Witnesses of Genuine Multipartite Entanglement and Nonlocal Measurement Back-action for Raman-scattering Quantum Systems

TL;DR

This work develops experimentally accessible criteria to certify genuine multipartite entanglement and nonlocal backaction in Raman-scattering–induced W-like states across N subsystems. It introduces two complementary witnesses: (i) a multipartite entanglement witness based on cross-correlators and number statistics with a depth parameter M, and (ii) a nonlocality/backaction witness that tests coherence of the single-excitation addition, each with quantified robustness to initial thermal occupations. The authors also present a practical photodetection scheme using linear optical networks and Stokes/anti-Stokes channels to prepare the W-like states and to extract all required observable statistics for the witnesses, including explicit guidance for measuring simple cross correlators and for implementing the nonlocality test. Overall, the framework enables scalable, measurement-focused verification of entanglement and nonlocality in Raman-scattering quantum systems, with concrete guidance for optomechanical and related platforms at finite temperature.

Abstract

Entanglement between remote quantum mechanical systems enables a range of quantum information tasks in communication, computation and distributed sensing. Large numbers of entangled subsystems also require experimentally accessible and practically feasible methods of verifying the genuine, i.e., simultaneous, entanglement of all subsystems. We have derived a class of entanglement witnesses suitable for $W$-states, which are states where a single excitation is coherently distributed across subsystems initially in their ground state or a state with low thermal occupation, e.g., via detection of a Raman-scattered photon. The entanglement is witnessed through violation of an inequality involving number statistics, which can be measured via detection of subsequent Raman-scattered photons. Unlike conventional, partially tomographic, witnesses, our method is experimentally accessible for both multipartite and continuous variable systems. The thermal robustness of the method is quantified by the initial thermal occupations for which violation occurs. As an alternative approach, we have derived an inequality which tests the nonlocal, or quantum coherent, nature of the photon measurement backaction which produces the $W$-state. Violation of this alternative inequality implies the entanglement of the resulting state given the assumption of a separable initial state, under less stringent thermal constraints than the general entanglement witness. Our results are applicable to all Raman-scattering systems which can exhibit sufficient degrees of quantum indistinguishability.

Figures (7)

• Figure 1: Every $N$-partite pure state (here with $N=6$) has some structure $S$ of entanglement characterized by disjoint combinations $C$ of modes (illustrated by color and connecting lines) whose modes are inseparable from each other, while modes belonging to different combinations are mutually separable. Structures which are equivalent under exchange of indices, i.e., which contain combinations of the same sizes, belong to the same class $\mathcal{C}$.
• Figure 2: Given $N=6$ modes and the assumption of maximally $M=3$ simultaneously entangled modes, selective application of the separability criterion \ref{['eq:abs2ineq']} allows the structures on the left to be treated as if they were one of the structures on the right, reducing the number of separable cross correlator pairs. Of the reducible structures, the topmost may be treated as either of the irreducible structures, while the second one may only be treated as the topmost irreducible structure. For the irreducible structures, no further reduction is appropriate, as combining any two combinations leads to entanglement of more than $M=3$ modes. Every choice of $N$ and $M$ leads to some collection of irreducible classes of structures. $N=6$, $M=3$ produces the two above.
• Figure 3: Thermal occupation numbers $n_\text{th.}$ which saturate the inequality \ref{['eq:threshold_sym_thermal']} (in red) evaluated for $M=N-1$, corresponding to genuine $N$-partite entanglement. For thermal occupations below these thresholds, violation of the entanglement witness \ref{['eq:absNMineq']} occurs, given a thermal $W$-like state. The corresponding values for an ensemble of two-level systems is shown in blue. It is clear that for increasing $N$, the values converge. This is because the $W$-like states for harmonic oscillators and two-level systems give identical number statistics in the limit $n_\text{th.} \to 0$, where no components of two or more excitations are present for a harmonic oscillator.
• Figure 4: Thermal occupation numbers $n_\text{th.}$ which saturate the nonlocality witness \ref{['eq:nonlocbound_distinguishM_thermal']} (in red) for $N$ harmonic oscillators in a thermal initial state, evaluated for $M=N-1$, corresponding to genuine $N$-partite nonlocality. The corresponding result for two-level systems is shown in blue. As opposed to the general entanglement witness (see Fig. \ref{['fig:thresholds_abswitness_true']}), where the harmonic oscillator and two-level system values quickly converge, here the two variants converge much slower. This is because adding a single excitation to a two-level system cannot occur if that system is already excited; an explicit dependence on the structure of the underlying Hilbert space is introduced with the assumption of a changing state. For the entanglement witness, on the other hand, only the proportion of the state which exceeds the first two levels (for harmonic oscillators) counts towards the difference, and quickly becomes negligible with decreasing $n_\text{th.}$.
• Figure 5: Diagram of the Raman-scattering processes of a coherently driven, linearly coupled oscillator-cavity system, with oscillator frequency $\omega_\text{osc.}$ and linewidth $\gamma$, and cavity resonance frequency $\omega_\text{opt.}$ and linewidth $\kappa$. Anti-Stokes photons from the red detuned ($D_{R}$) drive and Stokes photons from the blue detuned ($D_B$) drive can be separated by filters with linewidths $W_R, W_B \ll \omega_\text{osc.}$.