We relate Berry-phase control and retained entropy geometry to recoverable coherence, entanglement, and Clauser-Horne-Shimony-Holt (CHSH) correlations. For a depolarized Bell pair subject to recorded phase noise, we determine exact CHSH optima when Bob retains his record and when he compresses it with his qubit into one qubit by any deterministic quantum channel. Some records preserve a violation that every such compression loses. The retained transverse metric gives sharp resource intervals. Nevertheless, for every finite derivative order, two finite readouts of the same noise can match all pulled-back monotone-metric and relative-entropy derivatives at faithful diagonal inputs while giving opposite recovered CHSH outcomes. Complete transverse entropy response determines the reliability law. A finite transitionless Berry loop realizes the channel; zero-expectation conditional control errors give a quadratic reduced-channel accuracy certificate. Transport, thermal, and accelerated-detector applications specialize established channel criteria under explicit preparation and accuracy assumptions. These model-dependent certificates supply neither a universal entanglement lifetime nor a detector-independent acceleration bound.
We relate Berry-phase control and retained entropy geometry to recoverable coherence, entanglement, and Clauser-Horne-Shimony-Holt (CHSH) correlations. For a depolarized Bell pair subject to recorded phase noise, we determine exact CHSH optima when Bob retains his record and when he compresses it with his qubit into one qubit by any deterministic quantum channel. Some records preserve a violation that every such compression loses. The retained transverse metric gives sharp resource intervals. Nevertheless, for every finite derivative order, two finite readouts of the same noise can match all pulled-back monotone-metric and relative-entropy derivatives at faithful diagonal inputs while giving opposite recovered CHSH outcomes. Complete transverse entropy response determines the reliability law. A finite transitionless Berry loop realizes the channel; zero-expectation conditional control errors give a quadratic reduced-channel accuracy certificate. Transport, thermal, and accelerated-detector applications specialize established channel criteria under explicit preparation and accuracy assumptions. These model-dependent certificates supply neither a universal entanglement lifetime nor a detector-independent acceleration bound.
Reliable verification of entanglement is a central requirement for quantum networks. This paper presents a practical verification approach based on violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality. We derive tight mathematical bounds that relate the CHSH value to entanglement fidelity and introduce a statistical framework that optimizes resource usage while ensuring reliable certification. Our main contributions are: (i) fidelity bounds derived directly from the CHSH measure, which also enable nonlocality certification at sufficiently high fidelities; (ii) a sample-complexity analysis that quantifies the number of measurements required to achieve desired confidence levels for the CHSH measure and the entanglement fidelity; and (iii) verification protocols, some with rigorous mathematical guarantees and others with numerical evaluation. Using NetSquid, we develop a simulation framework that models diverse network conditions and enables systematic exploration of trade-offs in CHSH-based verification. This framework highlights the interplay between accuracy, efficiency, and operational parameters, providing concrete guidelines for deploying entanglement verification in resource-constrained quantum networks.
The key feature in correlations established by multi-party quantum entangled states is nonlocality. A quantity to measure the average nonlocality, distinguishing it from shared randomness and in a direct relation with no-signaling stochastic processes (which provide an operational interpretation of quantum correlations, without involving information transmission between the parties as to sustain causality), is proposed and resolved exhaustively for the quantum correlations established by a Clauser-Horne-Shimony-Holt setup (or CHSH box). The amount of nonlocality that is available in a CHSH box is measured by its proximity to the nearest Popescu-Rohrlich set of causal stochastic processes (aka a PR box) in the no-signaling polytope, related by polyhedral duality to Bell's correlation function. The proposed amount of average nonlocality is an entanglement monotone with a simple relation to concurrence. We provide the optimal setup vectors of a maximally nonlocal CHSH box for any entangled pair. The strongest nonlocality is the fraction [Formula: see text] of a PR box, attained by maximally entangled qubit pairs. The most economical causal stochastic process reproducing any maximally nonlocal CHSH box is developed. Data produced by a computer implementation of the simulator agrees with the quantum mechanical formulas.
