Abstract
Unextendible product bases (UPBs) play an important role in quantum information theory. However, very little is known about UPBs in Hilbert space of local dimension more than three. In this paper, we study the UPBs in qutrit-ququad system and find that there only exist six, seven and eight-state UPBs. We completely characterize the six-state and seven-state UPBs. For eight-state UPBs, seven classes of UPBs are found. As auxiliary results, we study the distinguishability of qutrit-ququad UPBs by separable measurements, and find that there exists a UPB that cannot be distinguished.
Highlights
Unextendible product bases (UPBs) play an important role in quantum information theory
It is well known that the members of any nontrivial unextendible product bases (UPBs) cannot be perfectly discriminated by local operations and classical communication (LOCC)[1], which exhibits the phenomenon “quantum nonlocality without entanglement”[2]
Every edge e ∈ E1 is represented by red solid line, and every edge e ∈ E2 is represented by black dotted line in orthogonality graphs
Summary
Unextendible product bases (UPBs) play an important role in quantum information theory. It should be noted that there does not exist a six-state UPB whose orthogonality graph contains multiple edges. We will prove that any UPB with seven states must have the only orthogonality graph, i.e., Fig. 3.
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