Finite Gaussian assistance protocols and a conic metric for extremizing spacelike vacuum entanglement

TL;DR

This work addresses the problem of quantifying and extracting pure entanglement resources between two disjoint Gaussian regions $A$ and $B$ by using classical Gaussian measurements in an external purification $C$, thereby bounding Gaussian entanglement of assistance $GEOA$ and Gaussian entanglement of formation $GEOF$. It introduces a direct protocol to construct a hierarchy of C-measurements that remove Gaussian noise $Y$ from $AB$ without divergences, and develops a multimode conic framework of double-cone volumes (DCVs) with an inter-DCV distance $\xi$ that serves as a necessary and sufficient entanglement quantifier. The key results for disjoint regions of the free scalar field vacuum show that the GEOA lower bound decays slower than two-point correlations (double-log behavior for certain regimes) while the GEOF upper bound decays exponentially in a manner mirroring negativity, and that a single $C$-measurement can maximize purified $AB$ entanglement. These findings provide tight, computable bounds via semidefinite cone geometry and MNF-type constructions, extendable to interacting theories and general Gaussian many-body states, and offer practical pathways for simulating or measuring large-scale quantum correlations in complex systems.

Abstract

In a pure Gaussian tripartition, a range of entanglement between two parties ($AB$) can be purified through classical communication of Gaussian measurements performed within the third ($C$). To begin, this work introduces a direct method to calculate a hierarchic series of projective $C$ measurements for the removal of any $AB$ Gaussian noise, circumventing divergences in prior protocols. Next, a multimode conic framework is developed for pursuing the maximum (Gaussian entanglement of assistance, GEOA) or minimum (Gaussian entanglement of formation, GEOF) pure entanglement that may be revealed or required between $AB$. Within this framework, a geometric necessary and sufficient entanglement condition emerges as a doubly-enclosed conic volume, defining a novel distance metric for conic optimization. Extremizing this distance for spacelike vacuum entanglement in the massless and massive free scalar fields yields (1) the highest known lower bound to GEOA, the first that decays slower than the two-point correlation functions and (2) the lowest known upper bound to GEOF, the first that decays exponentially mirroring the mixed $AB$ negativity. Furthermore, combination of the above with a generalization of previous partially-transposed noise filtering techniques allows calculation of a single $C$ measurement that maximizes the purified $AB$ entanglement. Beyond expectation that these behaviors of spacelike GEOA and GEOF persist in interacting theories, the present measurement and optimization techniques are applicable to physical many-body Gaussian states beyond quantum fields.

Figures (8)

