Qubit-qubit-qutrit quantum correlations in H to f f V

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Qubit-qubit-qutrit quantum correlations in H to f f V".

Mira: Here's the gist of the paper. The gist: The state produced in H → f f V decay carries rich quantum correlations, including genuine 2⊗2⊗3 entanglement and non-local magic, which are most pronounced in specific kinematic regions of the phase space.

Kai: Kinematics and Spin Quantum State The analysis focuses on the spin degrees of freedom of the final state spanning a Hilbert space H = C2 ⊗C2 ⊗C3, which is a 2⊗2⊗3 system whose quantum observables are studied in the following sections The process of primary interest is the three-body Higgs decay, h → τ −τ +Z, but in the analytic treatment we keep the masses and couplings general, specializing to (H, f, ¯f, V) = (h, τ −, τ +, Z) only when the observables are evaluated numerically The spin quantum state of the f¯fV system is given by ψ⟩ ∝ X hA,hB,hV MhAhBhV hA, hB, hV ⟩ Because M(2) and M(3) are proportional to the Yukawa coupling gHf f ∝ mf, the first diagram dominates in the limit mf ≪ mV, mH Exploiting this hierarchy, we expand the amplitudes in mf and retain terms up to linear order At mf = 0, the amplitude M(1) populates only the opposite-helicity subspace −+⟩, while M(2) and M(3) vanish The spin state takes the form ψ⟩ = 1/N ψ (0)⟩ + ε ψ (1)⟩ + · · · where N is the normalisation factor and the expansion parameter is ε ≡ √2mf (m2V − m2AB) mV (m2H − m2AB − m2V) For h → τ −τ +Z in the Standard Model, ε is small and remains nearly constant across the accessible phase space, ε ∈

3.14, 3.21: × 10−2 Entanglement Structure The final spin state of H → f¯fV carries quantum correlations at several structural levels, which we now dissect We first quantify the bipartite entanglement between a single particle and the remaining pair (one-to-other), then the entanglement between each pair of particles once the third is traced out (one-to-one), and finally the correlations that are irreducibly 2⊗2⊗3 (Bipartite Entanglement) The normalised I-concurrences quantifying one-to-other entanglement of the three possible partitions of A−B−V are given by CABV = q 2(1 − Tr ρ 2 A), CBAV = q 2(1 − Tr ρ 2 B), CV AB = q 3/2 (1 − Tr ρ 2 V) At zeroth order, the pointer structure of Eq.

Mira: (2.16) makes the reduced states immediate The two fermions are left in the diagonal states ρ (0) A = 1/⟨c 2⟩ c 2 R 0 0 c 2 L!, ρ (0) B = 1/⟨c 2⟩ c C L 0 0 c C R!, in the hA,B = +, − basis Evaluating the purities at this order yields the compact closed forms for the I-concurrences C (0) ABV = C (0) BAV = 2 cLcR /⟨c 2⟩ = ˆs, C(0) V AB = √3/2 sˆ r 1 − Q2 P2, with Q ≡ ⟨Ψ+Ψ−⟩ At this order C (0) ABV and C (0) BAV are fixed entirely by the chiral couplings, independently of the kinematics — a direct consequence of the GHZ-like structure of ψ (0)⟩ — and the smallness of ˆc 2

Eq. [Kai: (2.9)] renders them nearly maximal in the Standard Model, ≃ 0.988 (One-to-one Entanglement) The one-to-one entanglement is stored in the two-body reduced states obtained by tracing out the third party, ρAB = TrV ψ⟩⟨ψ, ρAV = TrB ψ⟩⟨ψ, ρBV = TrA ψ⟩⟨ψ, which describe a qubit–qubit pair (AB) and two qubit–qutrit pairs (AV, BV) For the 2 ⊗ 2 state ρAB, the convex roof can be evaluated exactly with Wootters’ formula

104: , CAB = max 0, λ1 − λ2 − λ3 − λ4, where λ1 ≥ λ2 ≥ λ3 ≥ λ4 are the square roots of the eigenvalues of ρAB ρ˜AB with ˜ρ = (σy ⊗ σy) ρ∗(σy ⊗ σy) (Genuine 2⊗2⊗3 Entanglement) The tripartite (f¯fV) state (2.14) is pure, therefore, to detect genuine 2⊗2⊗3 entanglement, we can use tools introduced by Miyake in his classification

