Absolutely maximally entangled pure states of multipartite quantum systems

TL;DR

This work surveys the landscape of absolutely maximally entangled (AME) states, emphasizing constructions beyond stabilizer/graph methods and analyzing entanglement properties of subsystems. It connects AME states to multi-unitary operators, quantum designs, and classical codes, illustrating how AME$(4,d)$ states emerge from orthogonal Latin squares and their quantum analogues, with notable non-stabilizer examples such as the golden AME$(4,6)$ state. The authors also explore LU equivalence classes, the link to quantum error correction via QMDS codes, and applications to parallel teleportation, entanglement swapping, and quantum secret sharing, culminating in holographic tensor-network perspectives through perfect tensors and HaPPY-type codes. The paper highlights new analytic and numeric constructions, advances in understanding reduced-subsystem entanglement, and ongoing open problems in higher dimensions and experimental realizations, underscoring the deep ties between entanglement structure, combinatorial designs, and quantum information processing.

Abstract

Absolutely maximally entangled (AME) pure states of a system composed of $N$ parties are distinguished by the property that for any splitting at least one partial trace is maximally mixed. Due to maximal possible correlations between any two selected subsystems these states have numerous applications in various fields of quantum information processing including multi-user teleportation, quantum error correction and secret sharing. We present an updated survey of various techniques to generate such strongly entangled states, including those going beyond the standard construction of graph and stabilizer states. Our contribution includes, in particular, analysis of the degree of entanglement of reduced states obtained by partial trace of AME projectors, states obtained by a symmetric superposition of GHZ states, an orthogonal frequency square representation of the "golden" AME state and an updated summary of the number of local unitary equivalence classes.

Figures (9)

• Figure 1: Four party state (\ref{['eq:psi_as_UAB']}). Three symmetric bi-partitions: ($a$) $AB|CD$, ($b$) $AC|BD$, and ($c$) $AD|BC$; and their associated marginal states in terms of the tensor $U\equiv U_{AB}$ and its rearrangements, see Eq. (\ref{['eq:3_rho_matrices']}). Wavy lines represent maximally entangled states between the respective particles. Figure borrowed from Suhail_Thesis.
• Figure 3: Entangling power $e_p(U)$ and gate-typicality $g_t(U)$ defined in (\ref{['eq:EP']}) and (\ref{['eq:GT']}) for two subsystems of local dimension: a) $d=2$, b) $d=3$ and c) $d=4$. Shown unitaries of size $d^2$ enjoy atypically large entangling power. Black stars mark the average values over the Haar measure: $\bar{e_p}=(d^2-1)/(d^2+1)$ and $\bar{g_t}=1/2$. Panel a) demonstrates that 2-unitary matrices of order $d^2=4$ do not exist as there are no matrices for which $e_p(U)=1$, while panel b) shows that for $d^2=9$ there are no dual unitaries (located along the upper dashed boundary line) in the neighborhood of the $2-$unitary permutation matrix $P_9$, in contrast to the case $P_{16}$ shown in panel c).
• Figure 4: Graphs corresponding to AME states of a) two, b) three and c) five qubits. Interestingly, a square graph with $4$ vertices does not represent AME(4,2) state, as it does not exist Higuchi_2000_twoCouples. Further examples of AME graphs are provided in Appendix \ref{['app:AMEtoolbox']}.
• Figure 5: An artistic visualization of the golden state QOLS(6), using cards of 6 different ranks and suits. The outcome of any two dice prepared in such a state determines the outcome of the remaining two dice. Note that a classical solution to Euler's problem would correspond to the array with only one card in each entry. For a full figure created by Paulina Rajchel-Mieldzioć see the original paper Rather_2022.
• Figure 6: Link structure illustrating 1-resistant 4-party states (left) and 2-resistant state (right) borrowed from burchardt2022thesis. Removing any single ring from the left configuration, renders the others in a Borromean configuration Karol-GeoQstates2006. This is analogous to AME$(4,d)$ states being 1-resistant: after removing (tracing away) any single subsystem, the remaining three are still entangled. However, if any two subsystems are traced out, the other two become separable. In contrast, a typical 4-party state is 2-resistant, as performing partial trace over any two particles produces an entangled state of the remaining two, analogous to the link structure on the right.