Quantum nonlocality, as an invaluable quantum resource, plays an indispensable role in processing numerous quantum information. Accurate characterization and effective detection of nonlocality have always been important and challenging topics in theoretical and experimental quantum information research. How to precisely identify and verify the phenomenon of quantum nonlocality in complex many-body quantum systems, and how to design more efficient detection methods for the nonlocality, have become urgent scientific issues that need to be addressed. This paper is dedicated to the detection of multipartite quantum nonlocality, with a focus on exploring how the Svetlichny inequality can be used to detect it. First, the maximum quantum violation of the Svetlichny inequality is discussed. Through construction, a quantum state p0 and a set of observables A0 are obtained, thereby achieving the maximum quantum violation of the Svetlichny inequality. It is also demonstrated how to construct other quantum states and sets of observables to achieve their maximum violation of the Svetlichny inequality, thereby clarifying that the quantum states and sets of observables that achieve the maximum quantum violation of the Svetlichny inequality are not unique. Second, in order to find more quantum states and sets of observables that violate the Svetlichny inequality, a corresponding Hamiltonian is constructed using the Svetlichny operator. This core issue of finding quantum states that violate the Svetlichny inequality is ingeniously transformed into solving the ground state of this Hamiltonian. Leveraging the powerful function approximation capability of neural networks, neural network quantum states are constructed. Two optimization algorithms, i.e. the Nelder-Mead simplex method and quantum variational Monte Carlo (VMC), are respectively adopted to optimize the network parameters in order to find the ground state energy and ground state of the Hamiltonian, thereby achieving a violation of the Svetlichny inequality and ultimately detecting nonlocal states. To ensure the efficiency and accuracy of the detection method, we conduct a comparative study of different optimization methods. By comparing the Nelder-Mead simplex method with the VMC method, we find that the VMC method is more suitable for nonlocality detection based on neural network quantum states in terms of efficiency and accuracy, providing reliable computational support for detecting many-body quantum nonlocality and the violation of the Svetlichny inequality. To verify the validity and universality of the proposed method, we detect the nonlocality of multipartite quantum pure states by using neural network quantum states and the VMC method under different Hamiltonians. The results indicate that this method successfully captures violations of the Svetlichny inequality in many-body quantum systems, thereby achieving effective detection of multipartite quantum nonlocality. This fully confirms the validity and universal potential of the VMC method in nonlocality detection based on neural network quantum states. This study not only verifies the theoretical and technical feasibility of detecting multipartite quantum nonlocality based on neural network quantum states and the VMC method, but also provides valuable new insights for detecting nonlocality. More importantly, it opens up a new research avenue for using neural networks to solve complex quantum many-body problems.

A linear network (LN), also called a chain network, consists of n+1 nodes (parties) labeled by A_1,… ,A_n+1 together with n sources labeled by S_1,… ,S_n . To explore the correlation of the LN, each party A_i performs m_i measurements labeled by x_i∈ [m_i] on its system and gets o_i outcomes denoted by a_i∈ [o_i] . The probabilities P(𝐚|𝐱)=P(a_1,… ,a_n+1|x_1,… ,x_n+1) form a tensor 𝐏= P(𝐚|𝐱) , called an n+1 -partite correlation tensor (CT) over the index set Δ _n+1. Particularly, when each party performs just one fixed measurement (also called without input setting), the resulted CT is said to be an n+1 -probability tensor (PT). In this work, we aim to characterize n-chain-locality of n+1 -CTs based on an LN. By introducing D-n-chain-locality (resp. C-n-chain-locality) of an n+1 -CT in light of the existence of a discrete (resp. continuous) n-chain-local hidden variable model, we prove that an n+1 -CT 𝐏 is D-n-chain-local if and only if it can be realized physically by n shared separable states and a set of local POVMs. Importantly, we also prove that C-n-chain-locality and D-n-chain-locality of an n+1 -CT are the same, so that we can call them n-chain-locality. The corresponding conclusions are obtained for n+1 -PTs, and the relationships between D-n-chain-local (resp. C-n-chain-local) CTs and D-n-chain-local (resp. C-n-chain-local) PTs are established. Lastly, we prove that the set consisting of all n-chain-local CTs over Δ _n+1 forms a nonconvex compact subset in the Hilbert space of all real tensors over Δ _n+1.

Recently, quantum-state texture has attracted increasing attention as a quantum resource with potential applications in quantum information processing. We derive tight lower and upper bounds relating the geometric texture of a superposition of n≥ 2 mutually orthogonal pure states to the textures of its constituent states and characterize the corresponding equality conditions. As an application, we determine the temperature-dependent attainable texture range of phase-generalized coherent Gibbs states in arbitrary dimensions. The coherent Gibbs state attains the lower bound at every positive temperature, whereas the upper bound is attained by appropriately choosing the relative phases of the energy-eigenbasis expansion coefficients of the phase-generalized coherent Gibbs state. For equally spaced energy levels, increasing the dimension lowers the scaled temperature at which the upper bound reaches unity, while the lower bound approaches its high-temperature limit of zero more slowly. These results provide a quantitative characterization of geometric texture under superposition and reveal how temperature, dimension and relative phases jointly determine the attainable texture range.

Quantum coherence is a fundamental resource in quantum information science and one of the most distinctive features of quantum mechanics. In this paper, we derive a tight upper bound for the coherence of the superposition composed of two orthogonal states using & ell;(1)-norm coherence measure, demonstrating that our upper bound is tighter than the one presented by Yue et al (2017 Sci. Rep. 7 4006). Furthermore, we extend the result to the superposition of n orthogonal states and establish the optimal upper bound for its coherence. Our results provide a more accurate estimate of the coherence of the superposition state.

Quantum steering is an important resource in quantum information processing, while Kirkwood–Dirac distribution plays a significant role in quantum information tasks. In this work, we discuss quantum steerability of a bipartite state shared by Alice and Bob using the Kirkwood–Dirac nonclassicality (KDNC) of Bob’s states after Alice performs a set of measurements. We use three l 1 -norm quantifiers of the KDNC of a qubit state with respect to three pairs of mutually unbiased bases to obtain a complementarity relation. Furthermore, for a d × 2 state and a measurement assemblage from Alice, we consider the total averaged KDNC of Bob’s states and then establish a steering inequality. A violation of the inequality exhibits the steerability of the state from Alice to Bob. An application to the Werner states shows the efficiency of our method.

Antidistinguishability, a key concept in quantum information, describes quantum states that can be excluded by null measurement outcomes, thus enabling indirect state certification through incompatible measurements. This work systematically explores the fundamental properties of quantum state antidistinguishability. First, we present several essential characteristics of antidistinguishable quantum states. Leveraging the Bloch representation, we establish a rigorous correspondence between quantum states and positive-operator-valued measures (POVMs). Through this approach, we prove that in qubit systems, real quantum states can be antidistinguished only by real POVMs, while complex states necessitate complex POVMs. However, the scenario becomes significantly more intricate in high-dimensional quantum systems. Finally, we derive the general forms for a three-state antidistinguishable set through real and complex POVMs, respectively, achieving a complete solution to the antidistinguishability problem within qubit systems.

Imaginarity plays an essential role in many quantum information processing tasks. Usually, real operations do not increase imaginarity measures. In present work, we aim to discuss freezing of imaginarity measures with real operations. It is proved that a real operation Phi freezes all imaginarity measures of a state rho if and only if it freezes the relative entropy measure of imaginarity of the state rho. In the qubit case, real operations that freeze imaginarity measures based on trace distance and relative entropy are characterized, respectively. Moreover, the freezing of imaginarity measure based on trace distance is not necessarily dilated into multipartite systems, while that based on the Tsallis relative entropy can be dilated into multipartite systems. Finally, we characterize real local operations that freeze the imaginarity measure based on trace distance of a special type of X-states in a multi-qubit system.

Maximally imaginary states are considered as those states that can be converted to any state of the system by real operations. Maximal imaginary-value states with respect to an imaginarity measure are defined as the states that attain the maximum under the imaginarity measure. In this work, we comprehensively characterize the forms of maximally imaginary states in quantum systems of arbitrary dimension by rigorously demonstrating that a quantum state is a maximally imaginary state if and only if it is a maximally imaginary-value state with respect to the relative entropy measure. Additionally, we prove the existence of maximally imaginary states within the set of quantum states sharing a fixed spectrum in a qubit system and derive their explicit forms.

