Entanglement transference and non-inertial quantum reference frames

Abstract

Given the recent interest in perspectival quantum reference frames (QRFs), we ask how quantum properties in the perspectival picture relate to their global, non-perspectival counterparts. It is instructive to establish this link, as most known results in quantum information theory are derived in the latter context. Specifically, we find sufficient conditions under which global entanglement decomposes into a combination of perspectival entanglement and coherence -- a phenomenon that we call entanglement transference. We apply this result to non-inertial QRFs, in particular, revisiting the problem of entanglement degradation. We find that entanglement degradation in the perspectival picture can be offset by an increase in coherence resources. The non-inertial problem may also provide clues to understanding perspectival QRFs in curved spacetime.

Figures (6)

• Figure 1: Schematic diagram of Alice, Rob, and anti-Rob's qubits. Alice (A) is inertial, while Rob (R) accelerates uniformly in region I. Another observer, anti-Rob ($\overline R$), travels along a complementary hyperbola in region II. Alice and Rob's particle detectors, which measure a Dirac field, are assumed to be sensitive to the same frequency.
• Figure 2: Visual representation of the even and odd parity states, $\ket{E}$ (blue) and $\ket{O}$ (orange) respectively. The tetrahedral arrangement is drawn in explicitly for $\ket{E}$. All vertices for a given parity state are separated by Hamming distance 2.
• Figure 3: A pictorial representation of entanglement transference relating the perspectival entanglement and coherence (top) to the global entanglement (bottom) for the case of $\mathcal{E}^{(A)}_{R,\overline R}+\mathcal{C}^{(A)}_{R}=\mathcal{E}_{\overline R,AR}$ and $\mathcal{E}^{(A)}_{R,\overline R}+\mathcal{C}^{(A)}_{\overline R}=\mathcal{E}_{R,A\overline R}$. The top diagram shows the perspective of Alice's qubit, with the entanglement $\mathcal{E}^{(A)}_{R,\overline R}$ depicted by the yellow dashed line connecting Rob and anti-Rob's qubits. The coherences $\mathcal{C}^{(A)}_{R}$ and $\mathcal{C}^{(A)}_{\overline R}$ are represented by blue and red dashed ovals around the Rob's and anti-Rob's qubits respectively. The bottom diagram shows two distinct bipartitions of the 3-qubit state $\ket{\psi}_{AR\overline R}$ for which entanglement is calculated. The entanglement between $\overline R$ and the joint system $AR$, $\mathcal{E}_{\overline R,AR}$, is shown in blue, while the entanglement between $R$ and the joint system $A\overline R$, $\mathcal{E}_{R,A\overline R}$, is shown in red. The yellow dashed quantity plus the blue (red) dashed quantity in the top diagram equals the blue (red) solid quantity in the bottom one.
• Figure 4: Comparison of entanglement entropy $\mathcal{E}$ plus subsystem (entropy of) coherence $\mathcal{C}$ from the perspectives of (a) anti-Rob, (b) Rob, and (c) Alice in the three-qubit system. Entanglement entropy from each perspective is depicted by the yellow dashed line. Blue colour corresponds to the clockwise cyclic direction, defined to be $A\rightarrow R\rightarrow\overline R\rightarrow A$, while red colour corresponds to the anti-clockwise cyclic direction, $A\rightarrow\overline R\rightarrow R\rightarrow A$. For example, in plot (a), the subsystem coherence of Alice from anti-Rob's perspective is given by the blue dashed line, since Alice is cyclically "downstream" of anti-Rob. If anti-Rob makes reference to Rob, who is in the reverse direction, then the corresponding subsystem coherence is the red dashed line. The totals $\mathcal{E}+\mathcal{C}$ are given in each plot by the solid lines, which follow the same colour convention. Observe that Eqs. \ref{['eq:pair1']}-\ref{['eq:pair3']} are satisfied, since red solid lines in a given panel are the same as the blue solid lines in the (cyclically) right adjacent panel.
• Figure 5: Comparison of linear entropy $\mathcal{E}$ plus subsystem $\ell^2$-norm coherence $\mathcal{C}$ from the perspectives of (a) anti-Rob, (b) Rob, and (c) Alice in the three-qubit system. Linear entropy from each perspective is depicted by the yellow dashed line. Blue colour corresponds to the clockwise cyclic direction, defined to be $A\rightarrow R\rightarrow\overline R\rightarrow A$, while red colour corresponds to the anti-clockwise cyclic direction, $A\rightarrow\overline R\rightarrow R\rightarrow A$. For example, in plot (a), the subsystem coherence of Alice from anti-Rob's perspective is given by the blue dashed line, since Alice is cyclically "downstream" of anti-Rob. If anti-Rob makes reference to Rob, who is in the reverse direction, then the corresponding subsystem coherence is the red dashed line. The totals $\mathcal{E}+\mathcal{C}$ are given in each plot by the solid lines, which follow the same colour convention. Observe that Eqs. \ref{['eq:pair1']}-\ref{['eq:pair3']} are satisfied, since red solid lines in a given panel are the same as the blue solid lines in the (cyclically) right adjacent panel.