Existence of absolutely maximally entangled pure states

Cite this problem as Problem 35.

Problem

For what number of systems and local dimension do exist pure states, which are maximally entangled across every bipartition?

Background

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.

This question goes back to the articles [2], [3], and [4]. The seminal article [4] by Higuchi and Sudbery on how entangled can two couples get has shown that there is no AME state of four qubits.

An AME state can be seen as a type of quantum error correcting code (QECC) [5]. Using the language of QECC, the question for the existence of AME(n,D) states is: for what parameters n and D is there a pure ((n, 1, \lfloor n/2\rfloor+1))_D code? here n is the number of subsystems, 1 is dimension of the code subspace, \lfloor n/2\rfloor+1 is the distance of the code, and D is the local dimension. This allows to translate tables such as [6]. Most known AME states arise from stabilizer or graph states [2, 6].

By a code propagation rule for pure codes, AME states yield to a family of quantum maximum distance separable codes (QMDS codes) with parameters ((n-s, D^s, \lfloor n/2\rfloor+1-s))_D for all s=0,..., \lfloor n/2 \rfloor (Theorem 20, [8]). For a ((n-1, D, \lfloor n/2\rfloor))_D this propagation rule is reversible to an ((n, 1, \lfloor n/2\rfloor+1))_D and thus an AME(n,D) state [9].

Partial results

Monogamy relations constrain the sharing of correlations present between subsystems of quantum states. It is thus 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 only existing AME states [5,10], all of which are graph respectively stabilizer states. However, for a local dimension D \geq 3 their existence is an ongoing question.

A recently solved case was that of a four six-level AME state [11]. This case was 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, it was known that AME states exist for all other number of levels D\neq 2,6 [1]. The construction of Rather et al. [11] for this state is particularly interesting, as it does not correspond to a stabilizer or classical code, but instead makes use of quantum Latin squares.

The website [12] keeps track of current results: AME states of 8 ququarts and of 7 quhex systems are among the smallest unresolved cases.

References

[1] Dardo Goyeneche, Daniel Alsina, José I. Latorre, Arnau Riera, and Karol Życzkowski, Phys. Rev. A 92, 032316 (2015).

[2] A.R. Calderbank, E.M. Rains, P.M. Shor, N.J.A. Sloane, IEEE Trans. Inf. Theory 44, 4 (1998).

[3] N. Gisin, H. Bechmann-Pasquinucci, Phys. Lett. A246, 1-6 (1998).

[4] A. Higuchi, A. Sudbery, Phys. Lett. A 273, 213-217 (2000).

[5] A.J. Scott, Phys. Rev. A 69, 052330 (2004).

[6] Markus Grassl. “Bounds on the minimum distance of linear codes and quantum codes.”, http://www.codetables.de. http://codetables.de/

[7] W. Helwig, W. Cui, A. Riera, J. I. Latorre, H.-K. Lo, Phys. Rev. A 86, 052335 (2012).

[8] E. Rains, IEEE Trans. Inf. Theory 44, 4 (1998).

[9] F. Huber, M Grassl, Quantum 4, 284 (2020).

[10] F. Huber, O. Gühne, Jens Siewert, Phys. Rev. Lett. 118, 200502 (2017).

[11] S.A. Rather, A. Burchardt, W. Bruzda, Rajchel-Mieldzioć, Arul Lakshminarayan, and Karol Życzkowski, Phys. Rev. Lett. 128, 080507 (2022).

[12] F. Huber and N. Wyderka, “Table of AME states”, http://www.tp.nt.uni-siegen.de/+fhuber/ame.html.