Nonstabilizerness Without Magic

• Nonstabilizerness without magic is defined as the operational nontriviality arising when stabilizer operations fail to fully distinguish orthogonal stabilizer states despite their classical simulability.
• This phenomenon exposes intrinsic limits in quantum state discrimination and data hiding protocols when only stabilizer (Clifford) operations are available.
• It serves as a robust tool for the unconditional verification of non-Clifford gates, highlighting a crucial gap between simulation capabilities and operational performance.

Nonstabilizerness without magic refers to scenarios in quantum information theory where operational limitations on classically simulatable quantum circuits, specifically stabilizer operations, lead to insights about quantum resources even when all the involved states contain no "magic" (i.e., are stabilizer states) (Kwon, 30 Sep 2025). This phenomenon exposes a resource-theoretic asymmetry: while stabilizer states are trivial to prepare and simulate classically, certain discrimination tasks become nontrivial when one is constrained to stabilizer operations alone. As a result, operational nonstabilizerness can manifest—despite the absence of genuine magic—purely as an artifact of limited operational capability. This has consequences for quantum data hiding, the no-cloning of stabilizer states, and unconditional verification of non-Clifford gates.

1. Foundations: Stabilizer Formalism and Magic

A quantum state is called a stabilizer state if it is the unique joint +1 eigenstate of a maximal abelian subgroup of the $N$-qubit Pauli group. Stabilizer operations comprise the Clifford group (generated by Hadamard, Phase, and CNOT gates), preparable stabilizer ancillae, computational basis measurements, and discarding ancillae. The Gottesman–Knill theorem ensures that circuits composed entirely of these elements are efficiently classically simulatable.

Whereas the presence of magic (nonstabilizerness in a resource-theoretic sense) requires states or operations outside the stabilizer polytope, the focus here is on operational phenomena—especially quantum state discrimination—that can arise entirely within the stabilizer theory under resource constraints.

2. Indistinguishability of Stabilizer States by Stabilizer Protocols

The central result is the construction of sets of mutually orthogonal stabilizer states—explicitly, the set

$$\{\,|{+}\,1\,0\rangle,\ |0\,+\,1\rangle,\ |1\,0\,+\rangle,\ |-\,1\,0\rangle,\ |0\,-\,1\rangle,\ |1\,0\,-\rangle\,\}$$

on three qubits—that cannot be perfectly distinguished using only stabilizer operations (Kwon, 30 Sep 2025). These states can be constructed by applying Clifford unitaries to computational basis states and are, by definition, classically efficiently preparable and simulable.

If unrestricted quantum operations are allowed (i.e., any projective measurement), these states are perfectly distinguishable; the measurement $\Pi_{\psi_\mu} = |\psi_\mu\rangle\langle\psi_\mu|$ enables this. However, if one is limited to stabilizer-computable measurements, all possible effects are Pauli projectors of the form

$\Pi_a^P = \frac{1 + (-1)^a P}{2}$

with $P \in \mathcal{P}_N$ and $a \in \{0,1\}$. The probability of outcome %%%%6%%%% on state $|\psi_\mu\rangle$ is always in $\{0,\frac{1}{2},1\}$ due to the spectrum of Pauli operators on stabilizer states.

Any informative stabilizer measurement necessarily disturbs the state, typically reducing mutual orthogonality among the post-measurement states, thereby making subsequent discrimination rounds increasingly imperfect. Explicit mutual information calculations demonstrate a gap: defining $I(\mu : \mathbf{a})$ as the mutual information between the identity label $\mu$ and the measurement record $\mathbf{a}$,

$I(\mu : \mathbf{a}) \leq \log_2 6 - \frac{1}{3}$

This strictly precludes perfect discrimination, as $\log_2 6$ bits would be required for perfect distinguishability among six states, and $1/3$ remain irreducibly hidden. Fano's inequality then upper bounds the success probability of any stabilizer protocol:

$p^{\text{succ}}_{\text{STAB}} \lesssim 0.9603$

3. Operational Resource Implications

3.1 Quantum Data Hiding With Magic-Free States

By leveraging indistinguishability, one may design quantum data-hiding protocols using only stabilizer resources. For example, mixing certain triples of the above stabilizer states into states $\rho_0$, $\rho_1$ that encode a classical bit, one obtains

$$\rho_b = \frac{1}{3} \sum_{|\psi_\mu\rangle \in S_b} |\psi_\mu\rangle\langle\psi_\mu|$$

where $S_0$, $S_1$ are complementary subsets from the six-state set. Any measurement restricted to stabilizer operations cannot distinguish between $\rho_0$ and $\rho_1$ perfectly, ensuring information hiding akin to "nonlocality without entanglement" in separable state data hiding scenarios.

