Distinguishability Transitions in Non-Unitary Boson Sampling Dynamics

Abstract

We discover novel transitions characterized by distinguishability of bosons in non-unitary dynamics with parity-time ($\mathcal{PT}$) symmetry. We show that $\mathcal{PT}$ symmetry breaking, a unique transition in non-Hermitian open systems, enhances regions in which bosons can be regarded as distinguishable. This means that classical computers can sample the boson distributions efficiently in these regions by sampling the distribution of distinguishable particles. In a $\mathcal{PT}$-symmetric phase, we find one dynamical transition upon which the distribution of bosons deviates from that of distinguishable particles, when bosons are initially put at distant sites. If the system enters a $\mathcal{PT}$-broken phase, the threshold time for the transition is suddenly prolonged, since dynamics of each boson is diffusive (ballistic) in the $\mathcal{PT}$-broken ($\mathcal{PT}$-symmetric) phase. Furthermore, the $\mathcal{PT}$-broken phase also exhibits a notable dynamical transition on a longer time scale, at which the bosons again become distinguishable. This transition, and hence the classical easiness of sampling bosons in long times, are true for generic postselected non-unitary quantum dynamics, while it is absent in unitary dynamics of isolated quantum systems. $\mathcal{PT}$ symmetry breaking can also be characterized by the efficiency of a classical algorithm based on the rank of matrices, which can (cannot) efficiently compute the photon distribution in the long-time regime of the $\mathcal{PT}$-broken ($\mathcal{PT}$-symmetric) phase.

Figures (8)

• Figure 1: (a) Schematic picture for the non-unitary boson sampling. Photons experience non-unitary dynamics by $U(t)\:(\neq[U^\dagger(t)]^{-1})$ through linear optical elements with loss effects, where we postselect cases for which all photons remain in the system. (b) Schematic phase diagram for sampling photon distributions with the algorithm that approximates photons to distinguishable particles. In blue and green regions, classical computers can efficiently sample the distribution of photons through the algorithm, where the latter is unique to non-unitary dynamics. Times $t_\mathrm{dis}^\mathrm{s}$ and $t_\mathrm{dis}^\mathrm{b}$ with $t_\mathrm{dis}^\mathrm{s}\ll t_\mathrm{dis}^\mathrm{b}$ are thresholds for the short-time dynamical distinguishability transition in the $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken phases, respectively. Time $t_\mathrm{max}$ is a threshold time for the long-time transition in the $\mathcal{PT}$-broken phase.
• Figure 2: Quasi-energy dispersion relations of $\tilde{H}(k)=i\log[\tilde{V}(k)]$ with $\theta_1=0.65\pi,\,\theta_2=0.25\pi$, (a) $e^\gamma=1$ where $\tilde{V}(k)$ is unitary, (b) $e^\gamma=1.2$ in the $\mathcal{PT}$-symmetric phase, and (c) $e^\gamma=1.5$ in the $\mathcal{PT}$-broken phase. Some quasi-energies have non-zero imaginary parts in the $\mathcal{PT}$-broken phase, while all quasi-energies are real in the $\mathcal{PT}$-symmetric phase.
• Figure 3: $L_1$-distance between the actual probability distribution of photons $P(\{n^\mathrm{in}\},\{n^\mathrm{out}\},t)$ in Eq. (\ref{['eq:probability-distribution']}) and that for distinguishable particles $P_\mathrm{dis}(\{n^\mathrm{in}\},\{n^\mathrm{out}\},t)$ in Eq. (\ref{['eq:probability-distribution_distinguishable']}). After the threshold time $t_\mathrm{dis}^\mathrm{s/b}$, the two distributions differ as $\eta(t)>\delta\:(=10^{-6})$, which defines the short-time distinguishability transition. We find that $t_\mathrm{dis}^\mathrm{b}$ for the $\mathcal{PT}$-broken phase is much larger than $t_\mathrm{dis}^\mathrm{s}$ for the $\mathcal{PT}$-symmetric phase. The blue asterisks, purple circles, and red squares respectively correspond to $e^\gamma=1,\,1.2$, and $1.5$. The lowest value around $10^{-14}$ is a numerical artifact. The number of photons is $n=3$, and the initial state is $\hat{b}^\dagger_{L/6,h}\hat{b}^\dagger_{L/2,v}\hat{b}^\dagger_{5L/6,v}\ket{0}$ with $L=300$. The rotation angles are $\theta_1=0.65\pi$ and $\theta_2=0.25\pi$, with which the threshold gain-loss parameter $\gamma_\mathcal{PT}$ for $\mathcal{PT}$ symmetry breaking becomes $e^{\gamma_\mathcal{PT}}\simeq1.22$.
• Figure 4: $L_1$-distance between $P(\{n^\mathrm{out}\},t)$ and $P_\mathrm{dis}(\{n^\mathrm{out}\},t)$. After the threshold time $t \geq t_\mathrm{max}$ in the $\mathcal{PT}$-broken phase, $\eta(t)<\delta$ holds and thus the classical algorithm based on $P_\mathrm{dis}(\{n^\mathrm{out}\},t)$ can efficiently sample $P(\{n^\mathrm{out}\},t)$, which defines the long-time distinguishability transition. The dashed lines show $e^{-2\Delta t}$, which confirms Eq. (\ref{['eq:critical-t_long-time']}). The blue, black, purples, yellow, green, and red symbols correspond to $e^\gamma=1,\,1.1,\,1.2,\,1.3,\,1.4$, and $1.5$, respectively. The initial condition and parameters are the same as those in Fig. \ref{['fig:difference_short']}. The long-time distinguishability transition is absent in the $\mathcal{PT}$-symmetric phase for $e^\gamma=1.1,1.2$ and unitary dynamics without loss for $e^\gamma=1$.
• Figure 5: Schematic picture for dynamics in which photons pass through optical elements. This figure corresponds to a sequence $G(+\gamma)SC(\theta)$ in terms of the matrices in Eq. (2). When photons with the polarization $v$ pass through the wave plate, which is described as the shallow purple rectangle, the superposition of $h$ and $v$ is realized with $\alpha=-\sin(\theta)$ and $\beta=\cos(\theta)$. The beam displacer, which corresponds to the thick green rectangle, causes the position shift of photons depending on their polarization; photons with $h$ and $v$ respectively go to the right and left. When photons enter the partially polarizing beam splitter, which is the blue square, photons with $v$ get into the environment with a probability $1-e^{-4\gamma}$, while photons with $h$ always go straight and remain in the system. If we carry out the postselection and focus only on the cases in which all photons are in the system, the creation operators for photons with $v$ acquire the additional factor $e^{-2\gamma}$. While unitary dynamics by $C(\theta)$ and $S$ is realized by photons passing through linear optical elements, couplings to environment and postselection are additionally needed for non-unitary dynamics by $G(\gamma)$.