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

TL;DR

This paper analytically compares real and complex Haar random quantum states by computing their spectral decompositions and trace distances, providing insights into their distinguishability and implications for quantum state designs.

Contribution

It introduces an exact analytical method to distinguish real from complex Haar random states and improves bounds related to quantum state designs and imaginarity testing.

Findings

Exact spectral decomposition of the density matrix for t copies.

Computed trace distance between real and complex Haar states.

Improved bounds on the number of copies needed for imaginarity testing.

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.