Stabilizer Rényi Entropy Without a Phase Reference

=== SUMMARY ===

Technical Summary: Stabilizer Rényi Entropy Without a Phase Reference

Problem Statement
Standard measures of quantum "magic" (non-stabilizerness), such as the Stabilizer Rényi Entropy (SRE), rely on the assumption that each qubit possesses a complete, synchronized Pauli frame. However, in various physical contexts—including helicity states of massless particles, unsynchronized qubit registers, energy eigenbases, and systems governed by superselection rules—the computational basis (e.g., $\sigma_z$ eigenstates) may be fixed, while the transverse Pauli axes remain unspecified up to local phase factors.

In such scenarios, the computational basis is defined only up to a phase for each qubit ($|+\rangle_i \to e^{i\theta_i}|+\rangle_i$). Standard SRE measures are generically not invariant under these relative rephasings. As demonstrated by the authors using a Bell state example, the second SRE ($M_2$) can vary continuously from 0 to $\ln(4/3)$ depending on the arbitrary choice of relative phases, rendering the measure ill-defined when the phase frame is unresolved.

Methodology
To address this, the authors define a phase-averaged Stabilizer Rényi Entropy ($\tilde{M}_q$). Instead of calculating the SRE for a specific phase convention and then averaging, they integrate the Pauli moments over the unresolved relative phases inside the logarithm.

For a pure state $|\psi\rangle$ of $k$ qubits, the novel invariant is defined as:
$$\tilde{M}_q(|\psi\rangle) = \frac{1}{1-q} \ln \left[ \prod_{i=1}^k \int_0^{2\pi} \frac{d\theta_i}{2\pi} \sum_P \frac{\langle \psi\{\theta_i\} | P | \psi\{\theta_i\} \rangle^{2q}}{2^k} \right]$$
where $|\psi\{\theta_i\}\rangle$ denotes the state with rephased basis vectors. The authors focus primarily on the second Rényi index ($q=2$).

Key Contributions and Properties
The paper establishes four primary properties of this new invariant:

1. Exact Invariance: $\tilde{M}_2$ is exactly invariant under all independent local $U(1)^k$ rephasings.
2. Vanishing Criterion: The invariant vanishes if and only if the state is a stabilizer state in every possible phase convention. This distinguishes it from locally minimized magic measures which optimize over the full local unitary group.
3. Closed-Form Reduction: For a broad class of states where the computational-basis components have pairwise-distinct flip patterns (i.e., for any two non-zero coefficients $c_i, c_j$, the bitwise XOR of their basis indices $s_i \oplus s_j$ is unique), the phase integration yields a closed-form expression dependent solely on the basis-state probabilities $|c_i|^2$:
   $$\tilde{M}_2 = -\ln \left[ 6 \left( \sum_{i=1}^n |c_i|^4 \right)^2 - 5 \sum_{i=1}^n |c_i|^8 \right]$$
   This eliminates the need to know the phases, relying only on populations measurable in a single basis.
4. Channel Ceilings: The authors derive the maximum possible value (ceiling) for $\tilde{M}_2$ for a system with $n$ non-zero probability components (channels).
   • For $n=2$, the global maximum is $\ln(7/5)$, occurring at unequal probabilities $|c_i|^2 = \frac{1}{2}(1 \pm \sqrt{3/7})$.
   • For $n \ge 3$, the global maximum occurs at equal probabilities ($|c_i|^2 = 1/n$) and is given by $\ln[n^3/(6n-5)]$.

Results: Application to Gluon Scattering
The authors apply this invariant to tree-level gluon scattering amplitudes, where helicity-basis phases are naturally unspecified. They analyze four- and five-point scattering processes:

• Four-Point Scattering ($2 \to 2$):

  • The invariant vanishes only at specific scattering angles ($\theta=0, \pi$).
  • At the "Bell point" ($\theta=\pi/2$), where the state becomes a Bell pair in a specific basis, the invariant lifts to $\ln(8/7)$, whereas the standard SRE in a fixed convention would vanish.
  • At specific angles ($\theta \approx 1.38$), the process saturates the two-channel ceiling of $\ln(7/5)$.

• Five-Point Scattering:

  • $3 \to 2$ Scattering: The invariant never vanishes in the studied configurations. When the three output probabilities coincide, it saturates the three-channel ceiling of $\ln(27/13) \approx 0.731$.
  • $2 \to 3$ Scattering (Same Helicity): The invariant is bounded between a "soft floor" and a "collinear ceiling."
    • Soft Limit: As one gluon becomes soft ($E_3 \to 0$), the invariant approaches a floor of $\ln(8/7)$. Contrary to expectations that soft radiation carries no magic, the residual invariant is carried entirely by the soft gluon.
    • Collinear Limit: The invariant approaches a ceiling of $\ln(184/125)$ in specific collinear limits.
    • The symmetric point ($x_3=x_4=x_5=2/3$) is a local minimum ($\approx 0.140$).

• Color Independence: A significant finding is that for all configurations studied, the color dependence of the amplitudes cancels exactly upon taking the absolute values required for the invariant. Consequently, the magic of the gluon final states is determined solely by kinematics, unlike the standard SRE which retains color-dependent phases.

Significance
The paper argues that a description of quantum magic is incomplete without specifying the reference frame. The proposed phase-invariant SRE provides a robust diagnostic for systems where the computational basis is fixed but transverse axes are undefined. It demonstrates that for a wide class of sparse supports, magic can be quantified purely by population statistics.

In the context of high-energy physics, the work connects bounds on quantum states to physical interaction constraints. It reveals that while some scattering processes (like $3 \to 2$) can attain their theoretical channel ceilings, others (like same-helicity $2 \to 3$) are kinematically restricted from reaching them. Furthermore, the exact cancellation of color factors suggests that the "magic" in gluon scattering is a kinematic property, independent of the specific color configuration. The authors suggest this framework could eventually lead to measurement protocols for magic in quantum registers using time-averaged Pauli moments, bridging the gap between theoretical diagnostics and experimental data where phase information is limited.