Reduced and Projected Two-Particle Entanglement at Finite Temperatures

We present a theory for two-particle entanglement production and detection in mesoscopic conductors at finite temperature. The entanglement of the density matrix projected out of the emitted many-body state differs from the entanglement of the reduced density matrix, detectable by current correlation measurements. Under general conditions reduced entanglement constitutes a witness for projected entanglement. Applied to the recent experiment [Neder et al., Nature (London) 448, 333 (2007)] on a fermionic Hanbury Brown Twiss two-particle interferometer we find that despite an appreciable entanglement production in the experiment, the detectable entanglement is close to zero.

Figure 1

(a) Schematic of the two-particle interferometer (2PI) [3] with beam splitters $C,D$ and biased (grounded) contacts 1,2 (3,4). Detector regions $A$ and $B$ (yellow shaded) contain beam splitters and grounded contacts $\pm$. (b) Amplitude-temperature plot for the 2PI with semitransparent splitters $C,D$. Above and to the left of the plotted lines the entanglement of the projected (reduced) density matrix is finite, $C_p>0(C_r>0)$, and a Bell inequality is violated, ${\Omega }_{Bp}^{\max }>2({\Omega }_{Br}^{\max }>2)$. The red ring denotes the parameters of the experiment [4], showing that while the emitted state is clearly entangled, $C_p>0$, it is barely detectable by current and current correlation measurements, $C_r\approx 0$.

Figure 2

(a) Entanglement production $C_p/\mathcal{N}$ (blue, transparent) and $C_r/\mathcal{N}$ (green, opaque) as functions of temperature $kT/eV$ and coherence $\gamma$ for the semitransparent 2PI. (b) Parameter $Q$ as a function of $kT/eV$ (blue line). Values 0.25, ${10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$ shown (gray lines).

Figure 3

Sketch of the general system. A mesoscopic scatterer is connected to $M$ biased and $N-M$ grounded source reservoirs and, via phase gates ${\varphi }_{A/B}$ and beam splitters, to grounded detector reservoirs $A\pm$, $B\pm$. The orbital basis states $|\pm {⟩}_{Ai/Bi}$ are displayed. For details see text.