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The perturbative method for quantum correlations

Sacha Cerf, Harold Ollivier

Abstract

The set $\mathcal{Q}$ of quantum correlations is the collection of all possible probability distributions on measurement outcomes achievable by space-like separated parties sharing a quantum state. Since the original work of Tsirelson [Tsirelson, Lett. Math. Phys. 4, 93 (1980)], this set has mainly been studied through the means algebraic and convex geometry techniques. We introduce a perturbative method using Lie-theoretic tools for the unitary group to analyze the response of the evaluations of Bell functionals under infinitesimal unitary perturbations of quantum strategies. Our main result shows that, near classical deterministic points, an $(n, 2, d)$ Bell operator decomposes into a direct sum of $(k, 2, d-1)$ Bell operators which we call \emph{subset games}. We then derive three key insights: (1) in the $(n, 2, 2)$ case, if $p_0$ is classically optimal, it remains locally optimal even among 2-dimensional quantum strategies, implying in turn that the boundary of $\mathcal{Q}$ is flat around classical deterministic points; (2) it suggests a proof strategy for Gisin's open problem on correlations in $\mathcal{Q}(D)$ unattainable by projective strategies of the same dimension; and (3) it establishes that the Ansatz dimension is a critical resource for learning in distributed scenarios, even when the optimal solution admits a low-dimensional representation.

The perturbative method for quantum correlations

Abstract

The set of quantum correlations is the collection of all possible probability distributions on measurement outcomes achievable by space-like separated parties sharing a quantum state. Since the original work of Tsirelson [Tsirelson, Lett. Math. Phys. 4, 93 (1980)], this set has mainly been studied through the means algebraic and convex geometry techniques. We introduce a perturbative method using Lie-theoretic tools for the unitary group to analyze the response of the evaluations of Bell functionals under infinitesimal unitary perturbations of quantum strategies. Our main result shows that, near classical deterministic points, an Bell operator decomposes into a direct sum of Bell operators which we call \emph{subset games}. We then derive three key insights: (1) in the case, if is classically optimal, it remains locally optimal even among 2-dimensional quantum strategies, implying in turn that the boundary of is flat around classical deterministic points; (2) it suggests a proof strategy for Gisin's open problem on correlations in unattainable by projective strategies of the same dimension; and (3) it establishes that the Ansatz dimension is a critical resource for learning in distributed scenarios, even when the optimal solution admits a low-dimensional representation.

Paper Structure

This paper contains 17 sections, 15 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Quantum correlations and nonlocality
  3. Nonlocal games
  4. Previous work
  5. Proposed approach and results
  6. Preliminaries
  7. Correlations
  8. Bell functional
  9. Local correlations, local strategies
  10. Quantum correlations, quantum strategies
  11. Extremal points, exposed points, and border
  12. Response of bell functionals under unitary perturbations
  13. Unitary perturbations of local deterministic strategies
  14. The (n, 2, 2) quantum set is flat around every local deterministic correlation
  15. Discussion and Open questions
  16. ...and 2 more sections

Key Result

Proposition 1

Let $\vec{U} \equiv (e^{-K_\rho}, e^{K_1}, \ldots, e^{K_n})$, where $K_i, 1 \le i \le n$ is a local skew-Hermitian operator, and $K_\rho$ is a skew-Hermitian operator on the shared state. Then, the value of the Bell functional on $\Sigma$ admits the following second order expansion. where we denoted $x \cdot \vec{K} \equiv \sum_{i=1}^n x_i K^{(i)}_i$, $\vert \vert \vec{K} \vert \vert^2 \equiv -\s

Figures (2)

  • Figure 1: Evolution of the evaluation of four Bell functionals under unitary perturbation of the optimal local deterministic strategy (Lemma \ref{['lem:locopt']}, see \ref{['app:functionals']} for more details) For each game, we draw $20$ trajectories by sampling a strategy $\Sigma (\rho, \Pi)$ realizing the best local deterministic correlation, (for all $i$, $(\Pi^{(i)}_{a_i \vert x_i})_{a_i \neq \tilde{a}_i(x_i)}$ is sampled from the Haar measure), and random perturbation generators $K_\rho$ and $(K_i^{(x)})_{i,x\geq 1}$, sampled with i.i.d $\mathcal{N}(0, 1)$ entries, skew-symmetrized and normalized. (Top) Evolution of the score $\vec{\beta}\cdot P(\vec{U}(t)\cdot\Sigma_0)$ as a function of the perturbation parameter $t$; the dashed red line indicates the classical score. (Bottom) Second-order variation $\partial^2_t [\vec{\beta} \cdot P(\vec{U}(t)\cdot\Sigma_0)]|_{t=0}$ estimated by finite differences for each trajectory (dots), with the mean indicated by a horizontal bar. All values are strictly negative for all four games, confirming Lemma \ref{['lem:locopt']} for CHSH and GHZ ($(n,2,2)$ scenarios), and suggesting that it may extend to the $(n,m,d)$ setting, with $m > 2$. Interestingly enough, the result holds for the Magic Square game, even though $d > 2$.
  • Figure 2: Illustration for the proof of Theorem \ref{['th:flat']}. Here, $p_{\tilde{a}}$ is the unique classical optimum for $\beta_\varepsilon$, and every point above $F_{\varepsilon}$ in $\partial \mathcal{Q}^P$ is better than $p_{\tilde{a}}$ for $\vec{\beta}_{\varepsilon}$. Thus, points too close from $p_{\tilde{a}}$ in $\partial \mathcal{Q}^P \cap H_1$ cannot be in $\mathcal{Q}_p(2)$, hence are not extremal.

Theorems & Definitions (30)

  • Definition 1
  • Proposition 1
  • proof
  • Corollary 1: First order conditions of optimality
  • proof
  • Definition 2: QP correlations and strategies
  • Lemma 1
  • proof
  • Lemma 2: Second order energy weighted sum rule for observables
  • proof
  • ...and 20 more