Absolutely Maximally Entangled States of Seven Qubits Do Not Exist

Abstract

Pure multiparticle quantum states are called absolutely maximally entangled if all reduced states obtained by tracing out at least half of the particles are maximally mixed. We provide a method to characterize these states for a general multiparticle system. With that, we prove that a sevenqubit state whose three-body marginals are all maximally mixed, or equivalently, a pure ((7, 1, 4))2 quantum error correcting code, does not exist. Furthermore, we obtain an upper limit on the possible number of maximally mixed three-body marginals and identify the state saturating the bound. This solves the seven-particle problem as the last open case concerning maximally entangled states of qubits.Introduction.-Multiparticle entanglement is central for the understanding of the possible quantum advantages in metrology or information processing. When investigating multiparticle entanglement as a resource, the question arises which quantum states are most entangled. For a pure multiparticle quantum state maximal entanglement is present across a bipartition if the smaller of the two corresponding reduced systems is maximally mixed. It is then a natural question to ask whether or not there exist quantum states for any number of parties n, such that all of its reductions to n/2 parties are maximally mixed [1,2]. If this is the case, maximal entanglement is present across all bipartitions and, accordingly, these states are also known as absolutely maximally entangled (AME) states [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. These states have been shown to be a resource for open-destination and parallel teleportation [10], for threshold quantum secret sharing schemes [12], and are a type of quantum error correcting codes [3].If the local dimension is chosen large enough, AME states always exist [12]. For qubits, however, the situation is only partially resolved. The three-qubit Greenberger-Horne-Zeilinger (GHZ) state is an AME state since all the single-qubit reduced states are maximally mixed. For four qubits it was shown that AME states do not exist [2] and best approximations of AME states (where not all reduced states are maximally mixed) have been presented [8]. Five-and six-qubit AME states are known [3,5,7]. These can be represented as graph states and correspond to additive or stabilizer codes used in quantum error correction [3,19]. For more than eight qubits, AME states do not exist [3,[20][21][22].Despite many attempts, the case of seven qubits remained unresolved. Numerical results give some evidence for the absence of an AME state [5][6][7]. By exhaustive search, it was shown that such a state could not have the form of a stabilizer state [19]. Nevertheless, some approximation has been presented by making many but not all three-body marginals maximally mixed [9,23].As shortly mentioned, AME states are a type of pure quantum error correcting codes (QECC), having the max-

MaximumN-body correlations do not in general imply genuine multipartite entanglement

“…We note the analogy of our problem of finding the maximum N -sector with that of deciding the existence of absolutely maximimally entangled (AME) states [8,11,21,36,37]: The region of GHZ dominance corresponds to 'AME state does not exist', while that of Bell dominance relates to 'AME does exist'. The line separating the two corresponds to the Scott bound [11,21].…”

“…An accurate analytical approximation for this line is found by equating S N (GHZ N d ) in Eq. (8) with S N (Bell N d ) = (d 2 − 1) N/2 (for even N ). Assuming d = γ −1 N , this leads to an equation that determines the parameter γ, e −γ = 1 − γ 2 , from which d ≃ 0.6275 · N .…”

Quantum codes from neural networks

“…Recently, it was proved in[HGS17] that an AME state on seven qubits cannot exist. This result completely settled the case of qubit AME states: they exist for = n 2, 3, 5, 6, and only for these n. Furthermore, there are constructions for certain combinations of parameters n d , ()[GBR04, HC13, Hel13, Goy+15, Goy+17].…”

Joint Schmidt-type decomposition for two bipartite pure quantum states

“…We will restrict our considerations to Hilbert spaces of equal dimension, dim H A = dim H B = d (this can always be done by extending the Hilbert space of lower dimension), because it makes the expressions more transparent. The Bloch representation of bipartite states is defined as follows (cf., e.g., [15][16][17][18][19][20][21][22]): Given an orthonormal basis of trace-free Hermitian matrices {h j } (with normalization Tr (h j h k ) = d δ jk and h 0 ≡ 1) we can expand any density operator ρ acting on…”