Quantum Information Processing, Vol. 3, No. 6, December 2004 (© 2004)
Unextendible Product Bases and Locally Unconvertible
Bound Entangled States
S. B. Bravyi
Received April 22, 2004; accepted August 3, 2004
Mutual convertibility of bound entangled states under local quantum operations
and classical communication (LOCC) is studied. We focus on states associated
with unextendible product bases (UPB) in a system of three qubits. A complete
classiﬁcation of such UPBs is suggested. We prove that for any pair of UPBs S
and T the associated bound entangled states ρ
cannot be converted to
each other by LOCC, unless S and T coincide up to local unitaries. More spe-
ciﬁcally, there exists a ﬁnite precision (S,T ) > 0 such that for any LOCC pro-
tocol mapping ρ
into a probabilistic ensemble (p
), the ﬁdelity between ρ
and any possible ﬁnal state ρ
)≤ 1− (S,T ).
KEY WORDS: unextendible product bases; bound entanglement; LOCC.
PACS: 03.65.Bz; 03.67.-a; 89.70+c.
1. INTRODUCTION AND SUMMARY OF RESULTS
One of the most challenging problems in the ﬁeld of quantum infor-
mation is to understand what transformations of multipartite entangled
states can be realized by local quantum operations and classical commu-
nication (LOCC). This problem has many important applications such as
construction of entanglement distillation protocols,
quantum data hid-
and separability check.
The following question is of particular
importance for all these applications. Given mixed states ρ
some ﬁxed composite quantum system, is it possible to convert ρ
by LOCC transformations? More speciﬁcally, we will be interested in
stochastic approximate convertibility. Recall, that an output of a LOCC
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