Characterizing Genuine Multilevel Entanglement

Entanglement of high-dimensional quantum systems has become increasingly important for quantum communication and experimental tests of nonlocality. However, many effects of high-dimensional entanglement can be simulated by using multiple copies of low-dimensional systems. We present a general theory to characterize those high-dimensional quantum states for which the correlations cannot simply be simulated by low-dimensional systems. Our approach leads to general criteria for detecting multilevel entanglement in multiparticle quantum states, which can be used to verify these phenomena experimentally.

Figure 1

Left: The four-dimensional maximally entangled state $|{\psi }_4⟩$ shared by the parties $A$ and $B$ directly decomposes in two entangled pairs of qubits shared by $A_1{B}_1$ and $A_2{B}_2$. Right: More generally, we ask whether a high-dimensional entangled state can be decomposed into pairs of entangled systems of smaller dimension, up to some local unitary operations. We show that this is not always possible and characterize those states carrying genuine multilevel entanglement.

Figure 2

Examples of weighted graph states. Left: The four-ququart chain-graph state from Eq. (14) can be encoded into a weighted graph state of eight qubits, see Eq. (16). Right: After application of the unitaries $U_{A_1{A}_2}$ and $U_{D_2{D}_1}$ the state exhibits decomposability with respect to the bipartitions $A|BCD$, $D|ABC$ and $AD|BC$ [see Eq. (17)] and thus the original ququart state is bidecomposable and not GMME.