Universal observable detecting all two-qubit entanglement and determinant-based separability tests

We construct a single observable, measurement of whose mean value on four copies of an unknown two-qubit state is sufficient to determine unambiguously whether the state is separable or entangled. In other words, there exists a universal collective entanglement witness detecting all two-qubit entanglement. The test is directly linked to a function that characterizes the entanglement quantitatively to some extent. This function is an entanglement monotone under so-called local pure operations and classical communication which preserve local dimensions. Moreover, it provides tight upper and lower bounds for negativity and concurrence. An elementary quantum computing device estimating unknown two-qubit entanglement is designed.

Figure 1

Plot of ${\pi }_2$ versus $N$ (or $C$) for randomly generated density matrices with bounds obtained in (5, 6). The bound ${\pi }_2\le (1∕2)(N+1)$ obtained from the geometric-arithmetic inequality applied to the absolute values of the eigenvalues of ${\varrho }^{\Gamma }$ is added. The reader is encouraged to consult [37].

Figure 2

Network determining entanglement properties of a two-qubit state by a single measurement of $⟨{\sigma }_z⟩$ on a control qubit. Here $\mid \varphi ⟩=(1∕\sqrt{23})(\sqrt{3}\mid 00⟩+\sqrt{6}\mid 01⟩+\sqrt{8}\mid 10⟩+\sqrt{6}\mid 11⟩)$; unitary $U_i$’s are combinations of SWAP operations such that $\text{Tr}\left(U_i{\varrho }^{\otimes i}\right)={\Pi }_i$. The state would be declared entangled if and only if the measurement yielded a result less than $-1∕23$.