Subsystems of Random Pure Quantum States are Random

Random quantum states play a crucial role in various fields, including quantum complexity growth, black holes, randomized benchmarking of quantum gates, quantum chaos, and the demonstration of quantum advantage. While the average entropy of a subsystem of a large system in a random pure quantum state is established, the randomness of these subsystems remains relatively unexplored. Partial measurement of a large quantum system can alter the quantum properties that a large quantum system originally possessed. For example, the entanglement of a Greenberger-Horne-Zeilinger state is destroyed by partial measurement. Here we demonstrate that a subsystem generated by partial measurement of a large system in a random pure quantum state remains in a random quantum state, unlike entangled states. To illustrate this, we first prepare a pure random quantum state from Haar random unitary operators, which are generated through the QR decomposition of a complex Gaussian random matrix or by implementing random quantum circuits. We then perform a partial projective measurement on a large system in a random pure state. The Shannon entropy for a quantum state is utilized to test whether a subsystem after partial measurement retains it characteristics as as a Haar-measured random quantum state. Our numerical calculations strongly support that a subsystem generated by partial measurement on a large system in a random pure state remain random. Notably, this randomness persists until the size of the subsystem is reduced to only a few qubits.