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arXiv:quant-ph/0408029 · Physical Review Letters

Hilbert’s 17th Problem and the Quantumness of States

Jarosław K. Korbicz, J. I. Cirac, Jan Wehr, Maciej Lewenstein

Published Apr 22, 2005• 41 citations
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Abstract

A state of a quantum system can be regarded as classical (quantum) with respect to measurements of a set of canonical observables if and only if there exists (does not exist) a well defined, positive phase-space distribution, the so called Glauber-Sudarshan P representation. We derive a family of classicality criteria that requires that the averages of positive functions calculated using P representation must be positive. For polynomial functions, these criteria are related to Hilbert's 17th problem, and have physical meaning of generalized squeezing conditions; alternatively, they may be interpreted as nonclassicality witnesses. We show that every generic nonclassical state can be detected by a polynomial that is a sum-of-squares of other polynomials. We introduce a very natural hierarchy of states regarding their degree of quantumness, which we relate to the minimal degree of a sum-of-squares polynomial that detects them.

DOI
10.1103/physrevlett.94.153601

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A state of a quantum system can be regarded as classical (quantum) with respect to measurements of a set of canonical observables if and only if there exists (does not exist) a well defined, positive phase-space distribution, the so called Glauber-Sudarshan P representation. We derive a family of classicality criteria that requires that the averages of positive functions calculated using P representation must be positive. For polynomial functions, these criteria are related to Hilbert's 17th problem, and have physical meaning of generalized squeezing conditions; alternatively, they may be interpreted as nonclassicality witnesses. We show that every generic nonclassical state can be detected by a polynomial that is a sum-of-squares of other polynomials. We introduce a very natural hierarchy of states regarding their degree of quantumness, which we relate to the minimal degree of a sum-of-squares polynomial that detects them.
The definition of nonclassical states in quantum optics by the nonpositivity of their Glauber–Sudarshan quasiprobability is investigated and it is shown that it hides some serious problems. It leads to a subdivision of squeezed thermal states into classical and nonclassical states which is difficult to interpret physically by some qualitatively different behaviour of the states. Nonclassical states are found in arbitrarily small neighbourhoods of every classical state that is illustrated by a very artificial modified thermal state. The observability of the criterion in comparison to that for nonclassicality of states determined by the nearest Hilbert–Schmidt distance to a class of reference states is discussed. The behaviour of the nonclassicality of states in models of phase-insensitive processes of damping and amplification is investigated and it is found that every nonclassical state eventually makes a transition to a classical state. However, this is not specific for the negativities or singularities of the Glauber–Sudarshan quasiprobability and is found in similar form for other quasiprobabilities, for example, for the Wigner quasiprobability. We discuss in quite general form some defects of the Glauber–Sudarshan quasiprobability if compared with classical distribution functions over the phase space, in particular the failure of an earlier advertised superposition formula.
In this paper, we use the characteristic function, i.e., the Fourier transform of the Glauber-Sudarshan phase-space distribution, to determine the degree of nonclassicality of a given state. This degree of nonclassicality quantifies the nonclassicality in terms of quantum superpositions. We demonstrate two ways to exactly find or to lower-bound the degree of nonclassicality by studying the properties of the characteristic functions. The developed criteria are applied to two examples of squeezed states undergoing a classical mixing or a nonclassical superposition with vacuum.
Necessary and sufficient conditions are derived that allow one to completely characterize the nonclassicality of an arbitrary quantum state of a harmonic oscillator. Starting from a theorem by Bochner we derive an infinite hierarchy of conditions for the characteristic functions of the quantum state. Following the hierarchy step by step, the nonclassicality can be classified with respect to increasing orders. The full hierarchy of conditions is completely equivalent to the most commonly accepted criterion for nonclassicality: the failure of the Glauber–Sudarshan P -function to be a probability density. Our reformulated version of the criterion is directly related to measurable quantities that can be observed in phase-sensitive measurements such as homodyne detection. It directly generalizes a criterion for nonclassicality (Vogel 2000 Phys. Rev. Lett. 84 1849) that has already been applied in experiments (Lvovsky and Shapiro 2002 Phys. Rev. A 65 033830). Examples of phase-insensitive quantum states are considered that exhibit no first-order nonclassical effects, but that are nonclassical in second order.
Genuinely quantum states of a harmonic oscillator may be distinguished from their classical counterparts by the Glauber-Sudarshan P representation-a state lacking a positive P function is said to be nonclassical.In this paper, we propose a general operational framework for studying nonclassicality as a resource in networks of passive linear elements and measurements with feed forward.Within this setting, we define new measures of nonclassicality based on the quantum fluctuations of quadratures, as well as the quantum Fisher information of quadrature displacements.These measures lead to fundamental constraints on the manipulation of nonclassicality, especially its concentration into subsystems, that apply to generic multimode non-Gaussian states.Special cases of our framework include no-go results in the concentration of squeezing and a complete hierarchy of nonclassicality for single-mode Gaussian states.
We experimentally examine the nonclassical character of a class of non-Gaussian states known as phase-diffused squeezed states. These states may show no squeezing effect at all and therefore provide an interesting example to test nonclassicality criteria. The characteristic function of the Glauber-Sudarshan representation (PPP function) proves to be a powerful tool to detect nonclassicality. Using this criterion we find that phase-diffused squeezed states are always nonclassical, even if the squeezing effect vanishes. Testing other criteria of nonclassicality based on higher-order squeezing and the positive semidefiniteness of special matrices of normally ordered moments, it is found that these criteria fail to reveal the nonclassicality for some of the prepared phase-diffused squeezed states.
Understanding quantum phenomena that go beyond classical concepts is a focus of modern quantum physics. Here, we show how the theory of non-negative polynomials emerging around Hilbert's 17th problem, can be used to optimally exploit data capturing the nonclassical nature of light. Specifically, we show that non-negative polynomials can reveal nonclassicality in data even when it is hidden from standard detection methods up to now. Moreover, the abstract language of non-negative polynomials also leads to a unified mathematical approach to nonclassicality for light and spin systems, allowing us to map methods for one to the other. Conversely, the physical problems arising also inspire several mathematical insights into characterization of non-negative polynomials.