Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Detecting genuine multipartite entanglement in multi-qubit devices with restricted measurements

Nicky Kai Hong Li, Xi Dai, Manuel H. Muñoz-Arias, Kevin Reuer, Marcus Huber, Nicolai Friis

TL;DR

The paper tackles scalable detection of genuine multipartite entanglement (GME) in large n-qubit devices when joint measurements are restricted. It introduces graph-state–based GME and k-inseparability criteria that require only O(n^2) stabilizer measurements with bounded weight m ≤ 2·max_{(i,j)∈E}(d(i)+d(j)), and employs SDP to further reduce measurement burden. Analytically, it derives graph-matching bounds for k-separable states and offers a fixed-k-partition variant, with extensive numerical tests on microwave-photonic graph states (cluster, ring, tree) under realistic noise; results show robustness to noise and the ability to bound state infidelity via certified k-inseparability, often outperforming existing witnesses under limited measurements. The work provides a practical entanglement benchmarking tool for scalable quantum devices and demonstrates how SDP can compensate for incomplete measurements, making high-fidelity graph-state certification feasible in current experimental platforms.

Abstract

Detecting genuine multipartite entanglement (GME) is a state-characterization task that benchmarks coherence and experimental control in quantum systems. Existing GME tests often require joint measurements on many qubits, posing experimental challenges for systems like time-bin encoded qubits and microwave photons from superconducting circuits, where qubit connectivity is limited or measurement noise grows with the number of jointly measured qubits. Here we introduce versatile GME and $k$-inseparability criteria applicable to any state, which only require measuring $O(n^2)$ out of $2^n$ (at most) $m$-body stabilizers of $n$-qubit target graph states, with $m$ upper-bounded by twice the underlying graph's maximum degree. For cluster or ring-graph states, only constant-weight stabilizers are needed. Using semidefinite programming, we further reduce both the number and weight of required stabilizers. Analytical and numerical results show that our criteria are noise-robust and can infer state infidelity from certified $k$-inseparability in microwave photonic graph states generated under realistic conditions.

Detecting genuine multipartite entanglement in multi-qubit devices with restricted measurements

TL;DR

The paper tackles scalable detection of genuine multipartite entanglement (GME) in large n-qubit devices when joint measurements are restricted. It introduces graph-state–based GME and k-inseparability criteria that require only O(n^2) stabilizer measurements with bounded weight m ≤ 2·max_{(i,j)∈E}(d(i)+d(j)), and employs SDP to further reduce measurement burden. Analytically, it derives graph-matching bounds for k-separable states and offers a fixed-k-partition variant, with extensive numerical tests on microwave-photonic graph states (cluster, ring, tree) under realistic noise; results show robustness to noise and the ability to bound state infidelity via certified k-inseparability, often outperforming existing witnesses under limited measurements. The work provides a practical entanglement benchmarking tool for scalable quantum devices and demonstrates how SDP can compensate for incomplete measurements, making high-fidelity graph-state certification feasible in current experimental platforms.

Abstract

Detecting genuine multipartite entanglement (GME) is a state-characterization task that benchmarks coherence and experimental control in quantum systems. Existing GME tests often require joint measurements on many qubits, posing experimental challenges for systems like time-bin encoded qubits and microwave photons from superconducting circuits, where qubit connectivity is limited or measurement noise grows with the number of jointly measured qubits. Here we introduce versatile GME and -inseparability criteria applicable to any state, which only require measuring out of (at most) -body stabilizers of -qubit target graph states, with upper-bounded by twice the underlying graph's maximum degree. For cluster or ring-graph states, only constant-weight stabilizers are needed. Using semidefinite programming, we further reduce both the number and weight of required stabilizers. Analytical and numerical results show that our criteria are noise-robust and can infer state infidelity from certified -inseparability in microwave photonic graph states generated under realistic conditions.
Paper Structure (23 sections, 4 theorems, 46 equations, 9 figures, 2 tables, 2 algorithms)

This paper contains 23 sections, 4 theorems, 46 equations, 9 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and notation
  3. GME & k-inseparability criteria
  4. SDP for incomplete measurements
  5. Experimental proposal and simulations
  6. Certifying GME/$k$-inseparability in simulated states
  7. Noise sensitivity of the GME/$k$-inseparability criteria
  8. Discussion
  9. Review of previous methods
  10. Proof of Theorem \ref{['theorem:kSepBound']} and Lemma \ref{['lemma:FixedkSepBound']}
  11. Proof of Observation \ref{['obs:gammaBetween0and1']}
  12. Algorithm for calculating the criterion upper bound
  13. A brief review of SDP
  14. Details of the SDP for incomplete measurements
  15. Theoretical examples
  16. ...and 8 more sections

Key Result

Lemma 2

Let $\{E_i\}_{i=1}^{d^2}$ be an orthonormal self-adjoint basis of $d\times d$ complex matrices (i.e., ${\rm Tr}(E_i E_j)=d\delta_{ij}$) and define $\Omega\subseteq \{1,\ldots,d^2\}$ such that $\frac{1}{2}\sqrt{\sum_{i\neq j\in\Omega}\langle\{E_i, E_j\}\rangle^2_\rho} \leqslant\mathcal{K}$. Then, for any state $\rho\in\mathcal{D}(\mathbb{C}^{d})$.

