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 seven-qubit 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.
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 seven-qubit 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.
A pure multipartite quantum state is called absolutely maximally entangled if all reductions of no more than half of the parties are maximally mixed. However, an n-qubit absolutely maximally entangled state exists only when n equals 2, 3, 5, and 6. A natural question arises when it does not exist: Which n-qubit pure state has the largest number of maximally mixed $\ensuremath{\lfloor}n/2\ensuremath{\rfloor}$-party reductions? We denote this number by Qex(n). It was shown that Qex(4)=4 by Higuchi and Sudbery [Phys. Lett. A 273, 213 (2000)] and Qex(7)=32 by Huber et al. [Phys. Rev. Lett. 118, 200502 (2017)]. In this paper, we give a general upper bound of Qex(n) by linking the well-known Tur\'an problem in graph theory and provide lower bounds using constructive and probabilistic methods. In particular, we show that Qex(8)=56, which is the third known value for this problem.
Huber et al. [Phys. Rev. Lett. 118 (2017) 200502] have proved that a seven-qubit state whose three-body marginal states are all maximally mixed does not exist. Here, we propose a method to build a maximally entangled state based on orthogonal arrays to construct maximally entangled seven-qubit states. Using this method, we not only determine that a seven-qubit state whose three-body marginals are all maximally mixed does not exist, but also find the condition for maximally entangled seven-qubit states. We consider that π ME = 19/140 is a condition for maximally entangled seven-qubit states. Furthermore, we derive three forms of maximally entangled seven-qubit states via orthogonal arrays.
Absolutely maximally entangled states require every reduction of at most half of the parties to be maximally mixed, a condition that is both rigid and often impossible for qubit systems. Previous work introduced the quantum extremal number, which maximizes the number of exactly maximally mixed half-body marginals, and determined the exact value Qex(8,4)=56. The present work develops a robust extension of this extremal problem. For a subsystem A, the marginal maximal-mixing defect is defined by DA=2∣A∣Tr(ρA2)−1=2∣A∣ρA−2∣A∣IA22, and Qex,εD(n,k) is defined as the maximum number of k-body marginals satisfying DA≤ε in an n-qubit pure state. This counting problem differs from approximate k-uniformity, which requires all k-body marginals to obey a common error bound.
For pure states on 4m qubits, the following local stability inequality is established: ∑i∈TDT∖{i}≥1(∣T∣=2m+1). It follows that, whenever ε<1/(2m+1), the hypergraph of ε-good 2m-subsets is K2m+1(2m)-free. Combined with the known exact eight-qubit construction, this yields the stability plateau Qex,εD(8,4)=56,0≤ε<51. For odd systems of 2k+1 qubits, the exact forbidden hypergraph Hk is used to derive explicit finite-error stability radii. In particular, Qex,εD(9,4)≤120 for 0≤ε<1/17. These results turn exact quantum Turán obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.
Abstract We present a generalized criterion for maximally entangled nine-qubit states, whose minimum averaged subsystem purity should be equal to 1/14. In this note, we prove that the absolutely maximally entangled state for nine qubits does not exist. In other words, we pronounce that the pure (9, 1, 5) 2 quantum error-correcting code does not exist. Further, we construct a new genuine nine-qubit maximally entangled states of 110,15 and 1 balanced purities equal to 1/16, 1/8 and 1/4, respectively. We found that the marginal density matrices for subsystems of 1,2,3- qubits all completely mixed in these states; therefore, maximally entangled nine-qubit state is a 3-uniform state.
Abstract A pure multipartite quantum state is called absolutely maximally entangled (AME), if all reductions obtained by tracing out at least half of its parties are maximally mixed. Maximal entanglement is then present across every bipartition. The existence of such states is in many cases unclear. With the help of the weight enumerator machinery known from quantum error correction and the shadow inequalities, we obtain new bounds on the existence of AME states in dimensions larger than two. To complete the treatment on the weight enumerator machinery, the quantum MacWilliams identity is derived in the Bloch representation. Finally, we consider AME states whose subsystems have different local dimensions, and present an example for a 2 × 3 × 3 × 3 system that shows maximal entanglement across every bipartition.
Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and, in particular, provide explicit closed form expressions for AME states of n parties with local dimension q a power of a prime for all q≥n−1. Building on this, we construct a generalization of the Bell-basis consisting of AME states and develop a stabilizer formalism for AME states. For every q≥n−1 prime we show how to construct QECCs that encode a logical qudit into a subspace spanned by AME states. Under a conjecture for which we provide numerical evidence, this construction produces a family of quantum error correcting codes [[n,1,n/2]]q for n even, saturating the quantum Singleton bound. We show that our conjecture is equivalent to the existence of an operator whose support cannot be decreased by multiplying it with stabilizer products and explicitly construct the codes up to n=8.