Absolutely Maximal Entanglement in Continuous Variables

TL;DR

The paper extends the notion of absolutely maximal entanglement (AME) from finite-dimensional systems to continuous-variable (CV) quantum systems, showing that AME is generic among infinitely squeezed Gaussian states and formalizing CV $k$-uniformity. It develops necessary and sufficient conditions on the adjacency matrix $A$ for CV cluster states to be $k$-uniform, providing explicit constructions from Cauchy, Vandermonde, totally positive, and real-block-code matrices, and then extends the framework to non-Gaussian Zak cluster states incorporating discrete Zak bases and GKP resources. The results enable a versatile CV entanglement resource landscape with applications to open-destination CV teleportation, CV quantum secret sharing, MAKD, holographic tensor networks, and multi-unitary circuits, with potential robustness to Gaussian noise. The work connects stabilizer formalisms, classical real-block-code theory, and tensor-network perspectives to advance CV quantum information processing and motivates experimental and hybrid Gaussian/non-Gaussian approaches.

Abstract

We explore absolutely maximal entanglement (AME) and k-uniformity in continuous-variable (CV) quantum systems, and show that-unlike in qudit systems-such entanglement is readily realizable in both Gaussian and non-Gaussian quantum states of multiple modes. We demonstrate that Gaussian CV cluster states are generically AME, rederiving the results of [Phys. Rev. Lett. 103, 070501 (2009)] from a generalized stabilizer formalism, and provide explicit constructions based on Cauchy, Vandermonde, totally positive, and real-block-code generator matrices. We further extend AME properties to a family of non-Gaussian states constructed from discrete Zak basis states that incorporate grid states (a.k.a., Gottesman-Kitaev-Preskill states) as non-Gaussian resources. Realizations of CV AME states enable open-destination multi-party CV teleportation, CV quantum secret sharing, CV majority-agreed key distribution, perfect-tensor networks on arbitrary geometries, and multi-unitary circuits. Our extension to non-Gaussian AME states may further provide robustness to Gaussian noise and benefits to quantum CV information processing.

Figures (2)

• Figure 1: (a) Tensor-network Pastawski_2015Hayden2016randHolographic depiction of $k$-uniformity, the condition that reduced states on any $k$ parties are maximally mixed (i.e., proportional to the identity). States with $k = \lfloor n/2 \rfloor$ are absolutely maximally entangled. (b) An $(n/2)$-uniform state on even-$n$ parties can be converted into a unitary by toggling half of its kets into bras. Uniformity translates into dual-unitarity (right diagram for $n=4$), i.e., contracting along "space-like" directions yields identity.
• Figure 2: Teleportation fidelity of coherent states in the two-mode protocol for cluster state matrices (a) $\left(0010000110000100\right)$, (b) $\left(1111123413610141020\right)$, (c) $\left(111111/21/41/811/41/161/6411/81/641/512\right)$, (d) $\left(0112101311032330\right)$ plotted against the uniform squeezing in decibels. Note that teleportation fidelity is dependent on the bipartition, as shown in Lupo2011Frustration. In our examples, the first two modes are chosen to be on one end, and the last two on the other. To achieve $0.999$ fidelity we require squeezing of (a) $33.007$ dB, (b) $52.712$ dB, (c) $47.710$ dB, (d) $48.582$ dB.

Theorems & Definitions (7)

• Definition 1
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof