Collider Spin-Density Reconstruction

• Collider spin-density reconstruction is a method that extracts complete quantum spin-state data from high-energy collisions using spin-density matrices.
• It employs advanced techniques like quantum tomography and automated helicity amplitude workflows to map observed particle distributions onto density matrices.
• This approach enables precise measurements of polarization, spin correlations, and entanglement, thereby testing new physics hypotheses in collider experiments.

Collider spin-density reconstruction refers to the systematic extraction and analysis of quantum spin-state information of particles produced in high-energy collider experiments, encoded as spin-density matrices. This formalism enables precise measurement of polarization, multipole moments, and quantum correlations for single particles and multipartite systems. Recent advances provide automated pipelines for density-matrix computation and quantum-tomographic analysis, facilitating studies of entanglement and probing new physics in collider data.

1. Theoretical Foundations: Spin-Density Matrices in Collider Physics

Spin-density matrices provide a complete quantum-mechanical description of the spin state of particles and systems at colliders. For a single spin-$j$ particle, the density operator $\rho$ encodes all statistical information accessible from repeated measurements. Concretely, for spin-½ (e.g., leptons, quarks), one works in the Pauli basis: $\rho = \frac{1}{2}[1 + \vec{P}\cdot\vec{\sigma}]$ where $\vec{P}$ is the polarization vector and $\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ are Pauli matrices.

For spin-3/2 particles, as detailed by Song & Chang (Song et al., 2019), the density matrix admits a Cartesian operator expansion using 4×4 matrices ($\Sigma_i$, $Z_{ij}$, $D_{ijk}$): $$\rho = \frac{1}{4}\big[\mathbf{1}_{4\times4} + \sum_{i=1}^3 S_i \Sigma_i + \sum_{i\leq j} S_{ij} Z_{ij} + \sum_{i\leq j\leq k} S_{ijk} D_{ijk}\big]$$ Subject to Hermiticity, trace normalization, and four linear constraints, $\rho$ encodes 15 independent physical multipoles ($t^k_q$) structurally equivalent to the spherical tensor decomposition: $\rho = \sum_{k=0}^3 \sum_{q=-k}^{k} t^k_q \, T^k_q$ where $T^k_q$ are the standard irreducible tensor operators for spin-3/2.

For two-particle systems (e.g., $t\bar t$, $\tau^+\tau^-$) the formalism extends via direct product bases ($\ket{h_A, h_B}$). The event-level production matrix is built from helicity amplitudes: $$R_{h_A h_B, h_A'h_B'} = \sum_\alpha \mathcal{M}_{h_A,h_B;\alpha}\mathcal{M}^{*}_{h_A',h_B';\alpha}$$ The normalized physical density matrix is $\rho = R/\text{Tr}\,R$, ensuring $\rho$ is Hermitian and positive semidefinite (Durupt et al., 20 Oct 2025, Li et al., 2015).

2. Methodologies for Spin-Density Reconstruction

Reconstruction relies on mapping observed final-state distributions to the density-matrix formalism:

Single Particle (Spin-½)

For leptons (e.g., $\tau$), the method proceeds:

• Measure the direction $\hat{p}_\tau$ by displaced vertex.
• Identify a spin-analyzer direction $\hat{h}$ from decay kinematics (triple product of momenta in 3-prong $\tau\to3\pi\nu$).
• Construct the polarimeter distribution via

$$\frac{d\Gamma}{d\Omega_{h}} \propto \text{Tr}[\rho(1+\alpha\,\vec{\sigma}\cdot\hat{h})]$$

with analyzing power $\alpha$ determined empirically.

Multi-Particle Systems

For $t\bar t$ or two $\tau$–leptons, construct joint weights: $$w(\hat{h}_1,\hat{h}_2) \approx 1 + \alpha_1 P_1^i \hat{h}_1^i + \alpha_2 P_2^j \hat{h}_2^j + \alpha_1\alpha_2 C^{ij} \hat{h}_1^i \hat{h}_2^j$$ where $C^{ij}=\text{Tr}[\rho\,(\sigma_i\otimes\sigma_j)]$ measures spin–correlations. Fit observed distributions to extract all polarization and correlation components.

Automated Helicity Amplitude Workflow

Modern event generators, e.g., MadGraph5_aMC@NLO, automate density-matrix extraction. For every process, helicity amplitudes are calculated, and event-level $R$ matrices are stored in LHE files, together with metadata specifying frame, quantization axis, and particle basis order. Python libraries (see analysis/) directly post-process LHE files to reconstruct normalized spin-density matrices per event and extract quantum-information observables (Durupt et al., 20 Oct 2025).

