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We show that the quantum Fisher information provides a sufficient condition to recognize multi-particle entanglement in a $N$ qubit state.
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The Fisher information is defined as F [ ρ ^ inp , J ^ n → ] ≡ ∫ d η 1 P ( η | θ ) ( d P ( η | θ ) d θ ) 2 F[\hat{\rho}_{\mathrm{inp}},\hat{J}_{\vec{n}}]\equiv\int\mathrm{d}\eta\frac{1}{P(\eta|\theta)}\big(\frac{\mathrm{d}P(\eta|\theta)}{\mathrm{d}\theta}\big)^{2} , where P ( η | θ ) ≡ Tr [ E ^ ( η ) ρ ^ o u t ( θ ) ] P(\eta|\theta)\equiv\mathrm{Tr}[\hat{E}(\eta)\hat{\rho}_{out}(\theta)] , E ^ ( η ) \hat{E}(\eta) is a positive operator satisfying ∫ d η E ^ ( η ) = 1 ^ \int\mathrm{d}\eta\hat{E}(\eta)=\hat{1} (unit operator) and ρ ^ out ( θ ) = e i θ J ^ n → ρ ^ inp e − i θ J ^ n → \hat{\rho}_{\mathrm{out}}(\theta)=e^{i\theta\hat{J}_{\vec{n}}}\hat{\rho}_{\mathrm{inp}}e^{-i\theta\hat{J}_{\vec{n}}} is the rotated state. The QFI is F Q [ ρ ^ inp , J ^ n → ] ≡ max E ^ ( η ) F [ ρ ^ inp , J ^ n → ] = 2 ∑ j , k ( p j − p k ) 2 p j + p k | ⟨ j | J ^ n → | k ⟩ | 2 F_{\mathrm{Q}}[\hat{\rho}_{\mathrm{inp}},\hat{J}_{\vec{n}}]\equiv\max_{\hat{E}(\eta)}F[\hat{\rho}_{\mathrm{inp}},\hat{J}_{\vec{n}}]=2\sum_{j,k}\frac{(p_{j}-p_{k})^{2}}{p_{j}+p_{k}}|\langle j|\hat{J}_{\vec{n}}|k\rangle|^{2} , where { | j ⟩ } \{|j\rangle\} is an orthonormal set of states which diagonalizes ρ ^ inp = ∑ j p j | j ⟩ ⟨ j | \hat{\rho}_{\mathrm{inp}}=\sum_{j}p_{j}|j\rangle\langle j| (with p j ≥ 0 p_{j}\geq 0 and ∑ j p j = 1 \sum_{j}p_{j}=1 ). The QFI can be always saturated by an optimal choice of E ^ ( η ) \hat{E}(\eta) [ 11 ]
Cited in the paper.
For pure states, this equation is solved by R ^ = i [ J ^ n → , ρ ^ inp ] \hat{R}=i[\hat{J}_{\vec{n}},\hat{\rho}_{\mathrm{inp}}] and we have ( Δ R ^ ) 2 = ( Δ J ^ n → ) 2 (\Delta\hat{R})^{2}=(\Delta\hat{J}_{\vec{n}})^{2} . In general, ( Δ R ^ ) 2 ≤ ( Δ J ^ n → ) 2 (\Delta\hat{R})^{2}\leq(\Delta\hat{J}_{\vec{n}})^{2} and the equality is obtained only for pure states. S. Boixo and A. Monras, Phys. Rev. Lett. 100
Cited in the paper.
L. Pezzé and A. Smerzi, in preparation
Cited in the paper.
The upper bound of Eq.( 6
Cited in the paper.
| j , μ ⟩ k → |j,\mu\rangle_{\vec{k}} are eigenstates of J ^ k → \hat{J}_{\vec{k}} with eigenvalues − j ≤ μ ≤ j -j\leq\mu\leq j
Cited in the paper.
We refer to Δ θ = α / N \Delta\theta=\alpha/N as the Heisenberg limit of phase sensitivity, provided, of course, that the prefactor α ≥ 1 \alpha\geq 1 does not depend on N N
Cited in the paper.
In two-mode approximation, the Hamiltonian of two independent BECs is H ^ = E c J ^ z 2 , \hat{H}=E_{c}\hat{J}_{z}^{2}, , where E c E_{c} is the charging energy proportional to the particle-particle scattering length. In has been shown that the previous model provides a good approximation of the dynamics of the system, see A.S. Sørensen, Phys. Rev. A 65
Cited in the paper.