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Physical Review A

Effectiveness of local unitary operations in n -local networks

Kaushiki Mukherjee

Published Mar 19, 2024• 1 citation
Publisher

Abstract

Analyzing the nonclassicality of correlations generated in $n$-local quantum networks is an important branch of study in quantum information theory. Speaking of nonclassicality, violation of an $n$-local inequality ensures non $n$-locality of corresponding network correlations. Being correlation-based, violation of any such inequality is experimentally feasible. The present work explores whether applying suitable local unitary operations can aid violation of any $n$-local inequality. Observations have ensured the utility of local unitary operations in enhancing the detection efficiency of bilocal inequality and $n$-local inequality (for $n=3,4$) in the star configuration. In an $n$-local quantum network, such operations correspond to local basis change applied by at least one party over its joint state of qubits received from the sources. In an $n$-local network, with all the parties performing any possible local unitary operation, correlations not violating an $n$-local inequality (compatible with the network considered), are first characterized. Such characterization is then exploited to provide examples of local unitary operations aiding violation of existing $n$-local inequalities for $n=2$, 3, and 4, specifically. Interestingly, there exist two qubit unitary operations, for which nontrilocal correlations can be simulated even when one of the sources distribute pure product state in star network. Quantum gates associated with the useful unitary operations are provided.

DOI
10.1103/physreva.109.032216

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Analyzing the nonclassicality of correlations generated in nnn-local quantum networks is an important branch of study in quantum information theory. Speaking of nonclassicality, violation of an nnn-local inequality ensures non nnn-locality of corresponding network correlations. Being correlation-based, violation of any such inequality is experimentally feasible. The present work explores whether applying suitable local unitary operations can aid violation of any nnn-local inequality. Observations have ensured the utility of local unitary operations in enhancing the detection efficiency of bilocal inequality and nnn-local inequality (for n=3,4n=3,4n=3,4) in the star configuration. In an nnn-local quantum network, such operations correspond to local basis change applied by at least one party over its joint state of qubits received from the sources. In an nnn-local network, with all the parties performing any possible local unitary operation, correlations not violating an nnn-local inequality (compatible with the network considered), are first characterized. Such characterization is then exploited to provide examples of local unitary operations aiding violation of existing nnn-local inequalities for n=2n=2n=2, 3, and 4, specifically. Interestingly, there exist two qubit unitary operations, for which nontrilocal correlations can be simulated even when one of the sources distribute pure product state in star network. Quantum gates associated with the useful unitary operations are provided.
Correlations in quantum networks with independent sources exhibit a completely novel form of nonclassicality in the sense that the nonlocality of such correlations can be demonstrated in fixed local input scenarios. Before the pioneering work of Renou et al. [Phys. Rev. Lett. 123, 140401 (2019)], the nonlocal feature of such network correlations was directly attributable to standard Bell nonlocality. In the above-mentioned reference, the authors provided some of the first examples of triangle network correlations, whose nonlocality cannot be deduced from Bell-CHSH nonlocality. To date, a complete characterization of such scenarios has yet to be provided. The present work characterizes correlations arising due to fixed local measurements in a triangle network under a source independence assumption. Precisely speaking, a set of criteria is framed in the form of Bell-type inequalities, each of which is necessarily satisfied by trilocal correlations. Possible quantum violation of at least one criterion from the set is analyzed, which in turn points out the utility of the set of criteria to detect nonlocality (nontrilocality) in quantum triangle networks. Interestingly, measurement on a local product state basis turns out to be sufficient to generate nontrilocal correlations in some quantum networks. Noise tolerance of the detection criteria is discussed followed by a generalization of the framework for demonstrating correlations in any nnn-sided polygon where nnn is finite.
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Source-independent quantum networks are considered as a natural generalization to the Bell scenario where we investigate the nonlocal properties of quantum states distributed and measured in a network. Considering the simplest network of entanglement swapping, recently Gisin et al. [Phys. Rev. A 96, 020304(R) (2017)] and Andreoli [New. J. Phys. 19, 113020 (2017)] independently provided a systematic characterization of the set of quantum states leading to violation of the so-called bilocality inequality. In this work, we consider the complexities in quantum networks with an arbitrary number of parties distributed in chain-shaped and star-shaped networks. We derive the maximal violation of the ``nnn-local'' inequality that can be achieved by arbitrary two-qubit states for such chain- and star-shaped networks. This would further provide a deeper understanding of quantum correlations in complex structures.
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The problem of characterizing classical and quantum correlations in networks is considered. Contrary to the usual Bell scenario, where distant observers share a physical system emitted by one common source, a network features several independent sources, each distributing a physical system to a subset of observers. In the quantum setting, the observers can perform joint measurements on initially independent systems, which may lead to strong correlations across the whole network. In this work, we introduce a technique to systematically map a Bell inequality to a family of Bell-type inequalities bounding classical correlations on networks in a star-configuration. Also, we show that whenever a given Bell inequality can be violated by some entangled state ρ , then all the corresponding network inequalities can be violated by considering many copies of ρ distributed in the star network. The relevance of these ideas is illustrated by applying our method to a specific multi-setting Bell inequality. We derive the corresponding network inequalities, and study their quantum violations.
The launch of a satellite capable of distributing entanglement through long distances and the first loophole-free violation of Bell inequalities are milestones indicating a clear path for the establishment of quantum networks. However, nonlocality in networks with independent entanglement sources has only been experimentally verified in simple tripartite networks, via the violation of bilocality inequalities. Here, by using a scalable photonic platform, we implement star-shaped quantum networks consisting of up to five distant nodes and four independent entanglement sources. We exploit this platform to violate the chained n-locality inequality and thus witness, in a device-independent way, the emergence of nonlocal correlations among the nodes of the implemented networks. These results open new perspectives for quantum information processing applications in the relevant regime where the observed correlations are compatible with standard local hidden variable models but are non-classical if the independence of the sources is taken into account.