Do Non-Local Quantum Resources in Scattering Depend Only on Distance From Entangling Gates?

A single distance sets the leading resource

The paper asks whether distinct non-local quantum resources produced in elastic scattering are independent diagnostics of quantum complexity. Its answer is that near S-matrices with zero entanglement power they are not: a wide class of non-local resource functions, including the entanglement power and non-local magic power, are equivalent up to overall normalization.

The leading dependence is not on the detailed interaction but on geometry. The resource scales with the squared projective Hilbert-Schmidt distance from the nearest non-entangling gate, so all the geometry is shared among resources and the resource dependence collapses into one number, the proportionality constant the authors call chi_R.

This is a stronger statement than expanding a resource around zero. A generic resource could grow at different rates along different directions; here the non-local resource Hessian matrix is proportional to the Hilbert-Schmidt metric and therefore has no directional dependence.

Closed form entanglement power for any spin J

Earlier work derived the entanglement power in terms of scattering phase shifts for rotationally invariant S-matrices with J = 1/2, 1, and 3/2. This paper derives a closed form solution valid for all J and all phase shifts, specializing the entanglement power of an arbitrary unitary to a rotationally invariant S-matrix.

The S-matrix is written as a sum over total-spin projectors, S(delta) = sum over mu of e^{2i delta_mu} P_mu, where P_mu projects onto the spin-mu sector with rank r_mu = 2mu + 1 and the real, dimensionless phase shift delta_mu is defined modulo pi. All dynamical information sits in the phase shifts; the projectors are fixed by representation theory.

The entanglement power is the average linear entropy of the outgoing state over independently sampled pure initial spin states, drawn from the Haar measure on CP^{d-1} with local qudit dimension d = 2J + 1. Using the general unitary formula, the authors express it through singular values of the realigned matrices R(S) and R(S SWAP), which for a rotationally invariant S-matrix reduce to closed-form sums over Wigner 6j symbols.

Identity and SWAP are the only zero-entanglement gates

For the rotationally invariant spin S-matrices considered here, the non-entangling class is restricted to identity and SWAP, up to an overall phase. These are the only two S-matrices with zero entanglement power, which makes them the natural reference points for a local expansion.

Perturbing around S_0 = I or S_0 = SWAP as S(tau) = S_0 e^{i tau K}, with K = sum over mu of epsilon_mu P_mu, the entanglement power to lowest order is Ep(S) = 2 tau^2 / (d+1)^2 times Tr K-tilde^2, plus terms of order tau^4. The phase-shift dependence enters only through Tr K-tilde^2.

The same region gives an exact expression for non-local magic. Writing the largest Schmidt weight of the outgoing state as lambda_0 = 1 - eta, the non-local magic is M_NL = 4 eta plus higher-order terms, while the linear entropy is 2 eta. To leading order the non-local magic is therefore twice the linear entropy, and the Haar-averaged non-local magic power equals twice the entanglement power.

Magic and entanglement differ only by normalization

Entanglement and magic are distinct measures of quantum complexity, and magic is particularly relevant to the complexity of fault-tolerant quantum computation. The paper shows that near identity and SWAP this distinction does not produce independent leading behavior: the two powers are simply related by a factor of two.

The non-local linear magic is defined by minimizing the linear stabilizer magic over arbitrary local unitaries acting separately on the two subsystems, then averaging over the ensemble of initial product states. Because local unitaries preserve the largest Schmidt weight 1 - eta, minimizing magic at leading order amounts to maximizing overlap with a product stabilizer, which is achieved by the largest Schmidt vector.

The authors note that the same leading result holds for the product stabilizer ensemble used in earlier work when d is prime, since for prime d the single-qudit stabilizer states form a 2-design and the linear entropy averaged over independent product stabilizers equals the Haar average.

A Hooke's law analogy for resource generation

The general result is stated as a theorem. For a non-negative resource function that is twice differentiable near the identity where it vanishes, and invariant under SWAP, global phases, and local unitaries, the resource of a rotationally invariant S-matrix near identity or SWAP takes the form R(S) = (1/2) chi_R ell^2 plus terms of order ell^2, where ell is the projective Hilbert-Schmidt distance from S_0.