Quantum imaginarity plays a crucial role in quantum information theory and represents one of the most distinctive features of quantum mechanics. In this paper, we derive a lower bound for the imaginarity of the superposition composed of two orthogonal states based on the geometric imaginarity measure. We further generalize the result to the superposition of n orthogonal states and establish a lower bound. These findings provide a more precise estimation of the imaginarity of superposition states.

Recently, a novel quantum resource termed "texture", describing the basis-dependent inhomogeneity of a quantum state, was first proposed by Parisio (2024) [22]. Quantum-state texture has been confirmed as an important physical resource in quantum information science. In this paper, we investigate the quantification of quantum-state texture and establish uncertainty relations for several candidate measures. First, we introduce two measures based on the weight and the Tsallis relative entropy. For the former, we provide an analytical lower bound, while for the latter we derive a closed-form expression. Next, we provide an analytical formula for the geometric measure, thereby enabling its efficient evaluation for arbitrary quantum states. Finally, we derive uncertainty relations for several measures of any bipartite state, where the measures based on the fidelity and geometry satisfy subadditivity, whereas ones based on the trace distance, Bures, Tsallis relative entropy and weight are subadditive only for product states.

The Kirkwood-Dirac (KD) distribution is a vital framework in quantum state characterization, which reveals nonclassical correlations through phase-space representations. In this work, we introduce trace-norm-based measures to assess the KD-nonclassicality of quantum states and derive the corresponding trade-off relations for qubit and qutrit systems. For a bipartite state shared by Alice and Bob and a set of measurements applied by Alice, the maximum value of the totally averaged quantum resource of Bob’s states is introduced with respect to a quantum resource quantifier. When the maximum value exceeds the upper bound in a trade-off relation, the bipartite state is said to exhibit nonlocal advantages of quantum resource (NAQR). We prove that a state exhibiting NAQR, such as nonlocal advantages of KD-nonclassicality (NAKDNC), is steerable from Alice to Bob. We demonstrate that NAKDNC of Werner states exhibit much more quantum steering than quantum coherence and quantum imaginarity do and also explore NAKDNC of the two-qutrit isotropic states. These findings emerge KD-nonclassicality as an independent nonclassical resource with operational relevance in quantum information protocols.

Quantum steering is an important resource in quantum information processing, while Kirkwood-Dirac distribution plays a significant role in quantum information tasks. In this work, we discuss quantum steerability of a bipartite state shared by Alice and Bob using the Kirkwood-Dirac nonclassicality (KDNC) of Bob's states after Alice performs a set of measurements. We use three l1-norm quantifiers of the KDNC of a qubit state with respect to three pairs of mutually unbiased bases to obtain a complementarity relation. Furthermore, for a d x 2 state and a measurement assemblage from Alice, we consider the total averaged KDNC of Bob's states and then establish a steering inequality. A violation of the inequality exhibits the steerability of the state from Alice to Bob. An application to the Werner states shows the efficiency of our method.

As an extension of classical probability distribution, the Kirkwood-Dirac distribution (KDD) was discussed by Kirkwood in 1933 and Dirac 1945, independently. Recently, it has been proved that nonclassical values (negative and non-real values) of the KDD have the ability of outperforming their classical counterparts in quantum computation, quantum measurement and so on. In this work, by dividing quantum states into KD-real (KD-free) and KD-imaginary (KD-resource) ones based on the KDD of a state, we establish a resource theory for KD-imaginarity with respect to a pair of bases $(A,B)$, {\color{blue}called the resource theory of Kirkwood-Dirac imaginarity. This theory} is different from the resource theory of imaginarity of quantum states with respect to one basis $A$, where the free states are those that have real density matrices under the basis $A$.

Quantum nonlocality represents correlations between subsystems of a composite quantum system, usually including Bell nonlocality, steerability, and entanglement. According to the hypothesis of quantum mechanics, states of a quantum system Q described by a d-dimensional Hilbert space 1-(Q are denoted by density operators acting on 1-(Q. Under a basis e for the Hilbert space 1-(A (R) 1-(B, every abstract density operator rho of the system AB corresponds to a density matrix rho e , which is a state of the d A d B-dimensional complex Hilbert space C d A (R) C d B . In this work, we discuss the consistency of quantum nonlocality of density operators rho and their corresponding density matrices rho e under the chosen basis e . It is proved that only when a basis e is a product one, a density operator rho is entangled (respectively, Bell nonlo cal, steerable) if and only if its density matrix rho e is entangled (respectively, Bell nonlo cal, steerable).

