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Nonstabilizerness without Magic: Classically Simulatable Quantum States That Are Indistinguishable by Classically Simulatable Quantum Circuits

Hyukjoon Kwon

TL;DR

The paper identifies nonstabilizerness without magic: a set of stabilizer states that cannot be perfectly discriminated by stabilizer operations, revealing a fundamental asymmetry in the magic resource theory. It derives a quantitative entropic bound $I(\mu : \boldsymbol{a}) \le \log_2 6 - \tfrac{1}{3}$ with $H(\mu|\boldsymbol{a}) = \tfrac{1}{3}$, and provides an explicit non-Clifford circuit based on controlled-controlled-Hadamard gates that achieves perfect discrimination. The authors generalize the construction to $3n$ qubits using a Boolean function with vanishing linear structure and prove minimality at $n=3$ for the six-state example. They discuss broad implications for data hiding, no-cloning of stabilizer states under stabilizer operations, and unconditional verification of nonstabilizerness, highlighting a deep operational contrast within quantum resource theories and fault-tolerant computing.

Abstract

Quantum state discrimination plays a central role in defining the possible and impossible operations through a restricted class of quantum operations. A seminal result by Bennett et al. [Phys. Rev. A 59, 1070 (1999)] demonstrates the existence of a set of mutually orthogonal separable quantum states that cannot be perfectly distinguished by local operations and classical communication, a phenomenon known as nonlocality without entanglement. We show that a parallel structure exists in the resource theory of magic: there exists a set of mutually orthogonal stabilizer states that cannot be perfectly distinguished by stabilizer operations, which consist of Clifford gates, measurements in the computational basis, and additional ancillary stabilizer states. This phenomenon, which we term 'nonstabilizerness without magic,' reveals a fundamental asymmetry between the preparation of classically efficiently simulatable stabilizer states and their discrimination, which cannot be performed by classically efficiently simulatable quantum circuits. We further discuss the implications of our findings for quantum data hiding, the no-cloning of stabilizer states, and unconditional verification of non-Clifford gates.

Nonstabilizerness without Magic: Classically Simulatable Quantum States That Are Indistinguishable by Classically Simulatable Quantum Circuits

TL;DR

The paper identifies nonstabilizerness without magic: a set of stabilizer states that cannot be perfectly discriminated by stabilizer operations, revealing a fundamental asymmetry in the magic resource theory. It derives a quantitative entropic bound with , and provides an explicit non-Clifford circuit based on controlled-controlled-Hadamard gates that achieves perfect discrimination. The authors generalize the construction to qubits using a Boolean function with vanishing linear structure and prove minimality at for the six-state example. They discuss broad implications for data hiding, no-cloning of stabilizer states under stabilizer operations, and unconditional verification of nonstabilizerness, highlighting a deep operational contrast within quantum resource theories and fault-tolerant computing.

Abstract

Quantum state discrimination plays a central role in defining the possible and impossible operations through a restricted class of quantum operations. A seminal result by Bennett et al. [Phys. Rev. A 59, 1070 (1999)] demonstrates the existence of a set of mutually orthogonal separable quantum states that cannot be perfectly distinguished by local operations and classical communication, a phenomenon known as nonlocality without entanglement. We show that a parallel structure exists in the resource theory of magic: there exists a set of mutually orthogonal stabilizer states that cannot be perfectly distinguished by stabilizer operations, which consist of Clifford gates, measurements in the computational basis, and additional ancillary stabilizer states. This phenomenon, which we term 'nonstabilizerness without magic,' reveals a fundamental asymmetry between the preparation of classically efficiently simulatable stabilizer states and their discrimination, which cannot be performed by classically efficiently simulatable quantum circuits. We further discuss the implications of our findings for quantum data hiding, the no-cloning of stabilizer states, and unconditional verification of non-Clifford gates.

Paper Structure

This paper contains 15 sections, 8 theorems, 25 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Stabilizer states and operations
  3. Nonstabilizerness without magic
  4. A set of stabilizer states indistinguishable by stabilizer operations
  5. Entropic analysis
  6. Explicit non-Clifford circuit for perfect discrimination
  7. $n$-qubit generalization
  8. Implications
  9. Applications
  10. Unconditional verification of a quantum circuit's nonstabilizerness
  11. Data hiding in magic
  12. Remarks
  13. Complete proof of Theorem \ref{['thm:main']}
  14. Complete proof of Theorem \ref{['thm:2qubit']}
  15. Complete proof of Theorem \ref{['thm:3nqubit']}

Key Result

Theorem 1

A set of mutually orthogonal stabilizer states cannot be perfectly discriminated by stabilizer operations. Here, $\left|0 \right>$ and $\left|1 \right>$ are computational basis states stabilized by $Z$ and $-Z$, respectively, and $\left|\pm \right> = \frac{1}{\sqrt{2}} (\left|0 \right> \pm \left|1 \right>)$ are stabilized by $\pm X$.

Figures (2)

  • Figure 1: Discrimination of a set of mutually orthogonal stabilizer states $\{ \left|\psi_\mu \right> \}_{\mu=1}^m$, generated by Clifford operations $U_\mu^{\rm CL}$, can be performed perfectly when all possible quantum circuits are allowed. However, when restricted to stabilizer operations, even with adaptivity, the discrimination probability may become strictly less than $1$.
  • Figure 2: Quantum circuit that perfectly distinguishes the states in Eq. \ref{['eq:stab_set']}.

Theorems & Definitions (11)

  • Theorem 1
  • Lemma 1: Informal
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Corollary 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 1 more