The quantum magic that entanglement misses

Imagine taking a quantum spin chain and applying exactly the same rotation to every spin. You have acted separately on each spin, without coupling spins that were previously separate. The entanglement across every cut stays the same.

Could a measure of quantum structure nevertheless change its long-distance power law? In recent work with Vishnu Jejjala, we show that stabilizer Rényi moments can do precisely this. To understand why, we can start with the quantum mechanics of a single spin.

Entanglement is only part of the story

A qubit has basis states |0⟩ and |1⟩, and can occupy a superposition such as |+⟩ = (|0⟩ + |1⟩)/√2. Two qubits can form the entangled Bell state (|00⟩ + |11⟩)/√2. Superposition and entanglement are both essential quantum ideas, but neither by itself tells us which computations are hard to simulate on an ordinary computer.

There is a large family of quantum circuits that can create entanglement and still be simulated efficiently. These circuits use Clifford gates: gates that turn products of Pauli operators into other products of Pauli operators under conjugation. Starting with |0⟩ states and using these gates produces stabilizer states. The Bell state is one of them.

To go beyond this restricted computational toolbox, we need an additional resource, called magic. For a pure state, having magic means being outside the stabilizer family. Even a single qubit can have it, although a single qubit cannot be entangled with itself.

That makes entanglement and magic different questions. Entanglement asks how a state is shared between subsystems. Magic asks about its relation to a specified stabilizer toolbox. Our work investigates how the second question appears in spatial correlations.

A fingerprint made of Pauli measurements

If you know spin-½, you already know the three Pauli matrices X, Y and Z. Their expectation values give the components of a qubit’s Bloch vector. For several qubits, their products play the same role: X ⊗ I, Z ⊗ Z, X ⊗ Y ⊗ I, and so on. These are called Pauli strings.

For a region A containing N qubits, let D = 2^N, and let P run over all 4^N Pauli strings, including the identity. Write ⟨P⟩ = Tr(ρ_AP), where ρ_A is the state of that region. We focus on the second stabilizer Rényi expression:

You do not need to calculate a huge sum to see the idea. Fourth powers give much more weight to a few large expectation values than to many small ones. This quantity probes how the state’s Pauli fingerprint is distributed. The logarithm makes it additive for independent systems. All logarithms here are natural.

For a pure stabilizer state, exactly D strings have expectation value +1 or −1; the rest have zero expectation. The expression in brackets is therefore 1, and the entropy is zero. A pure nonstabilizer state has a positive value.

For one qubit, try the real state |ψ_θ⟩ = cos(θ/2)|0⟩ + sin(θ/2)|1⟩. Its Pauli expectations are sin θ, 0 and cos θ. Rotating from θ = 0 to θ = π/4 changes the second stabilizer Rényi entropy from zero to ln(4/3), even though the state stays pure throughout.

The axes matter. A Clifford rotation permutes Pauli strings, possibly changing their signs, and leaves the sum unchanged. A generic rotation does not. If we rotate the state and the measurement axes together, nothing changes; our question keeps the measurement toolbox fixed while rotating the state.

A Bell pair gives a useful warning

So far the state was pure. A region of a spin chain usually is not: even when the whole chain is pure, tracing out the rest produces a mixed density matrix. Here we must be careful about what the expression measures.

Take a Bell pair and keep only its first qubit. That qubit has ρ_A = I/2, so its X, Y and Z expectations all vanish. Only the identity contributes to the sum. The result is M̂₂(A) = ln 2.

But the Bell pair is a stabilizer state, and I/2 is just an equal mixture of |0⟩ and |1⟩. That positive number cannot by itself diagnose magic. It records the mixedness caused by ignoring the partner.

We derive an exact general version of this observation. Whenever ρ_A comes from a pure stabilizer state,

S₂ is the ordinary second Rényi entropy. For a subsystem of a pure state, it measures entanglement with the rest. This is the entanglement baseline hidden inside the stabilizer expression.

Subtracting the baseline is useful, but even the remainder is not a faithful measure of magic for arbitrary mixed states: some mixtures of stabilizer states still give a positive remainder. In this post, “stabilizer moment” means the particular Pauli-based diagnostic we calculate. It should not be read as a count of magic resources inside an arbitrary mixed region.

Why bring in a conformal field theory?

Consider a line of spins with the Hamiltonian

The first term favours neighbouring spins aligned along X. The second favours alignment along Z. At the equal couplings written here, their competition puts the chain at a quantum critical point.

