Transitions of distinguishability and entanglement in monitored dynamics of bosons

The boson sampling problem has been extensively explored from the viewpoint of computational complexity and quantum supremacy since the probability distribution of bosons can be hard to compute through classical computers [1]. Here, whether or not computing the boson distribution is classically hard depends on physical features, such as quantum interference and the initial distances of bosons. This suggests that the computational complexity of boson distributions can be utilized for characterizing quantum phases and dynamics of bosonic systems. The experimental setting of the boson sampling, i.e. the quantum optical system where photons pass through various optical elements, has also gathered huge attention as a versatile platform for exploring non-unitary quantum dynamics governed by effective non-Hermitian Hamiltonians [2]. This is because the photon loss effect can be manipulated in such optical systems.

 

In this work, we characterize such non-unitary dynamics through the distinguishability of bosons, which is related to the computational complexity. We clarify that Parity and Time-reversal (PT) symmetry breaking accompanied by a real-complex transition for eigenvalues of non-Hermitian Hamiltonians, a transition unique to open systems, has huge effect on the dynamics of bosons [3]. PT-symmetry breaking enhances regions where bosons can be regarded as distinguishable particles and thus classical computers can easily compute distributions of bosons, in both short-time and long-time regimes, where the latter is unique to non-unitary dynamics. In addition, we find that scaling of entanglement entropy in the long run exhibits the volume law in the PT-symmetric phase and the log law in the PT-broken phase, with respect to the number of bosons [4]. Both results indicate that PT-symmetry breaking reduces the complexity and makes the bosonic system classical.

 

[1] S. Aaronson and A. Arkhipov, Theory of Computing 9, 143 (2013).

[2] L. Xiao et.al., Nat. Phys. 13, 1117 (2017).

[3] K. Mochizuki and R. Hamazaki, Phys. Rev. Research 5, 013177 (2023).

[4] D. Kagamihara, R. Kaneko, I. Danshita, and K. Mochizuki, in preparation.