Abstract The quantum bit error rate (QBER) is a central operational parameter in quantum communication, quantifying both channel noise and the potential presence of an eavesdropper. Here we establish that, for any two-qubit state, the QBER measured across two complementary bases serves as an operational witness of nonlocality and negative conditional entropy. By invoking the Horodecki criterion for violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality, we prove that an average QBER below approximately suffices to certify nonlocality. Moreover, applying entropic uncertainty relations, we show that QBER values below guarantee negative conditional entropy. We illustrate the framework with Werner and Gisin states and demonstrate its operational significance in the BBM92 QKD protocol, where parameter estimation simultaneously certifies nonlocality and negative conditional entropy. Our results therefore establish a direct connection between practical QKD parameter estimation and the certification of Bell nonlocality and negative conditional entropy.
We in this Letter derive analytic formulas of Bell correlations in terms of quantum probability statistics under the assumption of measuring outcome-independence. For a spin-1/2 singlet state we find analytically that the violations of Bell-type inequalities are really related to the quantum non-local correlations. However, the Bell and Clauser-Horne-Shimony-Holt (CHSH) inequalities are always satisfied for the spin-1 singlet states. More generally the quantum non-locality does not lead to the violation of Bell and CHSH inequalities for the integer-spin singlet since the non-local interference effects cancel each other by the quantum statistical-average. Such a cancellation no longer exists for the half-integer spin singlets due to the nontrivial Berry phase, and thus the relevant Bell-type inequalities can be violated. Specifically, our generic observations can be experimentally tested with the entangled photon-pairs. Our arguments are based on the spin-singlet states, but could be generalized to other bipartite quantum states.
For finite-dimensional binary systems and their bipartite composites, this paper establishes a conditional representation chain from a physical–observation dual-axis structure to quantum Bell correlations. An observational fact is jointly generated by a complete physical state and a complete observation state. Additivity and associativity of coherent responses, two-sided cancellation, and a full circle group of scalar phase transport select C among finite-dimensional unital associative real response algebras. Reversible mixing then determines a Hermitian metric up to an overall positive factor. Normalized two-channel responses form S3, and quotienting by their common phase yields the rank-one observation-event space and, with an outcome label, the pure observation fiber S3/U(1)=CP1≃S2. The universal bilinear composite of two independent systems is C4. Probabilities are first assigned within complete measurement contexts. Events with the same response filter are operationally interchangeable, so the local assignments glue along the observation axis into a positive orthogonally additive measure on the full projection lattice. Gleason's theorem then gives the unique trace representation p(P∣x)=Tr(ρxP), while composite marginal consistency returns the trace pairing to each local binary fiber. Every two-qubit correlation therefore has the universal form Eρ(a,b)=aTTρb. Joint rotational invariance and perfect equal-axis anticorrelation uniquely fix Tρ=−I3, producing the singlet cosine law E=−a⋅b=−cosθ. Finally, the local commutators in the squared CHSH operator confine all realizable correlations to 22. Fine-measurement completeness supplies saturating directions, while sharp preparation and reversible covariance make the singlet accessible. Thus CP1, the Born pairing, the cosine correlation law, and the Tsirelson bound arise within one chain linking phase structure, independent composition, and cross-context probability consistency.
Environmental records can be individually accurate while sharing the same errors. This preprint derives a sharp relation between retained Bell coherence, local record accuracy and conditional dependence among designated environmental readouts. For a balanced two-qubit state in the subspace spanned by |00⟩ and |11⟩, the full extremum reduces to a convex problem over R + 1 Hamming-weight probabilities and has an attaining quantum realization with an unobserved remainder. Its inverse gives the least measured conditional total correlation compatible with a CHSH observation and a common upper bound on both conditional errors of every record. The two-record inverse is explicit. At fixed accuracy, a fixed positive Bell excess requires dependence growing at least linearly with record count. Independent monitoring reaches the zero-dependence boundary in a Lindblad dephasing limit. A shared-memory Lindblad process improves record accuracy at constant Bell value and yields an exact onset of detectable dependence. The record contains a standalone PDF preprint and a supporting archive with the PDF, TeX and Markdown manuscript sources, computational supplement, reproducible Python source, calculated data, figures and build instructions. All results concern the stated theoretical models; no experimental dataset is used. The preprint has not undergone journal peer review. AI assistance is disclosed in the manuscript.