At criticality there is no finite correlation length cutting off the fluctuations. Correlations survive over arbitrarily large distances, decaying as power laws. The long-distance description is the Ising conformal field theory, or CFT. It lives in one space dimension and one time dimension. “Conformal” refers to an enlarged set of symmetries that includes changes of scale; those symmetries tightly constrain the correlations.

For our purposes, the most useful CFT fact is that a field’s scaling dimension Δ fixes its two-point decay: a correlator behaves as r^−2Δ. The Ising order field has Δ = 1/8, so the critical chain has ⟨X₀X_r⟩ proportional to r^−1/4 at large r.

We want to probe two separated regions A and B. Define T₂(A) as the sum of ⟨P⟩⁴ over all strings in A, and compare the joint sum with the separate sums:

For a product state of the two regions, the expectation values factorize and F₂ = 1. Thus F₂ − 1 isolates a departure from factorization in these moments. It is sensitive to correlations, including correlations that are not entanglement.

The chain’s spin-flip symmetry makes ⟨X⟩ = 0, while allowing ⟨X₀X_r⟩ to be nonzero: flipping both spins changes two signs, which cancel. For two single-site regions, this correlator enters T₂ to the fourth power. Its contribution therefore falls as (r^−1/4)⁴ = r⁻¹. In the unrotated chain, it is the leading term in F₂ − 1.

Rotate the state. Keep the measurements.

Now apply a rotation by π/4 about Y to every spin:

This is the halfway rotation in the qubit demonstration. For any region A, its density matrix changes by ρ_A → U_Aρ_AU_A^†. Unitary conjugation preserves eigenvalues. Therefore every subsystem entanglement spectrum is identical before and after the rotation, including all the ordinary Rényi entropies.

We can also rotate the Hamiltonian to H′ = UHU^†. Its energy spectrum is unchanged, and the rotated state is its ground state. This compares two equivalent descriptions of the critical Ising physics using the same fixed Pauli measurements.

The stabilizer moments respond differently. A direct calculation gives, for two single-site regions at large separation:

Why a different power? In the rotated state, some joint Pauli expectations contain a nonzero background together with a small, slowly decaying spin correlation. Pairs of terms have the form (b + c)⁴ + (b − c)⁴, where b approaches a nonzero constant and c is proportional to r^−1/4. Expanding gives 2b⁴ + 12b²c² + 2c⁴. The linear correction cancels, but the quadratic one survives. It falls as r^−1/2, slower than r⁻¹.

This is an exact asymptotic result from the Ising correlators. The rotation changes the exponent, not just the prefactor. Even the complete collection of subsystem entanglement spectra does not determine these spatial stabilizer moments. Their scaling also remembers how the state sits relative to the Pauli toolbox.

Do the lattice results survive at larger scales?

There is a subtlety in calling this a CFT result. Two single spins remain microscopic objects, even if their separation becomes enormous. To study macroscopic regions, we must also grow the number of sites in each interval.

Those are different limits. We can first move two fixed blocks far apart, or first grow both blocks while keeping their geometric ratio η fixed. A coefficient that is nonzero for every finite block could shrink to zero as the blocks grow. The fixed-block exponent alone does not settle the continuum answer.

For the critical Ising chain at index two, we derive a way to study much larger blocks. A block of 128 qubits has 4¹²⁸ Pauli strings—about 10⁷⁷. Enumerating them is hopeless. Using the model’s free-fermion structure, we rewrite the sums relevant to the leading charged coefficient in terms of Pfaffians, algebraic relatives of determinants. This lets us evaluate that coefficient up to 128 sites.

For equal blocks, the charged part of the normalized moment has the far-separation form C₂(ℓ)η^1/2. Here ℓ is the number of sites in a block. A candidate calculation in boundary CFT predicts a limiting coefficient of 1/4:

The sequence moves towards the prediction, and suitable extrapolations land close to 0.25. That is encouraging evidence, but identifying the lattice limit with this boundary calculation still requires matching the interval endpoints and controlling the order of limits. Notice that 1/4 here is an amplitude multiplying η^1/2; it is not the rotated two-site exponent discussed above.

We also derive a cancellation among neutral contributions, leaving terms tied to an interval endpoint. Numerically, the remaining linear neutral coefficient appears to fall inversely with block length. This helps explain how the continuum behaviour could differ from the fixed-site result, although a general bound is still missing.