Extremality of stabilizer states

Abstract

We investigate the extremality of stabilizer states to reveal their exceptional role in the space of all $n$-qubit/qudit states. We establish uncertainty principles for the characteristic function and the Wigner function of states, respectively. We find that only stabilizer states achieve saturation in these principles. Furthermore, we prove a general theorem that stabilizer states are extremal for convex information measures invariant under local unitaries. We explore this extremality in the context of various quantum information and correlation measures, including entanglement entropy, conditional entropy and other entanglement measures. Additionally, leveraging the recent discovery that stabilizer states are the limit states under quantum convolution, we establish the monotonicity of the entanglement entropy and conditional entropy under quantum convolution. These results highlight the remarkable information-theoretic properties of stabilizer states. Their extremality provides valuable insights into their ability to capture information content and correlations, paving the way for further exploration of their potential in quantum information processing.

Figures (1)

• Figure 1: We consider three different Fourier transforms: (1) using the characteristic function as the Fourier coefficients (2) using the Wigner function as the Fourier coefficients (3) applying the symplectic Fourier transform which will map the characteristic function to the Wigner function.

Theorems & Definitions (18)

• Theorem 1: Uncertainty principle for Pauli rank
• Theorem 2: Uncertainty principle for Wigner rank
• Proposition 3
• Theorem 4: General result
• Proposition 5: Monotonicity of entanglement entropy
• Proposition 6: Monotonicity of conditional entropy
• Theorem 7: Restatement of the uncertainty principle for Pauli rank
• proof
• Theorem 8: Restatement of the uncertainty principle for Wigner rank
• proof