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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.
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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.
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 $C_{12}^{2}+...+C_{1n}^{2}<C_{1(2...n)}^{2}<1$.
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 study the task of entanglement distillation in the one-shot setting under different classes of quantum operations which extend the set of local operations and classical communication (LOCC). Establishing a general formalism which allows for a straightforward comparison of their exact achievable performance, we relate the fidelity of distillation under these classes of operations with a family of entanglement monotones and the rates of distillation with a class of smoothed entropic quantities based on the hypothesis testing relative entropy. We then characterise exactly the one-shot distillable entanglement of several classes of quantum states and reveal many simplifications in their manipulation. We show in particular that the $varepsilon$-error one-shot distillable entanglement of any pure state is the same under all sets of operations ranging from one-way LOCC to separability-preserving operations or operations preserving the set of states with positive partial transpose, and can be computed exactly as a quadratically constrained linear program. We establish similar operational equivalences in the distillation of isotropic and maximally correlated states, reducing the computation of the relevant quantities to linear or semidefinite programs. We also show that all considered sets of operations achieve the same performance in environment-assisted entanglement distillation from any state.
In reply to Vaidmans Comment [arXiv:1304.6689], we show that his claim that our Protocol for Direct Counterfactual Quantum Communication [PRL 110, 170502 (2013), arXiv:1206.2042] is counterfactual only for one type of information bit is wrong.
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