86 research outputs found
Uniquely List Colorability of Complete Split Graphs
The join of null graph Om and complete graph Kn, denoted by S(m; n), is called a complete split graph. In this paper, we characterize unique list colorability of the graph G = S(m; n). We shall prove that G is uniquely 3-list colorable graph if and only if m>=4, n>=4 and m + n>=10, m(G)>=4 for every 1<=m<=5 and n>=6.The join of null graph Om and complete graph Kn, denoted by S(m; n), is called a complete split graph. In this paper, we characterize unique list colorability of the graph G = S(m; n). We shall prove that G is uniquely 3-list colorable graph if and only if m>=4, n>=4 and m + n>=10, m(G)>=4 for every 1<=m<=5 and n>=6
Entanglement Detection in the Stabilizer Formalism
We investigate how stabilizer theory can be used for constructing sufficient
conditions for entanglement. First, we show how entanglement witnesses can be
derived for a given state, provided some stabilizing operators of the state are
known. These witnesses require only a small effort for an experimental
implementation and are robust against noise. Second, we demonstrate that also
nonlinear criteria based on uncertainty relations can be derived from
stabilizing operators. These criteria can sometimes improve the witnesses by
adding nonlinear correction terms. All our criteria detect states close to
Greenberger-Horne-Zeilinger states, cluster and graph states. We show that
similar ideas can be used to derive entanglement conditions for states which do
not fit the stabilizer formalism, such as the three-qubit W state. We also
discuss connections between the witnesses and some Bell inequalities.Comment: 15 pages including 2 figures, revtex4; typos corrected, presentation
improved; to appear in PR
Multi-party entanglement in graph states
Graph states are multi-particle entangled states that correspond to
mathematical graphs, where the vertices of the graph take the role of quantum
spin systems and edges represent Ising interactions. They are many-body spin
states of distributed quantum systems that play a significant role in quantum
error correction, multi-party quantum communication, and quantum computation
within the framework of the one-way quantum computer. We characterize and
quantify the genuine multi-particle entanglement of such graph states in terms
of the Schmidt measure, to which we provide upper and lower bounds in graph
theoretical terms. Several examples and classes of graphs will be discussed,
where these bounds coincide. These examples include trees, cluster states of
different dimension, graphs that occur in quantum error correction, such as the
concatenated [7,1,3]-CSS code, and a graph associated with the quantum Fourier
transform in the one-way computer. We also present general transformation rules
for graphs when local Pauli measurements are applied, and give criteria for the
equivalence of two graphs up to local unitary transformations, employing the
stabilizer formalism. For graphs of up to seven vertices we provide complete
characterization modulo local unitary transformations and graph isomorphies.Comment: 22 pages, 15 figures, 2 tables, typos corrected (e.g. in measurement
rules), references added/update
Some Conclusion on Unique k
If a graph G admits a k-list assignment L such that G has a unique L-coloring, then G is called uniquely k-list colorable graph, or UkLC graph for short. In the process of characterizing UkLC graphs, the complete multipartite graphs K1*r,s(r,s∈N) are often researched. But it is usually not easy to construct the unique k-list assignment of K1*r,s. In this paper, we give some propositions about the property of the graph K1*r,s when it is UkLC, which provide a very significant guide for constructing such list assignment. Then a special example of UkLC graphs K1*r,s as a application of these propositions is introduced. The conclusion will pave the way to characterize UkLC complete multipartite graphs
Multiparticle entanglement purification for two-colorable graph states
We investigate multiparticle entanglement purification schemes which allow
one to purify all two colorable graph states, a class of states which includes
e.g. cluster states, GHZ states and codewords of various error correction
codes. The schemes include both recurrence protocols and hashing protocols. We
analyze these schemes under realistic conditions and observe for a generic
error model that the threshold value for imperfect local operations depends on
the structure of the corresponding interaction graph, but is otherwise
independent of the number of parties. The qualitative behavior can be
understood from an analytically solvable model which deals only with a
restricted class of errors. We compare direct multiparticle entanglement
purification protocols with schemes based on bipartite entanglement
purification and show that the direct multiparticle entanglement purification
is more efficient and the achievable fidelity of the purified states is larger.
We also show that the purification protocol allows one to produce private
entanglement, an important aspect when using the produced entangled states for
secure applications. Finally we discuss an experimental realization of a
multiparty purification protocol in optical lattices which is issued to improve
the fidelity of cluster states created in such systems.Comment: 22 pages, 8 figures; replaced with published versio
Uniquely list colorability of the graph Kn2 + Om
Given a list L(v) for each vertex v, we say that the graph G is L-colorable if there is a proper vertex coloring of G where each vertex v takes its color from L(v). The graph is uniquely k-list colorable if there is a list assignment L such that jL(v)j = k for every vertex v and the graph has exactly one L-coloring with these lists. In this paper, we characterize uniquely list colorability of the graph G = Kn2 + Om. We shall prove that if n = 2 then G is uniquely 3-list colorable if and only if m >= 9, if n = 3 and m >=1 then G is uniquely 3-list colorable, if n >=4 then G is uniquely k-list colorable with k =[m/2]+1, and if m>=n-1, entonce G es UnLC
ON GARLANDS IN x-UNIQUELY COLORABLE GRAPHS
a graph G is called x-uniquely colorable, if all its x-colorings induce the same partion of the vertex set into one-color components. For x-uniquely colorable graphs new bound of the number of vertex set partions into x + 1 cocliques is found. © 2019 gein p.a
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