Bell's inequalities are linear and apply for cases of two entangled bodies. In this work, we consider the case of entanglement among three bodies as previously discussed in [Renou, et al Phys. Rev. Lett., 123, 140 401 (2019)] and based on triangle network. By discussing the question whether a sparse probability tensor (SPT) can be represented by a discrete trilocal hidden variable model (D-triLHVM), we show that every SPT having a D-triLHVM satisfies a set of concrete equalities and a nonlinear inequality, which can be used to detect whether a D-triLHVM can describe the network completely. As an application, we re-explore the D-nontrilocality of the correlations studied by Renou et al and that of the triangle network with shared entangled pure states. We also leave open questions about the closednees of the set of all D-trilocal probability tensors and the description with a continuous trilocal hidden variable model.

The interest in quantum network correlations has surged, as Bell inequalities originating from a single source fall short of capturing the intricate many-body correlations within generic quantum networks. The introduction of a 3-grade m-star quantum network, a natural extension of star-shaped networks, features a hierarchy of nodes including A, m stars B (1), & mldr;, B (m) centered around A, and m stars C-1(j),& mldr;,C(m)(j )centered around each B (j ) , and m + m (2 )independent sources. The strong locality of this network has been explored, assuming that each party possesses only two observables. In this paper, we put forward a significant extension to the strong locality of 3-grade m-star quantum networks. Specifically, each C (j) (k) performs n dichotomic measurements, while A and each B j conduct 2 (n-1) dichotomic measurements. We establish a generalized strong locality inequality that holds for any value of n, and subsequently conduct a thorough analysis to determine the optimal quantum violation of established inequality. Through specific examples, we confirm the feasibility of achieving the optimal quantum violation of the generalized strong locality inequality for any n. We notice that for n > 3, a single copy of a two-qubit entangled state may not be sufficient to reveal the non-strong locality, whereas utilizing multiple copies can trigger this property.

Recently, a type of deterministic all-versus-nothing proofs of Bell nonlocality, induced from qubit non-stabilizer states, was proposed, diverging from the tradition where such proofs are typically derived from stabilizer states. Moreover, through the application of a specific map, one can derive certain trivial (dimensionally reducible) versions of such proofs for qudits (d d being even). Nevertheless, it remains unknown whether high-dimensional non-stabilizer states can induce nontrivial deterministic all-versus-nothing proofs of Bell nonlocality. In this study, we provide an example induced from a specific four-qudit non-stabilizer state (with d = 4), demonstrating the feasibility of constructing such proofs in high-dimensional scenarios.

Bell nonlocality is a special quantum nonlocality and has emerged as an important resource in quantum information processing tasks. Typically, there are two types of strategies for testing the Bell nonlocality: the Bell inequality method and the "all-versus-nothing proof", the later includes methods such as the GHZ argument and Hardy paradox. The objective of this work is to provide new methods for detecting Bell nonlocality by understanding Hardy-like paradoxes (HLPs) and Hardy-Bell inequalities (HBIs) as well as Hardy inequality. First, the weak HLP (WHLP) is established, and an HBI is obtained for tripartite correlation tensors (CTs) with two inputs and two outcomes. Based on an existing Hardy inequality for an n-qubit pure state, a Hardy inequality is proven for n-partite Bell local CTs. Second, the WHLP and HBI are proven for tripartite states in terms of conditional probabilities. Our WHLP is theoretically and practically easier to construct and more efficient in checking Bell nonlocality compared to the usual HLP.

Usually, the verification of Bell nonlocality involves two main approaches: violation of specific inequalities and utilization of no-inequality methods. In this paper, we continue to develop the inequality methods by deducing the so-called ‘Hardy-Bell inequalities (HBIs)’ and ‘fault-tolerant Hardy paradoxes (FTHPs)’ for correlation tensors (CTs) with two inputs and general outcomes. We prove that the HBIs are necessary conditions for a CT to be Bell local and one of the FTHPs is sufficient condition for a CT to be Bell nonlocal. We demonstrate the effectiveness of HBIs in determining the nonlocality of CTs or quantum states when the classical Hardy paradox does not appear or a Bell inequality is not violated. Consequently, our methods can be utilized to explore more correlations having Bell nonlocality. Based on the obtained results, we find a neighborhood of a Hardy nonlocal state, in which all states are all Bell nonlocal.