251 research outputs found
The Structure of Qubit Unextendible Product Bases
Unextendible product bases have been shown to have many important uses in
quantum information theory, particularly in the qubit case. However, very
little is known about their mathematical structure beyond three qubits. We
present several new results about qubit unextendible product bases, including a
complete characterization of all four-qubit unextendible product bases, which
we show there are exactly 1446 of. We also show that there exist p-qubit UPBs
of almost all sizes less than .Comment: 20 pages, 3 tables, 7 figure
The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases)
A long-standing open question asks for the minimum number of vectors needed
to form an unextendible product basis in a given bipartite or multipartite
Hilbert space. A partial solution was found by Alon and Lovasz in 2001, but
since then only a few other cases have been solved. We solve all remaining
bipartite cases, as well as a large family of multipartite cases.Comment: 17 pages, 4 figure
Generalized Multiplicative Domains and Quantum Error Correction
Given a completely positive map, we introduce a set of algebras that we refer
to as its generalized multiplicative domains. These algebras are
generalizations of the traditional multiplicative domain of a completely
positive map and we derive a characterization of them in the unital,
trace-preserving case, in other words the case of unital quantum channels, that
extends Choi's characterization of the multiplicative domains of unital maps.
We also derive a characterization that is in the same flavour as a well-known
characterization of bimodules, and we use these algebras to provide a new
representation-theoretic description of quantum error-correcting codes that
extends previous results for unitarily-correctable codes, noiseless subsystems
and decoherence-free subspaces.Comment: 14 page
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