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Quantum networks reveal quantum nonlocality
D. Cavalcanti, M. L. Almeida, V. Scarani, A. Acin
TL;DR
Determining whether a quantum state is nonlocal matters for foundational questions and information-processing applications, yet identifying nonlocal states remains difficult. The paper studies states distributed in networks and shows that one-way distillable entanglement yields nonlocal correlations, while some single-copy local states become nonlocal across copies.
Problem
Identifying nonlocal properties of even simple families of entangled states is difficult, despite nonlocal correlations being important to several information-processing applications.
Source: p. 1Method
The paper studies quantum states distributed as multiple copies across network configurations and proves nonlocality results within this framework.
Source: p. 1Results
Every one-way entanglement distillable state is a nonlocal resource, and examples show single-copy local states becoming nonlocal in networks or across copies.
Source: pp. 1, 3, 4Takeaways & Limitations
Quantum-state nonlocality depends on measurement context, and local single-copy isotropic states can form a nonlocal network in the overlap p > 0.64 and p < 0.66.
Source: pp. 3–4
Abstract
from arXiv · showhide
The results of local measurements on some composite quantum systems cannot be reproduced classically. This impossibility, known as quantum nonlocality, represents a milestone in the foundations of quantum theory. Quantum nonlocality is also a valuable resource for information processing tasks, e.g. quantum communication, quantum key distribution, quantum state estimation, or randomness extraction. Still, deciding if a quantum state is nonlocal remains a challenging problem. Here we introduce a novel approach to this question: we study the nonlocal properties of quantum states when distributed and measured in networks. Using our framework, we show how any one-way entanglement distillable state leads to nonlocal correlations. Then, we prove that nonlocality is a non-additive resource, which can be activated. There exist states, local at the single-copy level, that become nonlocal when taking several copies of it. Our results imply that the nonlocality of quantum states strongly depends on the measurement context.
A. Definition of genuine multipartite nonlocality.
Genuine multipartite nonlocality requires correlations that cannot be reproduced by any hybrid model splitting the parties into groups. The paper assumes no-signalling when defining these models.
A. Definition of genuine multipartite nonlocality.
- Hybrid models divide the parties into two groups, which may vary with the classical variable λ, and allow nonlocal correlations within each group.The two groups are correlated only through λ, so their nonlocal correlations are not shared by all parties.Source: A. Definition of genuine multipartite… · p. 5
- A distribution is genuinely N-partite nonlocal when no combination of hybrid models can reproduce it.A hybrid model may assign nonlocal correlations within a party group, with groups correlated only through a classical variable.Source: A. Definition of genuine multipartite… · p. 5
- The definition assumes no-signalling, requiring each term in the hybrid model to respect that principle.This has a natural operational meaning: parties cannot transmit information instantaneously, even when they share λ.Source: A. Definition of genuine multipartite… · p. 5
- Svetlichny inequalities are linear inequalities satisfied by hybrid local/nonlocal distributions and are used to detect genuine multipartite nonlocality.The paper notes that the definition is stronger than standard nonlocality, which may involve only a subset of parties.Source: A. Definition of genuine multipartite… · p. 5
B. Constructing Bell inequality for N-partite systems
The proof converts a violation by a post-selected nonlocal state into a standard Bell-inequality violation by the original N-party state. No-signalling enables the conversion.
B. Constructing Bell inequality for N-partite systems
- Local projections by k parties that produce a nonlocal state among the remaining parties imply that the original state is nonlocal.The proof constructs a Bell inequality for the original state from one violated by the post-measurement state.Source: B. Constructing Bell inequality for… · p. 5
- Post-selection yields a generalized Bell inequality because it conditions the remaining parties’ distribution on particular measurement outcomes.The inequality violated by the projected state can be expressed using the outcomes of the projecting parties’ measurements.Source: B. Constructing Bell inequality for… · p. 5
- Spacelike separation and no-signalling make the post-selection event independent of the remaining parties’ measurement choices, allowing conversion to a standard Bell inequality.The resulting standard inequality is violated by the original state and is satisfied by all local models.Source: B. Constructing Bell inequality for… · p. 5
C. Activation of nonlocality in the many-copy scenario.
The section shows how activation in network configurations can be converted into many-copy activation of quantum states. Correlated flags let the parties reconstruct the network with probability approaching one as copies increase.
C. Activation of nonlocality in the many-copy scenario.
- Correlated flags convert network-based activation into standard many-copy examples by letting parties reconstruct the activation network with probability approaching one as L increases.The construction is illustrated for genuine multipartite nonlocality and also applies to standard nonlocality.Source: C. Activation of nonlocality in the… · p. 6
- Two bipartite maximally entangled states in a Λ configuration define a genuinely tripartite nonlocal state, although each link connects only two parties.The correlations described for this state violate a Svetlichny-like inequality in the no-signalling framework.Source: C. Activation of nonlocality in the… · p. 6
- Flag measurements let Alice identify copies linking her to Bob and Charlie, after which the parties apply measurements producing the network’s genuine tripartite nonlocal correlations.The procedure can be implemented as one-shot local measurements; instances in which all flags agree occur with probability 1/2^(L−1).Source: C. Activation of nonlocality in the… · p. 6
- For the state σ, a finite number of copies suffices to obtain genuine tripartite nonlocal correlations, even though σ itself is not genuinely tripartite nonlocal.The resulting distribution can be made arbitrarily close to the target nonlocal distribution as the probability of equal flags decreases exponentially with L.Source: C. Activation of nonlocality in the… · p. 6
- A similar flag construction yields an (N + 1)-partite state τ that is local but whose sufficiently many copies are nonlocal.Increasing L allows τ⊗L to be transformed deterministically into a state arbitrarily close to the nonlocal star-configuration state.Source: C. Activation of nonlocality in the… · p. 6
D. Any two-outcome measurements can be simulated by von Neumann measurements.
Any dichotomic measurement can be simulated using a von Neumann measurement followed by outcome randomization. This extends projective-measurement locality to general two-outcome measurements.
D. Any two-outcome measurements can be simulated by von Neumann measurements.
- A dichotomic measurement can be simulated by measuring in the eigenbasis of its effects and randomly assigning an output based on the observed eigenvalue.For eigenvalue λ_i, the protocol outputs 0 with probability λ_i and 1 with probability 1 − λ_i.Source: D. Any two-outcome measurements can be… · p. 7
- A local model for projective measurements therefore also applies to general dichotomic measurements.In the activation example, isotropic states are local for dichotomic measurements when p < 0.66, while the network is nonlocal for p > 0.64.Source: D. Any two-outcome measurements can be… · p. 7
- The measurement simulation is relevant because the network’s nonlocality can be revealed using dichotomic measurements.The cited network threshold is p > 0.64, while single-copy locality for two-outcome measurements holds when p < 0.66.Source: D. Any two-outcome measurements can be… · p. 7