Experimentally accessible geometric measure for entanglement in $\mathit{N}$-qubit pure states

We present a multipartite entanglement measure for $\mathit{N}$-qubit pure states, using the norm of the correlation tensor which occurs in the Bloch representation of the state. We compute this measure for several important classes of $\mathit{N}$-qubit pure states such as Greenberger-Horne-Zeilinger and $\mathit{W}$ states and their superpositions. We compute this measure for interesting applications like the one-dimensional Heisenberg antiferromagnet. We use this measure to follow the entanglement dynamics of Grover’s algorithm. We prove that this measure possesses almost all the properties expected of a good entanglement measure, including monotonicity. Finally, we extend this measure to $\mathit{N}$-qubit mixed states via convex roof construction and establish its various properties, including its monotonicity. We also introduce a related measure which has all properties of the above measure and is also additive.

Figure 1

Variation of $E_{\mathcal{T}}\left(|\psi ⟩\right)$ [Eq. (10)] for $N=3$, expressed in units of ${\mathit{R}}_3$, with parameter $\mathit{p}$.

Figure 2

Variation of $R_N=E_{\mathcal{T}}\left(|\text{GHZ}⟩\right)$ [Eq. (11)] with number of qubits $\mathit{N}$.

Figure 3

Variation of $E_{\mathcal{T}}\left(|{\psi }_{s,\varphi }⟩\right)$, expressed in units of ${\mathit{R}}_{\mathit{N}}$, with the superposition parameter $\mathit{s}$, for $N=3$.

Figure 4

Variation of $E_{\mathcal{T}}\mathbf{\left(}|{\psi }_N\left(s\right)⟩\mathbf{\right)}$, (in units of ${\mathit{R}}_{\mathit{N}}$), with $\mathit{s}$, for $N=$ (a) 20 and (b) 100 (see text).

Figure 5

Variation of $E_{\mathcal{T}}\mathbf{\left(}|{\psi }_{W+\text{GHZ}}\left(s,\varphi \right)⟩\mathbf{\right)}$, expressed in units of ${\mathit{R}}_{\mathit{N}}$, with the superposition parameter $\mathit{s}$, for $N=3$.

Figure 6

Entanglement in Grover’s algorithm for six qubits as a function of number of iterations.