Figures (9)

  • Figure 1: Graphical illustration of what a $k$-partition subgraph $\overline{G}^{\space\raisebox{-2 pt}{\tiny{$(k)$}}}$ of the graph $G$ and the maximum cardinality matching $\overline{E}^{\space\raisebox{-2 pt}{\tiny{$(k)$}}}_\text{mcm}$ of $\overline{G}^{\space\raisebox{-2 pt}{\tiny{$(k)$}}}$ represent. In a 3-partition, we partition $G$ into three parts (highlighted in different colors). By removing edges that do not connect vertices in different regions—the two edges contained in the green region on the right-hand side of $G$—we obtain the subgraph $\overline{G}^{\space\raisebox{-2 pt}{\tiny{$(k\!=\!3)$}}}$. There are only two possible non-isomorphic maximal matchings $\overline{E}^{\space\raisebox{-2 pt}{\tiny{$(k\!=\!3)$}}}_\text{match}$ of $\overline{G}^{\space\raisebox{-2 pt}{\tiny{$(k\!=\!3)$}}}$, which are represented by the red edges. The top right matching has the most edges, making it the maximum-cardinality matching $\overline{E}^{\space\raisebox{-2 pt}{\tiny{$(k\!=\!3)$}}}_\text{mcm}$ of this particular 3-partition.
  • Figure 2: Cthulhu graphs: The parameter $r$ represents both the number of vertices in the "head" subgraph (red and purple vertices) and the degree of the central vertex in the star subgraph containing the "tentacles" (blue and purple vertices). These graphs are defined such that the "head" subgraph contains an $(r\!-\!1)$-vertex complete graph with two adjacent vertices being the leaves of the star subgraph.
  • Figure 3: Schematic and quantum circuit for cluster state generation. (a) Schematic of sequentially generated microwave photonic cluster state in two dimensions. The setup consists of a linear array of tunably interacting transmon qutrits. Arbitrary single- and two-qubit gates on the lowest two levels and the third level can be used for controlled emission of microwave photons. (b) Quantum circuit for creating $n_x\times n_y$ two-dimensional cluster state. In the circuit, $\mathrm{So}_i$ denotes the $i$'th source transmon, whereas $\mathrm{Ph}_i$ denotes the $i$'th generated photonic qubit. The double blue lines and two dots indicate the block of circuit to be repeated.
  • Figure 4: Certified $k$-inseparability and infidelity (to the ideal cluster state) of the simulated $5\!\times\!2$-qubit 2D cluster state for different noise parameters. In the x-axis, leakage errors, which are the dominant coherent errors, are varied. In the y-axis, the coherence times of the source transmons are varied. The parameter $\tau_\text{emit}/\tau_\text{coh}$ represents the photon emission versus coherence times ratio (see text for definition). The points corresponding to the experimental noise parameters from Ref. OsullivanEtAl2024 are marked with white squares.
  • Figure A.1: Illustration of an example corresponding to an optimal $k$-inseparability criterion with $\gamma\in(0,1)$. For a Cthulhu graph $G$ of $r=4$ (see Fig. \ref{['fig:CthulhuGraphTheory']} for the meaning of $r$), the optimal $r$-cuts of $R^\gamma_k\coloneqq \min_{\text{all }k\text{-cuts}}\!\left( \gamma |\overline{V}^{\space\raisebox{-2 pt}{\tiny{$(k)$}}}|+(1-\gamma)|\overline{E}^{\space\raisebox{-2 pt}{\tiny{$(k)$}}}_\text{mcm}|\right)$ for different values of $\gamma\in[0,1]$ are shown in the middle column where the color shadings represent partitioning of the graph into $r$ different parts. In general, for $r=4$ and $r\geqslant6$, the optimal $r$-cut for $\gamma\!\geqslant\!\left(\lfloor\frac{r}{2}\rfloor\!-\!1\right)\!/\lfloor\frac{r}{2}\rfloor$ (here $=\!\frac{1}{2}$) results in the "head" subgraph $\overline{G}^{\space\raisebox{-2 pt}{\tiny{$(r)$}}}_+$, whereas the optimal $r$-cut for $\gamma\!\leqslant\!\left(\lfloor\frac{r}{2}\rfloor\!-\!1\right)\!/\lfloor\frac{r}{2}\rfloor$ results in the "tentacles" subgraph $\overline{G}^{\space\raisebox{-2 pt}{\tiny{$(r)$}}}_-$. The transition of the optimal $r$-cut in $\gamma$ from 0 to 1 also leads to a transition in $R^\gamma_r$ [see Eq. \ref{['eq:Rgammaktransition']}].
  • ...and 4 more figures

Theorems & Definitions (7)

  • Lemma 2: Anticommutativity bound AsadianErkerHuberKlockl2016
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • proof : Proof of Theorem \ref{['theorem:kSepBound']} & Lemma \ref{['lemma:FixedkSepBound']}
  • Lemma 3: Slater's theorem for SDP Watrous2018