81: The Miyake classification and its relation to the 223-tangle are briefly summarised in Appendix C Arranging the twelve amplitudes into the 3 × 4 matrix A with rows labelled by hV and columns by the fermion configurations (hAhB) = (++, +−, −+, −−), one forms the four 3×3 minors mi (i = 1,..., 4), obtained by deleting the i-th column, and the 2 × 2 × 3 hyperdeterminant M = m1m4 − m2m3 Together with the local ranks (rA, rB, rV) of the single-party reduced states, the minors and the hyperdeterminant organise the pure 2 ⊗ 2 ⊗ 3 states into the SLOCC classes of Miyake’s classification In particular, the state is genuinely 2⊗2⊗3 entangled (class C1 or C2) if and only if at least one minor is non-zero Bell Inequalities To test the Bell nonlocality of the qubit–qubit–qutrit state of H → f¯fV, we employ three Bell-type operators All three derive from the 4 × 4 × 2 inequality of Wu and Zong

100: , which was proven to be tight in Ref.

Paper summary: Kai: We've just touched on the general idea behind this paper, which is about quantifying the quantum correlations in H to f f V decays. To recap, we’re dealing with a three-body system where the final state spins form a qubit qubit qutrit system.

Mira: And the core claim is that this setup produces genuine two by two by three entanglement and non-local magic that is most pronounced in certain kinematic regions of the phase space. The paper focuses on how this structure arises from the spin quantum state ψ⟩ ∝ X hA,hB,hV MhAhBhV hA, hB, hV⟩.

Lev: That structure seems to be built on an expansion parameter ε ≡ √2mf (m2V − m2AB) mV (m2H − m2AB − m2V) <ref:2607.26122#pg1>. Mira, what does that smallness of this parameter tell us about the physics driving the initial spin state?

Mira: It tells us that in the limit where mf is much smaller than mV and mH, the first diagram dominates. At mf = zero only one part of the opposite helicity subspace is populated by M(one), while M(two) and M(three) vanish.

Kai: So this hierarchy means that for many of these correlations, they are dominated by one specific process at low coupling limits. But the paper says epsilon remains nearly constant across the accessible phase space in the Standard Model case, staying in a range like

three point one four, three point two one: times ten-two.

Lev: If epsilon is nearly constant, that suggests that this structure isn't just a fluke happening at one extreme of the phase space; it’s a feature of the underlying decay dynamics itself. But what about the entanglement measures they calculate?

Mira: They quantify bipartite entanglement between one particle and the rest, one to one entanglement after tracing out a third party, and finally this irreducible two by two by three entanglement. For example, for the bipartite measures, at zeroth order they are fixed entirely by chiral couplings and kinematic structure.

Kai: So those initial bipartite measures, like C(zero) ABV = ˆs (zero), are nearly maximal in the Standard Model because of that GHZ-like structure of the ψ (zero)⟩ state.

Lev: And what about the one by one entanglement? They use Wootters’ formula for a two by two reduced state rho AB to find CAB. Does that calculation need any specific assumptions about the initial state preparation?

Mira: It needs assumptions about the initial spin alignment, but they show how to evaluate it exactly with Wootters’ formula using eigenvalues of rho AB AB, where = (sigma y sigma y) rho* (sigma y sigma y).

Kai: The point is that these calculations give us concrete values for the entanglement measures, which are then used to check against Bell inequalities, testing the non-locality of this qubit qubit qutrit system.

Conclusion: Kai: So we've covered a lot about this paper "Qubit-qubit-qutrit quantum correlations in H to f f V". The thesis is that the final state of a three body decay carries genuine two by two by three entanglement and non-local magic, which they find strongest in specific kinematic regions.

Mira: And the authors are using tools like Miyake's classification to organize these pure states into SLOCC classes, showing that genuine two⊗two⊗three entanglement is present if at least one minor is non-zero <ref:2607.26122#pg2,genuine 2⊗2⊗3 entanglement>. They also show that this quantum correlation violates classical bounds across the entire phase space for the three observables.

Lev: From an engineering view, what does all of this mean for building real hardware? It seems like we’re looking at a very specific signature we need to aim to detect in experiments.

Kai: It means that if we build detectors sensitive enough, and if our experimental setup can probe these specific kinematic regions where the correlations are strongest, we should be able to see this quantum behavior manifest in the spin correlations of the final state particles.

Mira: The implication is that these three-body decays provide a window into richer quantum information than just simple two-body systems. It suggests that the universe has ways of generating these complex, non-local correlations naturally in particle physics.

Lev: So, in simple terms, it means the paper shows that when you look at certain particle decays, you can find a signature of genuine three body quantum entanglement that exceeds what classical physics predicts.

Kai: Exactly. It’s a specific prediction for experimentalists to aim for when they're designing their next measurement setup.