Universal purification dynamics in real non-unitary quantum processes

Abstract

We study purification dynamics in monitored quantum processes governed by ensembles of quantum circuits in different random-matrix symmetry classes. We analyze the universal aspects that emerge away from the measurement induced phase transition and inside the volume/weak measurement phase and in the scaling limit of large time and Hilbert space dimension. We present two toy models that reveal two complementary visions and provide quantitative access to universal scaling: i) a discrete-time dynamic in which each time step corresponds to multiplication by a Gaussian random matrix; ii) weak continuous-time monitoring that induces a Dyson brownian motion of the eigenvalues of the density matrix. The first approach provides an algebraic characterization based on rotational invariance emerging in Kraus's operator space, focusing in particular on the unitary and orthogonal cases, respectively $β=2$ and $β=1$, with $β$ the Dyson random-matrix index. The second approach, on the other hand, allows for a unified treatment for any $β$, thanks to the mapping of the Fokker-Planck evolution of eigenvalues onto the Calogero-Sutherland integrable Hamiltonian diagonalized in terms of Jack polynomials. We provide explicit expressions for the universal decrease of Rényi entropies. We show that, approaching the universal scaling limit, numerical simulations of different models agree with each other and with our theoretical predictions. Our results clarify the existence of different classes of universality for the purification process in hybrid quantum systems, accessible in random circuit architectures and weak measurement protocols.

Figures (6)

• Figure 1: Products of independent random matrices from the Ginibre ensemble. Left: complex Ginibre matrices. By Wick's theorem, averaging at each time-step $i$ yields a sum over the permutations $\sigma_i\in \mathsf{S}_N$ of the $2N$ indices. Permutations are indeed depicted as loops connecting one index from the first batch of $N$ matrices to one from the second batch. Blueish rectangles represent the overlap matrix $G^{(\beta=2)}_{\sigma,\sigma'}$ Eq. \ref{['eq:grambeta2']}. Right: real Ginibre matrices. Averaging at each time-step $i$ yields a sum over pairings $p_i\in \mathsf{P}_N$ of the $2N$ matrix indices. Pairings are represented by loops connecting couples of indices in the batch of $2N$, while the Gram matrix elements $G^{(\beta=1)}_{p,p'}$\ref{['eq:grambeta1']} are represented by blueish rectangles.
• Figure 2: Graphical explanation for the overlap of pairing states, in the case of $|\mathsf{P}_2|=3$ pairings of $2N=4$ objects, dubbed $p_\mathbf{1},p_-,p_+$. Left: When interpreted as permutations in $\mathsf{S}_4$, the three considered pairings are 2 swaps apart from each other, from which $d_4(p,p')=2$. Right: A graphical justification of the Gram matrix elements $G_{p,p'}^{(\beta=1)}$ shows Eq. \ref{['eq:grambeta1']} for the case $N=2$.
• Figure 3: Comparison of the numerical data for the protocols discussed in Sec. \ref{['sec:numerics']}, at $q=256$, with the theoretical scaling functions obtained in Sec. \ref{['sec:small_x']}, in the perturbative expansion at small $x$. Both the BR - reweight - $N\to1$ average and the FM - independent random matrices (IRM) - $N\to 0$ averages are displayed. RM and PO refer to the discrete-time processes $(i)$ and $(ii)$ in Subsec. \ref{['subsec:simdiscrete']} respectively, namely, to the multiplication of independent random matrices, and to the protocol alternating projection and multiplication times an orthogonal matrix, while DWM refers to the discrete-time weak-measurement protocol, see Subsec. \ref{['subsection:continuous_num']}. Left: Sample-mean of the von Neumann entropy $\left\langle S_1 \right\rangle$, compared to the $O(x^4)$ prediction \ref{['eq:S1_theo_N1']} for $\mathcal{S}_{\rm VN}^{(1)}$ (dashed) and to the $O(x^2)$ prediction $\mathcal{S}_{\rm VN}^{(0)}$ Eq. \ref{['eq:S1_theo_N0']} (dotted). Right: Sample mean of the second Rényi entropy $\left\langle S_2 \right\rangle$, compared to the $O(x^6)$ predictions for $\mathcal{S}_{2}^{(1)}$ (dashed) and $\mathcal{S}_{\rm VN}^{(0)}$ (dotted) Eqs. \ref{['eq:S2_theo_N']}.
• Figure 4: Convergence of the numerical data for various models to universal curves. Top: shifted von Neumann entropy $\left\langle \tilde{S}_1 \right\rangle$ (see \ref{['eq:shifted']}). Curves merge at around $x \simeq 0.2$ where the effect of model-dependent and finite-size corrections becomes negligible. Bottom: shifted Rényi entropy $\left\langle \tilde{S}_2 \right\rangle$ (see \ref{['eq:shifted']}). The juxtaposition of numerical data is excellent already at small $x$. Left: BR - reweight - $N\to1$ averages. Right: FM - RM - $N\to 0$ averages.
• Figure 5: Next-to-leading-order terms in the von Neumann entropy $\left\langle \tilde{S}_1 \right\rangle$ Eq. \ref{['eq:shifted']}, via the extrapolation formula \ref{['eq:extrapolation']}. The convergence to the theoretical prediction $\tilde{\mathcal{S}}_{\rm VN}^{(1/0)} = \mathcal{S}_{\rm VN}^{(1/0)} - (1-\gamma_E) + \ln x$ (dashed) as given in Eq. \ref{['eq:S1_theo_N1']}, is very good for $x \gtrsim 0.1$. Moreover, it is particularly explicit for the continuous-time WM model in the region of $x\approx 0.05$, where the linear prediction alone is enough to capture the empirical behavior.