Minimal-support frontier for absolutely maximally entangled states
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Problem
Determine, for every integer \(d\geq 2\), the largest number \(\mathcal N(d)\) of \(d\)-level parties that admit an absolutely maximally entangled state of minimal computational-basis support. To make this precise, let \(m=\lfloor N/2\rfloor\) and write a normalized state as \(|\psi\rangle=\sum_{\boldsymbol{x}\in[d]^N}c_{\boldsymbol{x}} |\boldsymbol{x}\rangle\), where \([d]=\{0,\ldots,d-1\}\). The state is \(\operatorname{AME}(N,d)\) when every subsystem \(S\subseteq\{1,\ldots,N\}\) with \(|S|\leq m\) has reduced state
Condition (1) implies the computational-basis support bound
Minimal support means equality in (2), and the frontier to be determined is
Equivalently, for any alphabet \(\mathcal A\) of size \(d\), the existence event in (3) is characterized by
Thus (4) asks for general, possibly nonlinear, MDS codes rather than only linear codes [GAL+15], [Ber19].
Source
Bernal formulates the minimal-support AME frontier through its equivalent maximum-distance-separable-code problem and obtains only a conditional general determination [Ber19].
Progress
Goyeneche et al. proved the MDS equivalence (4), while Bernal proved that, for fixed \(d\), the admissible party numbers form the initial interval \(2\leq N\leq\mathcal N(d)\) [GAL+15], [Ber19]. This reduces the frontier problem to a maximum-length question for unrestricted \(d\)-ary MDS codes, which is not known in general.
Bernal obtained the exact small-alphabet values and the prime-power lower bound
\begin{equation} \begin{gathered} \mathcal N(2)=3,\quad \mathcal N(3)=4,\quad \mathcal N(4)=6, \quad \mathcal N(5)=6,\quad \mathcal N(6)=3,\quad \mathcal N(7)=8,\\ \mathcal N(d)\geq d+1 \quad\text{for every prime power }d\geq3. \end{gathered} \tag{5} \end{equation}The value \(\mathcal N(4)=6=d+2\) in (5) comes from the even-field exceptional MDS parameters at combinatorial dimension \(3\); consequently, equality with \(d+1\) cannot hold uniformly over prime powers [Ber19].
Let \(M(k,d)\) denote the maximum length of any, not necessarily linear, \(d\)-ary MDS code of cardinality \(d^k\). Bernal proved the conditional implication
\begin{equation} \left. \begin{gathered} d\geq8\text{ is a prime power},\qquad k_0=\left\lfloor\frac{d+2}{2}\right\rfloor,\\ M(k_0,d)=d+1 \end{gathered} \right\} \quad\Longrightarrow\quad \mathcal N(d)=d+1. \tag{6} \end{equation}The general MDS conjecture supplies the premise in (6). Its possible \(d+2\) exceptions for \(d=2^j\) and \(k\in\{3,d-1\}\) do not apply here because \(3<k_0<d-1\) for \(d\geq8\); the missing step is the conjecture for general nonlinear codes [Ber19].
Comment
The unresolved task is the exact existence frontier, not the classification of states realizing it. Support is measured in the fixed computational basis used in (2), and the equivalent coding problem allows arbitrary alphabets and nonlinear codes. The linear MDS conjecture alone therefore does not settle (3).