Self-Testing Graph States Permitting Bounded Classical Communication

TL;DR

The work addresses self-testing of graph states under bounded classical communication by introducing inflated graph states and a deflation procedure that allows certifying a target graph state even when devices can communicate within a fixed radius. The core method constructs reference experiments on inflated graphs and proves that any compatible physical experiment that reproduces the inflated correlations must be equivalent (up to a local isometry) to the inflated target, thereby recovering the original graph state after deflation. Robustness analyses show the self-testing remains reliable in the presence of noise, with explicit $\delta$-equivalence bounds that scale with $\sqrt{\epsilon}$. The results enable robust device-independent certification in communication-constrained settings and offer concrete REs for circular and honeycomb graph states, suggesting practical paths to scalable, bounded-communication quantum certification and potential circuit-depth separations in classical simulations.

Abstract

Self-testing identifies quantum states and correlations that exhibit nonlocality, distinguishing them, up to local transformations, from other quantum states. Due to their strong nonlocality, it is known that all graph states can be self-tested in the standard setting - where parties are not allowed to communicate. Recently it has been shown that graph states display nonlocal correlations even when bounded classical communication on the underlying graph is permitted, a feature that has found applications in proving a circuit-depth separation between classical and quantum computing. In this work, we develop self testing in the framework of bounded classical communication, and we show that certain graph states can be robustly self-tested even allowing for communication. In particular, we provide an explicit self-test for the circular graph state and the honeycomb cluster state - the latter known to be a universal resource for measurement based quantum computation. Since communication generally obstructs self-testing of graph states, we further provide a procedure to robustly self-test any graph state from larger ones that exhibit nonlocal correlations in the communication scenario.

Figures (7)

• Figure 1: Circuit implementing a SWAP gate between $\vert \psi \rangle$ and $\vert 0 \rangle$ using the Hadamard gate $H$ and Pauli operators $X,Z$.
• Figure 2: A graph $G$ with four vertices (rhs) is mapped to an exemplary inflated graph $G'$ for $d=2$ (lhs) by means of Def. \ref{['def:inflatedgraph']}, i.e., by adding $2d$ 'chain' vertices along the edges. The inverse mapping 'deflation' measures the chain vertices projectively (up to local corrections) to recover $\ket{G}$ from $\ket{G'}$. In Theorem \ref{['theo:main']}, the RE uses $\ket{G'}$ and local measurements to both prepare and self-test $\ket{G}$.
• Figure 3: A graph with four vertices (rhs) and its ($d=2$)-inflated graph (lhs) by means of Def. \ref{['def:inflatedgraph']}. Letters indicate non-trivial Pauli operators while empty nodes indicate Pauli identity in the Pauli product of the inflated generator element $f_{4}$ from Def. \ref{['def:infvstab']} (lhs), mimicking the generator element $g_{4}$ (rhs) on the corresponding graph state.
• Figure 4: Four panels of inflated graphs that address the ones in REs \ref{['def:refe1']} (a), REs \ref{['def:refe2']}, for $\vert V_{s} \vert >2$ (b), for $\vert V_{s} \vert = 2$ (d) and RE \ref{['def:refe3']} (c). The bold nodes depict power vertices. Within a panel, each graph depicts a measurement setting and its submeasurement. The measurements are those that challenge any $d$-LHV$^\ast$ model. The letters in black and light-blue font indicate he bases in the measurement setting. For the submeasurements, replace the letters in light-blue font by Pauli identity. Graphs with a circular arrow represent all measurements under invariant rotations of the power vertices. For RE \ref{['def:refe1']}, panel (a) shows $F_{v_{1}}$ with $f_{v_{1}}$ from Eq. \ref{['eq:infstabgenerator']} (upper left), while the circular arrow hints at $F_{v_{2}}$ with $f_{v_{2}}$, and $F_{v_{3}}$ with $f_{v_{3}}$, $M_{V_{c}}$ with $C_{V_{c}}$ (upper right) from Eq. \ref{['eq:measrefstab12']}, and $M_{v_{1}}^{(X)}$ (lower left) and $M_{v_{1}}^{(Z)}$ (lower right) with $C_{v_{1}}$ from Eq. \ref{['eq:measrefstab13']}, and the circular arrow hints at $M_{v_{2}}^{(X)}$, $M_{v_{2}}^{(Z)}$ with $C_{v_{2}}$, and $M_{v_{3}}^{(X)}$, $M_{v_{3}}^{(Z)}$ with $C_{v_{3}}$. For RE \ref{['def:refe2']} ($\vert N(v_{c}) \vert = 3$), panel (b) shows $\Tilde{F}_{v_{c}}$ with $f_{v_{c}}$ from Eq. \ref{['eq:infstabgenerator']} (left), and $M_{v_{2},v_{3}}$ with $C_{v_{2},v_{3}}$ from Eq. \ref{['eq:ref21']}, and the circular arrow hints at $M_{v_{1},v_{3}}$ with $C_{v_{1},v_{3}}$, $M_{v_{1},v_{2}}$ with $C_{v_{1},v_{2}}$. For RE \ref{['def:refe3']}, panel (c) shows, from top to bottom, $M_{i}$ with $I_{i}$ in Eq. \ref{['eq:ref3']} for $i = 1,2,3,4$ and $\Tilde{F}_{v_{l}}$ with $f_{v_{l}}$, $\Tilde{F}_{v_{r}}$ with $f_{v_{r}}$. For RE \ref{['def:refe2']} ($\vert N(v_{c}) \vert = 2$), panel (d) shows, from top to bottom, $\Tilde{F}_{v_{c}}$ with $f_{v_{c}}$ in Eq. \ref{['eq:infstabgenerator']}, $M_{v_{1}}$ with $C_{v_{1}}$ in Eq. \ref{['eq:ref22-1']}, $M_{v_{2}}$ with $C_{v_{2}}$ in Eq. \ref{['eq:ref22-2']}, $M_{v_{c}}$ with $C_{v_{c}}$ in Eq. \ref{['eq:ref22-3']}, $M_{2}^{(X)}$ and $M_{2}^{(Y)}$ with $C_{2}$ in Eq. \ref{['eq:ref22-4']}.
• Figure 5: We illustrate an excerpt of the honeycomb cluster. The dashed lines sketch the pattern that defines the $d$-inflated star graph, which we call tripoint star. The blue tripoint star is defined around vertex $u \in V_{hex_{1}}$ and the red tripoint star around vertex $v \in V_{hex_{2}}$, where $V_{hex_{1}}$ and $V_{hex_{2}}$ are the two hexagonal lattices that define the honeycomb lattice.

Theorems & Definitions (19)

• Definition 1: Submeasurements
• Definition 2: $\epsilon$-Simulation
• Definition 3: $\delta$-Equivalence
• Definition 4: Robust Self-testing
• Proposition 1
• Definition 5: Inflated graph
• Definition 6: Inflated Generator Element
• Definition 7: Induced subgraph
• Theorem 1
• Lemma 1