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arXiv:2604.17756 · Nuclear Physics A

Polarization, Maximal Concurrence, and Pure States in High-Energy Collisions

Yu-Xuan Liu, Wei Qi, Luo-Ting He, Bo-Wen Xiao

hep-phhep-exnucl-thquant-phSubmitted Apr 20, 2026• Announced Apr 21, 2026• 0 citations• Revised Sep 25, 2026 (v4)
v1 · Apr 20, 2026v2 · May 22, 2026v3 · Aug 20, 2026v4 · Sep 25, 2026
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Abstract

We establish a quantitative relation between local spin polarization and quantum entanglement in two-qubit systems by mathematically proving a closed-form upper bound on the concurrence at fixed local polarization magnitudes. The bound shows how the maximal concurrence is jointly constrained by the two local polarizations. We further demonstrate that this bound is saturated by pure states in certain cases with identical polarizations. As a concrete physical application, we consider the parity-violating process $e^+e^- \to Z^0 \to q\bar{q}$, which generates final-state spin polarization. We show that the maximal concurrence is attained in specific kinematic regions and is significantly reduced relative to the unpolarized case. These results establish a general, process-independent framework connecting local polarization, maximal entanglement, and pure states.

Comments
21 pages, 4 figures; V4: Additional numerical results and more details are provided

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We establish a quantitative relation between local spin polarization and quantum entanglement in two-qubit systems by mathematically proving a closed-form upper bound on the concurrence at fixed local polarization magnitudes. The bound shows how the maximal concurrence is jointly constrained by the two local polarizations. We further demonstrate that this bound is saturated by pure states in certain cases with identical polarizations. As a concrete physical application, we consider the parity-violating process e+e−→Z0→qqˉe^+e^- \to Z^0 \to q\bar{q}e+e−→Z0→qqˉ​, which generates final-state spin polarization. We show that the maximal concurrence is attained in specific kinematic regions and is significantly reduced relative to the unpolarized case. These results establish a general, process-independent framework connecting local polarization, maximal entanglement, and pure states.
Abstract For the maximal violation of all Bell inequalities by an arbitrary pure two-qudit state of any dimension, we derive a new lower bound expressed via the concurrence of this pure state. This new lower bound and the upper bound on the maximal Bell violation, found in Loubenets and Namkung (2022 J. Phys. A: Math. Theor. 55 285301) and also expressed via the concurrence, analytically quantify Bell nonlocality of a pure two-qudit state via its entanglement, in particular, prove explicitly that entanglement of a pure two-qudit state is necessary and sufficient for its Bell nonlocality. By re-visiting the pure two-qubit case, we also find and rigorously prove the new results on correlation properties of an arbitrary pure two-qubit state.
In the present paper, we examine the quantum entanglement for more general states of two-qubit system in the context of spin coherent states (SCSs). We consider the concurrence as a quantifier of entanglement and express it in terms of SCSs. We determine new set of maximally entangled conditions that provide the maximal amount of entanglement for certain values of the amplitudes of SCSs for the case of pure states. Finally, we examine the entanglement of a class of mixed states of the two qubits and provide the range in which the entanglement value is maximal with respect to the values of the amplitudes of SCSs.
In the case of bipartite two-qubit systems, we derive an analytical expression of bound Bell operator for any given pure state. Our result not only manifests some properties of Bell inequality, for example, which may be violated by any pure entangled state and only be maximally violated for a maximally entangled state, but also gives the explicit values of maximal violation for any pure state. Finally we point out that any mixed states which can produce maximal violation of Bell inequality must have a maximal concurrence value.
We analyze how a maximally entangled state of two qubits (e.g., the singlet ${\ensuremath{\psi}}_{s}$) is affected by the action of local channels described by completely positive maps E\mathcal{E}E. We analyze the concurrence and the purity of states ${\ensuremath{\varrho}}_{\mathcal{E}}=\mathcal{E}\ensuremath{\bigotimes}\mathcal{I}[{\ensuremath{\psi}}_{s}]$. Using the concurrence-versus-purity phase diagram we characterize local channels E\mathcal{E}E by their action on the singlet state ${\ensuremath{\psi}}_{s}$. We specify a region of the concurrence-versus-purity diagram that is achievable from the singlet state via the action of unital channels. We show that even the most general (including nonunital) local channels acting just on a single qubit of the original singlet state cannot generate the maximally entangled mixed states. We study in detail various time evolutions of the original singlet state induced by local Markovian semigroups. We show that the decoherence process is represented in the concurrence-versus-purity diagram by a line that forms the lower bound of the achievable region for unital maps. On the other hand, the depolarization process is represented by a line that forms the upper bound of the region of maps induced by unital maps.
There has recently been much discussion regarding entanglement transformations in terms of local filtering operations and whether the optimal entanglement for an arbitrary two-qubit state could be realized. We introduce an experimentally realizable scheme for manipulating the entanglement of an arbitrary state of two polarization-entangled qubits. This scheme is then used to provide some perspective to the mathematical concepts inherent in this field with respect to a laboratory environment. Specifically, we look at how to extract enhanced entanglement from systems with a fixed rank, and, in the case where the rank of the density operator for the state can be reduced, show how the state can be made arbitrarily close to a maximally entangled pure state. In this context we also discuss bounds on entanglement in mixed states.
Spin correlations offer a quantum-information perspective on the partonic final states produced in high-energy scattering. We discuss the spin-density matrix of a heavy quark-antiquark pair in two complementary small-xxx processes. In coherent diffractive production, color-singlet exchange enforces an unusually strong relation between entanglement and Bell nonlocality: a longitudinal photon creates a maximally entangled pair, whereas for a transverse photon the pair is generically both entangled and Bell nonlocal, with a model-independent point of maximal entanglement. In inclusive back-to-back production, the density matrix factorizes into a hard spin tensor and the unpolarized and linearly polarized Weizsäcker--Williams gluon distributions. The latter generates an azimuthal modulation and can increase the concurrence when the dijet relative momentum and imbalance are approximately orthogonal. Strikingly, in the saturation model considered, nonlinear effects wash out this modulation for q⊥≲3Qsq_\perp\lesssim 3Q_sq⊥​≲3Qs​, whereas the dilute BFKL limit, in which G2/G0→1G_2/G_0\to1G2​/G0​→1, yields a maximal modulation independent of the imbalance magnitude. These results connect quantum-information observables with the Pomeron and saturation physics.