Measuring multipartite entanglement efficiently by testing symmetries

Abstract

Recently, a technique known as quantum symmetry test has gained increasing attention for detecting bipartite entanglement in pure quantum states. In this work we show that, beyond qualitative detection, a family of well-defined measures of bipartite and multipartite entanglement can be obtained with symmetry tests. We propose and benchmark several efficient methods to estimate these measures, and derive near-optimal sampling strategies for each. Despite the nonlinearity of the methods, we demonstrate that the sampling error scales no worse than $O(N_{\mathrm{tot}}^{-1/2})$ with the total number of copies $N_{\mathrm{tot}}$, which suggests experimental feasibility. By exploiting symmetries we compute our measures for large number of copies, and derive the asymptotic decay exponents for relevant states in many-body systems. Using these results we identify tradeoffs between estimation complexity and sensitivity of the presented entanglement measures, oriented to practical implementations.

Figures (6)

• Figure 1: Circuit diagrams of (a) generalized SWAP test, (b) simultaneous moment estimation, (c) G-Bose symmetry test and (d) cyclic permutation test. Here $A$ is any gate mapping $\mathinner{|0\rangle}$ to a coherent equal superposition, $F$ is the qudit Fourier transform, $D$ is a full-cycle permutation and $\pi$ are the permutations in $\mathcal{G}_k$.
• Figure 2: Absolute sampling error in estimating multipartite $C_4^{2}(\mathinner{|\psi\rangle}, \mathcal{S})$. For each circuit and each total copy budget $N_{\mathrm{tot}}$, resulting errors are averaged over $1000$ 4-qubit Haar-random pure states (the individual errors are shown as scatter points; within each cluster the points share the same $N_{\mathrm{tot}}$ and are slightly offset horizontally for visual clarity). The empirical errors exhibit the scaling $\varepsilon \sim N_{\mathrm{tot}}^{-1/2}$.
• Figure 3: Values of $C_k^{2}(\mathinner{|\psi(\theta)\rangle},\mathcal{G})$ for $\theta\in\{\pi/8,\pi/6,\pi/4\}$, $\mathcal{G}_k=\mathcal{S}_k,\mathcal{C}_k,\mathcal{D}_k$ and $k=2,\cdots,50$. For symmetric projection onto $\mathcal{S}_k$, one observes a clear $\theta$-dependent exponential decay with $k$. In contrast, the differences for the cyclic and dihedral projections onto $\mathcal{C}_k$ and $\mathcal{D}_k$ are much more subtle (see zoom).
• Figure 4: Absolute error and logarithmic error in estimating $C_4^{\{0,1\}}(\mathinner{|\psi\rangle})$ and $C_4^{2}(\mathinner{|\psi\rangle})$. $S=\{0,1\}$ represents the subsystem consisting of the first two qubits. The numerical settings are the same as the ones in Fig. \ref{['fig:error']}. The empirical absolute errors have the scaling very close to $\sim N_{\mathrm{tot}}^{-1/2}$ for all three groups and both (a) bipartite and (c) multipartite cases. The empirical logarithmic errors also exhibit the scaling $\varepsilon \sim O( N_{\mathrm{tot}}^{-1/2})$, though with a factor larger than 1, as shown in (b,d).
• Figure 5: Absolute and logarithmic error in estimating (a,b) $C_4^{\{0,1\}}(\mathinner{|\psi\rangle})$ and (c,d) $C_4^{2}(\mathinner{|\psi\rangle})$ with respect to $k$, for $N_{\mathrm{tot}}=100000$ and $600000$, respectively. Other numerical settings are the same as the ones in Fig. \ref{['fig:error']}. We use the Newton-Girard method to extrapolate higher-order state moments from the estimates at $k=2,3,4$, thereby obtaining $C_k^{\{0,1\}}(\mathinner{|\psi\rangle})$ or $C_k^{2}(\mathinner{|\psi\rangle})$ for $k\geq 5$ without consuming additional state copies (right of the vertical dashed line).

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Lemma 4
• proof
• Lemma 5
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 9