Existence of absolutely maximally entangled pure states

Cite this problem as Problem 35.

Problem

Pure multiparticle quantum states are called absolutely maximally entangled (AME) or perfect tensors, if all reductions obtained by tracing out at least half of the parties are maximally mixed [1,2]. Thus, such states display maximal entanglement across every bipartition. It is then a natural question to ask for which number n of D-level quantum systems such states do exist, i.e. to determine the existence of AME(n,D) states for all parameters n and D.

AME states can be seen as a type of quantum error correcting codes (QECCs). Using the language of QECC, the question is: for what parameters n and D does a ((n, 1, \lfloor n/2\rfloor+1))_D code exist?

Partial results

As monogamy relations constrain the correlations present between subsystems of quantum states, it is not surprising that such states do not always exist. In fact, for qubits it has been shown that the cases of n=2,3,5, and 6 parties are in fact the only AME states [1,3], all of which are graph/stabilizer states. However, it is an ongoing question for which number of parties n and local dimension D \geq 3 such states exist. The “smallest” unknown case of existence is an AME state of four six-level systems, i.e. an AME(4,6). This case is particularly interesting, as a) due to the number six not being a prime power, many known AME and QECC constructions fail and b) while AME(4,2) is forbidden by monogamy relations, AME states exist for four systems having D=3,4,5,7, or 8 levels each [4].

References

[1] A.J. Scott, arXiv:quant-ph/0310137.

[2] Li et al., arXiv:1612.04504.

[3] FH et al., arXiv:1608.06228.

[4] Goyeneche et al., PRA 92, 032316 (2015).