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arXiv:2610.32319

Finite Nonsignalling Resources Suffice to Simulate Entangled Qubits

Ziao Tang, Xin Wang, Ge Bai

quant-phSubmitted Oct 23, 2026• Announced Oct 26, 2026• 0 citations
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Abstract

A longstanding question in Bell nonlocality is whether finite-alphabet nonsignalling resources can reproduce the continuum of measurement correlations of partially entangled qubits. Here we resolve this problem by showing that every fixed pure two-qubit state admits an exact simulation of all local projective-measurement statistics using shared randomness and finitely many Popescu-Rohrlich (PR) boxes, without communication or postselection. The boxes operate in parallel, with a worst-case bound independent of the measurement settings: at most $128\cot^4θ$ calls for the Schmidt state $\cosθ|00\rangle+\sinθ|11\rangle$, $0<θ<π/4$. The key is an exact finite-selection method that reduces input-dependent continuous sampling to a choice from a fixed-length shared random list, implemented through a finite PR-box network. The construction extends to arbitrary mixed two-qubit states and local binary-outcome POVMs. This reveals a fundamental separation between the continuity of quantum measurement correlations and the alphabet size of the resources needed to simulate them, shedding light on the resource-theoretic structure of Bell nonlocality.

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6+9 pages, 1 figure

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A longstanding question in Bell nonlocality is whether finite-alphabet nonsignalling resources can reproduce the continuum of measurement correlations of partially entangled qubits. Here we resolve this problem by showing that every fixed pure two-qubit state admits an exact simulation of all local projective-measurement statistics using shared randomness and finitely many Popescu-Rohrlich (PR) boxes, without communication or postselection. The boxes operate in parallel, with a worst-case bound independent of the measurement settings: at most 128cot⁡4θ128\cot^4θ128cot4θ calls for the Schmidt state cos⁡θ∣00⟩+sin⁡θ∣11⟩\cosθ|00\rangle+\sinθ|11\ranglecosθ∣00⟩+sinθ∣11⟩, 0<θ<π/40<θ<π/40<θ<π/4. The key is an exact finite-selection method that reduces input-dependent continuous sampling to a choice from a fixed-length shared random list, implemented through a finite PR-box network. The construction extends to arbitrary mixed two-qubit states and local binary-outcome POVMs. This reveals a fundamental separation between the continuity of quantum measurement correlations and the alphabet size of the resources needed to simulate them, shedding light on the resource-theoretic structure of Bell nonlocality.
A quantum system of any fixed dimension can be prepared in a continuum of states, yet it cannot be used to transmit an unlimited amount of classical information. Similarly, the correlations observed between measurement outcomes on separate parts of a shared quantum system can be stronger than classical correlations, but they cannot transmit information. These fundamental limitations suggest that the statistics observed from quantum communication and quantum correlations may admit a simulation using a finite amount of classical communication. This expectation is confirmed in the smallest nontrivial quantum dimension, with two classical bits being necessary and sufficient to exactly simulate qubit communication and all correlations between qubits. Despite significant efforts during the previous decades, this remained the only solved case. Here we resolve both problems for every quantum dimension. The solution reveals an unexpected qualitative transition starting at dimension four: no finite amount of classical communication can exactly simulate ququart communication nor all quantum correlations of two entangled ququarts, even with unlimited shared randomness. One might have expected this transition, if it existed, to appear already for qutrits. Instead, we construct an explicit protocol that exactly simulates qutrit communication using 357357357 classical bits, and consequently, all correlations of two entangled qutrits.
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Non-local quantum computation (NLQC) asks two parties to apply a joint operation to their quantum inputs using an entangled resource state and one round of simultaneous quantum communication. NLQC has applications across quantum information, including attacks on quantum position verification, communication complexity, and quantum gravity. Every bipartite unitary has an approximate NLQC protocol with finite entanglement, and exact protocols are known for several structured families. However, it has remained open whether every fixed unitary admits an exact protocol with finite-dimensional resources. We show that, even for two qubits, Haar-almost every unitary does not admit such a protocol, even when the resource state and local operations are tailored to the target. The proof first shows that smooth deformations of an exact protocol can only change its target by local unitaries. For a fixed architecture, semialgebraic geometry limits the exact protocols to finitely many connected families, so each architecture reaches only finitely many local-unitary orbits. Taking the union over all architectures, the exactly implementable unitaries form countably many local-unitary orbits. We also show that some natural targets do not admit a protocol: the controlled phase diag(1,1,1,eiθ)\mathrm{diag}(1,1,1,e^{iθ})diag(1,1,1,eiθ) has no exact protocol with finite-dimensional resources whenever eiθe^{iθ}eiθ is transcendental, for instance for θ=1θ= 1θ=1. Our results also apply to localizable quantum measurements, whose outcome is obtained from local measurements on inputs and a shared entangled resource state without communication. A rank-one projective measurement in a Haar-random basis is almost surely not localizable with a finite-dimensional resource state and finitely many local outcomes.
Bell nonlocality exemplifies the most profound departure of quantum theory from classical realism. Yet, the extent of nonlocality in quantum theory is intrinsically bounded, falling short of the correlations permitted by the relativistic causality (the no-signaling) principle. A paradigmatic example is the Popescu-Rohrlich correlation: two distant parties sharing arbitrary entanglement cannot achieve this correlation, though it can be simulated with classical communication between them. Here we show how such post-quantum correlations can instead be simulated using intrinsically non-signaling physical resources, and implement the proposed scheme using a quantum circuit on a four-qubit photonic platform. Unlike the conventional approaches, our method exploits dynamical correlations between distinct physical systems, with intrinsic randomness suppressing any signaling capacity. This enables the realization of post-quantum correlations both with and without entanglement. We also analyze how the simulation scheme extends to beyond quantum nonlocal correlations in multipartite systems. Our experimental demonstration using a photonic system establishes a versatile framework for exploring post-quantum correlations in both foundational settings and as a resource for computation and security applications.
Bell nonlocality is the key quantum resource in some device-independent quantum information processing. It is of great importance to study the efficient sharing of this resource. Unsharp measurements are widely used in sharing the nonlocality of an entangled state shared among several sequential observers. Recently, the authors in [Phys. Rev. Lett. 129, 230402 (2022)] showed that the Bell nonlocality of two-qubit pure states can be shared even when one only uses projective measurements and local randomness. We demonstrate that projective measurements are also sufficient for sharing the Bell nonlocality of arbitrary high-dimensional pure bipartite states. Our results promote further understanding of the nonlocality sharing of high-dimensional quantum states under projective measurements.
A central aspect of quantum information is that correlations between spacelike separated observers sharing entangled states cannot be reproduced by local hidden variable (LHV) models, a phenomenon known as Bell nonlocality. If one wishes to explain such correlations by classical means, a natural possibility is to allow communication between the parties. In particular, LHV models augmented with two bits of classical communication can explain the correlations of any two-qubit state. Would this still hold if communication is restricted to measurement outcomes? While in certain scenarios with a finite number of inputs the answer is yes, we prove that if a model must reproduce all projective measurements, then for any qubit-qudit state the answer is no. In fact, a qubit-qudit under projective measurements admits an LHV model with outcome communication if and only if it already admits an LHV model without communication. On the other hand, we also show that when restricted sets of measurements are considered (for instance, when the qubit measurements are in the upper hemisphere of the Bloch ball), outcome communication does offer an advantage. This exemplifies that trivial properties in standard LHV scenarios, such as deterministic measurements and outcome-relabelling, play a crucial role in the outcome communication scenario.