensional Gaussian states, for which we provide illustrating and instructive examples.
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Gaussian states have played an important role in the physics of continuous-variable quantum systems.They are appealing for the experimental ease with which they can be produced, and for their compact and elegant mathematical description.Nevertheless, many proposed quantum technologies require us to go beyond the realm of Gaussian states and introduce non-Gaussian elements.In this Tutorial, we provide a roadmap for the physics of non-Gaussian quantum states.We introduce the phase-space representations as a framework to describe the different properties of quantum states in continuous-variable systems.We then use this framework in various ways to explore the structure of the state space.We explain how non-Gaussian states can be characterized not only through the negative values of their Wigner function, but also via other properties such as quantum non-Gaussianity and the related stellar rank.For multimode systems, we are naturally confronted with the question of how non-Gaussian properties behave with respect to quantum correlations.To answer this question, we first show how non-Gaussian states can be created by performing measurements on a subset of modes in a Gaussian state.Then, we highlight that these measured modes must be correlated via specific quantum correlations to the remainder of the system to create quantum non-Gaussian or Wigner-negative states.On the other hand, non-Gaussian operations are also shown to enhance or even create quantum correlations.Finally, we demonstrate that Wigner negativity is a requirement to violate Bell inequalities and to achieve a quantum computational advantage.At the end of the Tutorial, we also provide an overview of several experimental realizations of non-Gaussian quantum states in quantum optics and beyond.
Abstract In this chapter we collect a number of key elements of quantum optics relevant for quantum information theory. We begin by discussing the quantization of the electromagnetic field in terms of harmonic oscillators with ladder operators and the corresponding Fock space. Based on this description we introduce Glauber’s coherent states and examine their properties as proper quantum analogues of the classical harmonic oscillator. We then study the description of states of light in a quantum-mechanical phase space by employing quasi-probability distributions such as the Wigner function. With these tools at hand we then focus on the family of Gaussian states and their compact description in terms of covariance matrices and vectors of first moments. We review the formalism of Gaussian operations, symplectic operations, the Williamson normal form, and the Bloch-Messiah decomposition. Finally, we take a look at entanglement of Gaussian states.
Conceptually, the phase space description of quantum states hinges on the completeness of the set of displacement operators, which can be proved in the form of the Fourier-Weyl relation between density matrices and characteristic functions. This relationship constitutes the bridge between phase space and Hilbert space descriptions, which is useful in several applications in the context of quantum information. This chapter shows that the marginal Wigner function along any phase space direction describes the probability distribution of the quantum measurement of the associated quadrature operator. Hence, as far as Gaussian states are concerned, the Wigner function provides one with a local, 'realistic' model to describe quadrature measurements. If one restricts to quadrature measurements, such systems may be mimicked by multivariate classical Gaussian distributions and will never show any signature of quantum non-locality, such as a violation of Bell or CHSH inequalities.
Gaussian bipartite states are basic tools for the realization of quantum information protocols with continuous variables. Their complete characterization is obtained by the reconstruction of the corresponding covariance matrix. Here we describe in detail and experimentally demonstrate a robust and reliable method to fully characterize bipartite optical Gaussian states by means of a single homodyne detector. We have successfully applied our method to the bipartite states generated by a sub-threshold type-II optical parametric oscillator which produces a pair of thermal cross-polarized entangled cw frequency degenerate beams. The method provides a reliable reconstruction of the covariance matrix and allows us to retrieve all the physical information about the state under investigation. These include observable quantities, such as energy and squeezing, as well as nonobservable ones such as purity, entropy, and entanglement. Our procedure also includes advanced tests for the Gaussianity of the state and, overall, represents a powerful tool to study the bipartite Gaussian state from the generation stage to the detection one.
The two-mode relative phase associated with Gaussian states plays an important role in quantum information processes in optical, atomic, and electronic systems. In this work, the origin and structure of the two-mode relative phase in pure Gaussian states is studied in terms of its dependences on the quadratures of the modes. This is done by constructing local canonical transformations to an associated two-mode squeezed state. The results are illustrated by studying the time dependence of the phase under a nonlocal unitary model evolution containing correlations between the modes. In a more general context, this approach may allow the two-mode phase to be studied in situations sensitive to different physical parameters within experimental configurations relevant to quantum information processing tasks.