65 research outputs found
Is DNA computing viable for 3-SAT problems?
AbstractAdleman reported how to solve a 7-vertex instance of the Hamiltonian path problem by means of DNA manipulations. After that a major goal of subsequent research is how to use DNA manipulations to solve NP-hard problems, especially 3-SAT problems. Lipton proposed DNA experiments on test tubes to solve 3-SAT problems. Liu et al. reported how to solve a simple case of 3-SAT using DNA computing on surfaces. Lipton's model of DNA computing is simple and intuitive for 3-SAT problems. The separate (or extract) operation, which is a key manipulation of DNA computing, only extracts some of the required DNA strands and Lipton thinks that a typical percentage might be 90. But it is unknown what would happen due to imperfect extract operation. Let p be the rate, where 0<p<1. Assume that for each distinct string s in a test tube, there are 10l (l=13 proposed by Adleman) copies of s and that extracting each of the required DNA strands is equally likely. Here, the present paper will report, no matter how large l is and no matter how close to 1 p is, there always exists a class of 3-SAT problems such that DNA computing error must occur. Therefore, DNA computing is not viable for 3-SAT
An entanglement measure for n-qubits
Recently, Coffman, Kundu, and Wootters introduced the residual entanglement
for three qubits to quantify the three-qubit entanglement in Phys. Rev. A 61,
052306 (2000). In Phys. Rev. A 65, 032304 (2007), we defined the residual
entanglement for qubits, whose values are between 0 and 1. In this paper,
we want to show that the residual entanglement for qubits is a natural
measure of entanglement by demonstrating the following properties. (1). It is
SL-invariant, especially LU-invariant. (2). It is an entanglement monotone.
(3). It is invariant under permutations of the qubits. (4). It vanishes or is
multiplicative for product states.Comment: 16 pages, no figure
Method for classifying multiqubit states via the rank of the coefficient matrix and its application to four-qubit states
We construct coefficient matrices of size 2^l by 2^{n-l} associated with pure
n-qubit states and prove the invariance of the ranks of the coefficient
matrices under stochastic local operations and classical communication (SLOCC).
The ranks give rise to a simple way of partitioning pure n-qubit states into
inequivalent families and distinguishing degenerate families from one another
under SLOCC. Moreover, the classification scheme via the ranks of coefficient
matrices can be combined with other schemes to build a more refined
classification scheme. To exemplify we classify the nine families of four
qubits introduced by Verstraete et al. [Phys. Rev. A 65, 052112 (2002)] further
into inequivalent subfamilies via the ranks of coefficient matrices, and as a
result, we find 28 genuinely entangled families and all the degenerate classes
can be distinguished up to permutations of the four qubits. We also discuss the
completeness of the classification of four qubits into nine families
SLOCC invariant and semi-invariants for SLOCC classification of four-qubits
We show there are at least 28 distinct true SLOCC entanglement classes for
four-qubits by means of SLOCC invariant and semi-invariants and derive the
number of the degenerated SLOCC classes for n-qubits.Comment: 22 pages, no figures, 9 tables, submit the paper to a journa
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