[Paper Review] Electron-hole entanglement in the Fermi sea

C. W. J. Beenakker|arXiv (Cornell University)|Aug 20, 2005

Advanced Physical and Chemical Molecular InteractionsChemistry2 references16 citations

TL;DR

This paper proposes that electron-hole pairs created via tunneling in a Fermi sea naturally form spin-entangled Bell states due to the Pauli exclusion principle and particle conservation, without requiring optical or interaction-based mechanisms. The key result is that the entanglement can be detected via noise measurements in a four-terminal geometry, with the Bell inequality violation reaching $\mathcal{B}_{\text{max}} = 2\sqrt{1 + \mathcal{C}^2}$ beyond the tunneling regime, confirming robust nonlocality in electronic systems.

ABSTRACT

I. Introduction (Preface, Exciton entanglers, Photon entanglers) II. Entanglement basics (Quantum versus classical correlations, Bell inequality, Entanglement measures for pure states, Entanglement measures for mixed states, Particle conservation, Phase reference) III. How to entangle free particles (Free bosons, Free fermions) IV. Spin versus orbital entanglement V. Entanglement detection by noise measurements (Tunneling regime, Beyond the tunneling regime, Full counting statistics) VI. Loss of entanglement by dephasing VII. Quantum entanglement pump VIII. Teleportation by electron-hole annihilation IX. Three-qubit entanglement X. The experimental challenge

Motivation & Objective

• To establish that single-particle tunneling in a metal can generate nonlocal electron-hole spin entanglement without requiring electron-electron interactions or optical excitation.
• To provide a theoretical framework for detecting such entanglement using full counting statistics and noise correlators in electronic circuits.
• To extend the validity of entanglement detection beyond the tunneling regime, where previous analyses were limited by approximations.
• To demonstrate that the concurrence and Bell inequality violation remain quantitatively linked, even in the non-tunneling regime, via a noise-based measurement protocol.

Proposed method

• Models electron-hole pair creation via voltage-biased tunneling across a barrier, where an electron tunnels from one reservoir to another, leaving behind a hole in the Fermi sea.
• Uses a two-channel scattering matrix formalism with transmission eigenvalues $\tau_1, \tau_2$ and unitary matrices $U_L, U_R, U_0$ to describe the scattering process.
• Derives the spin-spin correlator $C_{\bm{ab}}$ from the noise correlators and mean currents using the full counting statistics formalism.
• Expresses the correlator in terms of rotated spin measurement bases via unitary matrices $U_{\bm{a}}, U_{\bm{b}}$, enabling arbitrary measurement directions.
• Calculates the maximal Bell inequality violation $\mathcal{B}_{\text{max}}$ by optimizing over measurement settings, yielding $\mathcal{B}_{\text{max}} = 2\sqrt{1 + \mathcal{C}^2}$ beyond the tunneling regime.
• Introduces a four-terminal geometry to isolate outgoing currents from incoming ones, ensuring noise correlators depend only on the tunneling process and not on source currents.

Experimental results

Research questions

• RQ1 Can electron-hole pairs created by single-particle tunneling in a Fermi sea exhibit nonlocal quantum entanglement?
• RQ2 How does the entanglement strength, quantified by concurrence $\mathcal{C}$, relate to the violation of the CHSH Bell inequality in electronic systems?
• RQ3 Does the standard tunneling approximation limit the validity of entanglement detection, and can the framework be extended to the non-tunneling regime?
• RQ4 What is the role of noise measurements and full counting statistics in detecting entanglement in solid-state electronic systems?
• RQ5 Can the entanglement be robustly detected without requiring optical interfaces or strong electron-electron interactions?

Key findings

• Electron-hole pairs created by tunneling in a Fermi sea form a spin-entangled Bell state $2^{-1/2}(|\!\uparrow_h\uparrow_e\rangle + |\!\downarrow_h\downarrow_e\rangle)$, arising from the Pauli exclusion principle and particle conservation.
• In the tunneling regime ($\tau_1, \tau_2 \ll 1$), the maximal Bell parameter is $\mathcal{B}_{\text{max}} = 2\sqrt{1 + \kappa^2 \mathcal{C}^2}$, with $\kappa \approx 1$ and $\kappa \to 1$ in the limit of small transmissions.
• Beyond the tunneling regime, the maximal Bell violation simplifies to $\mathcal{B}_{\text{max}} = 2\sqrt{1 + \mathcal{C}^2}$, confirming the universal relation between concurrence and nonlocality.
• The spin-spin correlator $C_{\bm{ab}}$ is derived from noise correlators and mean currents using full counting statistics, valid even without the tunneling approximation.
• The framework allows detection of entanglement via measurable current noise in a four-terminal geometry, where only outgoing currents contribute to the noise correlators.
• The derivation shows that the entanglement is robust and detectable at finite temperature, with a critical temperature above which entanglement vanishes, as derived in Section III.2.