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Revealing hidden physical nonclassicality with nonnegative polynomials

Ties-A. Ohst, Benjamin Yadin, Birte Ostermann, Timo de Wolff, Otfried Gühne, Hai-Chau Nguyen

Abstract

Understanding quantum phenomena which go beyond classical concepts is a focus of modern quantum physics. Here, we show how the theory of nonnegative polynomials emerging around Hilbert's 17th problem, can be used to optimally exploit data capturing the nonclassical nature of light. Specifically, we show that nonnegative polynomials can reveal nonclassicality in data even when it is hidden from standard detection methods up to now. Moreover, the abstract language of nonnegative 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 characterisation of nonnegative polynomials.

Revealing hidden physical nonclassicality with nonnegative polynomials

Abstract

Understanding quantum phenomena which go beyond classical concepts is a focus of modern quantum physics. Here, we show how the theory of nonnegative polynomials emerging around Hilbert's 17th problem, can be used to optimally exploit data capturing the nonclassical nature of light. Specifically, we show that nonnegative polynomials can reveal nonclassicality in data even when it is hidden from standard detection methods up to now. Moreover, the abstract language of nonnegative 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 characterisation of nonnegative polynomials.
Paper Structure (12 sections, 9 theorems, 73 equations, 5 figures)

This paper contains 12 sections, 9 theorems, 73 equations, 5 figures.

Table of Contents

  1. Basic notations of quantum optics and nonclassicality of light
  2. Nonnegative polynomials and Reznick's hierarchy
  3. Witnesses of nonclassicality of light
  4. Witnesses of nonclassicality of light by nonnegative polynomials
  5. The method of moment matrices and SOS witnesses
  6. Nonclassicality beyond SOS with nonnegative polynomials
  7. Incorporating other experimental constraints
  8. Photon statistics and quadratures
  9. Witnesses by photon number probabilities
  10. Relationship between nonclassicality of light and that of spin systems
  11. A hierarchy derived from theorems by Pólya and Fejér-Riesz
  12. Finite rays for approximating the classical set from inside

Key Result

Theorem 1

Let $f\in \mathbb{R}[x_{1},\dots,x_{n}]$ be a polynomial such that $f(\mathbf{x}) > 0$ for all $\mathbf{x} \in \mathbb{R}^{n}\setminus \{0\}$. Then, there exists a number $b \in \mathbb{N}$ such that $f_b$ given by is a SOS.

Figures (5)

  • Figure 1: Sketch of our approach to nonclassicality detection with finite data. Given a quantum state, the expectation values $\langle a^{\dagger k} a^{l} \rangle$ up to degree $D$, i.e. $k+l \leq D$, are obtained experimentally. Analysis with nonclassicality witnesses by nonnegative polynomials of degree less than $D$ gives the exact characterisation of data which have nonclassical effects; the set of states detectable in this way is complementary to those that are fundamentally undetectable based on the given moment data. In contrast, the methods of sum of squares and moment matrix give an outer approximation to this set.
  • Figure 2: Nonclassicality indicated by the most negative expectation values $\langle W_M \rangle_{\rho^{\ast}}$, using the fixed Motzkin polynomial, among all states that are undetectable by SOS up to degree $6$ (solid blue), $8$ (dotted orange), $10$ (dash-dot green). The $x$-axis indicates the photon number cutoff of the state.
  • Figure 3: Detection of the $17$-qubit state in Eq. \ref{['eq:seventeen_qubits_state']} by the Pólya-Fejér-Riesz hierarchy until level $40$ (red squares). Negative values approaching the lower bound obtained by the finite rays approximation (blue dashed line) indicate optimal detection. In contrast, the SDPs by Reznick's hierarchy (until $b=9$) (green dash-dotted line) cannot detect the nonclassicality.
  • Figure 4: Plot of the expectation values of the observables $W_R$ and $W_{CL}$ corresponding to the Robinson and Choi-Lam polynomials minimized over all states that have nonnegative moment matrices of degree $6$ (solid blue), $8$ (dotted orange), $10$ (dashdot green). The $x$-axis corresponds to the maximal photon number in the support of the state.
  • Figure 5: Sketch of the recursive definition of the trigonometric polynomials in Theorem \ref{['thm:hybrid_hierarchy']} which. The trigonometric polynomials in the $b$'th layer of the "triangle" correspond to the polynomials whose nonnegativity is demanded in the $b$'th level of the hierarchy.

Theorems & Definitions (9)

  • Theorem 1: Reznick's Positivstellensatz; Reznick1995
  • Theorem 2: Gram matrix method; Choi:Lam:ReznickPowers:Woermann:GramMatrix
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Theorem 7: Pólya's theorem; Polya_original
  • Theorem 8: Ref. Dumitrescu2017
  • Theorem 9