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  • Open Access

Identifying entanglement phases with bipartite projected ensembles

Zi-Yong Ge1,* and Franco Nori1,2,†

  • 1Quantum Information Physics Theory Research Team, Center for Quantum Computing, RIKEN, Wako-shi, Saitama 351-0198, Japan
  • 2Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA

  • *Contact author: ziyong.ge@riken.jp
  • †Contact author: fnori@riken.jp
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Phys. Rev. Research 7, 043022 – Published 6 October, 2025

DOI: https://doi.org/10.1103/p4rx-c1fm

Abstract

We introduce bipartite projected ensembles (BPEs) for quantum many-body wave functions, which consist of pure states supported on two local subsystems, with each state associated with the outcome of a projective measurement of the complementary subsystem in a fixed local basis. We demonstrate that the corresponding ensemble-averaged entanglements (EAEs) between two subsystems can effectively identify entanglement phases. In volume-law entangled states, EAE converges to a nonzero value with increasing distance between subsystems. For critical systems, EAE exhibits power-law decay, and it decays exponentially for area-law systems. Thus, entanglement phase transitions can be viewed as a disordered-ordered phase transition. We also apply BPE and EAE to measured random Clifford circuits to probe measurement-induced phase transitions. We show that EAE not only serves as a witness to phase transitions, but also unveils additional critical phenomena properties, including dynamical scaling and surface critical exponents. Our findings provide an alternative approach to diagnosing entanglement laws, thus enhancing the understanding of entanglement phase transitions. Moreover, given the accessibility of measuring EAE in quantum simulators, our results hold promise for impacting quantum simulations.

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Phys. Rev. Research 7, 043022– Published 6 October, 2025
  • Received 15 August 2024
  • Revised 14 January 2025
  • Accepted 4 September 2025
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