3. Quantum-Information Analysis: Tomography and Observables

Reconstructed density matrices enable extensive quantum-tomographic analysis:

• Polarization vectors: $\vec{p}_A=\frac{1}{2}\vec{B}^A$, $\vec{p}_B=\frac{1}{2}\vec{B}^B$
• Correlation tensor: $C_{ij} = \text{Tr}[(\sigma_i\otimes\sigma_j)\rho]$
• Concurrence (two-qubit entanglement):

$$C(\rho) = \max\left[0,\lambda_1-\lambda_2-\lambda_3-\lambda_4\right]$$

where $\lambda_i$ are eigenvalues of a specified transformed matrix.

• Entanglement of formation, purity ($\gamma=\text{Tr}[\rho^2]$, normalized as $\mu$), PPT test and negativity, $D$-coefficients, and magic ($\tilde{M}_2$ stabilizer measure).

Tabular summary:

Observable	Symbol/Formula	Physics Extracted
Polarization vector	$\vec{P}$	Net spin orientation
Correlation tensor	$C_{ij}$	Spin–correlation coefficients
Concurrence	$C(\rho)$	Entanglement of pair
Purity	$\mu$	Mixedness of the state
$D$-coefficients	$D^{(1)},D^{(x)}$	Pairwise entanglement test
Magic	$\tilde{M}_2(\rho)$	Quantum non-stabilizer resource

Significance: This systematic machinery allows one to probe deep quantum features of collider events, including entanglement and information-theoretic measures previously inaccessible.

4. Experimental Implementation: Event Generation and Data Analysis

MC Event Generation

Density-matrix reconstruction is now integrated into event generators. For the example $pp \to t\bar t$ (Durupt et al., 20 Oct 2025):

• Generate events with generate p p > t t~ in MG5_aMC@NLO.
• Configure spin-density storage (particle_in_density_matrix, boost_choice, helicity_direction).
• Event records include unnormalized $R$; post-processing normalizes and diagonalizes to physical $\rho$.

Post-Processing and Fitting

Python libraries (LHEReader, Analysis) parse events, compute $\rho$ per event, and calculate ensemble averages. Observable histograms are accumulated for comparison with theoretical predictions or used in multivariate likelihood fits to extract density-matrix parameters.

Error Treatment and Corrections

Statistical precision is governed by the standard error formula (cf. Eq. (16) in (Li et al., 2015)): $\delta A = \sqrt{\frac{1-A^2}{N}}$ Corrections for initial-state radiation, beamstrahlung, and detector resolution are implemented through convolution with radiative kernels and Gaussian smearing, or via bin-by-bin response matrices to recover unsmeared $\rho$.

For example, in tau-pair studies (Li et al., 2015), ISR and detector-smearing reduce parameter fidelities by $\lesssim20\%$, and $\delta P_\tau \simeq 0.25$ is achievable with $N \simeq 1100$ events.

5. Physical Interpretation and Phenomenological Applications

Spin-density matrices provide direct access to the underlying production and decay mechanisms at colliders. The patterns of polarization, spin–correlation, and entanglement distinguish hypotheses for new particles (scalar, fermion, vector), reveal chiral couplings, and constrain quantum properties.

• In $e^+e^- \to h X$ (hadron production), the angular dependence of differential cross sections is expressed in terms of the $F_X(\theta)$ structure functions multiplying spin-density components (Song et al., 2019).
• For $\tau$-pair final states, the extracted $\rho$ matrix enables discrimination between scalar and fermion pair production via coefficients $A_s$ and $C^{ij}$ (Li et al., 2015).
• In $pp \to t\bar t$, quantum correlations (e.g., $C(\theta,\beta)$) and entanglement observables extracted from analytic Fano coefficients and per-event matrices agree to better than 1% between analytic computations and generator output (Durupt et al., 20 Oct 2025).

A plausible implication is that routine density-matrix tomography opens the path to exploiting quantum-information theory in collider phenomenology, including precision measurements and new searches for physics beyond the Standard Model.

6. Extensions, Limitations, and Future Outlook

Automated reconstruction pipelines currently support bi- and multipartite systems of qubits ($j=½$) and qutrits, with full control over reference frames and quantization axes. The inclusion of higher-spin ($j=1,3/2$) particles is tractable via expansion in tensor operators, as in Song & Chang (Song et al., 2019), but requires a full accounting of additional constraints linking Cartesian and spherical multipoles.

The computational workflow relies on the availability of helicity amplitudes at tree-level; extension to NLO or NNLO (incl. radiative corrections) is possible in principle, provided matrix-element information is retained.

Potential limitations include finite detector resolution, which degrades statistical precision. Correction methods, such as smearing-response inversion, are robust for moderate binning and sample size. Multi-particle entanglement witnesses are under active development; current tools quantify bipartite and some multipartite entanglement via stabilizer and correlation measures.

This suggests that further progress in spin-density reconstruction will integrate higher-order theoretical calculations, improved detector unfolding algorithms, and advanced entanglement diagnostics, enabling comprehensive quantum characterization of collider events.