Implementation of multiparticle quantum speed limits on observables

TL;DR

This work tackles multiparticle quantum speed limits on observables, addressing how entanglement and nonunitary dynamics can accelerate or decelerate the rate of change of an observable. The authors derive tighter bounds that separate coherent and incoherent contributions via quantum Fisher information, and identify optimal initial states that maximize the speed limit: $|+\rangle$, $|+\rangle^{\otimes N}$, and $(|H\rangle^{\otimes N}+|V\rangle^{\otimes N})/\sqrt{2}$, with maximum speeds $\pi$, $\sqrt{N}\pi$, and $N\pi$ respectively. They implement a high-precision two-photon (and limited open-system) experiment using a dephasing model realized with birefringent quartz crystals, achieving dense sampling over the evolution time $l$ and performing full state tomography to extract $a=\langle A\rangle$ and $|\dot{a}|$. The results confirm that multiparticle and entanglement can enhance quantum speeds, while nonunitary noise can both diminish and localize acceleration, and that the upper and lower bounds remain valid in these regimes. The findings provide a practical route to control the quantum speed of larger-scale systems and to characterize dynamic transients in complex quantum architectures.

Abstract

The energy-time uncertainty relation limits the maximum speed of quantum system evolution and is crucial for determining whether quantum tasks can be accelerated. However, multiparticle quantum speed limits have not been experimentally explored. In this work, we experimentally verify that both multiparticles and entanglement can accelerate the quantum speed on observables in two-particle systems based on ultrahigh precision control of quantum evolution time. Furthermore, we experimentally prove that the initial quantum state plays a critical role in the quantum speed limits of the entangled systems. In addition, we experimentally demonstrate that the upper bound and lower bound of the quantum speed are workable even in a nonunitary Markovian open system with two photons. The results obtained based on two-photon experiments have been shown to be generalizable to more particles. Our work facilitates the characterization of the dynamic transient properties of complex quantum systems and the control of the quantum speed of large-scale quantum systems.

Figures (3)

• Figure 1: Diagram and acceleration effect.(a) Diagram: multiparticles and entanglement accelerate the quantum speed. A quantum state with faster speed will evolve to a further quantum state simultaneously. (b) Blue, orange, dark green and red lines represent quantum speed limits with the $\left|{+}\right\rangle$, $\left|{++}\right\rangle$, $\left|{+++}\right\rangle$ and $\left|{++++}\right\rangle$ initial states, respectively. Here $\left|{+}\right\rangle=\left(\left|{H}\right\rangle+\left|{V}\right\rangle\right)/\sqrt{2}$. The speedup ratio is $\sqrt{N}$ for product $N$-particle qubit systems, which can be connected with the standard quantum limit. (c) Blue, yellow, light green, and pink lines represent quantum speed limits in frequency correlated systems with the $\left|{+}\right\rangle$, $\left|{\Phi^+}\right\rangle$, $\left|{\Phi_3^+}\right\rangle$, and $\left|{\Phi_4^+}\right\rangle$ initial states, respectively. Here $\left|{\Phi^+}\right\rangle=\left(\left|{HH}\right\rangle+\left|{VV}\right\rangle\right)/\sqrt{2}$, $\left|{\Phi_3^+}\right\rangle=\left(\left|{HHH}\right\rangle+\left|{VVV}\right\rangle\right)/\sqrt{2}$, $\left|{\Phi_4^+}\right\rangle=\left(\left|{HHHH}\right\rangle+\left|{VVVV}\right\rangle\right)/\sqrt{2}$. The speedup ratio is $N$ for entangled $N$-particle qubit systems, which can be connected with the Heisenberg limit. More quantum speed limits numerical results with different four-photon sources and different initial states are presented in Supplemental Material mySupplementalMaterial.
• Figure 2: Experimental setup for the quantum speed limits implemented by the two-photon dephasing model. Legend: PBS, polarizing beam splitter; $\beta$-BBO, beta barium borate; FC, fiber coupler; HWP, half-wave plate; QWP, quarter-wave plate; LMS, linear motorized stage; SPD, single photon detector. Entangled photon pairs are produced from $\beta$-BBO. We use filters (at the end of the light path) to control the frequency environment of the open system. QWP and HWP are used to control the polarization system of the open system. The open system evolves into a variable-length quartz crystal system, which can control the evolution time. The optic axes of quartz plates (pink colored) are set to $0^\circ$, while the optic axes of wedge-shaped quartz crystals are set to $90^\circ$. Then each photon may go through an optional quartz plate (green colored) whose optic axis is set to $45^\circ$ for simulating the noise in the nonunitary system. Finally, HWP, QWP, and PBS form the tomography process, and this process can fully extract polarization information within the open system.
• Figure 3: Expectation values $\bm{a}$ and quantum speeds $\bm{\left|\dot{a}\right|}$ on observables of the initial state.(a-c, h-k) represent expectation values $a$. The black line and green points represent the expectation value $a$ in theory and experiments. The error bars are hidden within the experimental points. (e-g, l-o) represent quantum speeds $\left|\dot{a}\right|$ and speed limits on observables of the initial state. The black line and green points represent the quantum speed $\left|\dot{a}\right|$ in theory and experiments, respectively. The blue and red areas indicate the forbidden areas beyond the upper bound and below the lower bound, respectively. Different column pairs except (d) represent different initial states, which are $\left|{+}\right\rangle$, $\left|{++}\right\rangle$, $\left|{\Phi^+}\right\rangle$, $\left|{P}\right\rangle$, $\left|{PP}\right\rangle$, $\left|{P}\right\rangle$, $\left|{PP}\right\rangle$, respectively. Here $\left|{+}\right\rangle=\left(\left|{H}\right\rangle+\left|{V}\right\rangle\right)/\sqrt{2}$, $\left|{\Phi^+}\right\rangle=\left(\left|{HH}\right\rangle+\left|{VV}\right\rangle\right)/\sqrt{2}$, and $\left|{P}\right\rangle$ is a special pure state. There are extra nonunitary noises after the evolution in (j, k, n, o). (d) lists the maximum quantum speeds in (e-g, l-o). According to (e-g) and (l, m), we can see that multiparticles and entanglement can improve quantum speed limits. While nonunitary noise can reduce quantum speed limits according to (l-o). Besides, all points which represent quantum speeds are lying in areas between blue and red areas, which means the tighter quantum speed limits on observables hold no matter in multiparticle or nonunitary systems.