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Register a new userStochastic local operations and classical communication properties of the n-qubit symmetric Dicke states
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Recently, several schemes for the experimental creation of Dicke states were described. In this paper, we show that all the $n$-qubit symmetric Dicke states with $l$ ($2leq lleq (n-2)$) excitations are inequivalent to the $% |GHZ>$ state or the $|W>$ state under SLOCC, that the even $n$% -qubit symmetric Dicke state with $n/2$ excitations is inequivalent to any even $n$-qubit symmetric Dicke state with $l eq n/2$ excitations under SLOCC, and that all the $n$-qubit symmetric Dicke states with $l$ ($2leq lleq (n-2)$) excitations satisfy Coffman, Kundu and Wootters generalized monogamy inequality $C_{12}^{2}+...+C_{1n}^{2}<C_{1(2...n)}^{2}<1$.
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We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for general n-qubit states. For two arbitrary pure n-qubit states connected via local operations, we establish an equation between the two coefficient matrices associated with the states. The rank of the coefficient matrix is preserved under SLOCC and gives rise to a simple way of partitioning all the pure states of n qubits into different families of entanglement classes, as exemplified here. When applied to the symmetric states, this approach reveals that all the Dicke states |l,n> with l=1, ..., [n/2] are inequivalent under SLOCC.
In a recent paper [Phys. Rev. A 76, 032304(2007)], Li et al. proposed the definition of the residual entanglement for n qubits by means of the Stochastic local operations and classical communication. Here we argue that their definition is not suitable for the case of odd-n qubits.
We construct $ell $-spin-flipping matrices from the coefficient matrices of pure states of $n$ qubits and show that the $ell $-spin-flipping matrices are congruent and unitary congruent whenever two pure states of $n$ qubits are SLOCC and LU equivalent, respectively. The congruence implies the invariance of ranks of the $ell $-spin-flipping matrices under SLOCC and then permits a reduction of SLOCC classification of n qubits to calculation of ranks of the $ell $-spin-flipping matrices. The unitary congruence implies the invariance of singular values of the $ell $-spin-flipping matrices under LU and then permits a reduction of LU classification of n qubits to calculation of singular values of the $ell $-spin-flipping matrices. Furthermore, we show that the invariance of singular values of the $ell $-spin-flipping matrices $Omega _{1}^{(n)}$ implies the invariance of the concurrence for even $n$ qubits and the invariance of the n-tangle for odd $n$ qubits. Thus, the concurrence and the n-tangle can be used for LU classification and computing the concurrence and the n-tangle only performs additions and multiplications of coefficients of states.
We have reviewed the comment in [3], posted on arXiv.org concerning our recent work in [1]. We reply to the comment in this paper.
Bipartite operations underpin both classical communication and entanglement generation. Using a superposition of classical messages, we show that the capacity of a two-qubit operation for error-free entanglement-assisted bidirectional classical communication can not exceed twice the entanglement capability. In addition we show that any bipartite two-qubit operation can increase the communication that may be performed using an ensemble by twice the entanglement capability.
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