The proof uses Schur's lemma twice. Rotational invariance forces the partial traces of the traceless generator to vanish, which restricts the relevant operator space to the irreducible adjoint tensor adjoint representation of U(d) x U(d). The Hessian then commutes with the local group action and, by Schur's lemma, must be proportional to the identity, giving B(X, Y) = chi_R Tr(XY).

The authors describe this as analogous to Hooke's law, with ell playing the role of displacement, chi_R a kind of stiffness, and R the stored energy. All geometry resides in ell and is shared among resources by construction, so the entire resource dependence sits in the single number chi_R.

Projective distance, not raw phase differences

The relevant distance is projective. Two S-matrices have equivalent action on Hilbert space if they differ only by an overall phase, and both the entanglement power and non-local magic are invariant under such phase changes. The Hilbert-Schmidt metric is therefore modified so that the overall phase becomes a null direction, pulling the metric back to the space of relative phases.

For J = 1 the space of S-matrices is parameterized by three phase shifts, each with period pi, forming a cube with opposite faces identified. Removing the common-phase direction projects onto the plane delta_0 + delta_1 + delta_2 = 0, and the Voronoi unit cell of this projective geometry is a hexagon with opposite edges identified.

The choice of coordinates matters for seeing the geometry. In coordinates that are not orthonormal with respect to the metric, the connection between distance and entanglement is obscured; in coordinates orthonormal with respect to the Hilbert-Schmidt metric, the equi-entanglement contours are constant-radius circles near identity and SWAP.

The proportionality holds only locally

The proportionality of a quantum resource to the Hilbert-Schmidt distance holds only locally and breaks down beyond order ell^2. The paper illustrates this with the J = 1 entanglement power across the Voronoi unit cell, where the full expression is a sum of six trigonometric terms in the relative coordinates x = 2(delta_1 - delta_0) and y = 2(delta_2 - delta_0).

The theorem applies to rotationally invariant S-matrices near identity or SWAP, with the s-wave assumed not to couple to higher partial waves so that the S-matrix is unitary. It is a statement about leading order in the distance from the nearest non-entangling gate, not a global identity across the full space of S-matrices.

The authors also note that the non-local magic definition includes an optimization over local unitary transformations that can only be performed exactly for J = 1/2, yet the universality result still applies for all J.

Entanglement minimization loses its special status

A motivating idea in this literature is that nature sits near entanglement-free regions of the resource landscape, suggesting that extremization of a quantum resource is a deeper principle from which accidental symmetries could be derived. The present work qualifies that picture for rotationally invariant scattering.

Near the non-entangling points, many non-local resources are proportional, so suppressing entanglement suppresses every such resource by the same factor. There is nothing special about entanglement minimization in this region: the same geometric distance controls them all, and the choice of resource only changes the overall normalization chi_R.

The paper frames this as the value of an operator-centric rather than state-centric view. Instead of computing the linear entropy for every initial state and integrating over the nonlinear state manifolds for each J, the authors specialize the operator formula using only SU(2) representation theory, which also exposed the universal form near identity and SWAP.

What matters most in this result

The most decision-relevant feature is the split between geometry and normalization. Near identity and SWAP, the squared projective Hilbert-Schmidt distance ell^2 carries all the shared geometric information, while the resource-dependent constant chi_R carries everything specific to entanglement power versus non-local magic power. Comparing resources in this regime therefore reduces to comparing their chi_R values.

The second feature is the boundary of that reduction. The equivalence is a leading-order statement in ell^2 and is established for rotationally invariant S-matrices near the two non-entangling gates, identity and SWAP, up to an overall phase. Away from that neighborhood, the J = 1 entanglement power shows the proportionality breaking down, so the single-distance picture should not be extended across the full phase-shift cell.

For readers weighing whether distinct quantum resources are independent diagnostics, the paper's answer is conditional: near the resource-free gates they are not independent, and the leading resource-generating capacity of rotationally invariant scattering has a common geometric origin independent of the details of the interaction.

Sources used in this answer