3.2 No-Cloning for Stabilizer States Under Stabilizer Operations

Suppose a hypothetical stabilizer circuit could perfectly clone any stabilizer state. Then, multiple copies could be used to perform perfect discrimination among the set of orthogonal stabilizer states, contradicting the indistinguishability bound. Thus, perfect cloning of stabilizer states under stabilizer operations is forbidden, even though the states themselves are "free" in a resource-theoretic sense. This is a strict, resource-limited version of the quantum no-cloning theorem.

3.3 Unconditional Verification of Non-Clifford Operations

The provable gap in discrimination power via stabilizer operations can be operationalized to verify non-Clifford gates unconditionally. A verifier can challenge a prover by providing a state from a set of indistinguishable stabilizer states; any prover that succeeds with probability higher than the stabilizer bound is necessarily using nonstabilizer (i.e., non-Clifford) resources. This protocol does not rely on computational complexity-theoretic assumptions, and thus provides a robust benchmark for demonstrating genuinely magic- or non-Clifford-enabled quantum computation.

4. Comparison With Nonlocality Without Entanglement

The above resource phenomena are formally analogous to "nonlocality without entanglement" [Bennett et al., PRA 59, 1070 (1999)], which demonstrates that certain sets of separable (i.e., unentangled) states cannot be reliably distinguished by LOCC protocols. Here, instead of LOCC, the restriction is to the Clifford/stabilizer paradigm; instead of separability, the resource limitation is on magic. The analogy underscores that the "free" set in a resource theory may exhibit substantial operational nontriviality even when its elements are devoid of the resource itself.

This parallel structure supports the emerging viewpoint that discrimination tasks serve as sensitive witnesses of operational nontriviality, transcending the boundary between "magic" and "non-magic" states.

5. Mathematical Formulation of the Discrimination Problem

In the stabilizer setting, each round of dynamical stabilization is characterized by a series of projective Pauli measurements:

$$\Pi_{a}^{P} = \frac{1 + (-1)^a P}{2}, \quad a \in \{0, 1\}$$

with Pauli $P$. The probabilities for outcome $a$ on $|\psi_\mu\rangle$ are thus:

$$p_{\mu}^{P}(a) = \langle \psi_\mu | \Pi_{a}^{P} | \psi_\mu \rangle \in \{0, 1/2, 1\}$$

Defining the mutual information between the label $\mu$ and all measurement outcomes $\mathbf{a}$ as

$I(\mu : \mathbf{a}) = H(\mu) - H(\mu|\mathbf{a})$

and noting $H(\mu) = \log_2 6$ for a uniform ensemble, the residual uncertainty $H(\mu|\mathbf{a}) \geq 1/3$ implies that the information deficit is fundamental—arising purely from the measurement disturbance-resource interplay in the stabilizer framework.

The upper bound on discrimination success probability follows from Fano’s inequality for a six-element uniform ensemble:

$$p^{\text{succ}}_{\text{STAB}} \leq 1 - \frac{H(\mu|\mathbf{a})}{\log_2 6}$$

6. Broader Consequences and Connections

The phenomenon of nonstabilizerness without magic promotes a more nuanced view of resource theories, highlighting that tasks such as state discrimination, data hiding, and circuit verification may experience resource-induced bottlenecks even when the relevant quantum states are "free" of magic.

This deepens the distinction between preparation/simulation and discrimination/verification tasks, indicating that classically simulatable quantum circuits are not universal with respect to all operational criteria. It motivates further resource-theoretic analyses distinguishing "structural" (state-based) from "operational" (task-based) nonstabilizerness, with immediate applications to cryptographic protocols, quantum verification, and foundational studies of quantum advantage.

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References:

• (Kwon, 30 Sep 2025) Nonstabilizerness without Magic: Classically Simulatable Quantum States That Are Indistinguishable by Classically Simulatable Quantum Circuits
• [Bennett et al., Phys. Rev. A 59, 1070 (1999)] Nonlocality without entanglement