Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed.These states are relevant for our understanding of multipartite entanglement, quantum information protocols, and the construction of quantum error correction codes.To our knowledge, the only known systematic construction of these quantum states is based on classical error correction codes.We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-uniform state constructed from the so-called maximum distance separable error correction codes.Furthermore, we use our method to construct several examples of absolutely maximally entangled states whose existence was open so far.
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.
A pure quantum state of
n parties associated with the Hilbert space $\CC^{d_1}\otimes \CC^{d_2}\otimes\cdots\otimes \CC^{d_n}$ is called
k-uniform if all the reductions to
k-parties are maximally mixed. The
n partite system is called homogenous if the local dimension
d1=d2=⋯=dn, while it is called heterogeneous if the local dimension are not all equal.
k-uniform sates play an important role in quantum information theory. There are many progress in characterizing and constructing
k-uniform states in homogeneous systems. However, the study of entanglement for heterogeneous systems is much more challenging than that for the homogeneous case. There are very few results known for the
k-uniform states in heterogeneous systems for
k>3. We present two general methods to construct
k-uniform states in the heterogeneous systems for general
k. The first construction is derived from the error correcting codes by establishing a connection between irredundant mixed orthogonal arrays and error correcting codes. We can produce many new
k-uniform states such that the local dimension of each subsystem can be a prime power. The second construction is derived from a matrix
H meeting the condition that
HA×Aˉ+HAˉ×AT has full rank for any row index set
A of size
k. These matrix construction can provide more flexible choices for the local dimensions, i.e., the local dimensions can be any integer (not necessarily prime power) subject to some constraints. Our constructions imply that for any positive integer
k, one can construct
k-uniform states of a heterogeneous system in many different Hilbert spaces.
A pure state of
N parties with local dimension
d is called a
k-uniform state if all the reductions to
k parties are maximally mixed. Based on the connections among
k-uniform states, orthogonal arrays and linear codes, we give general constructions for
k-uniform states. We show that when
d≥4k−2 (resp.
d≥2k−1) is a prime power, there exists a
k-uniform state for any
N≥2k (resp.
2k≤N≤d+1). Specially, we give the existence of
4,5-uniform states for almost every
N-qudits. Further, we generalize the concept of quantum information masking in bipartite systems given by [Modi \emph{et al.} {Phys. Rev. Lett. \textbf{120}, 230501 (2018)}] to
k-uniform quantum information masking in multipartite systems, and we show that
k-uniform states and quantum error-correcting codes can be used for
k-uniform quantum information masking.
Abstract 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 stabilizer 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 with the highest distance allowed by the quantum Singleton bound.
A pure quantum state of
N subsystems with
d levels each is called
k-multipartite maximally entangled state, which we call a
k-uniform state, if all its reductions to
k qudits are maximally mixed. These states form a natural generalization of
N-qudit Greenberger-Horne-Zeilinger states which belong to the class 1-uniform states. We establish a link between the combinatorial notion of orthogonal arrays and
k-uniform states and prove the existence of several classes of such states for
N-qudit systems. In particular, known Hadamard matrices allow us to explicitly construct 2-uniform states for an arbitrary number of
N>5 qubits. We show that finding a different class of 2-uniform states would imply the Hadamard conjecture, so the full classification of 2-uniform states seems to be currently out of reach. Furthermore, we establish links between the existence of
k-uniform states and classical and quantum error correction codes and provide a graph representation for such states.