• Figure 1: Diagram illustrating the assisted entanglement protocol applied to the scalar field vacuum. The global vacuum is partitioned into three spatial regions: two disjoint patches, $A$ and $B$, each of size $d$ and separated by distance $\tilde{r}$, with the external volume $C$. As depicted in the quantum circuit, classical communication of measurements within $C$ informs controlled single-mode displacements that identify pure entanglement resources shared between $A$ and $B$. The teal unitary gate defines the (entangled) measurement basis followed by single-mode measurements along the quadrature axes, $\phi$ or $\pi$. The present work calculates extremal entanglement purified in $AB$ corresponding to a tight lower(upper) bound on GEOA(GEOF), achieved by derivations of $U$ maximizing(minimizing) the multimode conic distance metric $\xi$ introduced in Sec. \ref{['sec:IIIA']}.
• Figure 2: Logarithmic negativity as a function of the separation $\tilde{r}$ between $AB$ in an infinite one-dimensional massless free lattice scalar field vacuum $(m, d) = \left( 10^{-10}, 10\right)$. Previous results of the logarithmic negativity upon volume tracing scalar1dextraNKnegativitysphereNKentsphere ($\mathcal{N}_{\sigma_{AB}}$, black), MNF-identified pure state gao2024partialtransposeguided and pure state identified through field-basis measurement NKvolumemeasure ($\mathcal{N}_{\sigma_{0}}^{\text{MNF}_{\mathcal{V}_{\mathcal{N}}}}$ and $\mathcal{N}_{\sigma_{0}}^{\phi}$, dark gray), and pure state identified through momentum-basis measurement NKvolumemeasure ($\mathcal{N}_{\sigma_{0}}^{\pi}$, light gray) are shown in all three panels for context. At this resolution, the field regions are separable at and beyond a dimensionless separation $\tilde{r}_{{ {\cal N} /}} = 98$. The light(dark) gray indicate previous lower(upper) bounds to the GEOA(GEOF), each of which are surpassed by results of this work presented in teal. (Left) Tight upper bounds on GEOF. In a log-scale plot, highlighting the exponential decay with increasing separation, $\mathcal{N}_{\sigma_{0}}^{\min \xi}$ depicts the logarithmic negativity of the pure state identified through minimizing distance $\xi$ between $DCV_+$ and $DCV_-$. (Right) Tight lower bounds on GEOA. In a linear-scale plot, highlighting the double-logarithmic decay with increasing separation, $\mathcal{N}_{\sigma_{0}}^{\text{MNF}_{\mathcal{V}_\pm}} = \mathcal{N}_{\sigma_{0}}^{\max \xi}$ depicts the logarithmic negativity of the pure state identified through maximizing $\xi$, or alternatively through the generalized MNF procedure (Appendix \ref{['sec:appcsec3']}).
• Figure 3: Schematic representation of the projective measurement procedures in $C$ realizing Gaussian noise decomposition $\sigma_{AB} = \sigma_{0} + Y$, with classical communication of the measurement results that control local displacement operators in $AB$. The top part of the diagram presents the iterative procedure that calculates profiles of projective measurements for each rank-one noise component, whereas the bottom part depicts an equivalent circuit simplification performing a single transformation in $C$. Since symplectic transformations are unitary in Hilbert space, the symplectic operations updating the covariance matrix, i.e., $\sigma \rightarrow S \sigma S^T$, are depicted in the form of unitary quantum circuitry. Each line of the quantum circuit represents a Gaussian continuous variable mode, where dashed lines indicate pure optical-vacuum modes that have decoupled after transformation into the local normal mode basis. The teal vertical lines with slash notation indicate the number of modes. In the presented scenario, $Y_1^{(1)}$ and $Y_1^{(3)}$ do not create new purities, necessitating a pure optical-vacuum mode in $S_{\langle\bar{p}|}^{(1)}$ and $S_{\langle\bar{p}|}^{(3)}$ as in Eq. \ref{['eq:pveccons']}, thus the number of impurities remains unchanged in the next iteration. However, $Y_1^{(2)}$ is chosen to generate a purity in $\sigma_{AB}$, thus a pure optical-vacuum mode is not utilized in $S_{\langle\bar{p}|}^{(2)}$ as in Eq. \ref{['eq:pveccons0']}.
• Figure 4: Classical communication procedure for projective measurements in $C$ that produces the $AB$ pure state lower-bounding the GEOA between disjoint regions of the 100-site OBC lattice scalar field vacuum, $(m, d, \tilde{r}) = (0,10, 20)$. After the measurement and communication of each $\phi$-$\pi$ collective profile pair, $j$, an additional $(1_A \times 1_B)$ pure core is generated in $AB$ with negativity $\mathcal{N}_{AB,j}$. Governing these profiles by the rank-one matrix decomposition provided by the $\mathcal{V}_{\pm}$-SOL MNF procedure allows maximization of $\mathcal{N}_{AB,1}$. As depicted by the diminishing size of profiles at later points in the measurement and communication sequence, and shown explicitly in the associated plot, the total pure negativity available in $AB$ after $\Lambda$ pairs, $\mathcal{N}_{AB}^{(\Lambda)} = \sum_{j = 1}^\Lambda \mathcal{N}_{AB,j}$, rapidly converges. A symplectic transformation, $S_{\{ \langle p| \} }$, localizing these profiles within the volume $C$ prior to measurement may be constructed following Eqs. \ref{['eq:gsreplace1']} and \ref{['eq:gsreplace2']}.
• Figure 5: Projections of $DCV$ to (left) plane $x'$-y that contains the semi-major axis of the ellipse and (right) plane $z'$-y that contains the semi-minor axis of the ellipse.