Exact distinguishability between real-valued and complex-valued Haar random quantum states

2025/07/22 by Nemoz, Tristan, Alléaume, Romain, Brown, Peter
#FOS: Physical sciences #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2507.16939

Abstract

Haar random states are fundamental objects in quantum information theory and quantum computing. We study the density matrix resulting from sampling t copies of a d-dimensional quantum state according to the Haar measure on the orthogonal group. In particular, we analytically compute its spectral decomposition. This allows us to compute exactly the trace distance between t-copies of a real Haar random state and t-copies of a complex Haar random state. Using this we show a lower-bound on the approximation parameter of real-valued state t-designs and improve the lower-bound on the number of copies required for imaginarity testing.

Citations

• Unitary designs in nearly optimal depth
• Random Real Valued and Complex Valued States Cannot be Efficiently Distinguished
• Efficient Quantum Pseudorandomness from Hamiltonian Phase States
• Simple constructions of linear-depth t-designs and pseudorandom unitaries
• Pseudorandom and Pseudoentangled States from Subset States
• Pseudorandomness from Subset States
• Quantum Pseudorandom Scramblers
• Pseudorandom unitaries are neither real nor sparse nor noise-robust
• Quantum Cryptography in Algorithmica
• Pseudorandom (Function-Like) Quantum State Generators: New Definitions and Applications
• Quantum Pseudoentanglement
• Cryptography from Pseudorandom Quantum States
• Scalable Pseudorandom Quantum States
• Predicting many properties of a quantum system from very few measurements
• (Pseudo) Random Quantum States with Binary Phase
• A blueprint for demonstrating quantum supremacy with superconducting qubits
• The Church of the Symmetric Subspace
• Quantum t-designs: t-wise independence in the quantum world
• Both Toffoli and Controlled-NOT need little help to do universal quantum computation

Related

• Random Real Valued and Complex Valued States Cannot be Efficiently Distinguished
• Haar-random and pretty good measurements for Bayesian state estimation
• Bounding the Sample Fluctuation for Pure States Certification with Local Random Measurement
• The power of a single Haar random state: constructing and separating quantum pseudorandomness
• Random measurement bases, quantum state distinction and applications to the hidden subgroup problem