A characterization of multipartite quantum states having
N subsystems, based on negativities of matrices obtained by selective partial transposition of the state operator, is proposed. The
K-way partial transpose with respect to a subsystem is constructed by imposing constraints involving the states of
K subsystems of the multipartite composite system. The
K-way negativity, defined as the negativity of
K-way partial transpose, quantifies the
K-way coherences of the composite system. For an
N-partite system the fraction of
K-way negativity $(2\ensuremath{\le}K\ensuremath{\le}N)$, contributing to global negativity, is obtained. The entanglement measures for a given state $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}$ are identified as the partial
K-way negativities of the corresponding canonical state.
A classification of N-partite states, based on K-way negativities (K=2 to N), is proposed. The K-way partial transpose with respect to a subsystem is defined so as to shift the focus to K-way coherences instead of K subsystems of the composite system. For an N-partite system, the fraction of K-way negativity contributing to global negativity, is obtained. After minimizing K-way negativities through local unitary qubit rotations, a combined analysis of 2-way, 3-way and global negativities is shown to provide distinct measures of genuine tripartite, W-state like and bipartite entanglement, for three qubit composite system. To illustrate the point, entanglement of three qubit GHZ class states, W-class states, three boson state and noisy states is analysed. While genuine N-partite entanglement of a composite system is generated by N-way coherences, N-partite entanglement in general can be present due to (K
We propose to quantify three-qubit entanglement using global negativity along with
K-way negativities, where
K=2 and 3. The principle underlying the definition of
K-way negativity for pure and mixed states of
N subsystems is a positive partial transpose sufficient condition. However,
K-way partial transpose with respect to a subsystem is defined so as to shift the focus to
K-way coherences instead of
K subsystems of the composite system. A quantum state of a three-qubit system is characterized by the coherences measured by global, two-way, and three-way negativities. For a canonical state of three-qubit system, entanglement measures for genuine tripartite entanglement, W-like entanglement, and bipartite entanglement can be related to two-way and three-way negativities.
We obtain, analytically, the global negativity, partial
K-way negativities
(K=2,3), Wootter's tangle, and three-tangle for the generic three-qubit canonical state. It is found that the product of global negativity and partial three-way negativity is equal to the three-tangle, while the partial two-way negativity is related to the tangle of qubit pairs. We also calculate similar quantities for the state canonical to a single-parameter
(0<q<1) pure state which is a linear combination of a Greenberger-Horne-Zeilinger state and a
W state. In this case, for
q=0.62685, the state has zero three-tangle and zero three-way negativity, having only
W-like entanglement. The difference between the product of global and partial three-way negativities and the three-tangle for a given state is a quantitative measure of two-qubit coherences transformed by unitary transformations on the canonical state into three-qubit coherences. The global negativity and partial
K-way negativities, obtained by selective partial transpositions on multiqubit state operators, satisfy inequalities which for three qubits are equivalent to the Coffman-Kundu-Wootter inequality.
Partial transposition of state operator is a well-known tool to detect quantum correlations between two parts of a composite system. In this paper, the global partial transpose (GPT) is linked to conceptually multipartite underlying structures in a state---the negativity fonts. If
K-way negativity fonts with nonzero determinants exist, then selective partial transposition of a pure state, involving
K of the
N qubits ($K\ensuremath{\leqslant}N$), yields an operator with negative eigenvalues, identifying
K-body correlations in the state. Expansion of GPT in terms of
K-way partially transposed (KPT) operators reveals the nature of intricate intrinsic correlations in the state. Classification criteria for multipartite entangled states based on the underlying structure of global partial transpose of canonical state are proposed. The number of
N-partite entanglement types for an
N-qubit system is found to be ${2}^{N\ensuremath{-}1}\ensuremath{-}N+2$, while the number of major entanglement classes is ${2}^{N\ensuremath{-}1}\ensuremath{-}1$. Major classes for three- and four-qubit states are listed. Subclasses are determined by the number and type of negativity fonts in canonical states.