Entanglement Harvesting from Quantum Field: Insights via the Partner Formula

TL;DR

The paper reformulates Simon's entanglement criterion for bipartite Gaussian detector modes in terms of the partner mode of one detector, linking entanglement harvesting to the spatial overlap between a detector's partner profile and the other detector's profile. It shows that overlap is a necessary condition but not sufficient, and derives a discriminant D that remains invariant under local transformations. In Unruh/Hawking-inspired scenarios, it demonstrates a no-go theorem: detectors built solely from Rindler or Milne annihilation operators cannot harvest entanglement from real Hawking-like particles in the early-stage or Unruh contexts. Exceptions arise when detectors include creation components (squeezing), allowing harvesting through vacuum fluctuations. The work provides a concrete bridge between the partner formalism, profile representations, and horizon physics, with implications for entanglement in black hole spacetimes and relativistic quantum information protocols.

Abstract

We examine the condition necessary for extracting entanglement from a quantum field through the use of two local modes A and B (detector modes). We show that Simon's entanglement criterion for the bipartite Gaussian state can be reformulated in terms of commutators between the canonical operators of the detector mode B and the partner mode P of the detector mode A. Using the profile representation of detector modes, we identify that harvesting is prohibited under certain specific conditions. According to analyses based on moving mirror models, Hawking radiation originates from the Milne modes at past null infinity, that reflect off at the mirror and ultimately transform into real particle modes. Drawing parallels between the Unruh effect and Hawking radiation, our findings indicate an absence of quantum correlations between ``real particles" emitted as Hawking radiation.

Figures (5)

• Figure 1: Setup of our problem. We introduce local modes A and B (detector modes) from the quantum field and investigate entanglement of the bipartite state AB. P denotes the partner mode of A that purifies mode A.
• Figure 2: Penrose diagram depicting one-half of the Minkowski spacetime. In region I, the solid lines denote surfaces of constant $\xi$, while the dashed lines indicate surfaces of constant $\eta$.
• Figure 3: The profile function $q_A(V)$ of the detector mode A placed in region I ($V>0$). Mode A is defined by $\hat{a}_A:=\int_0^\infty d\omega \,\omega^{3/2}e^{-\alpha (\omega-\omega_0)^2}\hat{a}^\text{I}_\omega$ (blue lines), and its partner mode $q_P(V)$ (red lines). The left panel shows the profile functions with $\alpha=50$, $\omega_0=1.5$. The right panel shows the profile functions with $\alpha=1$, $\omega_0=0.05$. In these plots, we choose $a=0.1$ as the acceleration of the Rindler observer.
• Figure 4: The left panel shows the profile functions $p_A^{(1)}(V)$ of detector modes (blue) and its partner mode $p_P^{(1)}(V)$ (red) with $a=1,\omega_-=0.1,\omega_+=1000,V_A=-3$. The right panel shows the profile functions $p_A^{(2)}(V)$ of detector modes (blue) and its partner mode $p_P^{(2)}(V)$ (red) with $L=1$, $\varepsilon=0.138$, $V_A=-3$. The green dashed line is the profile function of the detector mode with $\varepsilon=0$. Note that the partner mode is ill-defined with $\varepsilon=0$.
• Figure 5: Logarithmic negativity with type 1 detectors ($\omega$-top-hat weighting function) (blue) and with type 2 detectors (the sin-cos profile) (orange). Parameters are $a=1,\omega_-=0.1,\omega_+=50,V_A=-1,L=2\pi /(\omega_+-\omega_-)$, and $\varepsilon=0.02$.

Theorems & Definitions (2)

• Theorem 1: No-go theorem of harvesting
• Corollary 1: No-go theorem for vacuum fluctuation scenario