Stoquastic permutationally invariant Bell operators

Abstract

As Hermitian operators, many-body Bell operators can naturally be identified as many-body Hamiltonians. An important subclass of such Hamiltonians is the stoquastic class, characterized by having nonpositive off-diagonal matrix elements in a given basis. Interestingly, this property is shared by the permutationally invariant (PI) Bell operators underlying the largest Bell-correlation experiments to date. In this work, we explore the connection between many-body PI Bell operators and stoquasticity. We introduce the stoquasticity cone, which allows for a full characterization of the stoquastic parameter regimes for any PI Bell operator. We use this to show that PI Bell operators of the binary-input binary-output scenario consisting of at most three-body correlators can always be rendered stoquastic for any set of measurement parameters. Additionally, we also provide examples that use the stoquasticity cone to optimize for the quantum-classical gap. Numerical evidence suggests that the Bell operator used in the largest experiments to date is optimal with respect to stoquasticity. To the best of our knowledge, this work establishes the first connection between PI Bell operators and stoquasticity.

Figures (5)

• Figure 1: Schematic representation of the block structure of the permutationally invariant operator $B_{\mathrm{PI}}$. Here $J \in \{J_0, J_0 + 1, \ldots, n/2\}$ with $J_0 =0$ if $n$ is even and $J_0 =1/2$ if $n$ is odd
• Figure 2: The orange points denote values of measurement parameters $(\varphi,\theta)$ with a quantum violation above 1 and the black points denote measurement parameters without such a violation. The blue line denotes $\pi-\varphi = \theta$ and the red line denotes $\varphi = \theta$, which correspond to values that satisfy the stoquastic conditions of Eq. \ref{['eq:Ineq6StoqCond']}. For the blue line $(\pi-\varphi = \theta)$, we observe that the segment which extends into the orange region grows with increasing $n$.
• Figure 3: Plots of coefficients of eigenvectors from numerically optimized Bell operators of table \ref{['tb:OptGap']}. The blue dots represent $\psi_i^2$, where $\psi_i$ is the $i-$th element of the ground state eigenvector up to normalization. The red line represent the fitted normal distribution.
• Figure 4: Optimization sequence leading to the optimal Bell operator for $(\varphi,\theta)=(\pi/4,-\pi/4)$. For the first two optimization time steps, the domain was set to $[0,10^6]$ for the rays and $[-10^6,10^6]$ for the lines. For the last optimization step, the domain was increased to $[0,10^8]$ for the rays and $[-10^8,10^8]$ for the lines. The coefficients $c_1,\ldots,c_{13}$ correspond to rays and $c_{14}, c_{15}, c_{16}$ correspond to lines.
• Figure 5: Example of optimization sequence converging to $f_{n,\varphi,\theta}(\vec{c}) \leq 1$.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof